Introduction to quantum noise, measurement, and …optical–quantum optics communities. This review gives a pedagogical introduction to the physics of quantum noise and its connections

Introduction to quantum noise, measurement, and amplification

A. A. Clerk*

Department of Physics, McGill University, 3600 rue University Montréal, Quebec, CanadaH3A 2T8

M. H. Devoret

Department of Applied Physics, Yale University, P.O. Box 208284, New Haven,Connecticut 06520-8284, USA

S. M. Girvin

Department of Physics, Yale University, P.O. Box 208120, New Haven,Connecticut 06520-8120, USA

Florian Marquardt

Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center forTheoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37,D-80333 München, Germany

R. J. Schoelkopf

Department of Applied Physics, Yale University, P.O. Box 208284, New Haven,Connecticut 06520-8284, USA

Published 15 April 2010

The topic of quantum noise has become extremely timely due to the rise of quantum informationphysics and the resulting interchange of ideas between the condensed matter and atomic, molecular,optical–quantum optics communities. This review gives a pedagogical introduction to the physics ofquantum noise and its connections to quantum measurement and quantum amplification. Afterintroducing quantum noise spectra and methods for their detection, the basics of weak continuousmeasurements are described. Particular attention is given to the treatment of the standard quantumlimit on linear amplifiers and position detectors within a general linear-response framework. Thisapproach is shown how it relates to the standard Haus-Caves quantum limit for a bosonic amplifierknown in quantum optics and its application to the case of electrical circuits is illustrated, includingmesoscopic detectors and resonant cavity detectors.

DOI: 10.1103/RevModPhys.82.1155 PACS numbers: 72.70.m

CONTENTS

I. Introduction 1156

II. Quantum Noise Spectra 1159

A. Introduction to quantum noise 1159

B. Quantum spectrum analyzers 1161

III. Quantum Measurements 1162

A. Weak continuous measurements 1164

B. Measurement with a parametrically coupled

resonant cavity 1164

1. QND measurement of the state of a qubit

using a resonant cavity 1167

2. Quantum limit relation for QND qubit state

detection 1168

3. Measurement of oscillator position using a

resonant cavity 1169

IV. General Linear-Response Theory 1173

A. Quantum constraints on noise 1173

1. Heuristic weak-measurement noise

constraints 1173

2. Generic linear-response detector 1174

3. Quantum constraint on noise 1175

4. Evading the detector quantum noise

inequality 1176

B. Quantum limit on QND detection of a qubit 1177

V. Quantum Limit on Linear Amplifiers and Position

Detectors 1178

A. Preliminaries on amplification 1178

B. Standard Haus-Caves derivation of the quantum

limit on a bosonic amplifier 1179

C. Nondegenerate parametric amplifier 1181

1. Gain and added noise 1181

2. Bandwidth-gain trade-off 1182

3. Effective temperature 1182

D. Scattering versus op-amp modes of operation 1183

E. Linear-response description of a position detector 1185

1. Detector back-action 1185

2. Total output noise 1186

3. Detector power gain 1186

4. Simplifications for a quantum-ideal detector 1188*[email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 82, APRIL–JUNE 2010

0034-6861/2010/822/115554 ©2010 The American Physical Society1155

http://dx.doi.org/10.1103/RevModPhys.82.1155

5. Quantum limit on added noise and noisetemperature 1188

F. Quantum limit on the noise temperature of avoltage amplifier 1190

1. Classical description of a voltage amplifier 11912. Linear-response description 11923. Role of noise cross correlations 1193

G. Near quantum-limited mesoscopic detectors 11941. dc superconducting quantum interference

device amplifiers 11942. Quantum point contact detectors 11943. Single-electron transistors and

resonant-level detectors 1194H. Back-action evasion and noise-free amplification 1195

1. Degenerate parametric amplifier 11962. Double-sideband cavity detector 11963. Stroboscopic measurements 1197

VI. Bosonic Scattering Description of a Two-PortAmplifier 1198

A. Scattering versus op-amp representations 11981. Scattering representation 11982. Op-amp representation 11993. Converting between representations 1200

B. Minimal two-port scattering amplifier 12011. Scattering versus op-amp quantum limit 12012. Why is the op-amp quantum limit not

achieved? 1203VII. Reaching the Quantum Limit in Practice 1203

A. Importance of QND measurements 1203B. Power matching versus noise matching 1204

VIII. Conclusions 1204Acknowledgments 1205References 1205

I. INTRODUCTION

Recently several advances have led to a renewed in-terest in the quantum-mechanical aspects of noise in me-soscopic electrical circuits, detectors, and amplifiers.One motivation is that such systems can operate simul-taneously at high frequencies and at low temperatures,entering the regime where kBT. As such, quantumzero-point fluctuations will play a more dominant role indetermining their behavior than the more familiar ther-mal fluctuations. A second motivation comes from therelation between quantum noise and quantum measure-ment. There exists an ever-increasing number of experi-ments in mesoscopic electronics where one is forced tothink about the quantum mechanics of the detectionprocess, and about fundamental quantum limits whichconstrain the performance of the detector or amplifierused.

Given the above, we will focus in this review on dis-cussing what is known as the “standard quantum limit”SQL on both displacement detection and amplifica-tion. To preclude any possible confusion, it is worthwhileto state explicitly from the start that there is no limit tohow well one may resolve the position of a particle in aninstantaneous measurement. Indeed, in the typicalHeisenberg microscope setup, one would scatter pho-

tons off an electron, thereby detecting its position to anaccuracy set by the wavelength of photons used. Thefact that its momentum will suffer a large uncontrolledperturbation, affecting its future motion, is of no con-cern here. Only as one tries to further increase the res-olution will one finally encounter relativistic effects pairproduction that set a limit given by the Compton wave-length of the electron. The situation is obviously verydifferent if one attempts to observe the whole trajectoryof the particle. As this effectively amounts to measure-ments of both position and momentum, there has to be atrade-off between the accuracies of both, set by theHeisenberg uncertainty relation. This is enforced inpractice by the uncontrolled perturbation of the momen-tum during one position measurement adding to thenoise in later measurements, a phenomenon known as“measurement back-action.”

Just such a situation is encountered in “weak mea-surements” Braginsky and Khalili, 1992, where one in-tegrates the signal over time, gradually learning moreabout the system being measured; this review will focuson such measurements. There are many good reasonswhy one may be interested in doing a weak measure-ment, rather than an instantaneous, strong, projectivemeasurement. On a practical level, there may be limita-tions to the strength of the coupling between the systemand the detector, which have to be compensated by in-tegrating the signal over time. One may also deliberatelyopt not to disturb the system too strongly, e.g., to be ableto apply quantum feedback techniques for state control.Moreover, as one reads out an oscillatory signal overtime, one effectively filters away noise e.g., of a techni-cal nature at other frequencies. Finally, consider an ex-ample like detection of the collective coordinate of mo-tion of a micromechanical beam. Its zero-pointuncertainty ground-state position fluctuation is typi-cally on the order of the diameter of a proton. It is outof the question to reach this accuracy in an instanta-neous measurement by scattering photons of such asmall wavelength off the structure, since they would in-stead resolve the much larger position fluctuations of theindividual atoms comprising the beam and induce allkinds of unwanted damage, instead of reading out thecenter-of-mass coordinate. The same holds true forother collective degrees of freedom.

The prototypical example we discuss is that of a weakmeasurement detecting the motion of a harmonic oscil-lator such as a mechanical beam. The measurementthen actually follows the slow evolution of amplitudeand phase of the oscillations or, equivalently, the twoquadrature components, and the SQL derives from thefact that these two observables do not commute. It es-sentially says that the measurement accuracy will be lim-ited to resolving both quadratures down to the scale ofthe ground-state position fluctuations, within one me-chanical damping time. Note that, in special applica-tions, one might be interested only in one particularquadrature of motion. Then the Heisenberg uncertaintyrelation does not enforce any SQL and one may again

1156 Clerk et al.: Introduction to quantum noise, measurement, …

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

obtain unlimited accuracy, at the expense of renouncingall knowledge of the other quadrature.

Position detection by weak measurement essentiallyamounts to amplification of the quantum signal up to aclassically accessible level. Therefore, the theory ofquantum limits on displacement detection is intimatelyconnected to limits on how well an amplifier can work. Ifan amplifier does not have any preference for any par-ticular phase of the oscillatory signal, it is called “phasepreserving,” which is the case relevant for amplifyingand thereby detecting both quadratures equally well.1

We derive and discuss the SQL for phase-preserving lin-ear amplifiers Haus and Mullen, 1962; Caves, 1982.Quantum mechanics demands that such an amplifieradds noise that corresponds to half a photon added toeach mode of the input signal, in the limit of highphoton-number gain G. In contrast, for small gain, theminimum number of added noise quanta, 1−1/G /2,can become arbitrarily small as the gain is reduced downto 1 no amplification. One might ask, therefore,whether it should not be possible to evade the SQL bybeing content with small gains. The answer is no, sincehigh gains G1 are needed to amplify the signal to alevel where it can be read out or further amplified us-ing classical devices without their noise having any fur-ther appreciable effect, converting 1 input photon intoG1 output photons. According to Caves, it is neces-sary to generate an output that “we can lay our grubby,classical hands on” Caves, 1982. It is a simple exerciseto show that feeding the input of a first, potentially low-gain amplifier into a second amplifier results in an over-all bound on the added noise that is just the one ex-pected for the product of their respective gains.Therefore, as one approaches the classical level, i.e.,large overall gains, the SQL always applies in its simpli-fied form of half a photon added.

Unlike traditional discussions of the amplifier SQL,here we devote considerable attention to a generallinear-response approach based on the quantum relationbetween susceptibilities and noise. This approach treatsthe amplifier or detector as a black box with an inputport coupling to the signal source and an output port toaccess the amplified signal. It is more suited for mesos-copic systems than the quantum optics scattering-typeapproach, and it leads us to the quantum noise inequal-ity: a relation between the noise added to the output andthe back-action noise feeding back to the signal source.In the ideal case what we term a “quantum-limited de-tector”, the product of these two contributions reachesthe minimum value allowed by quantum mechanics. Weshow that optimizing this inequality on noise is a neces-sary prerequisite for having a detector achieve the quan-tum limit in a specific measurement task, such as linearamplification.

There are several motivations for understanding in

principle, and realizing in practice, amplifiers whosenoise reaches this minimum quantum limit. Achievingthe quantum limit on continuous position detection hasbeen one of the goals of many recent experiments onquantum electromechanical Cleland et al., 2002; Knobeland Cleland, 2003; LaHaye et al., 2004; Naik et al., 2006;Flowers-Jacobs et al., 2007; Etaki et al., 2008; Poggio etal., 2008; Regal et al., 2008 and optomechanical systemsArcizet et al., 2006; Gigan et al., 2006; Schliesser et al.,2008; Thompson et al., 2008; Marquardt and Girvin,2009. As we will show, having a near-quantum-limiteddetector would allow one to continuously monitor thequantum zero-point fluctuations of a mechanical resona-tor. It is also necessary to have a quantum-limited detec-tor is for such tasks as single-spin NMR detection Ru-gar et al., 2004, as well as gravitational wave detectionAbramovici et al., 1992. The topic of quantum-limiteddetection is also directly relevant to recent activity ex-ploring feedback control of quantum systems Wisemanand Milburn, 1993, 1994; Doherty et al., 2000; Korotkov,2001b; Ruskov and Korotkov, 2002; such schemes re-quire need a close-to-quantum-limited detector.

This review is organized as follows. We start in Sec. IIby providing a review of the basic statistical propertiesof quantum noise, including its detection. In Sec. III weturn to quantum measurements and give a basic intro-duction to weak continuous measurements. To makethings concrete, we discuss heuristically measurementsof both a qubit and an oscillator using a simple resonantcavity detector, giving an idea of the origin of the quan-tum limit in each case. Section IV is devoted to a morerigorous treatment of quantum constraints on noise aris-ing from general quantum linear-response theory. Theheart of the review is contained in Sec. V, where we givea thorough discussion of quantum limits on amplificationand continuous position detection. We also discuss vari-ous methods for beating the usual quantum limits onadded noise using back-action evasion techniques. Weare careful to distinguish two very distinct modes of am-plifier operation the “scattering” versus “op-amp”modes; we expand on this in Sec. VI, where we discussboth modes of operation in a simple two-port bosonicamplifier. Importantly, we show that an amplifier can bequantum limited in one mode of operation, but fail to bequantum limited in the other mode of operation. Finally,in Sec. VII we highlight a number of practical consider-ations that one must keep in mind when trying to per-form a quantum-limited measurement. Table I providesa synopsis of the main results discussed in the text aswell as definitions of symbols used.

In addition to the above, we have supplemented themain text with several pedagogical appendixes thatcover basic background topics. Particular attention isgiven to the quantum mechanics of transmission linesand driven electromagnetic cavities, topics that are espe-cially relevant given recent experiments making use ofmicrowave stripline resonators. These appendixes arecontained in a separate online-only supplementarydocument Clerk et al., 2009 see also http://arxiv.org/abs/0810.4729. In Table II, we list the contents of these

1In the literature this is often referred to as a “phase insensi-tive” amplifier. We prefer the term “phase preserving” to avoidany ambiguity.

1157Clerk et al.: Introduction to quantum noise, measurement, …


TABLE I. Table of symbols and main results.

Symbol Definition or result

General definitionsf Fourier transform of the function or operator ft, defined via f=−

dtfteit

„Note that for operators, we use the convention f†=− dtf†teit, implying f†= f−†

…

SFF Classical noise spectral density or power spectrum: SFF=−+dteitFtF0

SFF Quantum noise spectral density: SFF=−+dteitFtF0

SFF Symmetrized quantum noise spectral density SFF= 12 SFF+SFF−= 1

2−+dteitFt , F0

ABt General linear-response susceptibility describing the response of A to a perturbation that couples to B; in thequantum case, given by the Kubo formula ABt=−i /tAt , B0 Eq. 2.14

A Coupling constant dimensionless between measured system and detector or amplifier,e.g., V=AFtx, V=AxF,or V=Acza†a

M , Mass and angular frequency of a mechanical harmonic oscillatorxZPF Zero-point uncertainty of a mechanical oscillator, xZPF= /2M

0 Intrinsic damping rate of a mechanical oscillator due to coupling to a bath via V=AxF: 0= A2 /2MSFF−SFF− Eq. 2.12

c Resonant frequency of a cavity ,Qc Damping, quality factor of a cavity: Qc=c /

Sec. IITeff Effective temperature at a frequency for a given quantum noise spectrum, defined via SFF /SFF−

=exp /kBTeff Eq. 2.8Fluctuation-dissipation theorem relating the symmetrized noise spectrum to the dissipative part for anequilibrium bath: SFF= 1

2 coth /2kBTSFF−SFF− Eq. 2.16

Sec. III

Number-phase uncertainty relation for a coherent state: N 12 Eqs. 3.6 and G12

N Photon-number flux of a coherent beam

Imprecision noise in the measurement of the phase of a coherent beam

Fundamental noise constraint for an ideal coherent beam: SNNS= 14 Eqs. 3.8 and G21

Sxx0 Symmetrized spectral density of zero-point position fluctuations of a damped harmonic oscillator

Sxx,tot Total output noise spectral density symmetrized of a linear position detector, referred back to the oscillator

Sxx,add Added noise spectral density symmetrized of a linear position detector, referred back to the oscillator

Sec. IVx Input signal

F Fluctuating force from the detector, coupling to x via V=AxF

I Detector output signal

General quantum constraint on the detector output noise, back-action noise, and gain: SIISFF− SIF2

IF /221+ †SIF / /2‡ Eq. 4.11, where IFIF− †FI‡* and z= 1+z2− 1+ z2 /2„Note that 1+ z0 and =0 in most cases of relevance; see discussion around Eq. 4.17…

Complex proportionality constant characterizing a quantum-ideal detector: 2= SII / SFF and sinarg = /2 /SIISFF Eqs. 4.18 and I17

meas Measurement rate for a quantum nondemolition QND qubit measurement Eq. 4.24 Dephasing rate due to measurement back-action Eqs. 3.27 and 4.19

Constraint on weak, continuous QND qubit state detection: =meas/1 Eq. 4.25

Sec. VG Photon-number power gain, e.g., in Eq. 5.7

Input-output relation for a bosonic scattering amplifier: b†=Ga†+ F† Eq. 5.7



Appendixes. Note that, while some aspects of the topicsdiscussed in this review have been studied in the quan-tum optics and quantum dissipative systems communi-ties and are the subject of several comprehensive booksBraginsky and Khalili, 1992; Weiss, 1999; Gardiner andZoller, 2000; Haus, 2000, they are somewhat newer tothe condensed matter physics community; moreover,some of the technical machinery developed in thesefields is not directly applicable to the study of quantumnoise in quantum electronic systems. Finally, note thatwhile this article is a review, there is considerable new

material presented, especially in our discussion of quan-tum amplification see Secs. V.D and VI.

II. QUANTUM NOISE SPECTRA

A. Introduction to quantum noise

In classical physics, the study of a noisy time-dependent quantity invariably involves its spectral den-sity S. The spectral density tells us the intensity of thenoise at a given frequency and is directly related to the

TABLE I. Continued.

Symbol Definition or result

a2 Symmetrized field operator uncertainty for the scattering description of a bosonic amplifier: a2 12 a , a†

− a2

Standard quantum limit for the noise added by a phase-preserving bosonic scattering amplifier in the high-gainlimit, G1, where a2ZPF= 1

2 : b2 /G a2+ 12 Eq. 5.10

GP Dimensionless power gain of a linear position detector or voltage amplifiermaximum ratio of the powerdelivered by the detector output to a load, vs the power fed into signal source:GP= IF2 / 4 Im FFIm II Eq. 5.52For a quantum-ideal detector, in the high-gain limit: GP Im / 4 kBTeff /2 Eq. 5.57

Sxx,eq ,T Intrinsic equilibrium noise Sxx,eq ,T= coth /2kBT−Im xx Eq. 5.59Aopt Optimal coupling strength of a linear position detector which minimizes the added noise at frequency : Aopt

4 = SII / xx2SFF Eq. 5.64

Aopt Detector-induced damping of a quantum-limited linear position detector at optimal coupling; satisfies Aopt /0+Aopt= Im /1/GP= /4kBTeff1 Eq. 5.69Standard quantum limit for the added noise spectral density of a linear position detector valid at each frequency: Sxx,add limT→0 Sxx,eqw ,T Eq. 5.62Effective increase in oscillator temperature due to coupling to the detector back-action, for an ideal detector,with /kBTbathTeff: ToscTeff+0Tbath / +0→ /4kB+Tbath Eq. 5.70

Zin ,Zout Input and output impedances of a linear voltage amplifierZs Impedance of signal source attached to input of a voltage amplifierV Voltage gain of a linear voltage amplifier

Vt Voltage noise of a linear voltage amplifier „Proportional to the intrinsic output noise of the genericlinear-response detector Eq. 5.81…

It Current noise of a linear voltage amplifier „Related to the back-action force noise of the generic linear-responsedetector Eqs. 5.80…

TN Noise temperature of an amplifier defined in Eq. 5.74ZN Noise impedance of a linear voltage amplifier Eq. 5.77

Standard quantum limit on the noise temperature of a linear voltage amplifier: kBTN /2 Eq. 5.89

Sec. VI

Va Vb Voltage at the input output of the amplifier

Relation to bosonic mode operators: Eq. 6.2a

Ia Ib Current drawn at the input leaving the output of the amplifier

Relation to bosonic mode operators: Eq. 6.2bI Reverse current gain of the amplifier

s Input-output 22 scattering matrix of the amplifier Eq. 6.3Relation to op-amp parameters V ,I ,Zin ,Zout: Eqs. 6.7

V I Voltage current noise operators of the amplifier

Fa , Fb Input output port noise operators in the scattering description Eq. 6.3

Relation to op-amp noise operators V , I: Eq. 6.9



autocorrelation function of the noise.2 In a similar fash-ion, the study of quantum noise involves quantum noisespectral densities. These are defined in a manner thatmimics the classical case

Sxx = −

+

dteitxtx0 . 2.1

Here x is a quantum operator in the Heisenberg repre-sentation whose noise we are interested in, and the an-gular brackets indicate the quantum statistical averageevaluated using the quantum density matrix. Note thatwe use S throughout this review to denote the spec-tral density of a classical noise, while S will denote aquantum noise spectral density.

As a simple introductory example illustrating impor-tant differences from the classical limit, consider the po-sition noise of a simple harmonic oscillator having massM and frequency . The oscillator is maintained in equi-librium with a large heat bath at temperature T via someinfinitesimal coupling, which we ignore in consideringthe dynamics. The solutions of the Heisenberg equationsof motion are the same as for the classical case but withthe initial position x and momentum p replaced by thecorresponding quantum operators. It follows that theposition autocorrelation function is

Gxxt = xtx0 = x0x0cost

+ p0x01

Msint . 2.2

Classically the second term on the right-hand sideRHS vanishes because in thermal equilibrium x and pare uncorrelated random variables. As we will seeshortly for the quantum case, the symmetrized some-times called the “classical” correlator vanishes in ther-mal equilibrium, just as it does classically: xp+ px=0.

Note, however, that in the quantum case the canonicalcommutation relation between position and momentumimplies there must be some correlations between thetwo, namely, x0p0− p0x0= i. These correla-tions are easily evaluated by writing x and p in terms ofharmonic oscillator ladder operators. We find that inthermal equilibrium p0x0=−i /2 and x0p0=+i /2. Not only are the position and momentum cor-related, but their correlator is imaginary:3 This meansthat, despite the fact that the position is Hermitian ob-servable with real eigenvalues, its autocorrelation func-tion is complex and given from Eq. 2.2 by

Gxxt = xZPF2 nBe+it + nB + 1e−it , 2.3

where xZPF2 /2M is the RMS zero-point uncertainty

of x in the quantum ground state and nB is the Bose-Einstein occupation factor. The complex nature of theautocorrelation function follows from the fact that theoperator x does not commute with itself at differenttimes.

Because the correlator is complex it follows that thespectral density is not symmetric in frequency,

Sxx = 2xZPF2 nB +

+ nB + 1 − . 2.4

In contrast, a classical autocorrelation function is alwaysreal, and hence a classical noise spectral density is al-ways symmetric in frequency. Note that in the high-temperature limit kBT we have nBnB+1kBT /. Thus, in this limit Sxx becomes sym-metric in frequency as expected classically, and coincideswith the classical expression for the position spectraldensity cf. Eq. A12.

The Bose-Einstein factors suggest a way to under-stand the frequency asymmetry of Eq. 2.4: the positive-frequency part of the spectral density has to do withstimulated emission of energy into the oscillator and thenegative-frequency part of the spectral density has to dowith emission of energy by the oscillator. That is, thepositive-frequency part of the spectral density is a mea-sure of the ability of the oscillator to absorb energy,while the negative-frequency part is a measure of theability of the oscillator to emit energy. As we will see,this is generally true, even for nonthermal states. Figure1 shows this idea for the case of the voltage noise spec-tral density of a resistor see Appendix D.3 for moredetails. Note that the result Eq. 2.4 can be extended tothe case of a bath of many harmonic oscillators. As de-scribed in Appendix D a resistor can be modeled as aninfinite set of harmonic oscillators and from this modelthe Johnson or Nyquist noise of a resistor can be de-rived.

2For readers unfamiliar with the basics of classical noise, acompact review is given in Appendix A supplementarymaterial.

3Notice that this occurs because the product of two noncom-muting Hermitian operators is not itself a Hermitian operator.

TABLE II. Contents of online appendix material. Page num-bers refer to the supplementary material.

Section Page

A. Basics of classical and quantum noise 1B. Quantum spectrum analyzers: further details 4C. Modes, transmission lines and classicalinput-output theory

8

D. Quantum modes and noise of a transmission line 15E. Back-action and input-output theory for drivendamped cavities

18

F. Information theory and measurement rate 29G. Number phase uncertainty 30H. Using feedback to reach the quantum limit 31I. Additional technical details 34



B. Quantum spectrum analyzers

The qualitative picture described previously can beconfirmed by considering simple systems which act aseffective spectrum analyzers of quantum noise. The sim-plest such example is a quantum two-level system TLScoupled to a quantum noise source Aguado and Kou-wenhoven, 2000; Gavish et al., 2000; Schoelkopf et al.,2003. With the TLS described as a fictitious spin-1 /2particle with spin down spin up representing the

ground state excited state, its Hamiltonian is H0

= 01/2z, where 01 is the energy splitting betweenthe two states. The TLS is then coupled to an externalnoise source via an additional term in the Hamiltonian,

V = AFx, 2.5

where A is a coupling constant and the operator F rep-resents the external noise source. The coupling Hamil-

tonian V can lead to the exchange of energy between thetwo-level system and noise source and hence transitionsbetween its two eigenstates. The corresponding Fermigolden rule transition rates can be compactly expressed

in terms of the quantum noise spectral density of F,SFF,

↑ = A2/2SFF− 01 , 2.6a

↓ = A2/2SFF+ 01 . 2.6b

Here ↑ is the rate at which the qubit is excited from itsground to excited state and ↓ is the corresponding ratefor the opposite relaxation process. As expected,positive- negative- frequency noise corresponds to ab-sorption emission of energy by the noise source. Notethat if the noise source is in thermal equilibrium at tem-perature T, the transition rates of the TLS must satisfythe detailed balance relation ↑ /↓=e−01, where =1/kBT. This in turn implies that in thermal equilibriumthe quantum noise spectral density must satisfy

SFF+ 01 = e01SFF− 01 . 2.7

The more general situation is where the noise source isnot in thermal equilibrium; in this case, no general de-

tailed balance relation holds. However, if we are con-cerned only with a single particular frequency, then it isalways possible to define an “effective temperature” Tefffor the noise using Eq. 2.7, i.e.,

kBTeff

log†SFF/SFF− ‡. 2.8

Note that for a nonequilibrium system Teff will in gen-eral be frequency dependent. In NMR language, Teff willsimply be the “spin temperature” of our TLS spectrom-eter once it reaches steady state after being coupled tothe noise source.

Another simple quantum noise spectrometer is a har-monic oscillator frequency , mass M, and position xcoupled to a noise source see, e.g., Schwinger 1961and Dykman 1978. The coupling Hamiltonian is now

V = AxF = AxZPFa + a†F , 2.9

where a is the oscillator annihilation operator, F is theoperator describing the fluctuating noise, and A is again

a coupling constant. We see that −AF plays the role of afluctuating force acting on the oscillator. In complete

analogy to the previous section, noise in F at the oscil-lator frequency can cause transitions between the os-cillator energy eigenstates. The corresponding Fermigolden rule transition rates are again simply related tothe noise spectrum SFF. Incorporating these ratesinto a simple master equation describing the probabilityto find the oscillator in a particular energy state, onefinds that the stationary state of the oscillator is a Bose-Einstein distribution evaluated at the effective tempera-ture Teff defined in Eq. 2.8. Further, one can use themaster equation to derive a classical-looking equationfor the average energy E of the oscillator see Appen-dix B.2,

dE/dt = P − E , 2.10

where

P = A2/4MSFF + SFF− A2SFF/2M ,

2.11

= A2xZPF2 /2SFF − SFF− . 2.12

The two terms in Eq. 2.10 describe, heating and damp-ing of the oscillator by the noise source, respectively.The heating effect of the noise is completely analogousto what happens classically: a random force causes theoscillator’s momentum to diffuse, which in turn causesE to grow linearly in time at rate proportional to theforce noise spectral density. In the quantum case, Eq.2.11 indicates that it is the symmetric-in-frequency part

of the noise spectrum, SFF, which is responsible forthis effect, and which thus plays the role of a classical

noise source. This is another reason why SFF is often

-5 0 5 10-10

0

2

4

6

8

10

> 0absorptionby reservoir

< 0emissionby reservoir

h /kBT

SVV [ ]/2RkBT

FIG. 1. Quantum noise spectral density of voltage fluctuationsacross a resistor resistance R as a function of frequency atzero temperature dashed line and finite temperature solidline.



referred to as the “classical” part of the noise.4 In con-trast, we see that the asymmetric-in-frequency part ofthe noise spectrum is responsible for the damping. Thisalso has a simple heuristic interpretation: damping iscaused by the net tendency of the noise source to ab-sorb, rather than emit, energy from the oscillator.

The damping induced by the noise source may equiva-lently be attributed to the oscillator’s motion inducing anaverage value to F which is out of phase with x, i.e.,AFt=−Mxt. Standard quantum linear-responsetheory yields

AFt = A2 dtFFt − txt , 2.13

where we have introduced the susceptibility

FFt = − i/tFt,F0 . 2.14

Using the fact that the oscillator’s motion involves onlythe frequency , we thus have

= 2A2xZPF2 /†− Im FF‡ . 2.15

A straightforward manipulation of Eq. 2.14 for FFshows that this expression for is exactly equivalent toour previous expression, Eq. 2.12.

In addition to giving insight on the meaning of thesymmetric and asymmetric parts of a quantum noisespectral density, the above example also directly yieldsthe quantum version of the fluctuation-dissipation theo-rem Callen and Welton, 1951. As we saw earlier, if ournoise source is in thermal equilibrium, the positive- andnegative-frequency parts of the noise spectrum arestrictly related to one another by the condition of de-tailed balance cf. Eq. 2.7. This in turn lets us link theclassical symmetric-in-frequency part of the noise to thedamping i.e., the asymmetric-in-frequency part of thenoise. Setting =1/kBT and making use of Eq. 2.7, wehave

SFF SFF + SFF−

2

= 12 coth/2SFF − SFF−

= coth/2M

A2 . 2.16

Thus, in equilibrium, the condition that noise-inducedtransitions obey detailed balance immediately impliesthat noise and damping are related to one another viathe temperature. For T, we recover the more fa-miliar classical version of the fluctuation-dissipationtheorem

A2SFF = 2kBTM . 2.17

Further insight into the fluctuation-dissipation theoremis provided in Appendix C.3, where we discuss it in thesimple but instructive context of a transmission line ter-minated by an impedance Z.

We have thus considered two simple examples ofmethods of measure quantum noise spectral densities.Further details, as well as examples of other quantumnoise spectrum analyzers, are given in Appendix B.

III. QUANTUM MEASUREMENTS

Having introduced both quantum noise and quantumspectrum analyzers, we are now in a position to intro-duce the general topic of quantum measurements. Allpractical measurements are affected by noise. Certainquantum measurements remain limited by quantumnoise even though they use completely ideal apparatus.As we will see, the limiting noise here is associated withthe fact that canonically conjugate variables are incom-patible observables in quantum mechanics.

The simplest idealized description of a quantum mea-surement, introduced by von Neumann von Neumann,1932; Wheeler and Zurek, 1984; Bohm, 1989; Harocheand Raimond, 2006, postulates that the measurementprocess instantaneously collapses the system’s quantumstate onto one of the eigenstates of the observable to bemeasured. As a consequence, any initial superposition ofthese eigenstates is destroyed and the values of observ-ables conjugate to the measured observable are per-turbed. This perturbation is an intrinsic feature of quan-tum mechanics and cannot be avoided in anymeasurement scheme, be it of the projection type de-scribed by von Neumann or a rather weak continuousmeasurement to be analyzed further below.

To form a more concrete picture of quantum measure-ment, we begin by noting that every quantum measure-ment apparatus consists of a macroscopic “pointer”coupled to the microscopic system to be measured. Aspecific model is discussed by Allahverdyan et al.2001. This pointer is sufficiently macroscopic that itsposition can be read out “classically.” The interactionbetween the microscopic system and the pointer is ar-ranged so that the two become strongly correlated. Oneof the simplest possible examples of a quantum mea-surement is that of the Stern-Gerlach apparatus, whichmeasures the projection of the spin of an S=1/2 atomalong some chosen direction. What is really measured inthe experiment is the final position of the atom on thedetector plate. However, the magnetic field gradient inthe magnet causes this position to be perfectly corre-lated “entangled” with the spin projection so that thelatter can be inferred from the former. Suppose, for ex-ample that the initial state of the atom is the product ofa spatial wave function 0r centered on the entrance tothe magnet and a spin state that is the superposition ofup and down spins corresponding to the eigenstate of x,

4Note that with our definition F2=− d /2SFF. It is

common in engineering contexts to define so-called one-sidedclassical spectral densities, which are equal to twice ourdefinition.



0 = 1/2↑ + ↓0 . 3.1

After passing through a magnet with field gradient in thez direction, an atom with spin up is deflected upwardand an atom with spin down is deflected downward. Bythe linearity of quantum mechanics, an atom in a spinsuperposition state thus ends up in a superposition ofthe form

1 = 1/2↑+ + ↓− , 3.2

where r ±=1r±dz are spatial orbitals peaked in theplane of the detector. The deflection d is determined bythe device geometry and the magnetic field gradient.The z-direction position distribution of the particle foreach spin component is shown in Fig. 2. If d is suffi-ciently large compared to the wave packet spread then,given the position of the particle, one can unambigu-ously determine the distribution from which it came andhence the value of the spin projection of the atom. Thisis the limit of a strong “projective” measurement.

In the initial state one has 0x0=+1, but in thefinal state one has

1x1 = 12 −+ + +− . 3.3

For sufficiently large d the states ± are orthogonal andthus the act of z measurement destroys the spin coher-ence

1x1 → 0. 3.4

This is what we mean by projection or wave function“collapse.” The result of measurement of the atom po-sition will yield a random and unpredictable value of ± 1

2for the z projection of the spin. This destruction of thecoherence in the transverse spin components by a strongmeasurement of the longitudinal spin component is thefirst of many examples we will see of the Heisenberguncertainty principle in action. Measurement of onevariable destroys information about its conjugate vari-able. We study several examples in which we understandmicroscopically how it is that the coupling to the mea-surement apparatus causes the back-action quantumnoise which destroys our knowledge of the conjugatevariable.

In the special case where the eigenstates of the ob-servable we are measuring are also stationary states i.e.,energy eigenstates, a second measurement of the ob-servable would reproduce exactly the same result, thus

providing a way to confirm the accuracy of the measure-ment scheme. These optimal kinds of repeatable mea-surements are called quantum nondemolition QNDmeasurements Braginsky et al., 1980; Braginsky andKhalili, 1992, 1996; Peres, 1993. A simple examplewould be a sequential pair of Stern-Gerlach devices ori-ented in the same direction. In the absence of stray mag-netic perturbations, the second apparatus would alwaysyield the same answer as the first. In practice, one termsa measurement QND if the observable being measuredis an eigenstate of the ideal Hamiltonian of the mea-sured system i.e., one ignores any couplings betweenthis system and sources of dissipation. This is reason-able if such couplings give rise to dynamics on timescales longer than needed to complete the measurement.This point is elaborated in Sec. VII, where we discusspractical considerations related to the quantum limit.We also discuss in that section the fact that the repeat-ability of QND measurements is of fundamental practi-cal importance in overcoming detector inefficienciesGambetta et al., 2007.

A common confusion is to think that a QND measure-ment has no effect on the state of the system being mea-sured. While this is true if the initial state is an eigen-state of the observable, it is not true in general. Consideragain our example of a spin oriented in the x direction.The result of the first z measurement will be that thestate randomly and completely unpredictably collapsesonto one of the two z eigenstates: the state is indeedaltered by the measurement. However, all subsequentmeasurements using the same orientation for the detec-tors will always agree with the result of the first mea-surement. Thus QND measurements may affect thestate of the system, but never the value of the observ-able once it is determined. Other examples of QNDmeasurements include i measurement of the electro-magnetic field energy stored inside a cavity by determin-ing the radiation pressure exerted on a moving pistonBraginsky and Khalili, 1992, ii detection of the pres-ence of a photon in a cavity by its effect on the phase ofan atom’s superposition state Nogues et al., 1999;Haroche and Raimond, 2006, and iii the “dispersive”measurement of a qubit state by its effect on the fre-quency of a driven microwave resonator Blais et al.,2004; Wallraff et al., 2004; Lupascu et al., 2007, which isthe first canonical example described below.

In contrast to the above, in non-QND measurementsthe back-action of the measurement will affect the ob-servable being studied. The canonical example we con-sider is the position measurement of a harmonic oscilla-tor. Since the position operator does not commute withthe Hamiltonian, the QND criterion is not satisfied.Other examples of non-QND measurements include iphoton counting via photodetectors that absorb the pho-tons, ii continuous measurements where the observ-able does not commute with the Hamiltonian, thus in-ducing a time dependence of the measurement result,and iii measurements that can be repeated only after atime longer than the energy relaxation time of the sys-tem e.g., for a qubit, T1.

+d

zz

FIG. 2. Color online Schematic of position distributions of anatom in the detector plane of a Stern-Gerlach apparatus whosefield gradient is in the z direction. For small values of thedisplacement d described in the text, there is significant over-lap of the distributions and the spin cannot be unambiguouslyinferred from the position. For large values of d, the spin isperfectly entangled with position and can be inferred from theposition. This is the limit of strong projective measurement.



A. Weak continuous measurements

In discussing “real” quantum measurements, anotherkey notion to introduce is that of weak continuous mea-surements Braginsky and Khalili, 1992. Many measure-ments in practice take an extended time interval to com-plete, which is much longer than the “microscopic” timescales oscillation periods, etc. of the system. The rea-son may be quite simply that the coupling strength be-tween the detector and the system cannot be made arbi-trarily large, and one has to wait for the effect of thesystem on the detector to accumulate. For example, inour Stern-Gerlach measurement, suppose that we areonly able to achieve small magnetic field gradients andthat, consequently, the displacement d cannot be madelarge compared to the wave packet spread see Fig. 2.In this case the states ± would have nonzero overlapand it would not be possible to reliably distinguish them:we thus would have only a weak measurement. How-ever, by cascading together a series of such measure-ments and taking advantage of the fact that they areQND, we can eventually achieve an unambiguous strongprojective measurement: at the end of the cascade, weare certain of which z eigenstate the spin is in. Duringthis process, the overlap of ± would gradually fall tozero corresponding to a smooth continuous loss of phasecoherence in the transverse spin components. At the endof the process, the QND nature of the measurement en-sures that the probability of measuring z=↑ or ↓ willaccurately reflect the initial wave function. Note that it isonly in this case of weak continuous measurements thatit makes sense to define a measurement rate in terms ofa rate of gain of information about the variable beingmeasured, and a corresponding dephasing rate, the rateat which information about the conjugate variable is be-ing lost. We see that these rates are intimately relatedvia the Heisenberg uncertainty principle.

While strong projective measurements may seem tobe the ideal, there are many cases where one may inten-tionally desire a weak continuous measurement; as dis-cussed in the Introduction. There are many practical ex-amples of weak continuous measurement schemes.These include i charge measurements, where the cur-rent through a device e.g., quantum point contactQPC or single-electron transistor SET is modulatedby the presence or absence of a nearby charge, andwhere it is necessary to wait for a sufficiently long timeto overcome the shot noise and distinguish between thetwo current values, ii the weak dispersive qubit mea-surement discussed below, and iii displacement detec-tion of a nanomechanical beam e.g., optically or by ca-pacitive coupling to a charge sensor, where one looks atthe two quadrature amplitudes of the signal produced atthe beam’s resonance frequency.

Not surprisingly, quantum noise plays a crucial role indetermining the properties of a weak continuous quan-tum measurement. For such measurements, noise bothdetermines the back-action effect of the measurementon the measured system and how quickly information isacquired in the measurement process. Previously, we

saw that a crucial feature of quantum noise is the asym-metry between positive and negative frequencies; wefurther saw that this corresponds to the difference be-tween absorption and emission events. For measure-ments, another key aspect of quantum noise will be im-portant: as will be discussed extensively, quantummechanics places constraints on the noise of any systemcapable of acting as a detector or amplifier. These con-straints in turn place restrictions on any weak continu-ous measurement, and lead directly to quantum limitson how well one can make such a measurement.

In the remainder of this section, we give an introduc-tion to how one describes a weak continuous quantummeasurement, considering the specific examples of usingparametric coupling to a resonant cavity for QND detec-tion of the state of a qubit and the necessarily non-QND detection of the position of a harmonic oscillator.In the following section Sec. IV, we give a derivationof a very general quantum-mechanical constraint on thenoise of any system capable of acting as a detector, andshow how this constraint directly leads to the quantumlimit on qubit detection. Finally, in Sec. V, we turn to theimportant but slightly more involved case of a quantumlinear amplifier or position detector. We show that thebasic quantum noise constraint derived Sec. IV againleads to a quantum limit; here this limit is on how smallone can make the added noise of a linear amplifier.

Before leaving this section, it is worth pointing outthat the theory of weak continuous measurements issometimes described in terms of some set of auxiliarysystems which are sequentially and momentarily weaklycoupled to the system being measured see AppendixE. One then envisions a sequence of projective vonNeumann measurements on the auxiliary variables. Theweak entanglement between the system of interest andone of the auxiliary variables leads to a kind of partialcollapse of the system wave function more precisely, thedensity matrix, which is described in mathematicalterms not by projection operators, but rather by positiveoperator valued measures POVMs. We will not usethis and the related “quantum trajectory” language here,but direct the interested reader to the literature formore information on this important approach Peres,1993; Brun, 2002; Haroche and Raimond, 2006; Jordanand Korotkov, 2006.

B. Measurement with a parametrically coupled resonantcavity

A simple yet experimentally practical example of aquantum detector consists of a resonant optical or rfcavity parametrically coupled to the system being mea-sured. Changes in the variable being measured e.g., thestate of a qubit or the position of an oscillator shift thecavity frequency and produce a varying phase shift inthe carrier signal reflected from the cavity. This changingphase shift can be converted via homodyne interferom-etry into a changing intensity; this can then be detectedusing diodes or photomultipliers.



In this section, we analyze weak continuous measure-ments made using such a parametric cavity detector; thiswill serve as a good introduction to the more generalapproaches presented later. We show that this detector iscapable of reaching the “quantum limit” meaning that itcan be used to make a weak continuous measurement,as optimally as is allowed by quantum mechanics. This istrue for both the QND measurement of the state of aqubit and the non-QND measurement of the positionof a harmonic oscillator. Complementary analyses ofweak continuous qubit measurement are given by Ma-khlin et al. 2000, 2001 using a single-electron transis-tor and by Gurvitz 1997, Korotkov 2001b, Korotkovand Averin 2001, Pilgram and Büttiker 2002, andClerk et al. 2003 using a quantum point contact. Wefocus here on a high-Q cavity detector; weak qubit mea-surement with a low-Q cavity was studied by Johanssonet al. 2006.

It is worth noting the widespread usage of cavity de-tectors in experiment. One important current realizationis a microwave cavity used to read out the state of asuperconducting qubit Il’ichev et al., 2003; Blais et al.,2004; Izmalkov et al., 2004; Lupascu et al., 2004, 2005;Wallraff et al., 2004; Duty et al., 2005; Schuster et al.,2005; Sillanpää et al., 2005. Another class of examplesare optical cavities used to measure mechanical degreeof freedom. Examples of such systems include thosewhere one of the cavity mirrors is mounted on a canti-lever Arcizet et al., 2006; Gigan et al., 2006; Klecknerand Bouwmeester, 2006. Related systems involve afreely suspended mass Abramovici et al., 1992; Corbittet al., 2007, an optical cavity with a thin transparentmembrane in the middle Thompson et al., 2008, and,more generally, an elastically deformable whispering gal-lery mode resonator Schliesser et al., 2006. Systemswhere a microwave cavity is coupled to a mechanicalelement are also under active study Blencowe andBuks, 2007; Regal et al., 2008; Teufel et al., 2008.

We start our discussion with a general observation.The cavity uses interference and the wave nature of lightto convert the input signal to a phase-shifted wave. Forsmall phase shifts we have a weak continuous measure-ment. Interestingly, it is the complementary particle na-ture of light that turns out to limit the measurement. Aswe will see, it both limits the rate at which we can makea measurement via photon shot noise in the outputbeam and also controls the back-action disturbance ofthe system being measured due to photon shot noiseinside the cavity acting on the system being measured.These dual aspects are an important part of any weakcontinuous quantum measurement; hence, an under-standing of both the output noise i.e., the measurementimprecision and back-action noise of detectors will becrucial.

All of our discussion of noise in the cavity system willbe framed in terms of the number-phase uncertainty re-lation for coherent states. A coherent photon state con-tains a Poisson distribution of the number of photons,implying that the fluctuations in photon number obey

N2=N, where N is the mean number of photons. Fur-

ther, coherent states are overcomplete and states of dif-ferent phase are not orthogonal to each other; this di-rectly implies see Appendix G that there is anuncertainty in any measurement of the phase. For large

N, this is given by

2 = 1/4N . 3.5

Thus, large-N coherent states obey the number-phaseuncertainty relation

N = 12 , 3.6

analogous to the position-momentum uncertainty rela-tion.

Equation 3.6 can also be usefully formulated interms of noise spectral densities associated with themeasurement. Consider a continuous photon beam car-

rying an average photon flux N. The variance in thenumber of photons detected grows linearly in time andcan be represented as N2=SNNt, where SNN is thewhite-noise spectral density of photon flux fluctuations.On a physical level, it describes photon shot noise, and is

given by SNN= N.Consider now the phase of the beam. Any homodyne

measurement of this phase will be subject to the samephoton shot noise fluctuations discussed above see Ap-pendix G for more details. Thus, if the phase of thebeam has some nominal small value 0, the output signalfrom the homodyne detector integrated up to time t willbe of the form I=0t+0

t d, where is a noiserepresenting the imprecision in our measurement of 0due to the photon shot noise in the output of the homo-dyne detector. An unbiased estimate of the phase ob-tained from I is =I / t, which obeys =0. Further, onehas 2=S / t, where S is the spectral density of the white noise. Comparison with Eq. 3.5 yields

S = 1/4N . 3.7

The above results lead us to the fundamental wave-particle relation for ideal coherent beams,

SNNS = 12 . 3.8

Before we study the role that these uncertainty rela-tions play in measurements with high-Q cavities, con-sider the simplest case of reflection of light from a mir-ror without a cavity. The phase shift of the beam havingwave vector k when the mirror moves a distance x is2kx. Thus, the uncertainty in the phase measurementcorresponds to a position imprecision which can againbe represented in terms of a noise spectral density Sxx

I

=S /4k2. Here the superscript I refers to the fact thatthis is noise representing imprecision in the measure-ment, not actual fluctuations in the position. We alsoneed to worry about back-action: each photon hittingthe mirror transfers a momentum 2k to the mirror, sophoton shot noise corresponds to a random back-actionforce noise spectral density SFF=42k2SNN. Multiplying



these together, we have the central result for the productof the back-action force noise and the imprecision,

SFFSxxI = 2SNNS = 2/4, 3.9

or in analogy with Eq. 3.6

SFFSxxI = /2. 3.10

Not surprisingly, the situation considered here is as idealas possible. Thus, the RHS above is actually a lowerbound on the product of imprecision and back-actionnoise for any detector capable of significant amplifica-tion; we will prove this rigorously in Sec. IV.A Equation3.10 thus represents the quantum limit on the noise ofour detector. As we will see shortly, having a detectorwith quantum-limited noise is a prerequisite for reachingthe quantum limit on various different measurementtasks e.g., continuous position detection of an oscillatorand QND qubit state detection. Note that in general, agiven detector will have more noise than the quantum-limited value; we devote considerable effort later to de-termining the conditions needed to achieve the lowerbound of Eq. 3.10.

We now turn to the story of measurement using ahigh-Q cavity; it will be similar to the above discussion,except that we have to account for the filtering of thenoise by the cavity response. We relegate relevant calcu-lational details to Appendix E. The cavity is simply de-scribed as a single bosonic mode coupled weakly to elec-tromagnetic modes outside the cavity. The Hamiltonianof the system is given by

H = H0 + c1 + Aza†a + Henvt. 3.11

Here H0 is the unperturbed Hamiltonian of the systemwhose variable z which is not necessarily a position isbeing measured, a is the annihilation operator for thecavity mode, and c is the cavity resonance frequency inthe absence of the coupling A. We will take both A and

z to be dimensionless. The term Henvt describes the elec-tromagnetic modes outside the cavity, and their couplingto the cavity; it is responsible for both driving and damp-ing the cavity mode. The damping is parametrized byrate , which tells us how quickly energy leaks out of thecavity; we consider the case of a high-quality factor cav-ity, where Qcc /1.

Turning to the interaction term in Eq. 3.11, we seethat the parametric coupling strength A determines thechange in frequency of the cavity as the system variablez changes. We assume for simplicity that the dynamics ofz is slow compared to . In this limit the reflected phaseshift simply varies slowly in time adiabatically followingthe instantaneous value of z. We also assume that thecoupling A is small enough that the phase shifts are al-ways very small and hence the measurement is weak.Many photons will have to pass through the cavity be-fore much information is gained about the value of thephase shift and hence the value of z.

We first consider the case of a one-sided cavity whereonly one of the mirrors is semitransparent, the other

being perfectly reflecting. In this case, a wave incidenton the cavity say, in a one-dimensional 1D waveguidewill be perfectly reflected, but with a phase shift deter-mined by the cavity and the value of z. The reflectioncoefficient at the bare cavity frequency c is simplygiven by Walls and Milburn, 1994

r = − 1 + 2iAQcz/1 − 2iAQcz . 3.12

Note that r has unit magnitude because all incident pho-tons are reflected if the cavity is lossless. For weak cou-pling we can write the reflection phase shift as r=−ei,where

4QcAz = AcztWD. 3.13

We see that the scattering phase shift is simply the fre-quency shift caused by the parametric coupling multi-plied by the Wigner delay time Wigner, 1955

tWD = Im ln r/ = 4/ . 3.14

Thus the measurement-imprecision noise power for a

given photon flux N incident on the cavity is given by

SzzI = S/ActWD2. 3.15

The random part of the generalized back-action forceconjugate to z is, from Eq. 3.11,

Fz − H/z = − Acn , 3.16

where, since z is dimensionless, Fz has units of energy.Here n= n− n= a†a− a†a represents the photon-number fluctuations around the mean n inside the cavity.The back-action force noise spectral density is thus

SFzFz= Ac2Snn. 3.17

As shown in Appendix E, the cavity filters the photonshot noise so that at low frequencies , the numberfluctuation spectral density is simply

Snn = ntWD. 3.18

The mean photon number in the cavity is found to be

n= NtWD, where again N is the mean photon flux inci-dent on the cavity. From this it follows that

SFzFz= ActWD2SNN. 3.19

Combining this with Eq. 3.15 again yields the sameresult as Eq. 3.10 obtained without the cavity. Theparametric cavity detector used in this way is thus aquantum-limited detector, meaning that the product ofits noise spectral densities achieves the ideal minimumvalue.

We now examine how the quantum limit on the noiseof our detector directly leads to quantum limits on dif-ferent measurement tasks. In particular, we consider thecases of continuous position detection and QND qubitstate measurement.



1. QND measurement of the state of a qubit using a resonantcavity

Here we specialize to the case where the system op-erator z= z represents the state of a spin-1 /2 quantumbit. Equation 3.11 becomes

H = 1201z + c1 + Aza†a + Henvt. 3.20

We see that z commutes with all terms in the Hamil-tonian and is thus a constant of the motion assuming

that Henvt contains no qubit decay terms so that T1=and hence the measurement will be QND. From Eq.3.13 we see that the two states of the qubit producephase shifts ±0 where

0 = ActWD. 3.21

As 01, we need many reflected photons before weare able to determine the state of the qubit. This is adirect consequence of the unavoidable photon shotnoise in the output of the detector, and is a basic featureof weak measurements—information on the input is ac-quired only gradually in time.

Let It be the homodyne signal for the wave reflectedfrom the cavity integrated up to time t. Depending onthe state of the qubit the mean value of I will be I= ±0t, and the rms Gaussian fluctuations about themean will be I=St. As shown in Fig. 3 and discussedby in Makhlin et al. 2001, the integrated signal is drawnfrom one of two Gaussian distributions, which are betterand better resolved with increasing time as long as themeasurement is QND. The state of the qubit thus be-comes ever more reliably determined. The signal energyto noise energy ratio becomes

SNR = I2/ I2 = 02/St , 3.22

which can be used to define the measurement rate via

meas SNR/2 = 02/2S = 1/2Szz

I . 3.23

There is a certain arbitrariness in the scale factor of 2appearing in the definition of the measurement rate; thisparticular choice is motivated by precise information-theoretic grounds as defined, meas is the rate at whichthe “accessible information” grows, Appendix F.

While Eq. 3.20 makes it clear that the state of thequbit modulates the cavity frequency, we can easily re-write this equation to show that this same interactionterm is also responsible for the back-action of the mea-surement i.e., the disturbance of the qubit state by themeasurement process

H = /201 + 2Aca†az + ca

†a + Henvt. 3.24

We now see that the interaction can also be viewed asproviding a “light shift” i.e., ac Stark shift of the qubitsplitting frequency Blais et al., 2004; Schuster et al.,2005 which contains a constant part 2AnAc plus a ran-

domly fluctuating part 01=2Fz /, that depends on n= a†a, the number of photons in the cavity. During ameasurement, n will fluctuate around its mean and act asa fluctuating back-action “force” on the qubit. In thepresent QND case, noise in n= a†a cannot cause transi-tions between the two qubit eigenstates. This is the op-posite of the situation considered in Sec. II.B, where wewanted to use the qubit as a spectrometer. Despite thelack of any noise-induced transitions, there still is aback-action here, as noise in n causes the effective split-ting frequency of the qubit to fluctuate in time. For weakcoupling, the resulting phase diffusion leads tomeasurement-induced dephasing of superpositions inthe qubit Blais et al., 2004; Schuster et al., 2005 accord-ing to

e−i =exp− i0

t

d 01 . 3.25

For weak coupling the dephasing rate is slow and thuswe are interested in long times t. In this limit the integralis a sum of a large number of statistically independentterms and thus we can take the accumulated phase to beGaussian distributed. Using the cumulant expansion wethen obtain

e−i = exp−120

t

d 012= exp−

2

2SFzFzt . 3.26

Note also that the noise correlator above is naturallysymmetrized—the quantum asymmetry of the noiseplays no role for this type of coupling. Equation 3.26yields the dephasing rate

= 2/2SFzFz= 20

2SNN. 3.27

Using Eqs. 3.23 and 3.27, we find the interestingconclusion that the dephasing rate and measurementrates coincide,

/meas = 4/2SzzI SFzFz

= 4SNNS = 1. 3.28

As we will see and prove rigorously, this represents theideal quantum-limited case for QND qubit detection:the best one can do is measure as quickly as onedephases. In keeping with our earlier discussions, it rep-resents the enforcement of the Heisenberg uncertainty

I(t)

t

ΔI

FIG. 3. Color online Distribution of the integrated output forthe cavity detector It for the two different qubit states. Theseparation of the means of the distributions grows linearly intime, while the widths of the distributions grow only as t.



principle. The faster you gain information about onevariable, the faster you lose information about the con-jugate variable. Note that, in general, the ratio /measwill be larger than 1, as an arbitrary detector will notreach the quantum limit on its noise spectral densities.Such a nonideal detector produces excess back-actionbeyond what is required quantum mechanically.

In addition to the quantum noise point of view pre-sented above, there is a second complementary way inwhich to understand the origin of measurement-induceddephasing Stern et al., 1990, which is analogous to ourdescription of loss of transverse spin coherence in theStern-Gerlach experiment in Eq. 3.3. The measure-ment takes the incident wave, described by a coherentstate , to a reflected wave described by a phaseshifted coherent state r↑ · or r↓ ·, where r↑/↓ is thequbit-dependent reflection amplitude given in Eq. 3.12.Considering now the full state of the qubit-plus-detector,the measurement results in a state change:

12

↑ + ↓ →12

e+i01t/2↑ r↑ ·

+ e−i01t/2↓ r↓ · . 3.29

As r↑ · r↓ ·, the qubit has become entangled withthe detector: the above state cannot be written as aproduct of a qubit state times a detector state. To assessthe coherence of the final qubit state i.e., the relativephase between ↑ and ↓, one looks at the off-diagonalmatrix element of the qubit’s reduced density matrix,

↓↑ Trdetector↓↑ 3.30

=e+i01t/2r↑ · r↓ · 3.31

=e+i01t/2 exp− 21 − r↑*r↓ . 3.32

In Eq. 3.31 we have used the usual expression for theoverlap of two coherent states. We see that the measure-ment reduces the magnitude of ↑↓: this is dephasing.The amount of dephasing is directly related to the over-lap between the different detector states that resultwhen the qubit is up or down; this overlap can be

straightforwardly found using Eq. 3.32 and 2=N

= Nt, where N is the mean number of photons that havereflected from the cavity after time t. We have

exp− 21 − r↑*r↓ = exp− 2N0

2 exp− t ,

3.33

with the dephasing rate given by

= 202N 3.34

in complete agreement with the previous result in Eq.3.27.

2. Quantum limit relation for QND qubit state detection

We now return to the ideal quantum limit relation ofEq. 3.28. As stated previously, this is a lower bound:

quantum mechanics enforces the constraint that in aQND qubit measurement the best you can possibly do ismeasure as quickly as you dephase Devoret and Schoel-kopf, 2000; Korotkov and Averin, 2001; Makhlin et al.,2001; Averin, 2000b, 2003; Clerk et al., 2003,

meas . 3.35

While a detector with quantum-limited noise has anequality above, most detectors will be very far from thisideal limit, and will dephase the qubit faster than theyacquire information about its state. We provide a proofof Eq. 3.35 in Sec. IV.B; for now, we note that its heu-ristic origin rests on the fact that both measurement anddephasing rely on the qubit becoming entangled withthe detector. Consider again Eq. 3.29, describing theevolution of the qubit-detector system when the qubit isinitially in a superposition of ↑ and ↓. To say that wehave truly measured the qubit, the two detector statesr↑ and r↓ need to correspond to different values ofthe detector output i.e., phase shift in our example;this necessarily implies they are orthogonal. This in turnimplies that the qubit is completely dephased: ↑↓=0,just as we saw in Eq. 3.4 in the Stern-Gerlach example.Thus, measurement implies dephasing. The opposite isnot true. The two states r↑ and r↓ could in principlebe orthogonal without them corresponding to differentvalues of the detector output i.e., . For example, thequbit may have become entangled with extraneous mi-croscopic degrees of freedom in the detector. Thus, on aheuristic level, the origin of Eq. 3.35 is clear.

Returning to our one-sided cavity system, we see fromEq. 3.28 that the one-sided cavity detector reaches thequantum limit. It is natural to now ask why this is thecase: Is there a general principle in action here that al-lows the one-sided cavity to reach the quantum limit?The answer is yes: reaching the quantum limit requiresthat there is no “wasted” information in the detectorClerk et al., 2003. There should not exist any unmea-sured quantity in the detector which could have beenprobed to learn more about the state of the qubit. In thesingle-sided cavity detector, information on the state ofthe qubit is only available in that is, is entirely encodedin the phase shift of the reflected beam; thus, there is no“wasted” information, and the detector does indeedreach the quantum limit.

To make this idea of “no wasted information” moreconcrete, we now consider a simple detector that fails toreach the quantum limit precisely due to wasted infor-mation. Consider again a 1D cavity system where nowboth mirrors are slightly transparent. Now, a wave inci-dent at frequency R on one end of the cavity will bepartially reflected and partially transmitted. If the initialincident wave is described by a coherent state , thescattered state can be described by a tensor product ofthe reflected wave’s state and the transmitted wave’sstate,



→ r · t · , 3.36

where the qubit-dependent reflection and transmissionamplitudes r and t are given by Walls and Milburn,1994

t↓ = 1/1 + 2iAQc , 3.37

r↓ = 2iQcA/1 + 2iAQc , 3.38

with t↑= t↓* and r↑= r↓*. Note that the incident beam isalmost perfectly transmitted: t2=1−OAQc2.

Similar to the one-sided case, the two-sided cavitycould be used to make a measurement by monitoringthe phase of the transmitted wave. Using the expressionfor t above, we find that the qubit-dependent transmis-sion phase shift is given by

↑/↓ = ± 0 = ± 2AQc, 3.39

where again the two signs correspond to the two differ-ent qubit eigenstates. The phase shift for transmission isonly half as large as in reflection so the Wigner delaytime associated with transmission is

tWD = 2/ . 3.40

Upon making the substitution of tWD for tWD, the one-sided cavity Eqs. 3.15 and 3.17 remain valid. How-ever, the internal cavity photon-number shot noise re-mains fixed so that Eq. 3.18 becomes

Snn = 2ntWD. 3.41

which means that

Snn = 2NtWD2 = 2SNNtWD

2 3.42

and

SFzFz= 22A2c

2tWD2 SNN. 3.43

As a result the back-action dephasing doubles relative tothe measurement rate and we have

meas/ = 2SNNS = 12 . 3.44

Thus the two-sided cavity fails to reach the quantumlimit by a factor of 2.

Using the entanglement picture, we may again alter-natively calculate the amount of dephasing from theoverlap between the detector states corresponding tothe qubit states ↑ and ↓ cf. Eq. 3.31. We find

e−t = t↑t↓r↑r↓ 3.45

=exp− 21 − t↑*t↓ − r↑*r↓ . 3.46

Note that both the changes in the transmission and re-flection amplitudes contribute to the dephasing of thequbit. Using the above expressions, we find

t = 4022 = 402N = 402Nt = 2meast . 3.47

Thus, in agreement with the quantum noise result, thetwo-sided cavity misses the quantum limit by a factor of2.

Why does the two-sided cavity fail to reach the quan-tum limit? The answer is clear from Eq. 3.46: eventhough we are not monitoring it, there is information onthe state of the qubit available in the phase of the re-flected wave. Note from Eq. 3.38 that the magnitude ofthe reflected wave is weak A2, but unlike the trans-mitted wave the difference in the reflection phase asso-ciated with the two qubit states is large ± /2. The“missing information” in the reflected beam makes a di-rect contribution to the dephasing rate i.e., the secondterm in Eq. 3.46, making it larger than the measure-ment rate associated with measurement of the transmis-sion phase shift. In fact, there is an equal amount ofinformation in the reflected beam as in the transmittedbeam, so the dephasing rate is doubled. We thus have aconcrete example of the general principle connecting afailure to reach the quantum limit to the presence ofwasted information. Note that the application of thisprinciple to generalized quantum point contact detectorsis found in Clerk et al. 2003.

Returning to our cavity detector, we note in closingthat it is often technically easier to work with the trans-mission of light through a two-sided cavity, rather thanreflection from a one-sided cavity. One can still reachthe quantum limit in the two-sided cavity case if oneuses an asymmetric cavity in which the input mirror hasmuch less transmission than the output mirror. Mostphotons are reflected at the input, but those that enterthe cavity will almost certainly be transmitted. The priceto be paid is that the input carrier power must be in-creased.

3. Measurement of oscillator position using a resonant cavity

The qubit measurement discussed previously was anexample of a QND measurement: the back-action didnot affect the observable being measured. We now con-sider the simplest example of a non-QND measurement,namely, the weak continuous measurement of the posi-tion of a harmonic oscillator. The detector will again bea parametrically coupled resonant cavity, where the po-sition of the oscillator x changes the frequency of thecavity as per Eq. 3.11 see, e.g., Tittonen et al. 1999.Similarly to the qubit case, for a sufficiently weak cou-pling the phase shift of the reflected beam from the cav-ity will depend linearly on the position x of the oscillatorcf. Eq. 3.13; by reading out this phase, we may thusmeasure x. The origin of back-action noise is the same asbefore, namely, photon shot noise in the cavity. Now,however, this represents a random force which changesthe momentum of the oscillator. During the subsequenttime evolution these random force perturbations will re-appear as random fluctuations in the position. Thus themeasurement is not QND. This will mean that the mini-mum uncertainty of even an ideal measurement is largerby exactly a factor of 2 than the “true” quantum un-certainty of the position i.e., the ground-state uncer-tainty. This is known as the standard quantum limit onweak continuous position detection. It is also an ex-ample of a general principle that a linear phase-



preserving amplifier necessarily adds noise, and that theminimum added noise exactly doubles the output noisefor the case where the input is vacuum i.e., zero-pointnoise. A more general discussion of the quantum limiton amplifiers and position detectors will be presented inSec. V.

We start by emphasizing that we are speaking here ofa weak continuous measurement of the oscillator posi-tion. The measurement is sufficiently weak that the po-sition undergoes many cycles of oscillation before sig-nificant information is acquired. Thus we are not talkingabout the instantaneous position but rather the overallamplitude and phase, or more precisely the two quadra-ture amplitudes describing the smooth envelope of themotion,

xt = Xtcost + Ytsint . 3.48

One can easily show that, for an oscillator, the two

quadrature amplitudes X and Y are canonically conju-gate and hence do not commute with each other,

X,Y = i/M = 2ixZPF2 . 3.49

As the measurement is both weak and continuous, it will

yield information on both X and Y. As such, one is ef-fectively trying to simultaneously measure two incom-patible observables. This basic fact is intimately relatedto the property mentioned above, that even a com-pletely ideal weak continuous position measurement willhave a total uncertainty that is twice the zero-point un-certainty.

We are now ready to start our heuristic analysis ofposition detection using a cavity detector; relevant cal-culational details presented in Appendix E.3. Considerfirst the mechanical oscillator we wish to measure. Wetake it to be a simple harmonic oscillator of natural fre-quency and mechanical damping rate 0. For weakdamping, and at zero coupling to the detector, the spec-tral density of the oscillator’s position fluctuations isgiven by Eq. 2.4 with the delta function replaced by aLorentzian5

Sxx = xZPF2 nB

0

+2 + 0/22

+ nB + 10

−2 + 0/22 . 3.50

When we now weakly couple the oscillator to the cav-ity as per Eq. 3.11, with z= x /xZPF and drive the cav-ity on resonance, the phase shift of the reflected beamwill be proportional to x „i.e., t= d /dxxt…. Assuch, the oscillator’s position fluctuations will cause ad-

ditional fluctuations of the phase , over and above theintrinsic shot noise-induced phase fluctuations S. Weconsider the usual case where the noise spectrometerbeing used to measure the noise in i.e., the noise inthe homodyne current measures the symmetric-in-frequency noise spectral density; as such, it is thesymmetric-in-frequency position noise that we detect. Inthe classical limit kBT, this is given by

Sxx 12

Sxx + Sxx−

kBT

2M2

0

−2 + 0/22 . 3.51

If we ignore back-action effects, we expect to see thisLorentzian profile riding on top of the background im-precision noise floor; this is shown in Fig. 4.

Note that additional stages of amplification would alsoadd noise, and would thus further augment this back-ground noise floor. If we subtract off this noise floor, thefull width at half maximum of the curve will give thedamping parameter 0, and the area under the experi-mental curve,

−

d

2Sxx =

kBT

M2 , 3.52

measures the temperature. What the experimentalist ac-tually plots in making such a curve is the output of theentire detector-plus-following-amplifier chain. Impor-tantly, if the temperature is known, then the area of themeasured curve can be used to calibrate the coupling ofthe detector and the gain of the total overall amplifierchain see, e.g., LaHaye et al., 2004; Flowers-Jacobs etal., 2007. One can thus make a calibrated plot where themeasured output noise is referred back to the oscillatorposition.

5This form is valid only for weak damping because we areassuming that the oscillator frequency is still sharply defined.We have evaluated the Bose-Einstein factor exactly at fre-quency and have assumed that the Lorentzian centered atpositive negative frequency has negligible weight at negativepositive frequencies.

12

10

8

6

4

2

0

OutputN

oise

S(ω)(arb.units)

2.01.51.00.50.0Frequency

area prop. to Tmeasurementimprecision

ω / Ω

FIG. 4. Color online Spectral density of the symmetrizedoutput noise S of a linear position detector. The oscilla-tor’s noise appears as a Lorentzian on top of a noise floor i.e.,the measurement imprecision. As discussed in the text, thewidth of the peak is proportional to the oscillator damping rate0, while the area under the peak is proportional to tempera-ture. This latter fact can be used to calibrate the response ofthe detector.



Consider now the case where the oscillator is at zerotemperature. Equation 3.50 then yields for the symme-trized noise spectral density

Sxx0 = xZPF

2 0/2

−2 + 0/22 . 3.53

One might expect that one could see this Lorentziandirectly in the output noise of the detector i.e., the noise, above the measurement-imprecision noise floor.However, this neglects the effects of measurement back-action. From the classical equation of motion we expectthe response of the oscillator to the back-action forceF=Fz /xZPF cf. Eq. 3.16 at frequency to produce anadditional displacement x=xxF, wherexx is the mechanical susceptibility

xx 1

M

1

2 − 2 − i0. 3.54

These extra oscillator fluctuations will show up as addi-tional fluctuations in the output of the detector. For sim-plicity, we focus on this noise at the oscillator’s reso-nance frequency . As a result of the detector’s back-action, the total measured position noise i.e., inferredspectral density at the frequency is given by

Sxx,tot = Sxx0 +

xx2

2†SFF+ + SFF−‡

+12†Sxx

I + + SxxI −‡ 3.55

=Sxx0 + Sxx,add . 3.56

The first term here is just the intrinsic zero-point noiseof the oscillator,

Sxx0 = 2xZPF

2 /0 = xx . 3.57

The second term Sxx,add is the total noise added by themeasurement, and includes both the measurement im-precision Sxx

I SzzI xZPF

2 and the extra fluctuations caused

by the back-action. We stress that Sxx,tot corresponds to aposition noise spectral density inferred from the outputof the detector: one simply scales the spectral density of

total output fluctuations S,tot by d /dx2.Implicit in Eq. 3.57 is the assumption that the back-

action noise and the imprecision noise are uncorrelatedand thus add in quadrature. It is not obvious that this iscorrect, since in the cavity detector the back-action noiseand output shot noise are both caused by the vacuumnoise in the beam incident on the cavity. It turns out thatthere are indeed correlations, however, the symmetrized

i.e., classical correlator SF does vanish for our choiceof a resonant cavity drive. Further, Eq. 3.55 assumesthat the measurement does not change the damping rateof the oscillator. Again, while this will not be true for anarbitrary detector, it is the case here for the cavity de-tector when as we have assumed it is driven on reso-nance. Details justifying both these statements are given

in Appendix E; the more general case with nonzeronoise correlations and back-action damping is discussedin Sec. V.E.

Assuming we have a quantum-limited detector thatobeys Eq. 3.9 i.e., Sxx

I SFF=2 /4 and that the shotnoise is symmetric in frequency, the added position noisespectral density at resonance i.e., second term in Eq.3.56 becomes

Sxx,add = 2SFF +2

41

SFF . 3.58

Recall from Eq. 3.19 that the back-action noise is pro-portional to the coupling of the oscillator to the detectorand to the intensity of the drive on the cavity. The addedposition-uncertainty noise is plotted in Fig. 5 as a func-tion of SFF. We see that for high drive intensity the back-action noise dominates the position uncertainty, whilefor low drive intensity the output shot noise the lastterm in the equation above dominates.

The added noise and hence the total noise Sxx,totis minimized when the drive intensity is tuned so thatSFF is equal to SFF,opt, with

SFF,opt = /2xx = /2M0. 3.59

The more heavily damped is the oscillator, the less sus-ceptible it is to back-action noise and hence the higher isthe optimal coupling. At the optimal coupling strength,the measurement-imprecision noise and back-actionnoise each make equal contributions to the added noise,yielding

Sxx,add = /M0 = Sxx0 . 3.60

Thus, the spectral density of the added position noise isexactly equal to the noise power associated with the os-cillator’s zero-point fluctuations. This represents a mini-

10

8

6

4

2

Sxx

,add

[(x Z

PT)2

/γ0

]

0.1 1 10

Coupling SFF / SFF,opt

Total added position noiseMeasurement imprecisionContribution from backaction

FIG. 5. Color online Noise power of the added position noiseof a linear position detector, evaluated at the oscillator’s reso-nance frequency Sxx,add as a function of the magnitude ofthe back-action noise spectral density SFF. SFF is proportionalto the oscillator-detector coupling, and in the case of the cavitydetector is also proportional to the power incident on the cav-ity. The optimal value of SFF is given by SFF,opt=M0 /2 cf.Eq. 3.59. We have assumed that there are no correlationsbetween measurement-imprecision noise and back-actionnoise, as is appropriate for the cavity detector.



mum value for the added noise of any linear positiondetector, and is referred to as the standard quantumlimit on position detection. Note that this limit only in-volves the added noise of the detector, and thus hasnothing to do with the initial temperature of the oscilla-tor.

We emphasize that to reach the above quantum limiton weak continuous position detection one needs thedetector itself to be quantum limited, i.e., the productSFFSxx

I must be as small as is allowed by quantum me-chanics, namely, 2 /4. Having a quantum-limited detec-tor, however, is not enough: in addition, one must beable to achieve sufficiently strong coupling to reach theoptimum given in Eq. 3.59. Further, the measured out-put noise must be dominated by the output noise of thecavity, not by the added noise of following amplifierstages.

A related, stronger quantum limit refers to the total

inferred position noise from the measurement Sxx,tot.It follows from Eqs. 3.60 and 3.56 that at resonancethe smallest this can be is twice the oscillator’s zero-point noise

Sxx,tot = 2Sxx0 . 3.61

Half the noise here is from the oscillator itself, half isfrom the added noise of the detector. It is even morechallenging to reach this quantum limit: one needs toboth reach the quantum limit on the added noise andcool the oscillator to its ground state.

Finally, we emphasize that the optimal value of thecoupling derived above was specific to the choice ofminimizing the total position noise power at the reso-nance frequency. If a different frequency had been cho-sen, the optimal coupling would have been different;one again finds that the minimum possible added noisecorresponds to the ground-state noise at that frequency.It is interesting to ask what the total position noisewould be as a function of frequency, assuming that thecoupling has been optimized to minimize the noise atthe resonance frequency, and that the oscillator is ini-tially in the ground state. From our results above wehave

Sxx,tot = xZPF2 0/2

−2 + 0/22

+

2 xx2

xx+ xx

xZPF

2

01 + 3

0/22

−2 + 0/22 , 3.62

which is plotted in Fig. 6. Assuming that the detector isquantum limited, one sees that the Lorentzian peak risesabove the constant background by a factor of 3 when thecoupling is optimized to minimize the total noise powerat resonance. This represents the best one can do whencontinuously monitoring zero-point position fluctua-tions. Note that the value of this peak-to-floor ratio is adirect consequence of two simple facts which hold for an

optimal coupling at the quantum limit: i the totaladded noise at resonance back-action plus measure-ment imprecision is equal to the zero-point noise andii back-action and measurement imprecision makeequal contributions to the total added noise. Somewhatsurprisingly, the same maximum peak-to-floor ratio isobtained when one tries to continuously monitor coher-ent qubit oscillations with a linear detector which istransversely coupled to the qubit Korotkov and Averin,2001; this is also a non-QND situation. Finally, if oneonly wants to detect the noise peak as opposed to mak-ing a continuous quantum-limited measurement, onecould use two independent detectors coupled to the os-cillator and look at the cross correlation between thetwo output noises: in this case, there need not be anynoise floor Jordan and Büttiker, 2005a; Doiron et al.,2007.

In Table III, we give a summary of recent experimentswhich approach the quantum limit on weak continuousposition detection of a mechanical resonator. Note thatin many of these experiments the effects of detectorback-action were not seen. This could either be the re-sult of too low a detector-oscillator coupling or due tothe presence of excessive thermal noise. As shown, theback-action force noise serves to slightly heat the oscil-lator. If it is already at an elevated temperature due tothermal noise, this additional heating can be hard to re-solve.

In closing, we stress that this section has given only arudimentary introduction to the quantum limit on posi-tion detection. A complete discussion which treats the

2.0

1.5

1.0

0.5

0.01.20.80.40.0

Frequency ω/Ω

equilibriumnoise (T=0)

eq. + backaction noise

total noise

S xx,tot[ω]/S0xx[Ω]

FIG. 6. Color online Spectral density of measured positionfluctuations of a harmonic oscillator Sxx,tot as a function offrequency , for a detector which reaches the quantum limit atthe oscillator frequency . We have assumed that without thecoupling to the detector the oscillator would be in its groundstate. The y axis has been normalized by the zero-point posi-tion noise spectral density Sxx

0 , evaluated at =. Oneclearly sees that the total noise at is twice the zero-pointvalue, and that the peak of the Lorentzian rises a factor of 3above the background. This background represents the mea-surement imprecision and is equal to 1/2 of Sxx

0 .



important topics of back-action damping, effective tem-perature, noise cross correlation, and power gain isgiven in Sec. V.E.

IV. GENERAL LINEAR-RESPONSE THEORY

A. Quantum constraints on noise

In this section, we further develop the connection be-tween quantum limits and noise discussed previously, fo-cusing now on a more general approach. As before, weemphasize the idea that reaching the quantum limit re-quires a detector having quantum-ideal noise properties.The approach here is different from typical treatmentsin the quantum optics literature Gardiner and Zoller,2000; Haus, 2000, and uses nothing more than featuresof quantum linear response. Our discussion here will ex-pand upon Clerk et al. 2003 and Clerk 2004; some-what similar approaches to quantum measurement arealso discussed by Braginsky and Khalili 1992 andAverin 2003.

In this section, we start by heuristically sketching howconstraints on noise similar to Eq. 3.9 for the cavitydetector can emerge directly from the Heisenberg un-certainty principle. We then present a rigorous and gen-

eral quantum constraint on noise. We introduce both thenotion of a generic linear-response detector and the ba-sic quantum constraint on detector noise. Next, we dis-cuss how this noise constraint leads to the quantum limiton QND state detection of a qubit. The quantum limiton a linear amplifier or a position detector is discussedin the next section.

1. Heuristic weak-measurement noise constraints

As stressed in the introduction, there is no fundamen-tal quantum limit on the accuracy with which a givenobservable can be measured, at least not within theframework of nonrelativistic quantum mechanics. Forexample, one can, in principle, measure the position of aparticle to arbitrary accuracy in the course of a projec-tion measurement. However, the situation is differentwhen we specialize to continuous non-QND measure-ments. Such a measurement can be envisaged as a seriesof instantaneous measurements, in the limit where thespacing between the measurements t is taken to zero.Each measurement in the series has a limited resolutionand perturbs the conjugate variables, thereby affectingthe subsequent dynamics and measurement results. We

TABLE III. Synopsis of recent experiments approaching the quantum limit on continuous position detection of a mechanicalresonator. The second column corresponds to the best measurement-imprecision noise spectral density Sxx

I achieved in the experi-ment. This value is compared against the zero-point position noise spectral density Sxx

0 , calculated using the total measuredresonator damping which may include a back-action contribution. All spectral densities are at the oscillator’s resonance fre-quency . As discussed in the text, there is no quantum limit on how small one can make Sxx

I ; for an ideal detector, one needs totune the detector-resonator coupling so that Sxx

I = Sxx0 /2 in order to reach the quantum limit on position detection. The third

column presents the product of the measured imprecision noise unless otherwise noted and measured back-action noises, dividedby /2; this quantity must be one to achieve the quantum limit on the added noise.

Experiment

Mechanicalfrequency Hz

/ 2

Imprecisionnoise

zero-point noiseSxx

I / Sxx0

Detector noiseproducta

SxxI SFF / /2

Cleland et al. 2002 quantum point contact 1.5106 4.2104

Knobel and Cleland 2003 single-electron transistor 1.2108 1.8102

LaHaye et al. 2004 single-electron transistor 2.0107 5.4Naik et al. 2006 single-electron transistorif Sxx

I had been limited by SET shot noise2.2107 5.3b 8.1102

1.5101

Arcizet et al. 2006 optical cavity 8.1105 0.87Flowers-Jacobs et al. 2007 atomic point-contact 4.3107 29 1.7103

Regal et al. 2008 microwave cavity 2.4105 21Schliesser et al. 2008 optical cavity 4.1107 0.50Poggio et al. 2008 quantum point contact 5.2103 63Etaki et al. 2008 dc SQUID 2.0106 47Groblacher et al. 2009 optical cavity 9.5102 0.57Schliesser et al. 2009 optical cavity 6.5107 5.5 1.0102

Teufel et al. 2009 microwave cavity 1.0106 0.63

aA blank value in this column indicates that back-action was not measured in the experiment.bNote that back-action effects dominated the mechanical Q in this measurement, lowering it from 1.2105 to 4.2102. If one

compares the imprecision against the zero-point noise of the uncoupled mechanical resonator, one finds SxxI / Sxx

0 0.33.



discuss this for the example of a series of position mea-surements of a free particle.

After initially measuring the position with an accuracy x, the momentum suffers a random perturbation ofsize p / 2 x. Consequently, a second position mea-surement taking place a time t later will have an addi-tional uncertainty of size t p /mt /m x. Thus,when one is trying to obtain a good estimate of the po-sition by averaging several such measurements, it is notoptimal to make x too small, because otherwise thisadditional perturbation, called the back-action of themeasurement device, will become large. The back-actioncan be described as a random force F= p /t. A mean-ingful limit t→0 is obtained by keeping both x2t

Sxx and p2 /t SFF fixed. In this limit, the deviationsxt describing the finite measurement accuracy and thefluctuations of the back-action force F can be described

as white noise processes, xtx0= Sxxt and

FtF0= SFFt. The Heisenberg uncertainty relation

p x /2 then implies SxxSFF2 /4 Braginsky and

Khalili, 1992. Note that this is completely analogous tothe relation 3.9 we derived for the resonant cavity de-tector using the fundamental number-phase uncertaintyrelation. In this section, we derive rigorously more gen-eral quantum limit relations on noise spectral densitiesof this form.

2. Generic linear-response detector

To rigorously discuss the quantum limit, we start witha description of a detector that is as general as possible.To that end, we think of a detector as some physical

system described by some unspecified Hamiltonian Hdet

and some unspecified density matrix 0 which is timeindependent in the absence of coupling to the signalsource. The detector has both an input port, character-

ized by an operator F, and an output port, characterized

by an operator I see Fig. 7. The output operator I issimply the quantity that is read out at the output of thedetector e.g., the current in a single-electron transistor,or the phase shift in the cavity detector of the previous

section. The input operator F is the detector quantitythat directly couples to the input signal e.g., the qubitand that causes a back-action disturbance of the signalsource; in the cavity example of the previous section, we

had F= n, the cavity photon number. As we are inter-ested in weak couplings, we assume a simple bilinearform for the detector-signal interaction Hamiltonian

Hint = AxF . 4.1

Here the operator x which is not necessarily a positionoperator carries the input signal. Note that because xbelongs to the signal source, it necessarily commutes

with the detector variables I , F.We always assume the coupling strength A to be small

enough that we can accurately describe the output of thedetector using linear response.6 We thus have

It = I0 + A dtIFt − txt , 4.2

where I0 is the input-independent value of the detectoroutput at zero coupling and IFt is the linear-responsesusceptibility or gain of our detector. Note that Clerk etal. 2003 and Clerk 2004 denoted this gain coefficient. Using standard time-dependent perturbation theory

in the coupling Hint, one can easily derive Eq. 4.2, withIFt given by a Kubo-like formula,

IFt = − i/tIt,F00. 4.3

Here and in what follows the operators I and F areHeisenberg operators with respect to the detectorHamiltonian, and the subscript 0 indicates an expecta-tion value with respect to the density matrix of the un-coupled detector.

As discussed, there will be unavoidable noise in boththe input and output ports of our detector. This noise issubject to quantum-mechanical constraints, and its pres-ence is what limits our ability to make a measurement oramplify a signal. We thus need to quantitatively charac-terize the noise in both these ports. Recall from the dis-cussion in Sec. II.B that it is the symmetric-in-frequencypart of a quantum noise spectral density which plays arole akin to classical noise. We thus want to characterizethe symmetrized noise correlators of our detector de-noted as always with an overbar. Redefining these op-erators so that their average value is zero at zero cou-

pling i.e., F→ F− F0, I→ I− I0, we have

SFF 12−

dt eitFt,F00, 4.4a

SII 12−

dt eitIt, I00, 4.4b

6The precise condition for the breakdown of linear responsewill depend on specific details of the detector. For example, inthe cavity detector discussed in Sec. III.B, one would need thedimensionless coupling A to satisfy A1/Qcz to ensure thatthe nonlinear dependence of the phase shift on the signal zis negligible. This translates to the signal modulating the cavityfrequency by an amount much smaller than its linewidth .

input outputoperatordetector

signalsource

"load"

FIG. 7. Color online Schematic of a generic linear-responsedetector.



SIF 12−

dt eitIt,F00, 4.4c

where , indicates the anticommutator, SII represents

the intrinsic noise in the output of the detector, and SFFdescribes the back-action noise seen by the source of theinput signal. In general, there will be some correlationbetween these two kinds of noise; this is described by

the cross correlator SIF.Finally, we must also allow for the possibility that our

detector could operate in reverse i.e., with input andoutput ports playing opposite roles. We thus introducethe reverse gain FI of our detector. This is the responsecoefficient describing an experiment where we coupleour input signal to the output port of the detector i.e.,

Hint=AxI, and attempt to observe something at the in-

put port i.e., in Ft. In complete analogy to Eq. 4.2,one would then have

Ft = F0 + A dtFIt − txt 4.5

with

FIt = − i/tFt, I00. 4.6

Note that if our detector is in a time-reversal symmetric,thermal equilibrium state, then Onsager reciprocity re-lations would imply either IF=

FI* if I and F have the

same parity under time reversal or IF=−FI* if I and F

have the opposite parity under time reversal see, e.g.,Pathria 1996. Thus, if the detector is in equilibrium,the presence of gain necessarily implies the presence ofreverse gain. Nonzero reverse gain is also found in manystandard classical electrical amplifiers such as op-ampsBoylestad and Nashelsky, 2006.

The reverse gain is something that we must worryabout even if we are not interested in operating our de-tector in reverse. To see why, note that to make a mea-

surement of the output operator I, we must necessarilycouple to it in some manner. If FI0, the noise associ-ated with this coupling could in turn lead to additional

back-action noise in the operator F. Even if the reversegain did nothing but amplify vacuum noise entering theoutput port, this would heat up the system being mea-sured at the input port and hence produce excess back-action. Thus, the ideal situation is to have FI=0, imply-ing a high asymmetry between the input and output ofthe detector, and requiring the detector to be in a statefar from thermodynamic equilibrium. We note that al-most all mesoscopic detectors that have been studied indetail e.g., single-electron transistors and generalizedquantum point contacts have been found to have a van-ishing reverse gain: FI=0 Clerk et al., 2003. For thisreason, we often focus on the ideal but experimentallyrelevant situation where FI=0 in what follows.

Before proceeding, it is worth emphasizing that thereis a relation between the detector gains IF and FI and

the unsymmetrized I-F quantum noise correlator, SIF.This spectral density, which need not be symmetric infrequency, is defined as

SIF = −

dt eitItF00. 4.7

Using the definitions, one can easily show that

SIF = 12†SIF + SIF− *

‡ , 4.8a

IF − FI* = − i/†SIF − SIF− *‡ . 4.8b

Thus, while SIF represents the classical part of the I-Fquantum noise spectral density, the gains IF ,FI are de-termined by the quantum part of this spectral density.This also demonstrates that though the gains have anexplicit factor of 1/ in their definitions, they have awell-defined →0 limit, as the asymmetric-in-frequencypart of SIF vanishes in this limit.

3. Quantum constraint on noise

Despite having said nothing about the detector’sHamiltonian or state except that it is time indepen-dent, we can nonetheless derive a general quantum con-straint on its noise properties. Note first that for purelyclassical noise spectral densities one always has the in-equality

SIISFF − SIF2 0. 4.9

This simply expresses the fact that the correlation be-tween two different noisy quantities cannot be arbi-trarily large; it follows immediately from the Schwartzinequality. In the quantum case, this simple constraintbecomes modified whenever there is an asymmetry be-tween the detector’s gain and reverse gain. This asym-metry is parametrized by the quantity IF,

IF IF − FI*. 4.10

We show below that the following quantum noise in-equality involving symmetrized noise correlators is al-ways valid see also Eq. 6.36 in Braginsky and Khalili1996,

SIISFF − SIF2

IF2

21 + SIFIF/2

, 4.11

where

z = 1 + z2 − 1 + z2/2. 4.12

To interpret the quantum noise inequality Eq. 4.11,note that 1+ z0. Equation 4.11 thus implies that ifour detector has gain and does not have a perfect sym-metry between input and output i.e., IF

FI* , then it

must in general have a minimum amount of back-actionand output noise; moreover, these two noises cannot beperfectly anticorrelated. As shown in the following sec-tions, this constraint on the noise of a detector directly



leads to quantum limits on various different measure-ment tasks. Note that in the zero-frequency limit IF and

SIF are both real, implying that the term involving inEq. 4.11 vanishes. The result is a simpler-looking in-equality found elsewhere Averin, 2003; Clerk et al.,2003; Clerk, 2004.

While Eq. 4.11 may appear reminiscent of the stan-dard fluctuation-dissipation theorem, its origin is quitedifferent: in particular, the quantum noise constraint ap-plies irrespective of whether the detector is in equilib-rium. Equation 4.11 instead follows directly fromHeisenberg’s uncertainty relation applied to the fre-

quency representation of the operators I and F. In itsmost general form, the Heisenberg uncertainty relationgives a lower bound for the uncertainties of two observ-ables in terms of their commutator and their noise cor-relator Gottfried, 1966,

A2 B214 A,B2 + 1

4 A,B2. 4.13

Here we have assumed A= B=0. We now choose the

Hermitian operators A and B to be given by the cosine

transforms of I and F, respectively, over a finite timeinterval T,

A 2

T

−T/2

T/2

dt cost + It , 4.14a

B 2

T

−T/2

T/2

dt costFt . 4.14b

Note that we have phase shifted the transform of I rela-

tive to that of F by a phase . In the limit T→ we find,at any finite frequency 0

A2 = SII, B2 = SFF , 4.15a

A,B = 2 Re eiSIF , 4.15b

A,B = −

+

dt cost + It,F0

= i Re†ei„IF − FI*

…‡ . 4.15c

In the last equality, we have simply made use of theKubo formula definitions of the gain and reverse gaincf. Eqs. 4.3 and 4.6. As a consequence of Eqs.4.15a, 4.15b, and 4.15c, the Heisenberg uncertaintyrelation 4.13 directly yields

SIISFF ReeiSIF2 +2

4†Re ei

„IF

− FI*…‡

2. 4.16

Maximizing the RHS of this inequality over then yieldsthe general quantum noise constraint of Eq. 4.11.

With this derivation, we can now interpret the quan-tum noise constraint Eq. 4.11 as stating that the noiseat a given frequency given frequency in two observables

I and F is bounded by the value of their commutator at

that frequency. The fact that I and F do not commute isnecessary for the existence of linear response gainfrom the detector, but also means that the noise in both

I and F cannot be arbitrarily small. A more detailedderivation, yielding additional important insights, is de-scribed in Appendix I.1.

Given the quantum noise constraint of Eq. 4.11, wecan now very naturally define a quantum-ideal detectorat a given frequency as one which minimizes theleft-hand side LHS of Eq. 4.11—a quantum-ideal de-tector has a minimal amount of noise at frequency . Weare often interested in the ideal case where there is no

reverse gain i.e., measuring I does not result in addi-

tional back-action noise in F; the condition to have aquantum-limited detector thus becomes

SIISFF − SIF2

= IF2

21 + SIFIF/2

, 4.17

where z is given in Eq. 4.12. Again, as discussedbelow, in most cases of interest e.g., zero frequencyand/or large amplifier power gain the last term on theRHS will vanish. In the following, we demonstrate thatthe ideal noise requirement of Eq. 4.17 is necessary inorder to achieve the quantum limit on QND detection ofa qubit, or on the added noise of a linear amplifier.

Before leaving our general discussion of the quantumnoise constraint, it is worth emphasizing that achievingEq. 4.17 places a strong constraint on the properties ofthe detector. In particular, there must exist a tight con-nection between the input and output ports of the

detector—in a certain restricted sense, the operators I

and F must be proportional to one another see Eq. I13in Appendix I.1. As is discussed in Appendix I.1, thisproportionality immediately tells us that a quantum-ideal detector cannot be in equilibrium. The proportion-ality exhibited by a quantum-ideal detector is param-etrized by a single complex-valued number , whosemagnitude is given by

2 = SII/SFF . 4.18

While this proportionality requirement may seem purelyformal, it does have a simple heuristic interpretation; asdiscussed by Clerk et al. 2003, it may be viewed as aformal expression of the principle that a quantum-limited detector must not contain any wasted informa-tion cf. Sec. III.B.2.

4. Evading the detector quantum noise inequality

We now turn to situations where the RHS of Eq.4.11 vanishes, implying that there is no additionalquantum constraint on the noise of our detector beyondwhat exists classically. In such situations, one could havea detector with perfectly correlated back-action and out-



put noises i.e., SFFSII= SIF2, or even with a vanishing

back-action SFF=0. Perhaps not surprisingly, these situ-ations are not of much utility. As we now show, in caseswhere the RHS of Eq. 4.11 vanishes, the detector maybe low noise, but will necessarily be deficient in anotherimportant regard: it will not be good enough that we canignore the extra noise associated with the measurement

of the detector output I. As discussed, the reading out of

I will invariably involve coupling the detector output tosome other physical system. In the ideal case, this cou-pling will not generate any additional back-action on thesystem coupled to the detector’s input port. In addition,the signal at the detector output should be large enough

that any noise introduced in measuring I is negligible;we already came across this idea in our discussion of theresonant cavity detector see discussion following Eq.3.60. This means that we need our detector to trulyamplify the input signal, not simply reproduce it at theoutput with no gain in energy. As now shown, a detectorthat evades the quantum constraint of Eq. 4.11 bymaking the RHS of the inequality zero will necessarilyfail in one or both of the above requirements.

The most obvious case where the quantum noise con-straint vanishes is for a detector which has equal forwardand reverse gains, FI=

IF* . As mentioned, this relation

will necessarily hold if the detector is time-reversal sym-

metric and in equilibrium, and I and F have the sameparity under time reversal. In this case, the relativelylarge reverse gain implies that in the analysis of a givenmeasurement task, it is not sufficient to just consider thenoise of the detector: one must necessarily also considerthe noise associated with whatever system is coupled to

I to read out the detector output, as this noise will be fedback to the detector input port, causing additional back-action; we give an explicit example of this in the nextsubsection, where we discuss QND qubit detection.Even more problematically, when FI=

IF* , there is

never any amplification by the detector. As discussed inSec. V.E.3, the proper metric of the detector’s ability toamplify is its dimensionless power gain: What is thepower supplied at the output of the detector versus theamount of power drawn at the input from the signalsource? When FI=

IF* , one has negative feedback, with

the result that the power gain cannot be larger than 1see Eq. 5.53. There is thus no amplification whenFI=

IF* . Further, if one also insists that the noise con-

straint of Eq. 4.11 is optimized, then one finds thepower gain must be exactly 1; this is explicitly demon-strated in Appendix I.2. The detector thus will simplyact as a transducer, reproducing the input signal at theoutput without any increase in energy. We have here aspecific example of a more general idea that will be dis-cussed in Sec. V: if a detector acts only as a transducer, itneed not add any noise.

At finite frequencies, there is a second way to makethe RHS of the quantum noise constraint of Eq. 4.11vanish: one needs the quantity SIF / IF to be purely

imaginary, and larger in magnitude than /2. In thiscase, it would again be possible to have the LHS of thenoise constraint of Eq. 4.11 equal to 0. However, oneagain finds that in such a case the dimensionless powergain of the detector is at most equal to 1; it thus does notamplify. This is shown explicitly in Appendix I.2. Animportant related statement is that a quantum-limiteddetector with a large power gain must have the quantity

SIF /IF be real. Thus, at the quantum limit, correlationsbetween the back-action force and the intrinsic outputnoise fluctuations must have the same phase as the gainIF. As discussed in Sec. V.F, this requirement can beinterpreted in terms of the principle of no wasted infor-mation introduced in Sec. III.B.2.

B. Quantum limit on QND detection of a qubit

In Sec. III.B, we discussed the quantum limit on QNDqubit detection in the specific context of a resonant cav-ity detector. We now show how the full quantum noiseconstraint of Eq. 4.11 directly leads to this quantumlimit for an arbitrary weakly coupled detector. Similar toSec. III.B, we couple the input operator of our genericlinear-response detector to the z operator of the qubitwe wish to measure i.e., we take x= z in Eq. 4.1; wealso consider the QND regime, where z commutes withthe qubit Hamiltonian. As we saw in Sec. III.B, thequantum limit in this case involves the inequality meas, where meas is the measurement rate and is theback-action dephasing rate. For the latter quantity, wecan directly use the results of our calculation for thecavity system, where we found the dephasing rate wasset by the zero-frequency noise in the cavity photonnumber see Eq. 3.27. In complete analogy, the back-action dephasing rate will be determined here by the

zero-frequency noise in the input operator F of our de-tector,

= 2A2/2SFF. 4.19

We omit frequency arguments in this section, as it isalways the zero-frequency susceptibilities and spectraldensities that appear.

The measurement rate the rate at which informationon the state of qubit is acquired is also defined in com-plete analogy to what was done for the cavity detector.We imagine we turn the measurement on at t=0 andstart to integrate up the output It of our detector,

mt = 0

t

dtIt . 4.20

The probability distribution of the integrated outputmt will depend on the state of the qubit; for long times,we may approximate the distribution corresponding toeach qubit state as being Gaussian. Noting that we have

chosen I so that its expectation vanishes at zero cou-pling, the average value of mt corresponding to eachqubit state is in the long-time limit of interest



mt↑ = AIFt, mt↓ = − AIFt . 4.21

The variance of both distributions is, to leading order,independent of the qubit state,

m2t↑/↓ − mt↑/↓2 m2t↑/↓ = SIIt . 4.22

For the last equality above, we have taken the long-timelimit, which results in the variance of m being deter-mined completely by the zero-frequency output noise

SII=0 of the detector. The assumption here is that,due to the weakness of the measurement, the measure-ment time 1/meas will be much longer than the autocor-relation time of the detector’s noise.

We can now define the measurement rate, in completeanalogy to the cavity detector of the previous section cf.Eq. 3.23, by how quickly the resolving power of themeasurement grows,7

14 mt↑ − mt↓2/m2t↑ + m2t↓ meast .

4.23

This yields

meas = A2IF2/2SII. 4.24

Putting this all together, we find that the “efficiency”ratio =meas/ is given by

meas/ = 2IF2/4SIISFF. 4.25

In the case where our detector has a vanishing reversegain i.e., FI=0, the quantum-limit bound 1 followsimmediately from the quantum noise constraint of Eq.4.11. We thus see that achieving the quantum limit forQND qubit detection requires both a detector withquantum-ideal noise properties, as defined by Eq. 4.17,and a detector with a vanishing noise cross correlator:

SIF=0.If in contrast FI0, it would seem that it is possible

to have 1. This is of course an invalid inference: asdiscussed, FI0 implies that we must necessarily con-sider the effects of extra noise injected into the detec-

tor’s output port when one measures I, as the reversegain will bring this noise back to the qubit, causing extradephasing. The result is that one can do no better than=1. To see this explicitly, consider the extreme case

IF=FI and SII= SFF=0, and suppose we use a second

detector to read out the output I of the first detector.

This second detector has input and output operators F2,

I2; we also take it to have a vanishing reverse gain, sothat we do not have to also worry about how its outputis read out. Coupling of the detectors linearly in the

standard way i.e., Hint,2= IF2, gives the overall gain of

the two detectors in series as I2F2IF, while the back-

action driving the qubit dephasing is described by thespectral density FI2SF2F2

. Using the fact that our sec-ond detector must itself satisfy the quantum noise in-equality, we have

FI2SF2F2SI2I2

2/4I2F2IF2. 4.26

Thus, the overall chain of detectors satisfies the usualzero-reverse-gain quantum noise inequality, implyingthat we still have 1.

V. QUANTUM LIMIT ON LINEAR AMPLIFIERS ANDPOSITION DETECTORS

In the previous section we established the fundamen-tal quantum constraint on the noise of any system ca-pable of acting as a linear detector; we further showedthat this quantum noise constraint directly leads to thequantum limit on nondemolition qubit detection using aweakly coupled detector. In this section, we turn to themore general situation where our detector is a phase-preserving quantum linear amplifier: the input to the de-tector is described by some time-dependent operatorxt, which we wish to have amplified at the output ofour detector. As we see, the quantum limit in this case isa limit on how small one can make the noise added bythe amplifier to the signal. The discussion in this sectionboth furthers and generalizes the heuristic discussion ofposition detection using a cavity detector presented inSec. III.B.

In this section, we start by presenting a heuristic dis-cussion of quantum constraints on amplification. Wethen demonstrate explicitly how the previously dis-cussed quantum noise constraint leads directly to thequantum limit on the added noise of a phase-preservinglinear amplifier; we examine the cases of both a genericlinear position detector and a generic voltage amplifier,following the approach outlined by Clerk 2004. Wealso spend time explicitly connecting the linear-responseapproach used here to the bosonic scattering formula-tion of the quantum limit favored by the quantum opticscommunity Haus and Mullen, 1962; Caves, 1982; Gras-sia, 1998; Courty et al., 1999, paying particular attentionto the case of a two-port scattering amplifier. We will seethat there are some important subtleties involved in con-verting between the two approaches. In particular, thereexists a crucial difference between the case where theinput signal is tightly coupled to the input of the ampli-fier the case usually considered in the quantum opticscommunity, versus the case where, as in an ideal op-amp, the input signal is only weakly coupled to the inputof the amplifier the case usually considered in the solidstate community.

A. Preliminaries on amplification

What exactly does one mean by amplification? As wewill see Sec. V.E.3, a precise definition requires that theenergy provided at the output of the amplifier be much

7The factor of 1/4 here is purely chosen for convenience; weare defining the measurement rate based on the information-theoretic definition given in Appendix F. This factor of 4 isconsistent with the definition used in the cavity system.



larger than the energy drawn at the input of theamplifier—the power gain of the amplifier must belarger than 1. For the moment, however, we work withthe cruder definition that amplification involves makingsome time-dependent signal larger. To set the stage, wefirst consider an extremely simple classical analog of alinear amplifier. Imagine that the “signal” we wish toamplify is the coordinate xt of a harmonic oscillator;we can write this signal as

xt = x0cosSt + p0/MS sinSt . 5.1

Our signal has two quadrature amplitudes, i.e., the am-plitude of the cosine and sine components of xt. Toamplify this signal, we start at t=0 to parametricallydrive the oscillator by changing its frequency S periodi-cally in time: St=0+ sinPt, where we assume0. The well-known physical example is a swingwhose motion is being excited by effectively changingthe length of the pendulum at the right frequency andphase. For a “pump frequency” P equalling twice the“signal frequency,” P=2S, the resulting dynamics willlead to an amplification of the initial oscillator position,with the energy provided by the external driving,

xt = x0et cosSt + p0/MSe−t sinSt .

5.2

Thus, one of the quadratures is amplified exponentially,at a rate = /2, while the other one decays. In aquantum-mechanical description, this produces asqueezed state out of an initial coherent state. Such asystem is called a “degenerate parametric amplifier,”and we discuss its quantum dynamics in Sec. V.H. Wesee that such an amplifier, which amplifies only a singlequadrature, is not required quantum mechanically toadd any noise Caves et al., 1980; Caves, 1982; Braginskyand Khalili, 1992.

Can we now change this parametric amplificationscheme slightly in order to make both signal quadraturesgrow with time? It turns out that this is impossible aslong as we restrict ourselves to a driven system with asingle degree of freedom. The reason in classical me-chanics is that Liouville’s theorem requires phase-spacevolume to be conserved during motion. More formally,this is related to the conservation of Poisson brackets, orin quantum mechanics to the conservation of commuta-tion relations. Nevertheless, it is certainly desirable tohave an amplifier that acts equally on both quadraturesa so-called phase-preserving or phase-insensitive ampli-fier, since the signal’s phase is often not known before-hand. The way around the restriction created by Liou-ville’s theorem is to add more degrees of freedom, suchthat the phase-space volume can expand in both quadra-tures i.e., position and momentum of the interestingsignal degree of freedom, while being compressed inother directions. This is achieved most easily by couplingthe signal oscillator to another oscillator, the “idlermode.” The external driving now modulates the cou-pling between these oscillators, at a frequency that has

to equal the sum of the oscillators’ frequencies. The re-sulting scheme is called a phase-preserving nondegener-ate parametric amplifier see Sec. V.C.

Crucially, there is a price to pay for the introduction ofan extra degree of freedom: there will be noise associ-ated with the idler oscillator, and this noise will contrib-ute to the noise in the output of the amplifier. Classi-cally, one could make the noise associated with the idleroscillator arbitrarily small by simply cooling it to zerotemperature. This is not possible quantum mechanically;there are always zero-point fluctuations of the idler os-cillator to contend with. It is this noise which sets a fun-damental quantum limit for the operation of the ampli-fier. We thus have a heuristic accounting for theexistence of a quantum limit on the added noise of aphase-preserving linear amplifier: one needs extra de-grees of freedom to amplify both signal quadratures, andsuch extra degrees of freedom invariably have noise as-sociated with them.

B. Standard Haus-Caves derivation of the quantum limit on abosonic amplifier

We now make the ideas of the previous section moreprecise by sketching the standard derivation of thequantum limit on the noise added by a phase-preservingamplifier. This derivation is originally due to Haus andMullen 1962, and was both clarified and extended byCaves 1982; the amplifier quantum limit was also mo-tivated in a slightly different manner by Heffner 1962.8

While extremely compact, the Haus-Caves derivationcan lead to confusion when improperly applied; we dis-cuss this in Sec. V.D, as well as in Sec. VI, where weapply this argument carefully to the important case of atwo-port quantum voltage amplifier and discuss the con-nection to the general linear-response formulation ofSec. IV.

The starting assumption of this derivation is that boththe input and output ports of the amplifier can be de-scribed by sets of bosonic modes. If we focus on a nar-row bandwidth centered on frequency , we can de-scribe a classical signal Et in terms of a complexnumber a defining the amplitude and phase of the signalor equivalently the two quadrature amplitudes Hausand Mullen, 1962; Haus, 2000

Et iae−it − a*e+it . 5.3

In the quantum case, the two signal quadratures of Eti.e., the real and imaginary parts of at cannot be mea-sured simultaneously because they are canonically con-jugate; this is in complete analogy to a harmonic oscilla-tor see Eq. 3.48. As a result a ,a* must be elevated tothe status of photon ladder operators: a→ a, a*→a†.

Consider the simplest case where there is only a singlemode at both the input and output, with corresponding

8Note that Caves 1982 provided a discussion of why thederivation of the amplifier quantum limit given by Heffner1962 is not rigorously correct.



operators a and b.9 It follows that the input signal intothe amplifier is described by the expectation value a,while the output signal is described by b. Correspond-ingly, the symmetrized noise in both these quantities isdescribed by

a2 12 a, a† − a2, 5.4

with an analogous definition for b2.To derive a quantum limit on the added noise of the

amplifier, one uses two simple facts. First, both the inputand the output operators must satisfy the usual commu-tation relations

a, a† = 1, b,b† = 1. 5.5

Second, the linearity of the amplifier and the fact that itis phase preserving i.e., both signal quadratures are am-plified the same way implies a simple relation between

the output operator b and the input operator a,

b = Ga, b† = Ga†, 5.6

where G is the dimensionless photon-number gain of theamplifier. It is clear, however, that this expression cannotpossibly be correct as written because it violates the fun-

damental bosonic commutation relation b , b†=1. Weare therefore forced to write

b = Ga + F, b† = Ga† + F†, 5.7

where F is an operator representing additional noiseadded by the amplifier. Based on the discussion of the

previous section, we can anticipate what F represents: itis noise associated with the additional degrees of free-dom that must invariably be present in a phase-preserving amplifier.

As F represents noise, it has a vanishing expectationvalue; in addition, one also assumes that this noise is

uncorrelated with the input signal, implying F , a= F , a†=0 and Fa= Fa†=0. Insisting that b , b†=1thus yields

F,F† = 1 − G . 5.8

The question now becomes how small can we make

the noise described by F? From Eqs. 5.7, the noise atthe amplifier output b is given by

b2 = G a2 + 12 F,F†G a2 + 1

2 F,F†

G a2 + G − 1/2. 5.9

We have used here a standard inequality to bound the

expectation of F , F†. The first term here is simply the

amplified noise of the input, while the second term rep-resents the noise added by the amplifier. Note that ifthere is no amplification i.e., G=1, there need not beany added noise. However, in the more relevant case oflarge amplification G1, the added noise cannot van-ish. It is useful to express the noise at the output as anequivalent noise at the input by simply dividing out thephoton gain G. Taking the large-G limit, we have

b2/G a2 + 12 . 5.10

Thus, we have a simple demonstration that an amplifierwith a large photon gain must add at least half a quan-tum of noise to the input signal. Equivalently, the mini-mum value of the added noise is equal to the zero-pointnoise associated with the input mode; the total outputnoise referred to the input is at least twice the zero-point input noise. Note that both these conclusions areidentical to what we found in our analysis of the reso-nant cavity position detector in Sec. III.B.3. We discusslater how this conclusion can also be reached using thegeneral linear-response language of Sec. IV cf. Secs.V.E and V.F.

As discussed, the added noise operator F is associatedwith additional degrees of freedom beyond input andoutput modes necessary for phase-preserving amplifica-tion. To see this more concretely, note that every linearamplifier is inevitably a nonlinear system consisting ofan energy source and a “spigot” controlled by the inputsignal, which redirects the energy source partly to theoutput channel and partly to some other channels.Hence there are inevitably other degrees of freedom in-volved in the amplification process beyond the input andoutput channels. An explicit example is the quantumparametric amplifier, discussed in the next subsection.Further insights into amplifier-added noise and its con-nection to the fluctuation-dissipation theorem can be ob-tained by considering a simple model where a transmis-sion line is terminated by an effective negativeimpedance; we discuss this model in Appendix C.4.

To see explicitly the role of the additional degrees offreedom, note first that for G1 the RHS of Eq. 5.8 isnegative. Hence the simplest possible form for theadded noise is

F = G − 1d†, F† = G − 1d , 5.11

where d and d† represent a single additional mode of thesystem. This is the minimum number of additional de-grees of freedom that must inevitably be involved in theamplification process. Note that for this case the in-equality in Eq. 5.9 is satisfied as an equality, and theadded noise takes on its minimum possible value. If in-stead we have, say, two additional modes coupled in-equivalently

F = G − 1cosh d1† + sinh d2 , 5.12

it is straightforward to show that the added noise is in-evitably larger than the minimum. This again can be in-terpreted in terms of wasted information, as the extradegrees of freedom are not being monitored as part of

9To relate this to the linear-response detector of Sec. IV.A,one could naively write x, the operator carrying the input sig-nal as, x= a+ a†, and the output operator I as, I= b+ b† wediscuss how to make this correspondence in Sec. VI.



the measurement process and so information is beinglost.

C. Nondegenerate parametric amplifier

1. Gain and added noise

Before we start our discussion of the connection ofthe Haus-Caves formulation of the quantum limit to thegeneral linear-response approach of Sec. IV, it is usefulto consider a specific example. To that end, we analyzehere a nondegenerate parametric amplifier, a linearphase-preserving amplifier which reaches the quantumlimit on its added noise Louisell et al., 1961; Gordon etal., 1963; Mollow and Glauber, 1967a, 1967b and di-rectly realizes the ideas of the previous subsection. Onepossible realization Yurke et al., 1989 is a cavity withthree internal resonances that are coupled together by anonlinear element such as a Josephson junction whosesymmetry permits three-wave mixing. The three modesare called the pump, idler, and signal and their energylevel structure shown in Fig. 8 obeys P=I+S. Thesystem Hamiltonian is then

Hsys = PaP† aP + IaI

†aI + SaS†aS + iaS

†aI†aP

− aSaIaP† . 5.13

We have made the rotating wave approximation in thethree-wave mixing term, and without loss of generalitywe take the nonlinear susceptibility to be real andpositive. The system is driven at the pump frequencyand the three-wave mixing term permits a single pumpphoton to split into an idler photon and a signal photon.This process is stimulated by signal photons alreadypresent and leads to gain. A typical mode of operationwould be the negative-resistance reflection mode inwhich the input signal is reflected from a nonlinear cav-ity and the reflected beam extracted using a circulatorYurke et al., 1989; Bergeal et al., 2008.

The nonlinear equations of motion EOMs becometractable if we assume the pump has large amplitude andcan be treated classically by making the substitution

aP = Pe−iPt = Pe−iI+St, 5.14

where without loss of generality we take P to be realand positive. We note here the important point that if

this approximation were not valid, then our amplifierwould in any case not be the linear amplifier we seek.With this approximation we can hereafter ignore the dy-namics of the pump degree of freedom and deal with thereduced system Hamiltonian

Hsys = IaI†aI + SaS

†aS + iaS†aI

†e−iI+St

− aSaIe+iI+St , 5.15

where P. Transforming to the interaction repre-sentation we are left with the following time-independent quadratic Hamiltonian for the system

Vsys = iaS†aI

† − aSaI . 5.16

To get some intuitive understanding of the physics, wetemporarily ignore the damping of the cavity modes thatwould result from their coupling to modes outside thecavity. We now have a pair of coupled EOMs for the twomodes

aS = + aI†, aI

† = + aS, 5.17

for which the solutions are

aSt = coshtaS0 + sinhtaI†0 ,

5.18aI

†t = sinhtaS0 + coshtaI†0 .

We see that the amplitude in the signal channel growsexponentially in time and that the effect of the time evo-lution is to perform a simple unitary transformationwhich mixes aS with aI

† in such a way as to preserve thecommutation relations. Note the close connection withthe form found from general arguments in Eqs.5.7–5.11.

We may now tackle the full system which includes thecoupling between the cavity modes and modes externalto the cavity. Such a coupling is of course necessary inorder to feed the input signal into the cavity, as well asextract the amplified output signal. It will also result inthe damping of the cavity modes, which will cut off theexponential growth found above and yield a fixed ampli-tude gain. We present the main results in this section,relegating details to how one treats the bath modes so-called input-output theory Walls and Milburn, 1994 toAppendix E. Working in the standard Markovian limit,we obtain the following EOMs in the interaction repre-sentation:

aS = − S/2aS + aI† − SbS,in,

5.19aI

† = − I/2aI† + aS − IbI,in

† .

Here S and I are the respective damping rates of thecavity signal mode and the idler mode. The coupling toextra-cavity modes also lets signals and noise enter thecavity from the baths: this is described by the bosonic

operators bS,in and bI,in which drive the signal and idler

modes, respectively. bS,in describes both the input signalto be amplified and vacuum noise entering from the bath

ωP

ωI

ωS

FIG. 8. Color online Energy level scheme of the nondegen-erate phase-preserving parametric oscillator.



coupled to the signal mode, whereas bI,in simply de-scribes vacuum noise.10

Let us fix our attention on signals inside a frequencywindow centered on S hence zero frequency in theinteraction representation. For simplicity, we first con-sider the case where the signal bandwidth is almostinfinitely narrow i.e., much smaller than the dampingrate of the cavity modes. It then suffices to find thesteady state solution of these EOMs,

aS = 2/SaI† − 2/SbS,in, 5.20

aI† = 2/IaS − 2/IbI,in

† . 5.21

The output signal of the nondegenerate paramp is thesignal leaving the cavity signal mode and entering the

external bath modes; it is described by an operator bS,out.The standard input-output theory treatment of theextra-cavity modes Walls and Milburn, 1994, presentedin Appendix E, yields the simple relation cf. Eq. E37

bS,out = bS,in + SaS. 5.22

The first term corresponds to the reflection of the signaland noise incident on the cavity from the bath, while thesecond term corresponds to radiation from the cavitymode into the bath. Using this, we find that the outputsignal from the cavity is given by

bS,out =Q2 + 1

Q2 − 1bS,in +

2Q

Q2 − 1bI,in

† , 5.23

where Q2 /IS is proportional to the pump ampli-tude and inversely proportional to the cavity decayrates. We have to require Q2 1 to make sure that theparametric amplifier does not settle into self-sustainedoscillations, i.e., it works below threshold. Under thatcondition, we can define the photon-number gain G0 via

− G0 = Q2 + 1/Q2 − 1 , 5.24

such that

bS,out = − G0bS,in − G0 − 1bI,in† . 5.25

In the ideal case, the noise associated with bI,in , bS,in issimply vacuum noise. As a result, the input-output rela-tion Eq. 5.25 is precisely of the Haus-Caves form 5.11for an ideal quantum-limited amplifier. It demonstratesthat the nondegenerate parametric amplifier reaches thequantum limit for minimum added noise. In the limit oflarge gain the output noise referred to the input for avacuum input signal is precisely doubled.

2. Bandwidth-gain trade-off

The above results neglected the finite bandwidth of the input signal to the amplifier. The gain G0 given inEq. 5.24 is only the gain at precisely the mean signal

frequency S; for a finite bandwidth, we also need tounderstand how the power gain varies as a function offrequency over the entire signal bandwidth. As we see, aparametric amplifier suffers from the fact that as oneincreases the overall magnitude of the gain at the centerfrequency S e.g., by increasing the pump amplitude,one simultaneously narrows the frequency range overwhich the gain is appreciable. Heuristically, this is be-cause parametric amplification involves using the pumpenergy to decrease the damping and hence increase thequality factor of the signal mode resonance. This in-crease in quality factor leads to amplification, but it alsoreduces the bandwidth over which aS can respond to the

input signal bS,in.To deal with a finite signal bandwidth, one simply

Fourier transforms Eqs. 5.19. The resulting equationsare easily solved and substituted into Eq. 5.22, result-ing in a frequency-dependent generalization of theinput-output relation given in Eq. 5.25,

bS,out = − gbS,in − gbI,in† . 5.26

Here g is the frequency-dependent gain of the ampli-fier, and g satisfies g2= g2−1. In the rel-evant limit where G0= g021 i.e., large gain at thesignal frequency, one has to a good approximation

g =G0 − iS − I/I + S/D

1 − i/D, 5.27

with

D =1

G0

SI

S + I. 5.28

As always, we work in an interaction picture where thesignal frequency has been shifted to zero. D representsthe effective operating bandwidth of the amplifier. Com-ponents of the signal with frequencies in the rotatingframe D are strongly amplified, while componentswith frequencies D are not amplified at all, but canin fact be slightly attenuated. As already anticipated, theamplification bandwidth D becomes progressivelysmaller as the pump power and G0 are increased, withthe product G0D remaining constant. In a parametricamplifier increasing the gain via increasing the pumpstrength comes with a price: the effective operatingbandwidth is reduced.

3. Effective temperature

Recall that in Sec. II.B we introduced the concept ofan effective temperature of a nonequilibrium system,Eq. 2.8. As we will discuss, this concept plays an im-portant role in quantum-limited amplifiers; the degener-ate paramp gives us a first example of this. Returning tothe behavior of the paramp at the signal frequency, wenote that Eq. 5.25 implies that, even for vacuum inputto both the signal and idler ports, the output will containa real photon flux. To quantify this in a simple way, it is

10Note that the b operators are not dimensionless, as b†brepresents a photon flux see Appendix E.



useful to introduce temporal modes which describe theinput and output fields during a particular time intervalj t , j+1 t where j is an integer,

BS,in,j =1

t

j t

j+1 t

dbS,in , 5.29

with the temporal modes BS,out,j and BI,in,j defined analo-gously. These temporal modes are discussed in Appen-dix D.2, where we discuss the windowed Fourier trans-form see Eq. D18.

With the above definition, we find that the outputmode will have a real occupancy even if the input modeis empty,

nS,out = 0BS,out,j† BS,out,j0

= G00BS,in,j† BS,in,j0 + G0 − 10BI,in,jBI,in,j

† 0

= G0 − 1. 5.30

The dimensionless mode occupancy nS,out is best thoughtof as a photon flux per unit bandwidth see Eq. D26.This photon flux is equivalent to the photon flux thatwould appear in equilibrium at the very high effectivetemperature assuming large gain G0

Teff SG0. 5.31

This is an example of a more general principle, discussedin Sec. V.E.4: a high-gain amplifier must have associatedwith it a large effective temperature scale. Referring thistotal output noise back to the input, we have in thelimit G01

Teff/G0 = S/2 + S/2 =S

2+ TN. 5.32

This corresponds to the half photon of vacuum noiseassociated with the signal source, plus the added noise ofa half photon of our phase-preserving amplifier i.e., thenoise temperature TN is equal to its quantum-limitedvalue. Here the added noise is simply the vacuum noiseassociated with the idler port.

The above argument is merely suggestive that the out-put noise looks like an effective temperature. In fact, itis possible to show that the photon-number distributionof the output is precisely that of a Bose-Einstein distri-bution at temperature Teff. From Eq. 5.13 we see thatthe action of the paramp is to destroy a pump photonand create a pair of new photons, one in the signal chan-nel and one in the idler channel. Using the SU1,1 sym-metry of the quadratic hamiltonian in Eq. 5.16 it ispossible to show that, for vacuum input, the output ofthe paramp is a so-called “two-mode squeezed state” ofthe form Caves and Schumaker, 1985; Gerry, 1985;Knight and Buzek, 2004

out = Z−1/2ebS†bI

†0 , 5.33

where is a constant related to the gain and, to simplifythe notation, we have dropped the “out” labels on theoperators. The normalization constant Z can be workedout by expanding the exponential and using

bS†n0,0 = n!n,0 5.34

to obtain

out = Z−1/2n=0

nn,n 5.35

and hence

Z = 1/1 − 2 5.36

so the state is normalizable only for 2 1.Because this output is obtained by unitary evolution

from the vacuum input state, the output state is a purestate with zero entropy. In light of this, it is interesting toconsider the reduced density matrix obtain by tracingover the idler mode. The pure-state density matrix is

= outout = m,n=0

n,nnm

Zm,m . 5.37

If we now trace over the idler mode we are left with thereduced density matrix for the signal channel

S = TrIdler = nS=0

nS2nS

ZnS

1

Ze−SaS

†aS,

5.38

which is a pure thermal equilibrium distribution witheffective Boltzmann factor

e−S = 2 1. 5.39

The effective temperature can be obtained from the re-quirement that the signal mode occupancy is G0−1,

1/eS − 1 = G0 − 1, 5.40

which in the limit of large gain reduces to Eq. 5.31.This appearance of finite entropy in a subsystem even

when the full system is in a pure state is a purely quan-tum effect. Classically the entropy of a composite systemis at least as large as the entropy of any of its compo-nents. Entanglement among the components allows thislower bound on the entropy to be violated in a quantumsystem.11 In this case the two-mode squeezed state hasstrong entanglement between the signal and idler chan-nels since their photon numbers are fluctuating identi-cally.

D. Scattering versus op-amp modes of operation

We now begin to address the question of how thestandard Haus-Caves derivation of the amplifier quan-tum limit presented in Sec. V.B relates to the generallinear-response approach of Sec. IV. Recall that in Sec.III.B.3 we already used this latter approach to discussposition detection with a cavity detector, reaching simi-

11This paradox has prompted Charles Bennett to remark thata classical house is at least as dirty as its dirtiest room, but aquantum house can be dirty in every room and still perfectlyclean over all.



lar conclusions i.e., at best, the detector adds noiseequal to the zero-point noise. In that linear-response-based discussion, we saw that a crucial aspect of thequantum limit was the trade-off between back-actionnoise and measurement-imprecision noise. We saw thatreaching the quantum limit required both a detectorwith ideal noise, as well as an optimization of thedetector-oscillator coupling strength. Somewhat disturb-ingly, none of these ideas appeared explicitly in theHaus-Caves derivation; this can give the misleading im-pression that the quantum limit never has anything to dowith back-action. A further confusion comes from thefact that many detectors have input and outputs thatcannot be described by a set of bosonic modes. Howdoes one apply the above arguments to such systems?

The first step in resolving these seeming inconsisten-cies is to realize that there are really two different waysin which one can use a given amplifier or detector. Indeciding how to couple the input signal i.e., the signal tobe amplified to the amplifier, and in choosing whatquantity to measure, the experimentalist essentially en-forces boundary conditions; as now shown, there are ingeneral two distinct ways in which to do this. For con-creteness, consider the situation shown in Fig. 9: a two-port voltage amplifier where the input and output portsof the amplifier are attached to one-dimensional trans-mission lines see Appendix C for a quick review ofquantum transmission lines. As in the previous section,we focus on a narrow bandwidth signal centered about afrequency . At this frequency, there exists both a right-moving and a left-moving wave in each transmissionline. We label the corresponding amplitudes in the inputoutput line with ain ,aout bin ,bout, as per Fig. 9. Quan-tum mechanically these amplitudes become operators,much in the same way that we treated the mode ampli-tude a as an operator in the previous section. We ana-lyze this two-port bosonic amplifier in Sec. VI; here weonly sketch its operation to introduce the two differentamplifier operation modes. This will then allow us tounderstand the subtleties of the Haus-Caves quantumlimit derivation.

In the first kind of setup, the experimentalist arrangesthings so that ain, the amplitude of the wave incident onthe amplifier’s input port, is precisely equal to the signalto be amplified i.e., the input signal, irrespective of theamplitude of the wave leaving the input port i.e., aout.

Further, the output signal is taken to be the amplitude ofthe outgoing wave exiting the output of the amplifieri.e., bout, again irrespective of whatever might be enter-ing the output port see Table IV. In this situation, theHaus-Caves description of the quantum limit in the pre-vious section is almost directly applicable; we make thisprecise in Sec. VI. Back-action is indeed irrelevant, asthe prescribed experimental conditions mean that itplays no role. We call this mode of operation the scat-tering mode, as it is most relevant to time-dependentexperiments where the experimentalist launches a signalpulse at the input of the amplifier and looks at what exitsthe output port. One is usually only interested in thescattering mode of operation in cases where the sourceproducing the input signal is matched to the input of theamplifier: only in this case is the input wave ain perfectlytransmitted into the amplifier. As we see in Sec. VI, sucha perfect matching requires a relatively strong couplingbetween the signal source and the input of the amplifier;as such, the amplifier will strongly enhance the dampingof the signal source.

The second mode of linear amplifier operation is whatwe call the op-amp mode; this is the mode one usuallyhas in mind when thinking of an amplifier which isweakly coupled to the signal source, and will be the nextfocus. The key difference from the scattering mode isthat here the input signal is not simply the amplitude ofa wave incident on the input port of the amplifier; simi-larly, the output signal is not the amplitude of a waveexiting the output port. As such, the Haus-Caves deriva-tion of the quantum limit does not directly apply. For thebosonic amplifier discussed here the op-amp modewould correspond to using the amplifier as a voltage op-amp. The input signal would thus be the voltage at theend of the input transmission line. Recall that the volt-age at the end of a transmission line involves the ampli-tude of both left- and right-moving waves, i.e., VatReaint+aoutt. At first this might seem quite con-fusing: If the signal source determines Vat, does thismean it sets the values of both aint and aoutt? Doesnot this violate causality? The signal source enforces thevalue of Vat by simply changing aint in response tothe value of aoutt. While there is no violation of causal-ity, the fact that the signal source is dynamically re-sponding to what comes out of the amplifier’s input portimplies that back-action is indeed relevant.

The op-amp mode of operation is relevant to the typi-cal situation of weak coupling between the signal sourceand amplifier input. By weak coupling, we mean here

aout

ain

bin

bout

input line output line

amplifier

FIG. 9. Color online Schematic of a two-port bosonic ampli-fier. Both the inputs and outputs of the amplifier are attachedto transmission lines. The incoming and outgoing wave ampli-tudes in the input output transmission line are labeledain , aout bin , bout, respectively. The voltages at the end of thetwo lines Va , Vb are linear combinations of incoming and out-going wave amplitudes.

TABLE IV. Two different amplifier modes of operation.

ModeInput Signal

stOutput Signal

ot

Scattering st=aintain indep. of aout

ot=bouttbout indep. of bin

Op-amp st=Vatain depends on aout

ot=Vbtbout depends on bin



something stronger than just requiring that the amplifierbe linear: we require additionally that the amplifier doesnot appreciably change the dissipation of the signalsource. This is analogous to the situation in an ideal volt-age op-amp, where the amplifier input impedance ismuch larger than the impedance of the signal source. Westress that the op-amp mode and this limit of weak cou-pling is the relevant situation in most electrical measure-ments.

Thus, we see that the Haus-Caves formulation of thequantum limit is not directly relevant to amplifiers ordetectors operated in the usual op-amp mode of opera-tion. We clearly need some other way to describe quan-tum amplifiers used in this regime. As we demonstratedin the remainder of this section, the general linear-response approach of Sec. IV is exactly what is needed.To see this, expand the discussion of Sec. IV to includethe concepts of input and output impedance as well aspower gain. The linear-response approach will allow usto see similar to Sec. III.B.3 that reaching the quantumlimit in the op-amp mode does indeed require a trade-off between back-action and measurement imprecision,and requires use of an amplifier with ideal quantumnoise properties see Eq. 4.11. This approach also hasthe added benefit of being directly applicable to systemswhere the input and output of the amplifier are not de-scribed by bosonic modes.12 In Sec. VI, we return to thescattering description of a two-port voltage amplifierFig. 10, and show explicitly how an amplifier can bequantum limited when used in the scattering mode ofoperation, but miss the quantum limit when used in theop-amp mode of operation.

E. Linear-response description of a position detector

In this section, we examine the amplifier quantumlimit for a two-port linear amplifier in the usual weak-coupling, op-amp regime of operation. Our discussionhere will make use of the results obtained for the noise

properties of a generic linear-response detector in Sec.IV, including the fundamental quantum noise constraintof Eq. 4.11. For simplicity, we start with the problem ofcontinuous position detection of a harmonic oscillator.Our discussion will thus generalize the discussion of po-sition detection using a cavity detector given in Sec.III.B.3. We start with a generic detector as introducedin Sec. IV.A coupled at its input to the position x of aharmonic oscillator see Eq. 4.1.13 We want to under-stand the total output noise of our amplifier in the pres-ence of the oscillator, and more importantly how smallwe can make the amplifier’s contribution to this noise.The resulting lower bound is known as the standardquantum limit SQL on position detection, and is analo-gous to the quantum limit on the added noise of a volt-age amplifier discussed in Sec. V.F.

1. Detector back-action

We first consider the consequence of noise in the de-tector input port. As seen in Sec. II.B, the fluctuating

back-action force F acting on our oscillator will lead toboth damping and heating of the oscillator. To model theintrinsic i.e., detector-independent heating and damp-ing of the oscillator, we also assume that our oscillator iscoupled to an equilibrium heat bath. In the weak-coupling limit that we are interested in, one can uselowest-order perturbation theory in the coupling A to

describe the effects of the back-action force F on theoscillator. A full quantum treatment see Appendix I.4shows that the oscillator is described by an effective clas-sical Langevin equation,14

Mxt = − M2xt − M0xt + F0t

− MA2 dtt − txt − AFt . 5.41

The position xt in the above equation is not an opera-tor, but is simply a classical variable whose fluctuationsare driven by the fluctuating forces Ft and F0t. None-theless, the noise in x calculated from Eq. 5.41 corre-

sponds precisely to Sxx, the symmetrized quantum-mechanical noise in the operator x. The fluctuating forceexerted by the detector which represents the heatingpart of the back-action is described by AFt in Eq.5.41; it has zero mean, and a spectral density given by

A2SFF in Eq. 4.4a. The kernel t describes thedamping effect of the detector. It is given by the asym-

12Note that the Haus-Caves derivation for the quantum limitof a scattering amplifier has been generalized to the case offermionic operators Gavish et al., 2004.

13For consistency with previous sections, our coupling Hamil-tonian does not have a minus sign. This is different from theconvention of Clerk 2004, where the coupling Hamiltonian iswritten Hint=−Ax · F.

14Note that we have omitted a back-action term in this equa-tion which leads to small renormalizations of the oscillator fre-quency and mass. These terms are not important for the fol-lowing discussion, so we have omitted them for clarity; one canconsider M and in this equation to be renormalized quanti-ties. See Appendix I.4 for more details.

aout

ain

bin

bout

coldload


uin uoutvacuum noise

amplifier

circulator

FIG. 10. Illustration of a bosonic two-port amplifier used inthe scattering mode of operation. The signal is an incomingwave in the input port of the amplifier, and does not depend onwhat is coming out of the amplifier. This is achieved by con-necting the input line to a circulator and a “cold load” i.e., azero-temperature resistor: all that goes back toward thesource of the input signal is vacuum noise.



metric part of the detector’s quantum noise, as was de-rived in Sec. II.B see Eq. 2.12.

Equation 5.41 also describes the effects of an equi-librium heat bath at temperature T0 which models theintrinsic i.e., detector-independent damping and heat-ing of the oscillator. The parameter 0 is the dampingarising from this bath and F0 is the corresponding fluc-tuating force. The spectral density of the F0 noise is de-termined by 0 and T0 via the fluctuation-dissipationtheorem see Eq. 2.16. T0 and 0 have a simple physi-cal significance: they are the temperature and dampingof the oscillator when the coupling to the detector A isset to zero.

To make further progress, we recall from Sec. II.Bthat even though our detector will in general not be inequilibrium, we may nonetheless assign it an effectivetemperature Teff at each frequency see Eq. 2.8.The effective temperature of an out-of-equilibrium de-tector is simply a measure of the asymmetry of the de-tector’s quantum noise. We are often interested in thelimit where the internal detector time scales are muchfaster than the time scales relevant to the oscillator i.e.,−1 ,−1 ,0

−1. We may then take the →0 limit in theexpression for Teff, yielding

2kBTeff SFF0/M0 . 5.42

In this limit, the oscillator position noise calculated fromEq. 5.41 is given by

Sxx =1

M

20 + kB

2 −22 + 2 + 02

0T + Teff

0 + .

5.43

This is exactly what would be expected if the oscillatorwere only attached to an equilibrium Ohmic bath with a

damping coefficient !=0+ and temperature T= 0T+Teff /!.

2. Total output noise

The next step in our analysis is to link fluctuations inthe position of the oscillator as determined from Eq.5.41 to noise in the output of the detector. As dis-cussed in Sec. III.B.3, the output noise consists of theintrinsic output noise of the detector i.e.,“measurement-imprecision noise” plus the amplifiedposition fluctuations in the position of the oscillator. Thelatter contains both an intrinsic part and a term due tothe response of the oscillator to the back-action.

To start, imagine that we can treat both the oscillatorposition xt and the detector output It as classicallyfluctuating quantities. Using the linearity of the detec-tor’s response, we can then write Itotal, the fluctuatingpart of the detector’s output, as

Itotal = I0 + AIFx . 5.44

The first term I0 describes the intrinsic oscillator-independent fluctuations in the detector output, and

has a spectral density SII. If we scale this by IF2, we

have the measurement-imprecision noise discussed inSec. III.B.3. The second term corresponds to the ampli-fied fluctuations of the oscillator, which are in turn givenby solving Eq. 5.41,

x = − 1/M

2 −2 + i/QF0 − AF

xxF0 − AF , 5.45

where Q= / 0+ is the oscillator quality fac-tor. It follows that the spectral density of the total noisein the detector output is given classically by

SII,tot = SII + xxIF2A4SFF

+ A2SF0F0

+ 2A2 Re†xxIFSIF‡ . 5.46

Here SII, SFF, and SIF are the classical detector noisecorrelators calculated in the absence of any coupling tothe oscillator. Note importantly that we have included

the fact that the two kinds of detector noise in I and in

F may be correlated with one another.To apply the classically derived Eq. 5.46 to our quan-

tum detector-plus-oscillator system, we recall from Sec.II.B that symmetrized quantum noise spectral densitiesplay the role of classical noise. The LHS of Eq. 5.46thus becomes SII,tot, the total symmetrized quantum-mechanical output noise of the detector, while the RHSwill now contain the symmetrized quantum-mechanical

detector noise correlators SFF, SII, and SIF, defined as inEq. 4.4a. Though this may seem rather ad hoc, one caneasily demonstrate that Eq. 5.46 thus interpretedwould be quantum mechanically rigorous if the detectorcorrelation functions obeyed Wick’s theorem. Thus,quantum corrections to Eq. 5.46 will arise solely fromthe non-Gaussian nature of the detector noise correla-tors. We expect from the central limit, theorem that suchcorrections will be small in the relevant limit, where ismuch smaller that the typical detector frequencykBTeff /, and neglect these corrections in what fol-lows. Note that the validity of Eq. 5.46 for a specificmodel of a tunnel junction position detector has beenexplicitly verified by Clerk and Girvin 2004.

3. Detector power gain

Before proceeding, we need to consider our detectoronce again in isolation, and return to the fundamentalquestion of what we mean by amplification. To be ableto say that our detector truly amplifies the motion of theoscillator, it is not sufficient to simply say that the re-sponse function IF must be large note that IF is notdimensionless. Instead, true amplification requires thatthe power delivered by the detector to a following am-plifier be much larger than the power drawn by the de-tector at its input—i.e., the detector must have a dimen-sionless power gain GP much larger than 1. Asdiscussed, if the power gain was not large, we wouldneed to worry about the next stage in the amplification



of our signal, and how much noise is added in that pro-cess. Having a large power gain means that by the timeour signal reaches the following amplifier, it is so largethat the added noise of this following amplifier is unim-portant. The power gain is analogous to the dimension-less photon-number gain G that appears in the standardHaus-Caves description of a bosonic linear amplifiersee Eq. 5.7.

To make the above more precise, we start with theidea case of no reverse gain FI=0. We define the powergain GP of our generic position detector in a way thatis analogous to the power gain of a voltage amplifier.Imagine we drive the oscillator we are trying to measurewith a force 2FD cos t; this will cause the output of our

detector It to also oscillate at frequency . To opti-mally detect this signal in the detector output, we fur-ther couple the detector output I to a second oscillatorwith natural frequency , mass M, and position y: there

is a new coupling term in our Hamiltonian Hint =BI · y,where B is a coupling strength. The oscillations in Itwill now act as a driving force on the auxiliary oscillatory see Fig 11. We can consider the auxiliary oscillator yas a “load” we are trying to drive with the output of ourdetector.

To find the power gain, we need to consider both Pout,the power supplied to the output oscillator y from thedetector, and Pin, the power fed into the input of theamplifier. Consider first Pin. This is simply the time-averaged power dissipation of the input oscillator xcaused by the back-action damping . Using an over-bar to denote a time average, we have

Pin Mx2 = M2xx2FD2 . 5.47

Note that the oscillator susceptibility xx depends onboth the back-action damping and the intrinsic os-cillator damping 0 see Eq. 5.45.

Next, we need to consider the power supplied to theload oscillator y at the detector output. This oscillatorwill have some intrinsic detector-independent dampingld, as well as a back-action damping out. In the sameway that the back-action damping of the input oscilla-

tor x is determined by the quantum noise in F see Eqs.2.14, 2.13, and 2.12, the back-action damping of theload oscillator y is determined by the quantum noise in

the output operator I,

out = B2/M− Im II

= B2/MSII − SII− 2

, 5.48

where II is the linear-response susceptibility which de-

termines how I responds to a perturbation coupling to

I

II = −i

0

dtIt, I0eit. 5.49

As the oscillator y is being driven on resonance, therelation between y and I is given by y=yyIwith yy=−i†Mout‡−1. From conservation of en-ergy, we have that the net power flow into the outputoscillator from the detector is equal to the power dissi-pated out of the oscillator through the intrinsic dampingld. We thus have

Pout Mldy2

= Mld2yy2BAIFxxFD2

=1

M

ld

ld + out2 BAIFxxFD2. 5.50

Using the above definitions, we find that the ratio be-tween Pout and Pin is independent of 0, but depends onld,

Pout

Pin=

1

M22

A2B2IF2

outld/out

1 + ld/out2 . 5.51

We now define the detector power gain GP as thevalue of this ratio maximized over the choice of ld. Themaximum occurs for ld=out i.e., the load oscillatoris “matched” to the output of the detector, resulting in

GP maxPout

Pin =

1

4M22

A2B2IF2

out

=IF2

4 Im FFIm II. 5.52

In the last equality, we have used the relation betweenthe damping rates and out and the linear-response susceptibilities FF and II see Eqs.2.15 and 5.48. We thus find that the power gain is asimple dimensionless ratio formed by the three differentresponse coefficients characterizing the detector, and isindependent of the coupling constants A and B. As wesee in Sec. V.F, it is completely analogous to the powergain of a voltage amplifier, which is also determined bythree parameters: the voltage gain, the input impedance,and the output impedance. Note that there are otherimportant measures of power gain commonly in use inthe engineering community: we comment on these inSec. VII.B.

Finally, the above results are easily generalized to thecase where the detector’s reverse gain FI is nonvanish-ing. For simplicity, we present results for the case where

Ix yFA B

g

γ0 + γ Ag

Aλ Bgy

FD

γout + γld

FIG. 11. Color online Schematic of a generic linear-responseposition detector, where an auxiliary oscillator y is driven bythe detector output.



=ReIFFI / IF20, implying that there is no posi-tive feedback. Maximizing the ratio of Pout /Pin overchoices of ld now yields

GP,rev = 2GP/1 + 2GP + 1 + 4GP 1/ . 5.53

Here GP,rev is the power gain in the presence of reversegain, while GP is the zero-reverse gain power gain givenby Eq. 5.52. One can confirm that GP,rev is a monotonicincreasing function of GP, and is bounded by 1/. Asnoted in Sec. IV.A.4, if FI=

IF* , there is no additional

quantum noise constraint on our detector beyond whatexists classically i.e., the RHS of Eq. 4.11 vanishes.We now see explicitly that when FI=

IF* , the power gain

of our detector can be at most 1, as =1. Thus, whilethere is no minimum back-action noise required byquantum mechanics in this case, there is also no ampli-fication: at best, our detector would act as a transducer.Note further that if the detector has FI=

IF* and opti-

mizes the inequality of Eq. 4.11, then one can showGP,rev must be 1 see Appendix I.2: the detector is sim-ply a transducer. This is in keeping with the results ob-tained using the Haus-Caves approach, which also yieldsthe conclusion that a noiseless detector is a transducer.

4. Simplifications for a quantum-ideal detector

We now consider the important case where our detec-tor has no reverse gain allowing it to have a large powergain, and also has ideal quantum noise i.e., it satisfiesthe ideal noise condition of Eq. 4.17. Fulfilling thiscondition immediately places some powerful constraintson our detector.

First, note that we have defined in Eq. 5.42 the ef-fective temperature of our detector based on what hap-pens at the input port; this is the effective temperatureseen by the oscillator we are trying to measure. Wecould also consider the effective temperature of the de-tector as seen at the output i.e., by the oscillator y usedin defining the power gain. This output effective tem-perature is determined by the quantum noise in the out-

put operator I,

kBTeff,out /logSII+ /SII− . 5.54

For a general out-of-equilibrium amplifier, Teff,out doesnot have to be equal to the input effective temperatureTeff defined by Eq. 2.8. However, for a quantum-idealdetector, the effective proportionality between input andoutput operators see Eq. I13 immediately yields

Teff,out = Teff . 5.55

Thus, a detector with quantum-ideal noise necessarilyhas the same effective temperature at its input and itsoutput. This is all the more remarkable given that aquantum-ideal detector cannot be in equilibrium, andthus Teff cannot represent a real physical temperature.

Another important simplification for a quantum-idealdetector is the expression for the power gain. Using theproportionality between input and output operators cf.Eq. I13, one finds

GP =Im 2 coth2/2kBTeff + Re 2

2, 5.56

where is the parameter characterizing a quantum-limited detector in Eq. 4.18; recall that 2 deter-mines the ratio of SII and SFF. It follows immediatelythat for a detector with ideal noise to also have a largepower gain GP1, one absolutely needs kBTeff: alarge power gain implies a large effective detector tem-perature. In the large-GP limit, we have

GP Im

kBTeff

/22

. 5.57

Thus, the effective temperature of a quantum-ideal de-tector does more than just characterize the detectorback-action—it also determines the power gain.

Finally, an additional consequence of the large-GP,large Teff limit is that the gain IF and noise cross cor-

relator SIF are in phase: SIF /IF is purely real, up tocorrections that are as small as /Teff. This is shownexplicitly in Appendix I.3. Thus, we find that a largepower gain detector with ideal quantum noise cannothave significant out-of-phase correlations between itsoutput and input noises. This last point may be under-stood in terms of the idea of wasted information: if there

were significant out-of-phase correlations between I and

F, it would be possible to improve the performance ofthe amplifier by using feedback. We discuss this point in

Sec. VI. Note that because SIF /IF is real the last term inthe quantum noise constraint of Eq. 4.11 vanishes.

5. Quantum limit on added noise and noisetemperature

We now turn to calculating the noise added to oursignal i.e., xt by our generic position detector. Tocharacterize this added noise, it is useful to take the totalsymmetrized noise in the output of the detector andrefer it back to the input by dividing out the gain of thedetector,

Sxx,tot SII,tot/A2IF2. 5.58

Sxx,tot is simply the frequency-dependent spectraldensity of position fluctuations inferred from the outputof the detector. It is this quantity that will directly deter-mine the sensitivity of the detector–given a certain de-tection bandwidth, what is the smallest variation of x

that can be resolved? The quantity Sxx,tot will havecontributions from the intrinsic fluctuations of the inputsignal as well as a contribution due to the detector. We

first define Sxx,eq ,T to be the symmetrized equilib-rium position noise of our damped oscillator whosedamping is 0+ at temperature T,



Sxx,eq,T = coth/2kBT− Im xx , 5.59

where the oscillator susceptibility xx is defined inEq. 5.45. The total inferred position noise may then bewritten as

Sxx,tot 0/0 + · Sxx,eq,T0 + Sxx,add .

5.60

In the usual case where the detector noise can be ap-proximated as being white, this spectral density will con-sist of a Lorentzian sitting atop a constant noise floorsee Fig. 6. The first term in Eq. 5.60 represents posi-tion noise arising from the fluctuating force F0t asso-ciated with the intrinsic detector-independent dissipa-tion of the oscillator see Eq. 5.41. The prefactor ofthis term arises because the strength of the intrinsicLangevin force acting on the oscillator is proportional to0, not to 0+.

The second term in Eq. 5.60 represents the addedposition noise due to the detector. It has contributions

both from the detector’s intrinsic output noise SII from

the detector’s back-action noise SFF, and may be writtenas

Sxx,add =SII

IF2A2 + A2xx2SFF

+2 ReIF

* xx*SIF

IF2. 5.61

For clarity, we have omitted writing the explicit fre-quency dependence of the gain IF, susceptibility xx,and noise correlators; they should all be evaluated at thefrequency . Note that the first term on the RHS corre-sponds to the measurement-imprecision noise of our de-

tector, SxxI .

We can now finally address the quantum limit on theadded noise in this setup. As discussed in Sec. V.D, theHaus-Caves derivation of the quantum limit cf. Sec.V.B is not directly applicable to the position detectorwe are describing here; nonetheless, we may use its re-sult to guess what form the quantum limit will take here.The Haus-Caves argument told us that the added noiseof a phase-preserving linear amplifier must be at least aslarge as the zero-point noise. We thus anticipate that, ifour detector has a large power gain, the spectral density

of the noise added by the detector i.e., Sxx,add mustbe at least as large as the zero-point noise of ourdamped oscillator,

Sxx,add limT→0

Sxx,eqw,T = Im xx . 5.62

We now show that the bound above is rigorously correctat each frequency .

The first step is to examine the dependence of the

added noise Sxx,add as given by Eq. 5.61 on thecoupling strength A. If we ignore for a moment thedetector-dependent damping of the oscillator, the situa-

tion is the same as the cavity position detector of Sec.III.B.3: there is an optimal value of the couplingstrength A which corresponds to a trade-off between im-precision noise and back-action i.e., the first and second

terms in Eq. 5.61. We thus expect Sxx,add to attain aminimum value at an optimal choice of coupling A=Aopt where both these terms make equal contributionssee Fig. 5. Defining "=arg xx, we thus have thebound

Sxx,add 2xx SIISFF

IF2

+ReIF

* e−i"SIF

IF2 , 5.63

where the minimum value at frequency is achievedwhen

Aopt2 = SII/IFxx2SFF . 5.64

Using the inequality X2+Y22 XY we see that this

value serves as a lower bound on Sxx,add even in thepresence of detector-dependent damping. In the casewhere the detector-dependent damping is negligible, theRHS of Eq. 5.63 is independent of A, and thus Eq.5.64 can be satisfied by simply tuning the couplingstrength A; in the more general case where there isdetector-dependent damping, the RHS is also a functionof A through the response function xx, and it mayno longer be possible to achieve Eq. 5.64 by simplytuning A.15

While Eq. 5.63 is certainly a bound on the added

displacement noise Sxx,add, it does not in itself repre-sent the quantum limit. Reaching the quantum limit re-quires more than simply balancing the detector back-action and intrinsic output noises i.e., the first two termsin Eq. 5.61; one also needs a detector with quantum-ideal noise properties, that is a detector which satisfies Eq.(4.17). Using the quantum noise constraint of Eq. 4.11to further bound Sxx,add, we obtain

Sxx,add 2xxIF

IF

221 + 2SIF

IF + SIF2

+ReIF

* e−i"SIF

IF , 5.65

where the function z is defined in Eq. 4.12. The

minimum value of Sxx,add in Eq. 5.65 is nowachieved when one has both an optimal coupling i.e.,

15Note that, in the heuristic discussion of position detectionusing a resonant cavity detector in Sec. III.B.3, these concernsdid not arise as there was no back-action damping.



Eq. 5.64 and a quantum-limited detector, that is onewhich satisfies Eq. 4.11 as an equality.

Next, we consider the relevant case where our detec-tor is a good amplifier and has a power gain GP1over the width of the oscillator resonance. As discussed,

this implies that the ratio SIF /IF is purely real, up tosmall /kBTeff corrections see Sec. IV.A.4 and Appen-dix I.3 for more details. This in turn implies that

2SIF /IF=0; we thus have

Sxx,add 2xx22

+ SIF

IF2

+cos†"‡SIF

IF . 5.66

Finally, as there is no further constraint on SIF /IF be-yond the fact that it is real, we can minimize the expres-

sion over its value. The minimum Sxx,add is achievedfor a detector whose cross correlator satisfies

SIF/IFoptimal = − /2 cot " , 5.67

with the minimum value given by

Sxx,addmin = Im xx = limT→0

Sxx,eq,T , 5.68

where Sxx,eq ,T is the equilibrium contribution to

Sxx,tot defined in Eq. 5.59. Thus, in the limit of alarge power gain, we have that at each frequency theminimum displacement noise added by the detector isprecisely equal to the noise arising from a zero-temperature bath. This conclusion is irrespective of thestrength of the intrinsic detector-independent oscillatordamping.

We have thus derived the amplifier quantum limit inthe context of position detection for a two-port ampli-fier used in the op-amp mode of operation. Though wereached a conclusion similar to that given by the Haus-Caves approach, the linear-response, quantum noise ap-proach used is quite different. This approach makes ex-plicitly clear what is needed to reach the quantum limit.We find that to reach the quantum limit on the added

displacement noise Sxx,add with a large power gain,one needs 1 a quantum-limited detector, that is, a de-tector which satisfies the ideal noise condition of Eq.4.17, and hence the proportionality condition of Eq.I13; 2 a coupling A which satisfies Eq. 5.64; and 3a detector cross correlator SIF that satisfies Eq. 5.67.

Recall that condition 1 is identical to what is re-quired for quantum-limited detection of a qubit; it israther demanding, and requires that there is no wastedinformation about the input signal in the detector whichis not revealed in the output Clerk et al., 2003. Alsonote that cot " changes quickly as a function of fre-

quency across the oscillator resonance, whereas SIF willbe roughly constant; condition 2 thus implies that it

will not be possible to achieve a minimal Sxx,add

across the entire oscillator resonance. A more reason-

able goal is to optimize Sxx,add at resonance, =. As

xx is imaginary, Eq. 5.67 tells us that SIF should bezero. Assuming we have a quantum-limited detectorwith a large power gain kBTeff, the remainingcondition on the coupling A Eq. 5.64 may be writtenas

Aopt0 + Aopt

= Im

1

2GP=

4kBTeff. 5.69

As AA2 is the detector-dependent damping ofthe oscillator, we thus have that to achieve the quantum-

limited value of Sxx,add with a large power gain, oneneeds the intrinsic damping of the oscillator to be muchlarger than the detector-dependent damping. Thedetector-dependent damping must be small enough tocompensate the large effective temperature of the detec-tor; if the bath temperature satisfies /kBTbathTeff, Eq. 5.69 implies that at the quantum limit thetemperature of the oscillator will be given by

Tosc · Teff + 0 · Tbath/ + 0 → /4kB + Tbath.

5.70

Thus, at the quantum limit and for large Teff, the detec-tor raises the oscillator’s temperature by /4kB.16 Asexpected, this additional heating is only half the zero-point energy; in contrast, the quantum-limited value of

Sxx,add corresponds to the full zero-point result, as italso includes the contribution of the intrinsic outputnoise of the detector.

Finally, we return to Eq. 5.65; this is the constraint

on the added noise Sxx,add before we assumed ourdetector to have a large power gain, and consequently alarge Teff. Note crucially that if we did not require alarge power gain, then there need not be any addednoise. Without the assumption of a large power gain, the

ratio SIF /IF can be made imaginary with a large magni-

tude. In this limit, 1+ 2SIF /IF→0: the quantum con-straint on the amplifier noises e.g., the RHS of Eq.4.11 vanishes. One can then easily use Eq. 5.65 to

show that the added noise Sxx,add can be zero. Thisconfirms a general conclusion that we have seen severaltimes now see Secs. IV.A.4 and V.B: if a detector doesnot amplify i.e., the power gain is unity, it need notproduce any added noise.

F. Quantum limit on the noise temperature of a voltageamplifier

We now turn our attention to the quantum limit onthe added noise of a generic linear voltage amplifier

16If in contrast our oscillator was initially at zero temperaturei.e., Tbath=0, one finds that the effect of the back-action atthe quantum limit and for GP1 is to heat the oscillator to atemperature /kB ln 5.



used in the op-amp mode of operation see, e.g., De-voret and Schoelkopf 2000. For such amplifiers, theadded noise is usually expressed in terms of the “noisetemperature” of the amplifier; we define this conceptand demonstrate that, when appropriately defined, thisnoise temperature must be bigger than / 2kB, where is the signal frequency. Though the voltage amplifier isclosely analogous to the position detector treated previ-ously, its importance makes it worthy of a separate dis-cussion. As in the previous section, our discussion herewill use the general linear-response approach. In con-trast, in Sec. VI, we present the bosonic scattering de-scription of a two-port voltage amplifier, a descriptionsimilar to that used in formulating the Haus-Caves proofof the amplifier quantum limit. We will then be in a goodposition to contrast the linear-response and scatteringapproaches and will see there that the scattering and theop-amp modes of operation discussed here are notequivalent. We stress that the general treatment pre-sented here can also be applied directly to the systemdiscussed in Sec. VI.

1. Classical description of a voltage amplifier

We begin by recalling the standard schematic descrip-tion of a voltage amplifier see Fig. 12. The input volt-age to be amplified vint is produced by a circuit whichhas a Thevenin-equivalent impedance Zs, the source im-pedance. We stress that we are considering the op-ampmode of amplifier operation, and thus the input signaldoes not correspond to the amplitude of a wave incidentupon the amplifier see Sec. V.D. The amplifier itselfhas an input impedance Zin and an output impedanceZout, as well as a voltage gain coefficient V: assuming nocurrent is drawn at the output i.e., Zload→ in Fig. 12,the output voltage Voutt is simply V times the voltageacross the input terminals of the amplifier.

The added noise of the amplifier is usually repre-sented by two noise sources placed at the amplifier in-

put. There is both a voltage noise source Vt in serieswith the input voltage source and a current noise source

It in parallel with the input voltage source Fig. 12.The voltage noise produces a fluctuating voltage Vtspectral density SVV, which simply adds to the sig-nal voltage at the amplifier input, and is amplified at theoutput; as such, it is completely analogous to the intrin-sic detector output noise SII of our linear-response de-tector. In contrast, the current noise source of the volt-

age amplifier represents back-action: this fluctuatingcurrent spectral density SII flows back across theparallel combination of the source impedance and am-plifier input impedance, producing an additional fluctu-ating voltage at its input. The current noise is thus analo-gous to the back-action noise SFF of our generic linear-response detector.

Putting the above together, the total voltage at theinput terminals of the amplifier is

vin,tott =Zin

Zin + Zsvint + Vt −

ZsZin

Zs + ZinIt

vint + Vt − ZsIt . 5.71

In the second line, we have taken the usual limit of anideal voltage amplifier which has an infinite input im-pedance i.e., the amplifier draws zero current. Thespectral density of the total input voltage fluctuations isthus

SVV,tot = Svinvin + SVV,add . 5.72

Here Svinvinis the spectral density of the voltage fluctua-

tions of the input signal vint and SVV,add is the amplifi-er’s contribution to the total noise at the input

SVV,add = SVV + Zs2SII − 2 ReZs*SVI . 5.73

For clarity, we have dropped the frequency index for thespectral densities appearing on the RHS of this equa-tion.

It is useful now to consider a narrow bandwidth inputsignal at a frequency , and ask the following question:if the signal source was simply an equilibrium resistor ata temperature T0, how much hotter would it have to beto produce a voltage noise equal to SVV,tot? The re-sulting increase in the source temperature is defined asthe noise temperature TN of the amplifier and is aconvenient measure of the amplifier’s added noise. It isstandard among engineers to define the noise tempera-ture with the assumption that the initial temperature ofthe resistor T0. One may then use the classical ex-pression for the thermal noise of a resistor, which yieldsthe definition

2 Re ZskBTN SVV,tot . 5.74

Writing Zs= Zsei", we have

2kBTN =1

cos "SVV

Zs+ ZsSII − 2 Ree−i"SVI .

5.75

It is clear from this expression that TN will have a mini-mum as a function of Zs. For Zs too large, the back-action current noise of the amplifier will dominate TN,while for Zs too small, the voltage noise of the amplifieri.e., its intrinsic output noise will dominate. The situa-tion is completely analogous to that of the position de-tector of the last section; there we needed to optimizethe coupling strength A to balance back-action and in-trinsic output noise contributions and thus minimize the

input output

Zin Zout

ZS

vin

ZL

IL Vout

λV

λI

V~

I~

FIG. 12. Schematic of a linear voltage amplifier, including areverse gain I. V and I represent the standard voltage andcurrent noises of the amplifier, as discussed in the text. Thecase with reverse gain is discussed in Sec. VI.



total added noise. Optimization of the source impedancethus yields a completely classical minimum bound onTN,

kBTN SVVSII − Im SVI2 − Re SVI, 5.76

where the minimum is achieved for an optimal sourceimpedance that satisfies

Zsopt = SVV/SII ZN, 5.77

sin "opt = − Im SVI/SVVSII . 5.78

The above equations define the so-called noise imped-ance ZN. We stress again that the discussion so far in thissection has been completely classical.

2. Linear-response description

It is easy to connect the classical description of a volt-age amplifier to the quantum-mechanical description ofa generic linear-response detector; in fact, all that isneeded is a “relabeling” of the concepts and quantitiesintroduced in Sec. V.E when discussing a linear positiondetector. Thus, the quantum voltage amplifier will be

characterized by both an input operator Q and an out-

put operator Vout; these play the roles of F and I in the

position detector, respectively. Vout represents the out-

put voltage of the amplifier, while Q is the operatorwhich couples to the input signal vint via a couplingHamiltonian

Hint = vintQ . 5.79

In more familiar terms, Iin−dQ /dt represents the cur-rent flowing into the amplifier.17 The voltage gain of ouramplifier V will again be given by the Kubo formula of

Eq. 4.3, with the substitutions F→Q , I→ Vout we as-sume these substitutions throughout this section.

We can now easily relate the fluctuations of the inputand output operators to the noise sources used to de-scribe the classical voltage amplifier. As usual, symme-

trized quantum noise spectral densities S will play therole of the classical spectral densities S appearing in

the classical description. First, as the operator Q repre-sents a back-action force, its fluctuations correspond to

the amplifier’s current noise It,

SII ↔ 2SQQ . 5.80

Similarly, the fluctuations in the operator Vout, when re-ferred back to the amplifier input, will correspond to the

voltage noise Vt discussed above,

SVV ↔ SVoutVout/V2. 5.81

A similar correspondence holds for the cross correlatorof these noise sources,

SVI ↔ + iSVoutQ/V. 5.82

To proceed, we need to identify the input and outputimpedances of the amplifier, and then define its powergain. The first step in this direction is to assume that the

output of the amplifier Vout is connected to an externalcircuit via a term

Hint = qouttVout, 5.83

where iout=dqout /dt is the current in the external circuit.We may now identify the input and output impedancesof the amplifier in terms of the damping at the input andoutput. Use of the Kubo formulas for conductance andresistance yields cf. Eq. 2.12 and Eq. 5.48, with the

substitutions F→Q and I→ V

1/Zin = iQQ , 5.84

Zout = VV/− i , 5.85

i.e., Iin= 1/Zinvin and V=Zoutiout,where the subscript indicates the Fourier transform ofa time-dependent expectation value.

We consider throughout this section the case of noreverse gain, QVout

=0. We can define the power gain GPexactly as in Sec. V.E.3 for a linear position detector. GPis defined as the ratio of the power delivered to a loadattached to the amplifier output divided by the powerdrawn by the amplifier, maximized over the impedanceof the load. One finds

GP = V2/4 ReZoutRe1/Zin . 5.86

Expressing this in terms of the linear-response coeffi-cients VV and QQ, we obtain an expression that is com-pletely analogous to Eq. 5.52 for the power gain for aposition detector

GP = V2/4 Im QQ Im VV. 5.87

Finally, we define again the effective temperature Teffof the amplifier via Eq. 2.8, and define a quantum-limited voltage amplifier as one that satisfies the ideal-noise condition of Eq. 4.17. For such an amplifier, thepower gain will be determined by the effective tempera-ture via Eq. 5.56.

Turning to the noise, we calculate the total symme-trized noise at the output port of the amplifier followingthe same argument used to get the output noise of theposition detector cf. Eq. 5.46. As we did in the clas-

17Note that one could have instead written the coupling

Hamiltonian in the more traditional form Hintt="tIin,where "=dtvint is the flux associated with the input volt-age. The linear-response results we obtain are exactly thesame. We prefer to work with the charge Q in order to beconsistent with the rest of the text.



sical approach, we assume that the input impedance ofthe amplifier is much larger than the source impedance:ZinZs; we test this assumption for consistency at theend of the calculation. Focusing only on the amplifiercontribution to this noise as opposed to the intrinsicnoise of the input signal, and referring this noise backto the amplifier input, we find that the symmetrizedquantum noise spectral density describing the added

noise of the amplifier SVV,add satisfies the same equa-tion we found for a classical voltage amplifier, Eq. 5.73,with each classical spectral density S replaced by thecorresponding symmetrized quantum spectral density

S as per Eqs. 5.80–5.82.It follows that the amplifier noise temperature will

again be given by Eq. 5.75, and that the optimal noisetemperature after optimizing over the source imped-ance will be given by Eq. 5.76. Whereas classicallynothing more could be said, quantum mechanically wenow get a further bound from the quantum noise con-straint of Eq. 4.11 and the requirement of a largepower gain. The latter requirement tells us that the volt-

age gain V and the cross correlator SVout Q must bein phase cf. Sec. IV.A.4 and Appendix I.3. This in turn

means that SVI must be purely imaginary. In this case,the quantum noise constraint may be rewritten as

SVVSII − Im SVI2 /22. 5.88

Using these results in Eq. 5.76, we find the ultimatequantum limit on the noise temperature,18

kBTN /2 5.89

As for the position detector, reaching the quantum limithere is not simply a matter of tuning the coupling i.e.,tuning the source impedance Zs to match the noise im-pedance, cf. Eq. 5.77 and 5.78; one also needs tohave an amplifier with ideal quantum noise, that is, anamplifier satisfying Eq. 4.17.

Finally, we need to test our initial assumption thatZs Zin, taking Zs to be equal to its optimal valueZN. Using the proportionality condition of Eq. I13 andthe fact that we are in the large-power-gain limitGP1, we find

ZNRe Zin

=

Im

4kBTeff=

1

2GP 1.

5.90

It follows that ZN Zin in the large-power-gain, large-effective-temperature regime of interest, thus justifyingthe form of Eq. 5.73. Equation 5.90 is analogous tothe case of the displacement detector, where we foundthat reaching the quantum limit on resonance requiredthe detector-dependent damping to be much weakerthan the intrinsic damping of the oscillator cf. Eq.5.69.

Thus, similarly to the situation of the displacementdetector, the linear-response approach allows us both toderive rigorously the quantum limit on the noise tem-perature TN of an amplifier and to state conditions thatmust be met to reach this limit. To reach the quantum-limited value of TN with a large power gain, one needsboth a tuned source impedance Zs and an amplifier thatpossesses ideal noise properties cf. Eqs. 4.17 and Eq.I13.

3. Role of noise cross correlations

Before leaving the topic of a linear voltage amplifier,we pause to note the role of cross correlations in currentand voltage noise in reaching the quantum limit. First,note from Eq. 5.78 that in both the classical and quan-tum treatments the noise impedance ZN of the amplifierwill have a reactive part i.e., Im ZN0 if there areout-of-phase correlations between the amplifier’s currentand voltage noises i.e., if Im SVI0. Thus, if such cor-relations exist, it will not be possible to minimize thenoise temperature and hence reach the quantum limit,if one uses a purely real source impedance Zs.

More significantly, note that the final classical expres-sion for the noise temperature TN explicitly involves thereal part of the SVI correlator cf. Eq. 5.76. In contrast,

we have shown that in the quantum case Re SVI must bezero if one wishes to reach the quantum limit while hav-ing a large power gain see Appendix I.3; as such, thisquantity does not appear in the final expression for theminimal TN. It also follows that to reach the quantumlimit while having a large power gain, an amplifier can-not have significant in-phase correlations between itscurrent and voltage noise.

This last statement can be given a heuristic explana-tion. If there are out-of-phase correlations between cur-rent and voltage noise, we can easily make use of theseby appropriately choosing our source impedance. How-ever, if there are in-phase correlations between currentand voltage noise, we cannot use these simply by tuningthe source impedance. We could, however, have usedthem by implementing feedback in our amplifier. Thefact that we have not done this means that these corre-lations represent a kind of missing information; as a re-sult, we must necessarily miss the quantum limit. In Sec.VI.B, we explicitly give an example of a voltage ampli-fier which misses the quantum limit due to the presence

18Note that our definition of the noise temperature TN con-forms with that of Devoret and Schoelkopf 2000 and mostelectrical engineering texts, but is slightly different from that ofCaves 1982. Caves assumed the source is initially at zero tem-perature i.e., T0=0, and consequently used the full quantumexpression for its equilibrium noise. In contrast, we have as-sumed that kBT0. The different definition of the noisetemperature used by Caves leads to the result kBTN / ln 3 as opposed to our Eq. 5.89. We stress that thedifference between these results has nothing to do with phys-ics, but only with how one defines the noise temperature.



of in-phase current and voltage fluctuations; we showhow this amplifier can be made to reach the quantumlimit by adding feedback in Appendix H.

G. Near quantum-limited mesoscopic detectors

Having discussed the origin and precise definition ofthe quantum limit on the added noise of a linear, phase-preserving amplifier, we now provide a review of workexamining whether particular detectors are able in prin-ciple to achieve this ideal limit. We focus on the op-ampmode of operation discussed in Sec. V.D, where the de-tector is only weakly coupled to the system producingthe signal to be amplified. As repeatedly stressed, reach-ing the quantum limit in this case requires the detectorto have quantum-ideal noise, as defined by Eq. 4.17.Heuristically, this corresponds to the general require-ment of no wasted information: there should be no otherquantity besides the detector output that could be moni-tored to provide information on the input signal Clerket al., 2003. We have already given one simple but rel-evant example of a detector which reaches the amplifierquantum limit: the parametric cavity detector, discussedin Sec. III.B. Here we turn to other more complex de-tectors.

1. dc superconducting quantum interference device amplifiers

The dc superconducting quantum interference deviceSQUID is a detector based on a superconducting ringhaving two Josephson junctions. It can in principle beused as a near quantum-limited voltage amplifier orflux-to-voltage amplifier. Theoretically, this was investi-gated using a quantum Langevin approach Koch et al.,1981; Danilov et al., 1983, as well as more rigorously byusing perturbative techniques Averin, 2000b and map-pings to quantum impurity problems Clerk, 2006. Ex-periments on SQUIDS have also confirmed their poten-tial for near-quantum-limited operation. Mück et al.2001 were able to achieve a noise temperature TN ap-proximately 1.9 times the quantum-limited value at anoperating frequency of =2519 MHz. Working atlower frequencies appropriate to gravitational wave de-tection applications, Vinante et al. 2001 were able toachieve a TN approximately 200 times the quantum-limited value at a frequency =21.6 kHz; more re-cently, the same group achieved a TN approximately tentimes the quantum limit at a frequency =21.6 kHz Falferi et al., 2008. In practice, it can be dif-ficult to achieve the theoretically predicted quantum-limited performance due to spurious heating caused bythe dissipation in the shunt resistances used in theSQUID. This effect can be significantly ameliorated,however, by adding cooling fins to the shunts Wellstoodet al., 1994.

2. Quantum point contact detectors

A quantum point contact QPC is a narrow conduct-ing channel formed in a two-dimensional gas. The cur-rent through the constriction is sensitive to nearby

charges, and thus the QPC acts as a charge-to-currentamplifier. It has been shown theoretically that the QPCcan achieve the amplifier quantum limit, both in the re-gime where transport is due to tunneling Gurvitz, 1997,and in regimes where the transmission is not smallAleiner et al., 1997; Levinson, 1997; Korotkov andAverin, 2001; Pilgram and Büttiker, 2002; Clerk et al.,2003. Experimentally, QPCs are in widespread use asdetectors of quantum dot qubits. The back-actiondephasing of QPC detectors was studied by Buks et al.1998 and Sprinzak et al. 2000; good agreement wasfound with the theoretical prediction, confirming thatthe QPC has quantum-limited back-action noise.

3. Single-electron transistors and resonant-level detectors

A metallic single-electron transistor SET consists ofa small metallic island attached via tunnel junctions tolarger source and drain electrodes. Because ofCoulomb-blockade effects, the conductance of a SET issensitive to nearby charges, and hence it acts as a sensi-tive charge-to-current amplifier. Considerable work hasinvestigated whether metallic SETs can approach thequantum limit in various different operating regimes.Theoretically, the performance of a normal-metal SETin the sequential tunneling regime was studied by Shnir-man and Schön 1998, Devoret and Schoelkopf 2000,Makhlin et al. 2000, Aassime et al. 2001, and Johans-son et al. 2002, 2003. In this regime, where transport isvia a sequence of energy-conserving tunnel events, oneis far from optimizing the quantum noise constraint ofEq. 4.17, and hence one cannot reach the quantumlimit Shnirman and Schön, 1998; Korotkov, 2001b. Ifone instead chooses to work with a normal-metal SET inthe cotunneling regime a higher-order tunneling pro-cess involving a virtual transition, then one can indeedapproach the quantum limit Averin, 2000a; van denBrink, 2002. However, by virtue of being a higher-orderprocess, the related currents and gain factors are small,impinging on the practical utility of this regime of opera-tion. It is worth noting that while most theory on SETsassume a dc voltage bias, to enhance bandwidth, experi-ments are usually conducted using the rf-SET configura-tion Schoelkopf et al., 1998, where the SET changesthe damping of a resonant LC circuit. Korotkov andPaalanen 1999 showed that this mode of operation fora sequential tunneling SET increases the measurement-imprecision noise by approximately a factor of 2. Themeasurement properties of a normal-metal, sequential-tunneling rf-SET including back-action were studiedexperimentally by Turek et al. 2005.

Measurement using superconducting SETs has alsobeen studied. Clerk et al. 2002 showed that so-calledincoherent Cooper-pair tunneling processes in a super-conducting SET can have a noise temperature which isapproximately a factor of 2 larger than the quantum-limited value. The measurement properties of supercon-ducting SETs biased at a point of incoherent Cooper-pair tunneling have been probed recently in experimentThalakulam et al., 2004; Naik et al., 2006.



The quantum measurement properties of phase-coherent noninteracting resonant level detectors havealso been studied theoretically Averin, 2000b; Clerkand Stone, 2004; Mozyrsky et al., 2004; Gavish et al.,2006. These systems are similar to metallic SET, exceptthat the central island only has a single level as opposedto a continuous density of states, and Coulomb-blockade effects are typically neglected. These detectorscan reach the quantum limit in the regime where thevoltage and temperature are smaller than the intrinsicenergy broadening of the level due to tunneling. Theycan also reach the quantum limit in a large-voltage re-gime that is analogous to the cotunneling regime in ametallic SET Averin, 2000b; Clerk and Stone, 2004.The influence of dephasing processes on such a detectorwas studied by Clerk and Stone 2004.

H. Back-action evasion and noise-free amplification

Having discussed in detail quantum limits on phase-preserving linear amplifiers i.e., amplifiers which mea-sure both quadratures of a signal equally well, we nowreturn to the situation discussed in Sec. V.A: imagine wewish only to amplify a single quadrature of some time-dependent signal. For this case, there need not be anyadded noise from the measurement. Unlike the case ofamplifying both quadratures, Liouville’s theorem doesnot require the existence of any additional degrees offreedom when amplifying a single quadrature: phasespace volume can be conserved during amplificationsimply by contracting the unmeasured quadrature seeEq. 5.2. As no extra degrees of freedom are needed,there need not be any extra noise associated with theamplification process.

Alternatively, single-quadrature detection can take aform similar to a QND measurement, where the back-action does not affect the dynamics of the quantity beingmeasured Thorne et al., 1978; Braginsky et al., 1980;Caves et al., 1980; Caves, 1982; Braginsky and Khalili,1992; Bocko and Onofrio, 1996. For concreteness, con-sider a high-Q harmonic oscillator with position xt andresonant frequency . Its motion may be written interms of quadrature operators defined as in Eq. 3.48,

xt = Xtcost + + Ytsint + . 5.91

Here xt is the Heisenberg-picture position operator ofthe oscillator. The quadrature operators can be writtenin terms of the Schrödinger-picture oscillator creationand destruction operators as

Xt = xZPFceit+ + c†e−it+ , 5.92a

Yt = − ixZPFceit+ − c†e−it+ . 5.92b

As previously discussed, the two quadrature ampli-

tude operators X and Y are canonically conjugate cf.Eq. 3.49. Making a measurement of one quadrature

amplitude, say X, will thus invariably lead to back-

action disturbance of the other conjugate quadrature Y.

However, due to the dynamics of a harmonic oscillator,this disturbance will not affect the measured quadratureat later times. One can already see this from the classicalequations of motion. Suppose our oscillator is driven bya time-dependent force Ft which has appreciable band-width only near . We may write this as

Ft = FXtcost + + FYtsint + , 5.93

where FXt, FYt are slowly varying compared to .Using the fact that the oscillator has a high-quality fac-tor Q= /, one can easily find the equations of motion,

dXt/dt = − /2Xt − FYt/2m , 5.94a

dYt/dt = − /2Yt + FXt/2m . 5.94b

Thus, as long as FYt and FXt are uncorrelated andsufficiently slow, the dynamics of the two quadraturesare completely independent; in particular, if Y is subjectto a narrow-bandwidth, noisy force, it is of no conse-quence to the evolution of X. An ideal measurement ofX will result in a back-action force having the form ofEq. 5.93 with FYt=0, implying that Xt will be com-pletely unaffected by the measurement.

Not surprisingly, if one can measure and amplify X

without any back-action, there need not be any addednoise due to the amplification. In such a setup, the onlyadded noise is the measurement-imprecision noise asso-ciated with intrinsic fluctuations of the amplifier output.These may be reduced in principle to an arbitrarily smallvalue by simply increasing the amplifier gain e.g., byincreasing the detector-system coupling: in an idealsetup, there is no back-action penalty on the measuredquadrature associated with this increase.

The above conclusion can lead to what seems like acontradiction. Imagine we use a back-action evadingamplifier to make a “perfect” measurement of X i.e.,negligible added noise. We would then have no uncer-tainty as to the value of this quadrature. Consequently,we would expect the quantum state of our oscillator tobe a squeezed state, where the uncertainty in X is muchsmaller than xZPF. However, if there is no back-actionacting on X, how is the amplifier able to reduce its un-certainty? This seeming paradox can be fully resolved byconsidering the conditional aspects of an ideal single-quadrature measurement, where one considers the stateof the oscillator given a particular measurement historyRuskov et al., 2005; Clerk et al., 2008.

It is worth stressing that the possibility of amplifying asingle quadrature without back-action and hence with-out added noise relies crucially on our oscillator resem-bling a perfect harmonic oscillator: the oscillator Q mustbe large, and nonlinearities which could couple the twoquadratures must be small. In addition, the envelope ofthe nonvanishing back-action force FXt must have anarrow bandwidth. One should further note that a veryhigh-precision measurement of X will produce a verylarge back-action force FX. If the system is not nearly



perfectly harmonic, then the large amplitude impartedto the conjugate quadrature Y will inevitably leak backinto X.

Amplifiers or detectors that treat the two signalquadratures differently are known in the quantum opticsliterature as phase sensitive; we prefer the designationphase nonpreserving since they do not preserve thephase of the original signal. Such amplifiers invariablyrely on some internal clock i.e., an oscillator with awell-defined phase which breaks time-translation in-variance and picks out the phase of the quadrature thatwill be amplified i.e., the choice of used to define thetwo quadratures in Eq. 5.91; we see this explicitly inwhat follows. This leads to an important caveat: even ina situation where the interesting information is in asingle signal quadrature, to benefit from using a phase-nonpreserving amplifier, we must know in advance theprecise phase of this quadrature. If we do not know thisphase, we either have to revert to a phase-preservingamplification scheme and thus be susceptible to addednoise or we would have to develop a sophisticated andhigh speed quantum feedback scheme to dynamicallyadapt the measurement to the correct quadrature in realtime Armen et al., 2002. In what follows, we make theabove ideas concrete by considering a few examples ofquantum phase-nonpreserving amplifiers.19

1. Degenerate parametric amplifier

Perhaps the simplest example of a phase nonpreserv-ing amplifier is the degenerate parametric amplifier; theclassical version of this system was described in Sec. V.Asee Eq. 5.2. The setup is similar to the nondegenerateparametric amplifier discussed in Sec. V.C, except thatthe idler mode is eliminated, and the nonlinearity con-verts a single pump photon into two signal photons atfrequency S=P /2. As we now show, the resulting dy-namics causes one signal quadrature to be amplifiedwhile the other is attenuated, in such a way that it is notnecessary to add extra noise to preserve the canonicalcommutation relations.

The system Hamiltonian is

Hsys = PaP† aP + SaS

†aS + iaS†aS

†aP − aSaSaP† .

5.95

If the pump is treated classically as before, the analog ofEq. 5.16 is

Vsys = i/2aS†aS

† − aSaS , 5.96

where /2=P, and the analog of Eq. 5.19 is

aS = − S/2aS + aS† − SbS,in. 5.97

The dimensionless quadrature operators correspond-ing to the signal mode are

xS = 1/2aS† + aS, yS = i/2aS

† − aS , 5.98

which obey xS , yS= i. We can define quadrature opera-

tors XS,in/out, YS,in/out corresponding to the input and out-put fields in a completely analogous manner.

The steady state solution of Eq. 5.97 for the outputfields becomes

XS,out = GXS,in, YS,out = YS,in/G , 5.99

where the number gain G is given by

G = + S/2/ − S/22. 5.100

We thus see clearly that the amplifier treats the twoquadratures differently. One quadrature is amplified andthe other attenuated, with the result that the commuta-tion relation can be preserved without the necessity ofextra degrees of freedom and added noise. Note that thelarge-amplitude pump mode has played the role of aclock in the degenerate paramp: it is the phase of thepump that picks out which quadrature of the signal willbe amplified.

Before ending our discussion here, it is important tostress that while the degenerate parametric amplifier isphase sensitive and has no added noise, it is not an ex-ample of back-action evasion see Caves et al. 1980,footnote on p. 342. This amplifier is operated in thescattering mode of amplifier operation, a mode whereas discussed in Sec. V.D back-action is not relevant atall. Recall that in this mode of operation the amplifierinput is perfectly impedance matched to the signalsource, and the input signal is simply the amplitude of anincident wave on the amplifier input. This mode of op-eration necessarily requires a strong coupling between

the input signal and the amplifier input i.e., bS,in. Ifone instead tried to weakly couple the degenerate para-metric amplifier to a signal source, and operate it in theop-amp mode of operation cf. Sec. V.D, one finds thatthere is indeed a back-action disturbance of the mea-sured quadrature. We have yet another example whichdemonstrates that one must be careful to distinguish theop-amp and scattering modes of amplifier operation.

2. Double-sideband cavity detector

We now turn to a simple but experimentally relevantdetector that is truly back-action evading. We take asour input signal the position x of a mechanical oscillator.The amplifier setup we consider is almost identical tothe cavity position detector discussed in Sec. III.B.3: weagain have a single-sided resonant cavity whose fre-quency depends linearly on the oscillator’s position, withthe Hamiltonian given by Eq. 3.11 with z= x /xZPF.We showed in Sec. III.B.3 and Appendix E.3 that, bydriving the cavity on resonance, it could be used to makea quantum-limited position measurement: one can oper-ate it as a phase-preserving amplifier of the mechanicaloscillator’s position and achieve the minimum possible

19One could in principle generalize the linear-response ap-proach of Sec. V to deal with phase-nonpreserving detectors.However, as such detectors are not time-translational invari-ant, such a description becomes rather cumbersome and is notparticularly helpful. We prefer instead to present concreteexamples.



amount of added noise. To use the same system to makea back-action free measurement of one oscillatorquadrature only, one simply uses a different cavity drive.Instead of driving at the cavity resonance frequency c,one drives at the two sidebands associated with the me-chanical motion i.e., at frequencies c±, where is asalways the frequency of the mechanical resonator. Aswe see, such a drive results in an effective interactionthat couples the cavity to only one quadrature of theoscillator’s motion. This setup was first proposed as ameans of back-action evasion by Braginsky et al. 1980;further discussion can be found in Caves et al. 1980 andBraginsky and Khalili 1992, as well as in Clerk et al.2008, which gives a fully quantum treatment and con-siders conditional aspects of the measurement. In whatfollows, we sketch the operation of this system followingClerk et al. 2008; details are provided in Appendix E.4.

We start by requiring that our system be in the “good-cavity” limit, where c is the damping of thecavity mode; we also require the mechanical oscillatorto have a high-Q factor. In this regime, the two side-bands associated with the mechanical motion at c±are well separated from the main cavity resonance at c.Making a single-quadrature measurement requires thatone drives the cavity equally at the two sideband fre-

quencies. The amplitude of the driving field bin enteringthe cavity will be chosen to have the form

bint = −iN

4eie−ic−t − e−ie−ic+t

=N

2sint + e−ict. 5.101

Here N is the photon-number flux associated with thecavity drive see Appendix E for more details on how toproperly include a drive using input-output theory.Such a drive could be produced by taking a signal at thecavity resonance frequency and amplitude-modulating itat the mechanical frequency.

To understand the effect of this drive, note that itsends the cavity both photons with frequency c− andphotons with frequency c+. The first kind of drivephoton can be converted to a cavity photon if a quantumis absorbed from the mechanical oscillator; the secondkind of drive photon can be converted to a cavity pho-ton if a quantum is emitted to the mechanical oscillator.The result is that we can create a cavity photon by eitherabsorbing or emitting a mechanical oscillator quantum.If there is a well-defined relative phase of ei2 betweenthe two kinds of drive photons, we would expect thedouble-sideband drive to yield an effective cavity-oscillator interaction of the form

VeffNa†eic + e−ic† + H.c. 5.102a

Na + a†X. 5.102b

This is exactly what is found in a full calculation seeAppendix E.4. Note that we have written the interac-tion in an interaction picture in which the fast oscilla-tions of the cavity and oscillator operators have beenremoved. In the second line, we have made use of Eqs.5.92 to show that the effective interaction involves only

the X oscillator quadrature.We thus see from Eq. 5.102b that the cavity is

coupled only to the oscillator X quadrature. As shownrigorously in Appendix E.3, the result is that the systemonly measures and amplifies this quadrature: the light

leaving the cavity has a signature of X but not of Y.Further, Eq. 5.102b implies that the cavity operatorNa+ a† will act as a noisy force on the Y quadrature.While this will cause a back-action heating of Y, it willnot affect the measured quadrature X. We thus have atrue back-action-evading amplifier: the cavity outputlight lets one measure X free from any back-action ef-fect. Note that, in deriving Eq. 5.102a, we have usedthe fact that the cavity operators have fluctuations in anarrow bandwidth : the back-action force noise isslow compared to the oscillator frequency. If this werenot the case, we could still have a back-action heating ofthe measured X quadrature. Such effects, arising from anonzero ratio /, are treated in Clerk et al. 2008.

Finally, as there is no back-action on the measured X

quadrature, the only added noise of the amplificationscheme is measurement-imprecision noise e.g., shotnoise in the light leaving the cavity. This added noisecan be made arbitrarily small by increasing the gain ofthe detector by, for example, increasing the strength of

the cavity drive N. In a real system where /M is non-zero, the finite bandwidth of the cavity number fluctua-tions leads to a small back-action on the X. As a result,one cannot make the added noise arbitrarily small, astoo large a cavity drive will heat the measured quadra-ture. Nonetheless, for a sufficiently small ratio /M,one can still beat the standard quantum limit on theadded noise Clerk et al., 2008.

3. Stroboscopic measurements

With sufficiently high bandwidth it should be possibleto do stroboscopic measurements synchronized with theoscillator motion, which could allow one to go below thestandard quantum limit in one quadrature of motionCaves et al., 1980; Braginsky and Khalili, 1992. To un-derstand this idea, imagine an extreme form of phase-sensitive detection in which a Heisenberg microscopemakes a strong high-resolution measurement whichprojects the oscillator onto a state of well-defined posi-tion X0 at time t=0,



X0t =

n=0

ane−in+1/2tn , 5.103

where the coefficients obey an= n X0. Because the po-sition is well defined the momentum is extremely uncer-tain. Equivalently, the momentum kick delivered by theback-action of the microscope makes the oscillator mo-mentum uncertain. Thus the wave packet quicklyspreads out and the position uncertainty becomes large.However, because of the special feature that the har-monic oscillator levels are evenly spaced, we can seefrom Eq. 5.103 that the wave packet re-assembles itselfprecisely once each period of oscillation because eint

=1 for every integer n. At half periods, the packet re-assembles at position −X0. Hence stroboscopic mea-surements made once or twice per period will be back-action evading and can go below the standard quantumlimit. The only limitations will be the finite anharmonic-ity and damping of the oscillator. Note that the possibil-ity of using mesoscopic electron detectors to performstroboscopic measurements has recently received atten-tion Jordan and Büttiker, 2005b; Ruskov et al., 2005.

VI. BOSONIC SCATTERING DESCRIPTION OF ATWO-PORT AMPLIFIER

In this section, we return again to the topic of Sec.V.F, quantum limits on a quantum voltage amplifier. Wenow discuss the physics in terms of the bosonic voltageamplifier first introduced in Sec. V.D. Recall that in thatsection we demonstrated that the standard Haus-Cavesderivation of the quantum limit was not directly relevantto the usual weak-coupling op-amp mode of amplifieroperation, a mode where the input signal is not simplythe amplitude of a wave incident on the amplifier. In thissection, we expand upon that discussion, giving an ex-plicit discussion of the differences between the op-ampdescription of an amplifier presented in Sec. V.E, andthe scattering description often used in the quantum op-tics literature Grassia, 1998; Courty et al., 1999. We seethat what one means by back-action and added noise arenot the same in the two descriptions! Further, eventhough an amplifier may reach the quantum limit whenused in the scattering mode i.e., its added noise is assmall as allowed by commutation relations, it can none-theless fail to achieve the quantum limit when used inthe op-amp mode. Finally, the discussion here will alsoallow us to highlight important aspects of the quantumlimit not easily discussed in the more general context ofSec. IV.

A. Scattering versus op-amp representations

In the bosonic scattering approach, a generic linearamplifier is modeled as a set of coupled bosonic modes.To make matters concrete, we consider the specific caseof a voltage amplifier with distinct input and outputports, where each port is a semi-infinite transmission linesee Fig. 9. We start by recalling that a quantum trans-

mission line can be described as a set of noninteractingbosonic modes see Appendix D for a quick review.Denoting the input transmission line with an a and theoutput transmission line with a b, the current and volt-age operators in these lines may be written as

Vqt = 0

d

2Vqe−it + H.c. , 6.1a

Iqt = q0

d

2Iqe−it + H.c. , 6.1b

with

Vq =

2Zqqin + qout , 6.2a

Iq =

2Zqqin − qout . 6.2b

Here q can be equal to a or b, and we have a=1,b=−1. The operators ain, aout are bosonic annihi-lation operators; ain describes an incoming wave inthe input transmission line i.e., incident on the ampli-fier having frequency , while aout describes an out-

going wave with frequency . The operators bin and

bout describe analogous waves in the output transmis-

sion line. We can think of Va as the input voltage to our

amplifier and Vb as the output voltage. Similarly, Ia is the

current drawn by the amplifier at the input and Ib is thecurrent drawn at the output of the amplifier. Finally, ZaZb is the characteristic impedance of the input outputtransmission line.

As we have seen, amplification invariably requires ad-ditional degrees of freedom. Thus, to amplify a signal ata particular frequency , there will be 2N bosonicmodes involved, where the integer N is necessarilylarger than 2. Four of these modes are simply thefrequency- modes in the input and output lines i.e.,

ain, aout, bin, and bout. The remaining 2N−2 modes describe auxiliary degrees of freedom in-volved in the amplification process; these additionalmodes could correspond to frequencies different fromthe signal frequency . The auxiliary modes can also bedivided into incoming and outgoing modes. It is thusconvenient to represent them as additional transmissionlines attached to the amplifier; these additional linescould be semi-infinite or could be terminated by activeelements.

1. Scattering representation

In general, our generic two-port bosonic amplifier willbe described by an NN scattering matrix, which deter-mines the relation between the outgoing and incomingmode operators. The form of this matrix is constrainedby the requirement that the output modes obey theusual canonical bosonic commutation relations. It is con-



venient to express the scattering matrix in a form whichonly involves the input and output lines explicitly,

aout

bout = s11 s12

s21 s22 ain

bin + Fa

Fb .

6.3

Here Fa and Fb are each an unspecified linearcombination of the incoming auxiliary modes introducedabove. They thus describe noise in the outgoing modesof the input and output transmission lines which arisesfrom the auxiliary modes involved in the amplificationprocess. Note the similarity between Eqs. 6.3 and 5.7for the simple one-port bosonic amplifier considered inSec. V.B.

In the quantum optics literature, one typically viewsEq. 6.3 as the defining equation of the amplifier; wecall this the scattering representation of our amplifier.The representation is best suited to the scattering modeof amplifier operation described in Sec. V.D. In thismode of operation, the experimentalist ensures thatain is precisely equal to the signal to be amplified,irrespective of what is coming out of the amplifier. Simi-larly, the output signal from the amplifier is the ampli-

tude of the outgoing wave in the output line, bout. If

we focus on bout, we have precisely the same situation asdescribed in Sec. 5.10, where we presented the Haus-Caves derivation of the quantum limit see Eq. 5.7. Itfollows that in the scattering mode of operation the ma-trix element s21 represents the gain of our amplifier atfrequency , s212 the corresponding “photon-

number gain,” and Fb the added noise operator of the

amplifier. The operator Fa represents the back-actionnoise in the scattering mode of operation; this back-action has no effect on the added noise of the amplifierin the scattering mode.

Similar to Sec. V.B, one can now apply the standardargument of Haus and Mullen 1962 and Caves 1982to our amplifier. This argument tells us that since the“out” operators must have the same commutation rela-

tions as the “in” operators, the added noise Fb cannotbe arbitrarily small in the large-gain limit i.e., s211.Note that this version of the quantum limit on the addednoise has nothing to do with back-action. As discussed,this is perfectly appropriate for the scattering mode ofoperation, as in this mode the experimentalist ensuresthat the signal going into the amplifier is completely in-dependent of whatever is coming out of the amplifier.This mode of operation could be realized in time-dependent experiments, where a pulse is launched at theamplifier. This mode is not realized in most weak-coupling amplification experiments, where the signal tobe amplified is not identical to an incident wave ampli-tude.

2. Op-amp representation

In the usual op-amp amplifier mode of operation de-scribed in Sec. IV, the input and output signals are notsimply incoming and outgoing wave amplitudes; thus,the scattering representation is not an optimal descrip-tion of our amplifier. The system we are describing hereis a voltage amplifier: thus, in the op-amp mode, theexperimentalist would ensure that the voltage at the end

of the input line Va is equal to the signal to be ampli-fied, and would read out the voltage at the end of the

output transmission line Vb as the output of the ampli-fier. From Eq. 6.1a, we see that this implies that theamplitude of the wave going into the amplifier, ain, willdepend on the amplitude of the wave exiting the ampli-fier, aout.

Thus, if we want to use our amplifier as a voltageamplifier, we want to find a description that is more tai-lored to our needs than the scattering representation ofEq. 6.3. This can be found by simply re-expressing thescattering matrix relation of Eq. 6.3 in terms of volt-ages and currents. The result will be what we term theop amp representation of our amplifier, a representationwhich is standard in the discussion of classical amplifierssee, e.g., Boylestad and Nashelsky 2006. In this rep-

resentation, one views Va and Ib as inputs to the ampli-

fier: Va is set by whatever we connect to the amplifier

input, while Ib is set by whatever we connect to the am-plifier output. In contrast, the outputs of our amplifier

are the voltage in the output line, Vb, and the current

drawn by the amplifier at the input, Ia. Note that thisinterpretation of voltages and currents is identical to theway we viewed the voltage amplifier in the linear-response and quantum noise treatment of Sec. V.F.

Using Eqs. 6.1a and 6.1b and suppressing fre-quency labels for clarity, Eq. 6.3 may be written explic-itly in terms of the voltages and current in the input

Va , Ia and output Vb , Ib transmission lines,

Vb

Ia

= V − Zout

1

ZinI Va

Ib

+ V · V

I . 6.4

The coefficients in the above matrix are completely de-termined by the scattering matrix of Eq. 6.3 see Eqs.6.7 below; moreover, they are familiar from the dis-cussion of a voltage amplifier in Sec. V.F. V is thevoltage gain of the amplifier, I is the reverse currentgain of the amplifier, Zout is the output impedance, andZin is the input impedance. The last term on the RHS ofEq. 6.4 describes the two familiar kinds of amplifier

noise. V is the usual voltage noise of the amplifier re-

ferred back to the amplifier input, while I is the usualcurrent noise of the amplifier. Recall that in this stan-

dard description of a voltage amplifier see Sec. V.F Irepresents the back-action of the amplifier: the systemproducing the input signal responds to these current



fluctuations, resulting in an additional fluctuation in the

input signal going into the amplifier. Similarly, VV rep-resents the intrinsic output noise of the amplifier: thiscontribution to the total output noise does not dependon properties of the input signal. Note that we are using

a sign convention where a positive Ia indicates a cur-rent flowing into the amplifier at its input, while a posi-

tive Ib indicates a current flowing out of the amplifier

at its output. Also note that the operators Va and Ib onthe RHS of Eq. 6.4 will have noise; this noise is en-tirely due to the systems attached to the input and out-put of the amplifier, and as such should not be includedin what we call the added noise of the amplifier.

Additional important properties of our amplifier fol-low immediately from quantities in the op-amp repre-sentation. As discussed in Sec. V.E, the most importantmeasure of gain in our amplifier is the dimensionlesspower gain. This is the ratio between power dissipatedat the output to that dissipated at the input, taking theoutput current IB to be VB /Zout,

GP V2

4Zin

Zout1 +

VI

2Zin

Zout−1

. 6.5

Another important quantity is the loaded input im-pedance: what is the input impedance of the amplifier inthe presence of a load attached to the output? In thepresence of reverse current gain I0, the input imped-ance will depend on the output load. Taking the loadimpedance to be Zload, some simple algebra yields

1/Zin,loaded = 1/Zin + IV/Zload + Zout . 6.6

It is of course undesirable to have an input impedancewhich depends on the load. Thus, we see yet again that itis undesirable to have appreciable reverse gain in ouramplifier cf. Sec. IV.A.2.

3. Converting between representations

After some straightforward algebra we now expressthe op-amp parameters appearing in Eq. 6.4 in termsof the scattering matrix appearing in Eq. 6.3,

V = 2Zb

Za

s21

D, 6.7a

I = 2Zb

Za

s12

D, 6.7b

Zout = Zb1 + s111 + s22 − s12s21

D, 6.7c

1

Zin=

1

Za

1 − s111 − s22 − s12s21

D, 6.7d

where all quantities are evaluated at the same frequency, and D is defined as

D = 1 + s111 − s22 + s12s21. 6.8

Further, the voltage and current noises in the op-amprepresentation are simple linear combinations of the

noises Fa and Fb appearing in the scattering representa-tion,

V

Za · I = 2Za −

12

1 + s11

2s21

s22 − 1

D−

s12

DFa

Fb

. 6.9

Again, all quantities above are evaluated at frequency .Equation 6.9 immediately leads to an important con-

clusion and caveat: what one calls the ‘‘back-action’’ and‘‘added noise’’ in the scattering representation i.e., Faand Fb are not the same as the ‘‘back-action’’ and‘‘added noise’’ defined in the usual op-amp representa-

tion. For example, the op-amp back-action I does not in

general coincide with Fa, the back-action in the scatter-ing picture. If we are indeed interested in using our am-plifier as a voltage amplifier, we are interested in thetotal added noise of our amplifier as defined in the op-amp representation. As we saw in Sec. V.F cf. Eq.

5.71, this quantity involves both the noises I and V.We thus see explicitly something discussed in Sec. V.D: itis dangerous to make conclusions about how an ampli-fier behaves in the op-amp mode of operation based onits properties in the scattering mode of operation. As wesee, even though an amplifier is ideal in the scatteringmode i.e., Fa as small as possible, it can nonethelessfail to reach the quantum limit in the op-amp mode ofoperation.

In what follows, we calculate the op-amp noises V and

I in a minimal bosonic voltage amplifier, and show ex-plicitly how this description is connected to the moregeneral linear-response treatment of Sec. V.F. However,before proceeding, it is worth noting that Eqs. 6.7a,6.7b, 6.7c, and 6.7d are themselves completely con-sistent with linear-response theory. Using linear re-sponse, one would calculate the op-amp parameters V,I, Zin, and Zout using Kubo formulas cf. Eqs. 5.84 and5.85 and the discussion following Eq. 5.79. These in

turn would involve correlation functions of Ia and Vbevaluated at zero coupling to the amplifier input andoutput. Zero coupling means that there is no input volt-age to the amplifier i.e., a short circuit at the amplifier

input, Va=0 and there is nothing at the amplifier outputdrawing current i.e., an open circuit at the amplifier

output, Ib=0. Equation 6.4 tells us that in this case Vb

and Ia reduce to the noise operators VV and I, respec-

tively. Using the fact that the commutators of Fa and Fbare completely determined by the scattering matrix cf.Eq. 6.3, we verify explicitly in Appendix I.5 that theKubo formulas yield the same results for the op-amp



gains and impedances as Eqs. 6.7a, 6.7b, 6.7c, and6.7d above.

B. Minimal two-port scattering amplifier

1. Scattering versus op-amp quantum limit

In this section we demonstrate that an amplifier whichis ideal and minimally complex when used in the scatter-ing operation mode fails, when used as a voltage op-amp, to have a quantum-limited noise temperature. Thesystem we look at is similar to the amplifier consideredby Grassia 1998, though our conclusions are somewhatdifferent than those found there.

In the scattering representation, one might guess thatan ideal amplifier would be one where there are no re-flections of signals at the input and output, and no wayfor incident signals at the output port to reach the input.In this case, Eq. 6.3 takes the form

aout

bout = 0 0

G 0 ain

bin + Fa

Fb

, 6.10

where we have defined Gs21. All quantities aboveshould be evaluated at the same frequency ; for clarity,we omit the explicit dependence of quantitiesthroughout this section.

Turning to the op-amp representation, the aboveequation implies that our amplifier has no reverse gain,and that the input and output impedances are simplygiven by the impedances of the input and output trans-mission lines. From Eqs. 6.7, we have

V = 2ZbG/Za, 6.11a

I = 0, 6.11b

Zout = Zb, 6.11c

1/Zin = 1/Za. 6.11d

We immediately see that our amplifier looks less ideal asan op-amp. The input and output impedances are thesame as those of the input and output transmission line.However, for an ideal op-amp, we would have likedZin→ and Zout→0.

Also of interest are the expressions for the amplifiernoises in the op-amp representation

V

Za · I = − 2Za1

2−

1

2G

1 0Fa

Fb

. 6.12

As s12=0, the back-action noise is the same in both theop-amp and scattering representations: it is determined

completely by the noise operator Fa. However, the volt-age noise i.e., the intrinsic output noise involves both

Fa and Fb. We thus have the unavoidable consequence

that there will be correlations in I and V. Note that frombasic linear-response theory we know that there must be

some correlations between I and V if there is to be gaini.e., V is given by a Kubo formula involving these op-erators, see Eq. 4.3.

To make further progress, we note again that commu-

tators of the noise operators Fa and Fb are completelydetermined by Eq. 6.10 and the requirement that theoutput operators obey canonical commutation relations.We thus have

Fa,Fa† = 1, 6.13a

Fb,Fb† = 1 − G , 6.13b

Fa,Fb = Fa,Fb† = 0. 6.13c

We will be interested in the limit of a large powergain, which requires G1. A minimal solution to theabove equations would be to have the noise operatorsdetermined by two independent i.e., mutually commut-ing auxiliary input mode operators uin and vin

† ,

Fa = uin, 6.14

Fb = G − 1vin† . 6.15

Further, to minimize the noise of the amplifier, we takethe operating state of the amplifier to be the vacuum forboth these modes. With these choices, our amplifier is inexactly the minimal form described by Grassia 1998: aninput and output line coupled to a negative resistancebox and an auxiliary “cold load” via a four-port circula-tor see Fig. 13. The negative resistance box is nothingbut the single-mode bosonic amplifier discussed in Sec.V.B; an explicit realization of this element would be theparametric amplifier discussed in Sec. V.C. The ‘‘coldload’’ is a semi-infinite transmission line which modelsdissipation due to a resistor at zero temperature i.e., itsnoise is vacuum noise, cf. Appendix D.

aout

ain

bin

bout

ideal1-portamp.

coldload


cin cout=G1/2cin+(G-1)

1/2v†in

uin uout

vin

vacuum noise

vacuum noisevout

FIG. 13. Schematic of a minimal two-port amplifier whichreaches the quantum limit in the scattering mode of operation,but misses the quantum limit when used as a weakly coupledop-amp. See text for further description.



Note that within the scattering picture one would con-clude that our amplifier is ideal: in the large-gain limit,

the noise added by the amplifier to bout corresponds to asingle quantum at the input

Fb,Fb†/G = G − 1/Gvin

† , vin → 1. 6.16

This, however, is not the quantity which interests us: aswe want to use this system as a voltage op-amp, we wantto know if the noise temperature defined in the op-amppicture is as small as possible. We are also usually inter-ested in the case of a signal that is weakly coupled to ouramplifier; here weak coupling means that the input im-pedance of the amplifier is much larger than the imped-ance of the signal source i.e., ZinZs. In this limit, theamplifier only slightly increases the total damping of thesignal source.

To address whether we can reach the op-amp quan-tum limit in the weak-coupling regime, we can make useof the results of the general theory presented in Sec. V.F.In particular, we need to check whether the quantumnoise constraint of Eq. 5.88 is satisfied, as this is a pre-requisite for reaching the weak-coupling quantumlimit. Thus, we need to calculate the symmetrized spec-tral densities of the current and voltage noises, and theircross correlation. It is easy to confirm from the defini-tions of Eqs. 6.1a and 6.1b that these quantities takethe form

SVV = V, V†/4 − , 6.17a

SII = I, I†/4 − , 6.17b

SVI = V, I†/4 − . 6.17c

The expectation values here are over the operating stateof the amplifier; we have chosen this state to be thevacuum for the auxiliary mode operators uin and vin tominimize the noise.

Taking s211, and using Eqs. 6.14 and 6.15, wehave

SVV = Zauu + vv/4 = Za/2, 6.18a

SII = uu/Za = /Za, 6.18b

SVI = uu/2 = /2, 6.18c

where we have defined

ab ab† + b†a 6.19

and have used the fact that there cannot be any correla-tions between the operators u and v in the vacuum statei.e., uv†=0.

It follows immediately from the above equations thatour minimal amplifier does not optimize the quantumnoise constraint of Eq. 5.88,

SVVSII − Im SVI2 = 2 /22. 6.20

The noise product SVVSII is precisely twice the quantum-limited value. As a result, the general theory of Sec. V.Ftells us that if one couples an input signal weakly to thisamplifier i.e., ZsZin, it is impossible to reach thequantum limit on the added noise. Thus, while our am-plifier is ideal in the scattering mode of operation cf.Eq. 6.16, it fails to reach the quantum limit when usedin the weak-coupling, op-amp mode of operation. Ouramplifier’s failure to have ideal quantum noise alsomeans that if we tried to use it to do QND qubit detec-tion, the resulting back-action dephasing would be twiceas large as the minimum required by quantum mechan-ics cf. Sec. IV.B.

One might object to the above conclusions based onthe classical expression for the minimal noise tempera-ture, Eq. 5.76. Unlike the quantum noise constraint ofEq. 5.88, this equation also involves the real part of

SVI, and is optimized by our minimal amplifier. However,this does not mean that one can achieve a noise tem-perature of /2 at weak coupling. Recall from Sec. V.Fthat in the usual process of optimizing the noise tem-perature one starts by assuming the weak-coupling con-dition that the source impedance Zs is much smallerthan the amplifier input impedance Zin. One then findsthat to minimize the noise temperature, Zs should betuned to match the noise impedance of the amplifier

ZNSVV / SII. However, in our minimal bosonic ampli-fier, it follows from Eqs. 6.18 that ZN=Zin /2Zin: thenoise impedance is on the order of the input impedance.Thus, it is impossible to match the source impedance tothe noise impedance while at the same time satisfyingthe weak-coupling condition ZsZin.

Despite its failings, our amplifier can indeed yield aquantum-limited noise temperature in the op-amp modeof operation if we no longer insist on a weak coupling tothe input signal. To see this explicitly, imagine we con-nected our amplifier to a signal source with source im-pedance Zs. The total output noise of the amplifier, re-ferred back to the signal source, will now have the form

Vtot = − ZsZa/Zs + ZaI + V . 6.21

Note that this classical-looking equation can be rigor-ously justified within the full quantum theory if onestarts with a full description of the amplifier and thesignal source e.g., a parallel LC oscillator attached inparallel to the amplifier input. Plugging in the expres-

sions for I and V, we find

Vtot =

2 ZsZa

Zs + Za 2

Za

uin − Zauin − vin†

=Za

2Zs − Za

Zs + Zauin − vin

† . 6.22

Thus, if one tunes Zs to Za=Zin, the mode uin does notcontribute to the total added noise, and one reaches the



quantum limit. Physically speaking, by matching the sig-nal source to the input line the back-action noise de-

scribed by Fa= uin does not feed back into the input ofthe amplifier. Note that to achieve this matching explic-itly requires one to be far from weak coupling. HavingZs=Za means that when we attach the amplifier to thesignal source, we dramatically increase the damping ofthe signal source.

2. Why is the op-amp quantum limit not achieved?

Returning to the more interesting case of a weakamplifier-signal coupling, one might still be puzzled as towhy our seemingly ideal amplifier misses the quantumlimit. While the mathematics behind Eq. 6.20 is fairlytransparent, it is also possible to understand this resultheuristically. To that end, note again that the amplifier

noise cross correlation SIV does not vanish in the large-gain limit cf. Eq. 6.18c. Correlations between the twoamplifier noises represent a kind of information, as bymaking use of them, we can improve the performance ofthe amplifier. It is easy to take advantage of out-of-phase

correlations between I and V i.e., Im SVI by simply tun-ing the phase of the source impedance cf. Eq. 5.75.However, one cannot take advantage of in-phase noise

correlations i.e., Re SVI as easily. To take advantage ofthe information here, one needs to modify the amplifieritself. By feeding back some of the output voltage to theinput, one could effectively cancel out some of the back-

action current noise I and thus reduce the overall mag-

nitude of SII. Hence, the unused information in the cross

correlator Re SVI represents a kind of wasted informa-tion: had we made use of these correlations via a feed-back loop, we could have reduced the noise temperatureand increased the information provided by our amplifier.

The presence of a nonzero Re SVI thus corresponds towasted information, implying that we cannot reach thequantum limit. Recall that within the linear-response ap-proach we were able to prove rigorously that a large-gain amplifier with ideal quantum noise must have

Re SVI=0 cf. the discussion following Eq. 5.57; thus, a

nonvanishing Re SVI rigorously implies that one cannotbe at the quantum limit. In Appendix H, we give anexplicit demonstration of how feedback may be used toutilize these cross correlations to reach the quantumlimit.

Finally, yet another way of seeing that our amplifierdoes not reach the quantum limit in the weak-couplingregime is to realize that this system does not have awell-defined effective temperature. Recall from Sec. V.Fthat a system with ideal quantum noise i.e., one thatsatisfies Eq. 5.88 as an equality necessarily has thesame effective temperature at its input and output portscf. Eq. 5.55. Here that implies the requirement

V2SVV/Zout = ZinSII 2kBTeff. 6.23

In contrast, our minimal bosonic amplifier has very dif-ferent input and output effective temperatures

2kBTeff,in = ZinSII = /2, 6.24

2kBTeff,out = V2 · SVV/Zout = 2G . 6.25

This large difference in effective temperatures meansthat it is impossible for the system to possess ideal quan-tum noise, and thus it cannot reach the weak-couplingquantum limit.

While it implies that one is not at the quantum limit,the fact that Teff,inTeff,out can nonetheless be viewed asan asset. From a practical point of view, a large Teff,in canbe dangerous. Even though the direct effect of the largeTeff,in is offset by an appropriately weak coupling to theamplifier see Eq. 5.70 and following discussion, thislarge Teff,in can also heat up other degrees of freedom ifthey couple strongly to the back-action noise of the am-plifier. This can in turn lead to unwanted heating of theinput system. As Teff,in is usually constant over a broadrange of frequencies, this unwanted heating effect canbe quite bad. In the minimal amplifier discussed here,this problem is circumvented by having a small Teff,in.The only price that is paid is that the added noise will be2 the quantum limit value. We discuss this issue furtherin Sec. VII.

VII. REACHING THE QUANTUM LIMIT IN PRACTICE

A. Importance of QND measurements

The fact that QND measurements are repeatable is offundamental practical importance in overcoming detec-tor inefficiencies Gambetta et al., 2007. A prototypicalexample is the electron shelving technique Nagourneyet al., 1986; Sauter et al., 1986 used to measure trappedions. A related technique is used in present implemen-tations of ion-trap-based quantum computation. Herethe extremely long-lived hyperfine state of an ion isread out via state-dependent optical fluorescence. Withproperly chosen circular polarization of the exciting la-ser, only one hyperfine state fluoresces and the transitionis cycling; that is, after fluorescence the ion almost al-ways returns to the same state it was in prior to absorb-ing the exciting photon. Hence the measurement isQND. Typical experimental parameters Wineland et al.,1998 allow the cycling transition to produce N106

fluorescence photons. Given the photomultiplier quan-tum efficiency and typically small solid angle coverage,only a small number nd will be detected on average. Theprobability of getting zero detections ignoring darkcounts for simplicity and hence misidentifying the hy-perfine state is P0=e−nd. Even for a very poor overalldetection efficiency of only 10−5, we still have nd=10 andnearly perfect fidelity F=1−P00.999 955. It is impor-tant to note that the total time available for measure-ment is not limited by the phase coherence time T2 of



the qubit or by the measurement-induced dephasingKorotkov, 2001a; Makhlin et al., 2001; Schuster et al.,2005; Gambetta et al., 2006, but rather only by the rateat which the qubit makes real transitions between mea-surement z eigenstates. In a perfect QND measure-ment there is no measurement-induced state mixingMakhlin et al., 2001 and the relaxation rate 1/T1 isunaffected by the measurement process.

B. Power matching versus noise matching

In Sec. V, we saw that an important part of reachingthe quantum limit on the added noise of an amplifierwhen used in the op-amp mode of operation is to op-timize the coupling strength to the amplifier. For a posi-tion detector, this condition corresponds to tuning thestrength of the back-action damping to be muchsmaller than the intrinsic oscillator damping cf. Eq.5.69. For a voltage amplifier, this condition corre-sponds to tuning the impedance of the signal source tobe equal to the noise impedance cf. Eq. 5.77, an im-pedance that is much smaller than the amplifier’s inputimpedance cf. Eq. 5.90.

In this section, we make the simple point that optimiz-ing the coupling i.e., source impedance to reach thequantum limit is not the same as what one would do tooptimize the power gain. To understand this, we need tointroduce another measure of power gain commonlyused in the engineering community, the available powergain GP,avail. For simplicity, we discuss this quantity inthe context of a linear voltage amplifier, using the nota-tions of Sec. V.F; it can be analogously defined for theposition detector of Sec. V.E. GP,avail tells us how muchpower we are providing to an optimally matched outputload relative to the maximum power we could in prin-ciple have extracted from the source. This is in markedcontrast to the power gain GP, which was calculated us-ing the actual power drawn at the amplifier input.

For the available power gain, we first consider Pin,avail.This is the maximum possible power that could be deliv-ered to the input of the amplifier, assuming we opti-mized both the value of the input impedance Zin and theload impedance Zload while keeping Zs fixed. For sim-plicity, we take all impedances to be real in our discus-sion. In general, the power drawn at the input of theamplifier is Pin=vin

2 Zin / Zs+Zin2. Maximizing this overZin, we obtain the available input power Pin,avail,

Pin,avail = vin2 /4Zs. 7.1

The maximum occurs for Zin=Zs.The output power supplied to the load Pout

=vload2 /Zload is calculated as before, keeping Zin and Zs

distinct. One has

Pout =vout

2

Zload Zload

Zout + Zload2

7.2

=2vin

2

Zout Zin

Zin + Zs2 Zload/Zout

1 + Zload/Zout2 .

As a function of Zload, Pout is maximized when Zload=Zout,

Pout,max =2vin

2

4Zout Zin

Zin + Zs2

. 7.3

The available power gain is now defined as

GP,avail Pout,max

Pin,avail=2Zs

Zout Zin

Zin + Zs2

= GP4Zs/Zin

1 + Zs/Zin2 . 7.4

We see that GP,avail is strictly less than or equal to thepower gain GP; equality is achieved only when Zs=Zini.e., when the source impedance is “power matched” tothe input of the amplifier. The general situation whereGP,avail GP indicates that we are not drawing as muchpower from the source as we could, and hence the actualpower supplied to the load is not as large as it could be.

Consider now a situation where we have achieved thequantum limit on the added noise. This necessarilymeans that we have “noise matched,” i.e., taken Zs to beequal to the noise impedance ZN. The available powergain in this case is

GP,avail 2ZN/Zout 2GPGP. 7.5

We have used Eq. 5.90, which tells us that the noiseimpedance is smaller than the input impedance by alarge factor of 1/ 2GP. Thus, as reaching the quantumlimit requires the use of a source impedance muchsmaller than Zin, it results in a dramatic drop in theavailable power gain compared to the case where wepower match i.e., take Zs=Zin. In practice, one mustdecide whether it is more important to minimize theadded noise or maximize the power provided at the out-put of the amplifier: one cannot do both at the sametime.

VIII. CONCLUSIONS

In this review, we have given an introduction to quan-tum limits for position detection and amplification, lim-its that are tied to fundamental constraints on quantumnoise correlators. We end by emphasizing notable cur-rent developments and pointing out future perspectivesin the field.

As emphasized, much of our discussion has been di-rectly relevant to the measurement of mechanical nan-oresonators, a topic attracting considerable recent atten-tion. These nanoresonators are typically studied bycoupling them either to electrical often superconduct-ing circuits or to optical cavities. A key goal is toachieve quantum-limited continuous position detectioncf. Sec. V.E; current experiments are coming tantaliz-ingly close to this limit cf. Table III. Although the abil-ity to follow the nanoresonator’s motion with a precision



set by the quantum limit is in principle independentfrom being at low temperatures, it becomes interestingonly when the systems are near their ground state; onecould then, e.g., monitor the oscillator’s zero-point fluc-tuations cf. Sec. III.B.3. Given the comparatively smallvalues of mechanical frequencies mostly less than 1GHz, this often calls for the application of nonequilib-rium cooling techniques which exploit back-action to re-duce the effective temperature of the mechanical device,a technique that has been demonstrated recently in bothsuperconducting circuits and optomechanical setups seeMarquardt and Girvin 2009 for a review.

The ability to perform quantum-limited position de-tection will in turn open up many new interesting av-enues of research. Among the most significant is thepossibility of quantum feedback control Wiseman andMilburn, 1993, 1994, where one uses the continuouslyobtained measurement output to tailor the state of themechanical resonator. The relevant theoretical frame-work is that of quantum conditional evolution and quan-tum trajectories see, e.g., Brun, 2002; Jacobs and Steck,2006, where one tracks the state of a measured quan-tum system in a particular run of the experiment. Appli-cation of these ideas has only recently been explored incondensed matter contexts Korotkov, 1999, 2001b;Goan and Milburn, 2001; Goan et al., 2001; Oxtoby et al.,2006, 2008; Bernád et al., 2008. Fully understanding thepotential of these techniques, as well as differences thatoccur in condensed matter versus atomic physics con-texts, remains an active area of research. Other impor-tant directions in nanomechanics include the possibilityof detecting quantum jumps in the state of a mechanicalresonator via QND measurement of its energy Santa-more, Doherty, and Cross, 2004; Santamore, Goan, et al.,2004; Jayich et al., 2008; Thompson et al., 2008, as wellas the possibility of making back-action-evading mea-surements cf. Sec. V.H. Back-action evasion using amicrowave cavity detector coupled to a nanomechanicalresonator was recently reported Hertzberg et al., 2010.

Another area distinct from nanomechanics whererapid progress is being made is the readout of solid statequbits using microwave signals sent through cavitieswhose transmission properties are controlled by the qu-bit. At the moment, one is close to achieving good-fidelity single-shot QND readout, which is a prerequisitefor a large number of applications in quantum informa-tion processing. The gradually growing informationabout the qubit state is extracted from the measurednoisy microwave signal trace, leading to a correspondingcollapse of the qubit state. This process can also be de-scribed by conditional quantum evolution and quantumtrajectories.

A promising method for superconducting qubit read-out currently employed is a so-called latching measure-ment, where the hysteretic behavior of a strongly drivenanharmonic system e.g., a Josephson junction is ex-ploited to toggle between two states depending on thequbit state Siddiqi et al., 2004; Lupascu et al., 2006.Although this is then no longer a linear measurementscheme and is therefore distinct from what was dis-

cussed in this review, it can be turned into a linear am-plifier for a sufficiently weak input signal. An interestingand important open question is whether such a setup canreach the quantum limit on linear amplification.

Both qubit detection and mechanical measurementsin electrical circuits would benefit from quantum-limitedon-chip amplifiers. Such amplifiers are now being devel-oped using the tools of circuit quantum electrodynamics,employing Josephson junctions or SQUIDs coupled tomicrowave transmission line cavities Bergeal et al.,2008; Castellanos-Beltran et al., 2008. Such an amplifierhas already been used to perform continuous positiondetection with a measurement imprecision below theSQL level Teufel et al., 2009.

ACKNOWLEDGMENTS

This work was supported in part by NSERC, the Ca-nadian Institute for Advanced Research CIFAR, NSAunder ARO Contract No. W911NF-05-1-0365, the NSFunder Grant No. DMR-0653377 and No. DMR-0603369,the David and Lucile Packard Foundation, the W. M.Keck Foundation, and the Alfred P. Sloan Foundation.We acknowledge the support and hospitality of the As-pen Center for Physics where part of this work was car-ried out. F.M. acknowledges support via DIP, GIF, andthrough SFB 631, SFB/TR 12, the Nanosystems Initia-tive Munich, and the Emmy-Noether program. S.M.G.acknowledges the support of the Centre for AdvancedStudy at the Norwegian Academy of Science and Letterswhere part of this work was carried out.

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Introduction to Quantum Noise, Measurement and Amplification: OnlineAppendices

A.A. Clerk,1 M.H. Devoret,2 S.M. Girvin,3 Florian Marquardt,4 and R.J. Schoelkopf21Department of Physics, McGill University, 3600 rue UniversityMontreal, QC Canada H3A 2T8∗2Department of Applied Physics, Yale UniversityPO Box 208284, New Haven, CT 06520-82843Department of Physics, Yale UniversityPO Box 208120, New Haven, CT 06520-81204Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-Universitat MunchenTheresienstr. 37, D-80333 Munchen, Germany

(Dated: July 2, 2009)

Contents

A. Basics of Classical and Quantum Noise 11. Classical noise correlators 12. The Wiener-Khinchin Theorem 23. Square law detectors and classical spectrum analyzers 3

B. Quantum Spectrum Analyzers: Further Details 41. Two-level system as a spectrum analyzer 42. Harmonic oscillator as a spectrum analyzer 53. Practical quantum spectrum analyzers 6

a. Filter plus diode 6b. Filter plus photomultiplier 7c. Double sideband heterodyne power spectrum 7

C. Modes, Transmission Lines and ClassicalInput/Output Theory 81. Transmission lines and classical input-output theory 82. Lagrangian, Hamiltonian, and wave modes for a

transmission line 103. Classical statistical mechanics of a transmission line 114. Amplification with a transmission line and a negative

resistance 13

D. Quantum Modes and Noise of a Transmission Line 151. Quantization of a transmission line 152. Modes and the windowed Fourier transform 163. Quantum noise from a resistor 17

E. Back Action and Input-Output Theory for DrivenDamped Cavities 181. Photon shot noise inside a cavity and back action 192. Input-output theory for a driven cavity 203. Quantum limited position measurement using a cavity

detector 244. Back-action free single-quadrature detection 28

F. Information Theory and Measurement Rate 291. Method I 292. Method II 30

G. Number Phase Uncertainty 30

H. Using feedback to reach the quantum limit 311. Feedback using mirrors 31

∗Electronic address: [email protected]

2. Explicit examples 323. Op-amp with negative voltage feedback 33

I. Additional Technical Details 341. Proof of quantum noise constraint 342. Proof that a noiseless detector does not amplify 363. Simplifications for a quantum-limited detector 364. Derivation of non-equilibrium Langevin equation 375. Linear-response formulas for a two-port bosonic amplifier37

a. Input and output impedances 38b. Voltage gain and reverse current gain 38

6. Details for the two-port bosonic voltage amplifier withfeedback 39

References 40

Appendix A: Basics of Classical and Quantum Noise

1. Classical noise correlators

Consider a classical random voltage signal V (t). Thesignal is characterized by zero mean 〈V (t)〉 = 0, andautocorrelation function

GV V (t− t′) = 〈V (t)V (t′)〉 (A1)

whose sign and magnitude tells us whether the voltagefluctuations at time t and time t′ are correlated, anti-correlated or statistically independent. We assume thatthe noise process is stationary (i.e., the statistical proper-ties are time translation invariant) so that GV V dependsonly on the time difference. If V (t) is Gaussian dis-tributed, then the mean and autocorrelation completelyspecify the statistical properties and the probability dis-tribution. We will assume here that the noise is due tothe sum of a very large number of fluctuating chargesso that by the central limit theorem, it is Gaussian dis-tributed. We also assume that GV V decays (sufficientlyrapidly) to zero on some characteristic correlation timescale τc which is finite.

The spectral density of the noise as measured by aspectrum analyzer is a measure of the intensity of thesignal at different frequencies. In order to understand

2

the spectral density of a random signal, it is useful todefine its ‘windowed’ Fourier transform as follows:

VT [ω] =1√T

∫ +T/2

−T/2dt eiωtV (t), (A2)

where T is the sampling time. In the limit T τc theintegral is a sum of a large number N ≈ T

τcof random

uncorrelated terms. We can think of the value of theintegral as the end point of a random walk in the complexplane which starts at the origin. Because the distancetraveled will scale with

√T , our choice of normalization

makes the statistical properties of V [ω] independent ofthe sampling time T (for sufficiently large T ). Noticethat VT [ω] has the peculiar units of volts

√secs which is

usually denoted volts/√

Hz.The spectral density (or ‘power spectrum’) of the noise

is defined to be the ensemble averaged quantity

SV V [ω] ≡ limT→∞

〈|VT [ω]|2〉 = limT→∞

〈VT [ω]VT [−ω]〉 (A3)

The second equality follows from the fact that V (t) isreal valued. The Wiener-Khinchin theorem (derived inAppendix A.2) tells us that the spectral density is equalto the Fourier transform of the autocorrelation function

SV V [ω] =∫ +∞

−∞dt eiωtGV V (t). (A4)

The inverse transform relates the autocorrelation func-tion to the power spectrum

GV V (t) =∫ +∞

−∞

dω

2πe−iωtSV V [ω]. (A5)

We thus see that a short auto-correlation time impliesa spectral density which is non-zero over a wide range offrequencies. In the limit of ‘white noise’

GV V (t) = σ2δ(t) (A6)

the spectrum is flat (independent of frequency)

SV V [ω] = σ2 (A7)

In the opposite limit of a long autocorrelation time, thesignal is changing slowly so it can only be made up outof a narrow range of frequencies (not necessarily centeredon zero).

Because V (t) is a real-valued classical variable, it natu-rally follows that GV V (t) is always real. Since V (t) is nota quantum operator, it commutes with its value at othertimes and thus, 〈V (t)V (t′)〉 = 〈V (t′)V (t)〉. From this itfollows that GV V (t) is always symmetric in time and thepower spectrum is always symmetric in frequency

SV V [ω] = SV V [−ω]. (A8)

As a prototypical example of these ideas, let us con-sider a simple harmonic oscillator of mass M and fre-quency Ω. The oscillator is maintained in equilibrium

with a large heat bath at temperature T via some in-finitesimal coupling which we will ignore in consideringthe dynamics. The solution of Hamilton’s equations ofmotion are

x(t) = x(0) cos(Ωt) + p(0)1MΩ

sin(Ωt)

p(t) = p(0) cos(Ωt)− x(0)MΩ sin(Ωt), (A9)

where x(0) and p(0) are the (random) values of the po-sition and momentum at time t = 0. It follows that theposition autocorrelation function is

Gxx(t) = 〈x(t)x(0)〉 (A10)

= 〈x(0)x(0)〉 cos(Ωt) + 〈p(0)x(0)〉 1MΩ

sin(Ωt).

Classically in equilibrium there are no correlations be-tween position and momentum. Hence the second termvanishes. Using the equipartition theorem 1

2MΩ2〈x2〉 =12kBT , we arrive at

Gxx(t) =kBT

MΩ2cos(Ωt) (A11)

which leads to the spectral density

Sxx[ω] = πkBT

MΩ2[δ(ω − Ω) + δ(ω + Ω)] (A12)

which is indeed symmetric in frequency.

2. The Wiener-Khinchin Theorem

From the definition of the spectral density in Eqs.(A2-A3) we have

SV V [ω] =1T

∫ T

0

dt

∫ T

0

dt′ eiω(t−t′)〈V (t)V (t′)〉

=1T

∫ T

0

dt

∫ +2B(t)

−2B(t)

dτ eiωτ 〈V (t+ τ/2)V (t− τ/2)〉

(A13)

where

B (t) = t if t < T/2= T − t if t > T/2.

If T greatly exceeds the noise autocorrelation time τcthen it is a good approximation to extend the bound B(t)in the second integral to infinity, since the dominant con-tribution is from small τ . Using time translation invari-ance gives

SV V [ω] =1T

∫ T

0

dt

∫ +∞

−∞dτ eiωτ 〈V (τ)V (0)〉

=∫ +∞

−∞dτ eiωτ 〈V (τ)V (0)〉 . (A14)

3

This proves the Wiener-Khinchin theorem stated inEq. (A4).

A useful application of these ideas is the following.Suppose that we have a noisy signal V (t) = V + η(t)which we begin monitoring at time t = 0. The integratedsignal up to time t is given by

I(T ) =∫ T

0

dt V (t) (A15)

and has mean

〈I(T )〉 = V T. (A16)

Provided that the integration time greatly exceeds theautocorrelation time of the noise, I(T ) is a sum of a largenumber of uncorrelated random variables. The centrallimit theorem tells us in this case that I(t) is gaussiandistributed even if the signal itself is not. Hence theprobability distribution for I is fully specified by its meanand its variance

〈(∆I)2〉 =∫ T

0

dtdt′ 〈η(t)η(t′)〉. (A17)

From the definition of spectral density above we have thesimple result that the variance of the integrated signalgrows linearly in time with proportionality constant givenby the noise spectral density at zero frequency

〈(∆I)2〉 = SV V [0]T. (A18)

As a simple application, consider the photon shot noiseof a coherent laser beam. The total number of photonsdetected in time T is

N(T ) =∫ T

0

dt N(t). (A19)

The photo-detection signal N(t) is not gaussian, butrather is a point process, that is, a sequence of delta func-tions with random Poisson distributed arrival times andmean photon arrival rate N . Nevertheless at long timesthe mean number of detected photons

〈N(T )〉 = NT (A20)

will be large and the photon number distribution will begaussian with variance

〈(∆N)2〉 = SNN T. (A21)

Since we know that for a Poisson process the variance isequal to the mean

〈(∆N)2〉 = 〈N(T )〉, (A22)

it follows that the shot noise power spectral density is

SNN (0) = N . (A23)

Since the noise is white this result happens to be valid atall frequencies, but the noise is gaussian distributed onlyat low frequencies.

3. Square law detectors and classical spectrum analyzers

Now that we understand the basics of classical noise,we can consider how one experimentally measures a clas-sical noise spectral density. With modern high speeddigital sampling techniques it is perfectly feasible to di-rectly measure the random noise signal as a function oftime and then directly compute the autocorrelation func-tion in Eq. (A1). This is typically done by first per-forming an analog-to-digital conversion of the noise sig-nal, and then numerically computing the autocorrelationfunction. One can then use Eq. (A4) to calculate thenoise spectral density via a numerical Fourier transform.Note that while Eq. (A4) seems to require an ensembleaverage, in practice this is not explicitly done. Instead,one uses a sufficiently long averaging time T (i.e. muchlonger than the correlation time of the noise) such thata single time-average is equivalent to an ensemble aver-age. This approach of measuring a noise spectral densitydirectly from its autocorrelation function is most appro-priate for signals at RF frequencies well below 1 MHz.

For microwave signals with frequencies well above 1GHz, a very different approach is usually taken. Here, thestandard route to obtain a noise spectral density involvesfirst shifting the signal to a lower intermediate frequencyvia a technique known as heterodyning (we discuss thismore in Sec. B.3.c). This intermediate-frequency signalis then sent to a filter which selects a narrow frequencyrange of interest, the so-called ‘resolution bandwidth’.Finally, this filtered signal is sent to a square-law detector(e.g. a diode), and the resulting output is averaged overa certain time-interval (the inverse of the so-called ‘videobandwidth’). It is this final output which is then takento be a measure of the noise spectral density.

It helps to put the above into equations. Ignoring forsimplicity the initial heterodyning step, let

Vf [ω] = f [ω]V [ω] (A24)

be the voltage at the output of the filter and the inputof the square law detector. Here, f [ω] is the (ampli-tude) transmission coefficient of the filter and V [ω] is theFourier transform of the noisy signal we are measuring.From Eq. (A5) it follows that the output of the squarelaw detector is proportional to

〈I〉 =∫ +∞

−∞

dω

2π|f [ω]|2SV V [ω]. (A25)

Approximating the narrow band filter centered on fre-quency ±ω0 as1

|f [ω]|2 = δ(ω − ω0) + δ(ω + ω0) (A26)

1 A linear passive filter performs a convolution Vout(t) =R +∞−∞ dt′ F (t − t′)Vin(t′) where F is a real-valued (and causal)

function. Hence it follows that f [ω], which is the Fourier trans-form of F , obeys f [−ω] = f∗[ω] and hence |f [ω]|2 is symmetricin frequency.

4

we obtain

〈I〉 = SV V (−ω0) + SV V (ω0) (A27)

showing as expected that the classical square law detectormeasures the symmetrized noise power.

We thus have two very different basic approaches forthe measurement of classical noise spectral densities: forlow RF frequencies, one can directly measure the noiseautocorrelation, whereas for high microwave frequencies,one uses a filter and a square law detector. For noisesignals in intermediate frequency ranges, a combinationof different methods is generally used. The whole storybecomes even more complicated, as at very high frequen-cies (e.g. in the far infrared), devices such as the so-called ‘Fourier Transform spectrometer’ are in fact basedon a direct measurement of the equivalent of an auto-correlation function of the signal. In the infrared, visibleand ultraviolet, noise spectrometers use gratings followedby a slit acting as a filter.

Appendix B: Quantum Spectrum Analyzers: Further Details

1. Two-level system as a spectrum analyzer

In this sub-appendix, we derive the Golden Rule tran-sition rates Eqs. (2.6) describing a quantum two-level sys-tem coupled to a noise source (cf. Sec. II.B). Our deriva-tion is somewhat unusual, in that the role of the contin-uum as a noise source is emphasized from the outset. Westart by treating the noise F (t) in Eq. (2.5) as being aclassically noisy variable. We assume that the couplingA is under our control and can be made small enoughthat the noise can be treated in lowest order perturba-tion theory. We take the state of the two-level system tobe

|ψ(t)〉 =(αg(t)αe(t)

). (B1)

In the interaction representation, first-order time-dependent perturbation theory gives

|ψI(t)〉 = |ψ(0)〉 − i

~

∫ t

0

dτ V (τ)|ψ(0)〉. (B2)

If we initially prepare the two-level system in its groundstate, the amplitude to find it in its excited state at timet is from Eq. (B2)

αe = − iA~

∫ t

0

dτ 〈e|σx(τ)|g〉F (τ),

= − iA~

∫ t

0

dτ eiω01τF (τ). (B3)

Since the integrand in Eq. (B3) is random, αe is a sumof a large number of random terms; i.e. its value is theendpoint of a random walk in the complex plane (as dis-cussed above in defining the spectral density of classical

noise). As a result, for times exceeding the autocorre-lation time τc of the noise, the integral will not growlinearly with time but rather only as the square root oftime, as expected for a random walk. We can now com-pute the probability

pe(t) ≡ |αe|2 =A2

~2

∫ t

0

∫ t

0

dτ1dτ2 e−iω01(τ1−τ2)F (τ1)F (τ2)

(B4)which we expect to grow quadratically for short timest < τc, but linearly for long times t > τc. Ensembleaveraging the probability over the random noise yields

pe(t) =A2

~2

∫ t

0

∫ t

0

dτ1dτ2 e−iω01(τ1−τ2) 〈F (τ1)F (τ2)〉

(B5)Introducing the noise spectral density

SFF (ω) =∫ +∞

−∞dτ eiωτ 〈F (τ)F (0)〉, (B6)

and utilizing the Fourier transform defined in Eq. (A2)and the Wiener-Khinchin theorem from Appendix A.2,we find that the probability to be in the excited stateindeed increases linearly with time at long times,2

pe(t) = tA2

~2SFF (−ω01) (B7)

The time derivative of the probability gives the transitionrate from ground to excited states

Γ↑ =A2

~2SFF (−ω01) (B8)

Note that we are taking in this last expression the spec-tral density on the negative frequency side. If F were astrictly classical noise source, 〈F (τ)F (0)〉 would be real,and SFF (−ω01) = SFF (+ω01). However, because as wediscuss below F is actually an operator acting on the en-vironmental degrees of freedom,

[F (τ), F (0)

]6= 0 and

SFF (−ω01) 6= SFF (+ω01).Another possible experiment is to prepare the two-level

system in its excited state and look at the rate of decayinto the ground state. The algebra is identical to thatabove except that the sign of the frequency is reversed:

Γ↓ =A2

~2SFF (+ω01). (B9)

We now see that our two-level system does indeed act as aquantum spectrum analyzer for the noise. Operationally,

2 Note that for very long times, where there is a significant de-pletion of the probability of being in the initial state, first-orderperturbation theory becomes invalid. However, for sufficientlysmall A, there is a wide range of times τc t 1/Γ for whichEq. B7 is valid. Eqs. (2.6a) and (2.6b) then yield well-definedrates which can be used in a master equation to describe the fulldynamics including long times.

5

we prepare the system either in its ground state or in itsexcited state, weakly couple it to the noise source, andafter an appropriate interval of time (satisfying the aboveinequalities) simply measure whether the system is nowin its excited state or ground state. Repeating this pro-tocol over and over again, we can find the probabilityof making a transition, and thereby infer the rate andhence the noise spectral density at positive and nega-tive frequencies. Naively one imagines that a spectrom-eters measures the noise spectrum by extracting a smallamount of the signal energy from the noise source andanalyzes it. This is not the case however. There mustbe energy flowing in both directions if the noise is to befully characterized.

We now rigorously treat the quantity F (τ) as a quan-tum Heisenberg operator which acts in the Hilbert spaceof the noise source. The previous derivation is unchanged(the ordering of F (τ1)F (τ2) having been chosen cor-rectly in anticipation of the quantum treatment), andEqs. (2.6a,2.6b) are still valid provided that we interpretthe angular brackets in Eq. (B5,B6) as representing aquantum expectation value (evaluated in the absence ofthe coupling to the spectrometer):

SFF (ω) =∫ +∞

−∞dτ eiωτ

∑α,γ

ραα 〈α|F (τ)|γ〉〈γ|F (0)|α〉.

(B10)Here, we have assumed a stationary situation, wherethe density matrix ρ of the noise source is diagonal inthe energy eigenbasis (in the absence of the coupling tothe spectrometer). However, we do not necessarily as-sume that it is given by the equilibrium expression. Thisyields the standard quantum mechanical expression forthe spectral density:

SFF (ω) =∫ +∞

−∞dτ eiωτ

∑α,γ

ραα ei~ (εα−εγ)τ |〈α|F |γ〉|2

= 2π~∑α,γ

ραα |〈α|F |γ〉|2δ(εγ − εα − ~ω).(B11)

Substituting this expression into Eqs. (2.6a,2.6b), we de-rive the familiar Fermi Golden Rule expressions for thetwo transition rates.

In standard courses, one is not normally taught thatthe transition rate of a discrete state into a continuumas described by Fermi’s Golden Rule can (and indeedshould!) be viewed as resulting from the continuum act-ing as a quantum noise source which causes the am-plitudes of the different components of the wave func-tion to undergo random walks. The derivation presentedhere hopefully provides a motivation for this interpreta-tion. In particular, thinking of the perturbation (i.e. thecoupling to the continuum) as quantum noise with asmall but finite autocorrelation time (inversely relatedto the bandwidth of the continuum) neatly explains whythe transition probability increases quadratically for veryshort times, but linearly for very long times.

It it is important to keep in mind that our expressionsfor the transition rates are only valid if the autocorrela-tion time of our noise is much shorter that the typicaltime we are interested in; this typical time is simply theinverse of the transition rate. The requirement of a shortautocorrelation time in turn implies that our noise sourcemust have a large bandwidth (i.e. there must be largenumber of available photon frequencies in the vacuum)and must not be coupled too strongly to our system. Thisis true despite the fact that our final expressions for thetransition rates only depend on the spectral density atthe transition frequency (a consequence of energy con-servation).

One standard model for the continuum is an infinitecollection of harmonic oscillators. The electromagneticcontinuum in the hydrogen atom case mentioned above isa prototypical example. The vacuum electric field noisecoupling to the hydrogen atom has an extremely shortautocorrelation time because the range of mode frequen-cies ωα (over which the dipole matrix element couplingthe atom to the mode electric field ~Eα is significant) isextremely large, ranging from many times smaller thanthe transition frequency to many times larger. Thus, theautocorrelation time of the vacuum electric field noise isconsiderably less than 10−15s, whereas the decay time ofthe hydrogen 2p state is about 10−9s. Hence the inequal-ities needed for the validity of our expressions are veryeasily satisfied.

2. Harmonic oscillator as a spectrum analyzer

We now provide more details on the system describedin Sec. II.B, where a harmonic oscillator acts as a spec-trometer of quantum noise. We start with the couplingHamiltonian givein in Eq. (2.9). In analogy to the TLSspectrometer, noise in F at the oscillator frequency Ωcan cause transitions between its eigenstates. We as-sume both that A is small, and that our noise sourcehas a short autocorrelation time, so we may again useperturbation theory to derive rates for these transitions.There is a rate for increasing the number of quanta inthe oscillator by one, taking a state |n〉 to |n+ 1〉:

Γn→n+1 =A2

~2

[(n+ 1)x2

ZPF

]SFF [−Ω] ≡ (n+ 1)Γ↑

(B12)As expected, this rate involves the noise at −Ω, as en-ergy is being absorbed from the noise source. Similarly,there is a rate for decreasing the number of quanta in theoscillator by one:

Γn→n−1 =A2

~2

(nx2

ZPF

)SFF [Ω] ≡ nΓ↓ (B13)

This rate involves the noise at +Ω, as energy is beingemitted to the noise source.

Given these transition rates, we may immediately writea simple master equation for the probability pn(t) that

6

there are n quanta in the oscillator:

d

dtpn = [nΓ↑pn−1 + (n+ 1)Γ↓pn+1]

− [nΓ↓ + (n+ 1)Γ↑] pn (B14)

The first two terms describe transitions into the state |n〉from the states |n + 1〉 and |n − 1〉, and hence increasepn. In contrast, the last two terms describe transitionsout of the state |n〉 to the states |n+ 1〉 and |n− 1〉, andhence decrease pn. The stationary state of the oscillatoris given by solving Eq. (B14) for d

dtpn = 0, yielding:

pn = e−n~Ω/(kBTeff )(

1− e−~Ω/(kBTeff ))

(B15)

where the effective temperature Teff [Ω] is defined inEq. (2.8). Eq. (B15) describes a thermal equilibriumdistribution of the oscillator, with an effective oscillatortemperature Teff [Ω] determined by the quantum noisespectrum of F . This is the same effective temperaturethat emerged in our discussion of the TLS spectrum an-alyzer. As we have seen, if the noise source is in thermalequilibrium at a temperature Teq, then Teff [Ω] = Teq.In the more general case where the noise source is notin thermal equilibrium, Teff only serves to characterizethe asymmetry of the quantum noise, and will vary withfrequency 3.

We can learn more about the quantum noise spectrumof F by also looking at the dynamics of the oscillator.In particular, as the average energy 〈E〉 of the oscillatoris just given by 〈E(t)〉 =

∑∞n=0 ~Ω

(n+ 1

2

)pn(t), we can

use the master equation Eq. (B14) to derive an equa-tion for its time dependence. One thus finds Eq. (2.10).By demanding d〈E〉/dt = 0 in this equation, we findthat the combination of damping and heating effectscauses the energy to reach a steady state mean valueof 〈E〉 = P/γ. This implies that the finite ground stateenergy 〈E〉 = ~Ω/2 of the oscillator is determined via thebalance between the ‘heating’ by the zero-point fluctua-tions of the environment (described by the symmetrizedcorrelator at T = 0) and the dissipation. It is possible totake an alternative but equally correct viewpoint, whereonly the deviation 〈δE〉 = 〈E〉 − ~Ω/2 from the groundstate energy is considered. Its evolution equation

d

dt〈δE〉 = 〈δE〉(Γ↑ − Γ↓) + Γ↑~Ω (B16)

only contains a decay term at T = 0, leading to 〈δE〉 → 0.

3. Practical quantum spectrum analyzers

As we have seen, a ‘quantum spectrum analyzer’ canin principle be constructed from a two level system (or

3 Note that the effective temperature can become negative if thenoise source prefers emitting energy versus absorbing it; in thepresent case, that would lead to an instability.

a harmonic oscillator) in which we can separately mea-sure the up and down transition rates between statesdiffering by some precise energy ~ω > 0 given by thefrequency of interest. The down transition rate tells usthe noise spectral density at frequency +ω and the uptransition rate tells us the noise spectral density at −ω.While we have already discussed experimental implemen-tation of these ideas using two-level systems and oscilla-tors, similar schemes have been implemented in other sys-tems. A number of recent experiments have made use ofsuperconductor-insulator-superconductor junctions (Bil-langeon et al., 2006; Deblock et al., 2003; Onac et al.,2006) to measure quantum noise, as the current-voltagecharacteristics of such junctions are very sensitive tothe absorption or emission of energy (so-called photon-assisted transport processes). It has also been suggestedthat tunneling of flux in a SQUID can be used to measurequantum noise (Amin and Averin, 2008).

In this subsection, we discuss additional methods forthe detection of quantum noise. Recall from Sec. A.3 thatone of the most basic classical noise spectrum analyzersconsists of a linear narrow band filter and a square lawdetector such as a diode. In what follows, we will considera simplified quantum treatment of such a device where wedo not explicitly model a diode, but instead focus on theenergy of the filter circuit. We then turn to various noisedetection schemes making use of a photomultiplier. Wewill show that depending on the detection scheme used,one can measure either the symmetrized quantum noisespectral density S[ω], or the non-symmetrized spectraldensity S[ω].

a. Filter plus diode

Using the simple treatment we gave of a harmonic os-cillator as a quantum spectrum analyzer in Sec. B.2, onecan attempt to provide a quantum treatment of the clas-sical ‘filter plus diode’ spectrum analyzer discussed inSec. A.3. This approach is due to Lesovik and Loosen(1997) and Gavish et al. (2000). The analysis starts bymodeling the spectrum analyzer’s resonant filter circuitas a harmonic oscillator of frequency Ω weakly coupledto some equilibrium dissipative bath. The oscillator thushas an intrinsic damping rate γ0 Ω, and is initially ata finite temperature Teq. One then drives this dampedoscillator (i.e. the filter circuit) with the noisy quantumforce F (t) whose spectrum at frequency Ω is to be mea-sured.

In the classical ‘filter plus diode’ spectrum analyzer,the output of the filter circuit was sent to a square lawdetector, whose time-averaged output was then taken asthe measured spectral density. To simplify the analy-sis, we can instead consider how the noise changes theaverage energy of the resonant filter circuit, taking thisquantity as a proxy for the output of the diode. Sureenough, if we subject the filter circuit to purely classicalnoise, it would cause the average energy of the circuit

7

〈E〉 to increase an amount directly proportional to theclassical spectrum SFF [Ω]. We now consider 〈E〉 in thecase of a quantum noise source, and ask how it relates tothe quantum noise spectral density SFF [Ω].

The quantum case is straightforward to analyze usingthe approach of Sec. B.2. Unlike the classical case, thenoise will both lead to additional fluctuations of the filtercircuit and increase its damping rate by an amount γ(c.f. Eq. (2.12)). To make things quantitative, we let neq

denote the average number of quanta in the filter circuitprior to coupling to F (t), i.e.

neq =1

exp(

~ΩkBTeq

)− 1

, (B17)

and let neff represent the Bose-Einstein factor associatedwith the effective temperature Teff [Ω] of the noise sourceF (t),

neff =1

exp(

~ΩkBTeff [Ω]

)− 1

. (B18)

One then finds (Gavish et al., 2000; Lesovik and Loosen,1997):

∆〈E〉 = ~Ω · γ

γ0 + γ(neff − neq) (B19)

This equation has an extremely simple interpretation:the first term results from the expected heating effectof the noise, while the second term results from thenoise source having increased the circuit’s damping byan amount γ. Re-expressing this result in terms of thesymmetric and anti-symmetric in frequency parts of thequantum noise spectral density SFF [Ω], we have:

∆〈E〉 =SFF (Ω)−

(neq + 1

2

)(SFF [Ω]− SFF [−Ω])

2m (γ0 + γ)(B20)

We see that ∆〈E〉 is in general not simply proportionalto the symmetrized noise SFF [Ω]. Thus, the ‘filter plusdiode’ spectrum analyzer does not simply measure thesymmetrized quantum noise spectral density. We stressthat there is nothing particularly quantum about this re-sult. The extra term on the RHS of Eq. (B20) simplyreflects the fact that coupling the noise source to the fil-ter circuit could change the damping of this circuit; thiscould easily happen in a completely classical setting. Aslong as this additional damping effect is minimal, thesecond term in Eq. (B20) will be minimal, and our spec-trum analyzer will (to a good approximation) measurethe symmetrized noise. Quantitatively, this requires:

neff neq. (B21)

We now see where quantum mechanics enters: if the noiseto be measured is close to being zero point noise (i.e.

neff → 0), the above condition can never be satisfied, andthus it is impossible to ignore the damping effect of thenoise source on the filter circuit. In the zero point limit,this damping effect (i.e. second term in Eq. (B20)) willalways be greater than or equal to the expected heatingeffect of the noise (i.e. first term in Eq. (B20)).

b. Filter plus photomultiplier

We now turn to quantum spectrum analyzers involvinga square law detector we can accurately model– a photo-multiplier. As a first example of such a system, consider aphotomultiplier with a narrow band filter placed in frontof it. The mean photocurrent is then given by

〈I〉 =∫ +∞

−∞dω |f [ω]|2r[ω]SVV[ω], (B22)

where f is the filter (amplitude) transmission functiondefined previously and r[ω] is the response of the pho-todetector at frequency ω, and SVV represents the elec-tric field spectral density incident upon the photodetec-tor. Naively one thinks of the photomultiplier as a squarelaw detector with the square of the electric field repre-senting the optical power. However, according to theGlauber theory of (ideal) photo-detection (Gardiner andZoller, 2000; Glauber, 2006; Walls and Milburn, 1994),photocurrent is produced if, and only if, a photon isabsorbed by the detector, liberating the initial photo-electron. Glauber describes this in terms of normal or-dering of the photon operators in the electric field auto-correlation function. In our language of noise power atpositive and negative frequencies, this requirement be-comes simply that r[ω] vanishes for ω > 0. Approximat-ing the narrow band filter centered on frequency ±ω0 asin Eq. (A26), we obtain

〈I〉 = r[−ω0]SVV[−ω0] (B23)

which shows that this particular realization of a quantumspectrometer only measures electric field spectral densityat negative frequencies since the photomultiplier neveremits energy into the noise source. Also one does notsee in the output any ‘vacuum noise’ and so the output(ideally) vanishes as it should at zero temperature. Ofcourse real photomultipliers suffer from imperfect quan-tum efficiencies and have non-zero dark current. Notethat we have assumed here that there are no additionalfluctuations associated with the filter circuit. Our re-sult thus coincides with what we found in the previoussubsection for the ‘filter plus diode’ spectrum analyzer(c.f. Eq. (B20), in the limit where the filter circuit is ini-tially at zero temperature (i.e. neq = 0).

c. Double sideband heterodyne power spectrum

At RF and microwave frequencies, practical spectrome-ters often contain heterodyne stages which mix the initial

8

frequency down to a lower frequency ωIF (possibly in theclassical regime). Consider a system with a mixer and lo-cal oscillator at frequency ωLO that mixes both the uppersideband input at ωu = ωLO+ωIF and the lower sidebandinput at ωl = ωLO − ωIF down to frequency ωIF. Thiscan be achieved by having a Hamiltonian with a 3-wavemixing term which (in the rotating wave approximation)is given by

V = λ[aIFala†LO + a†IFa

†l aLO] + λ[a†IFaua

†LO + aIFa

†uaLO](B24)

The interpretation of this term is that of a Raman pro-cess. Notice that there are two energy conserving pro-cesses that can create an IF photon which could thenactivate the photodetector. First, one can absorb an LOphoton and emit two photons, one at the IF and one atthe lower sideband. The second possibility is to absorban upper sideband photon and create IF and LO photons.Thus we expect from this that the power in the IF chan-nel detected by a photomultiplier would be proportionalto the noise power in the following way

I ∝ S[+ωl] + S[−ωu] (B25)

since creation of an IF photon involves the signal sourceeither absorbing a lower sideband photon from the mixeror the signal source emitting an upper sideband photoninto the mixer. In the limit of small IF frequency thisexpression would reduce to the symmetrized noise power

I ∝ S[+ωLO] + S[−ωLO] = 2S[ωLO] (B26)

which is the same as for a ‘classical’ spectrum analyzerwith a square law detector (c.f. Appendix A.3). For equi-librium noise spectral density from a resistance R0 de-rived in Appendix D we would then have

SVV[ω] + SVV[−ω] = 2R0~|ω|[2nB(~|ω|) + 1], (B27)

Assuming our spectrum analyzer has high inputimpedance so that it does not load the noise source, thisvoltage spectrum will determine the output signal of theanalyzer. This symmetrized quantity does not vanish atzero temperature and the output contains the vacuumnoise from the input. This vacuum noise has been seenin experiment. (Schoelkopf et al., 1997)

Appendix C: Modes, Transmission Lines and ClassicalInput/Output Theory

In this appendix we introduce a number of importantclassical concepts about electromagnetic signals whichare essential to understand before moving on to the studyof their quantum analogs. A signal at carrier frequencyω can be described in terms of its amplitude and phaseor equivalently in terms of its two quadrature amplitudes

s(t) = X cos(ωt) + Y sin(ωt). (C1)

We will see in the following that the physical oscillationsof this signal in a transmission line are precisely the sinu-soidal oscillations of a simple harmonic oscillator. Com-parison of Eq. (C1) with x(t) = x0 cosωt+(p0/Mω) sinωtshows that we can identify the quadrature amplitude Xwith the coordinate of this oscillator and thus the quadra-ture amplitude Y is proportional to the momentum con-jugate to X. Quantum mechanically, X and Y becomeoperators X and Y which do not commute. Thus theirquantum fluctuations obey the Heisenberg uncertaintyrelation.

Ordinarily (e.g., in the absence of squeezing), the phasechoice defining the two quadratures is arbitrary and sotheir vacuum (i.e. zero-point) fluctuations are equal

XZPF = YZPF. (C2)

Thus the canonical commutation relation becomes

[X, Y ] = iX2ZPF. (C3)

We will see that the fact that X and Y are canoni-cally conjugate has profound implications both classicallyand quantum mechanically. In particular, the action ofany circuit element (beam splitter, attenuator, amplifier,etc.) must preserve the Poisson bracket (or in the quan-tum case, the commutator) between the signal quadra-tures. This places strong constraints on the propertiesof these circuit elements and in particular, forces everyamplifier to add noise to the signal.

1. Transmission lines and classical input-output theory

We begin by considering a coaxial transmission linemodeled as a perfectly conducting wire with inductanceper unit length of ` and capacitance to ground per unitlength c as shown in Fig. 1. If the voltage at position xat time t is V (x, t), then the charge density is q(x, t) =cV (x, t). By charge conservation the current I and thecharge density are related by the continuity equation

∂tq + ∂xI = 0. (C4)

The constitutive relation (essentially Newton’s law) givesthe acceleration of the charges

`∂tI = −∂xV. (C5)

We can decouple Eqs. (C4) and (C5) by introducing leftand right propagating modes

V (x, t) = [V→ + V←] (C6)

I(x, t) =1Zc

[V→ − V←] (C7)

where Zc ≡√`/c is called the characteristic impedance

of the line. In terms of the left and right propagatingmodes, Eqs. (C4) and C5 become

vp∂xV→ + ∂tV

→ = 0 (C8)vp∂xV

← − ∂tV← = 0 (C9)

9

where vp ≡ 1/√`c is the wave phase velocity. These

equations have solutions which propagate by uniformtranslation without changing shape since the line is dis-persionless

V→(x, t) = Vout(t−x

vp) (C10)

V←(x, t) = Vin(t+x

vp), (C11)

where Vin and Vout are arbitrary functions of their argu-ments. For an infinite transmission line, Vout and Vin arecompletely independent. However for the case of a semi-infinite line terminated at x = 0 (say) by some system S,these two solutions are not independent, but rather re-lated by the boundary condition imposed by the system.We have

V (x = 0, t) = [Vout(t) + Vin(t)] (C12)

I(x = 0, t) =1Zc

[Vout(t)− Vin(t)], (C13)

from which we may derive

Vout(t) = Vin(t) + ZcI(x = 0, t). (C14)

Zc , vp

0 dx

V

I

Vin

Vout

V (x,t)

V (x,t)

I(x,t)V(x,t)

a)

c)

L

a

b)L L

C C C

FIG. 1 a) Coaxial transmission line, indicating voltages andcurrents as defined in the main text. b) Lumped elementrepresentation of a transmission line with capacitance per unitlength c = C/a and inductance per unit length ` = L/a. c)Discrete LC resonator terminating a transmission line.

If the system under study is just an open circuit sothat I(x = 0, t) = 0, then Vout = Vin, meaning that theoutgoing wave is simply the result of the incoming wavereflecting from the open circuit termination. In generalhowever, there is an additional outgoing wave radiatedby the current I that is injected by the system dynamicsinto the line. In the absence of an incoming wave we have

V (x = 0, t) = ZcI(x = 0, t), (C15)

indicating that the transmission line acts as a simple re-sistor which, instead of dissipating energy by Joule heat-ing, carries the energy away from the system as propa-gating waves. The fact that the line can dissipate energydespite containing only purely reactive elements is a con-sequence of its infinite extent. One must be careful withthe order of limits, taking the length to infinity beforeallowing time to go to infinity. In this way the outgoingwaves never reach the far end of the transmission line andreflect back. Since this is a conservative Hamiltonian sys-tem, we will be able to quantize these waves and make aquantum theory of resistors (Caldeira and Leggett, 1983)in Appendix D. The net power flow carried to the rightby the line is

P =1Zc

[V 2out(t)− V 2

in(t)]. (C16)

The fact that the transmission line presents a dissipa-tive impedance to the system means that it causes damp-ing of the system. It also however opens up the possibilityof controlling the system via the input field which par-tially determines the voltage driving the system. Fromthis point of view it is convenient to eliminate the outputfield by writing the voltage as

V (x = 0, t) = 2Vin(t) + ZcI(x = 0, t). (C17)

As we will discuss in more detail below, the first termdrives the system and the second damps it. FromEq. (C14) we see that measurement of the outgoing fieldcan be used to determine the current I(x = 0, t) injectedby the system into the line and hence to infer the systemdynamics that results from the input drive field.

As a simple example, consider the system consisting ofan LC resonator shown in Fig. (1 c). This can be viewedas a simple harmonic oscillator whose coordinate Q is thecharge on the capacitor plate (on the side connected toL0). The current I(x = 0, t) = Q plays the role of thevelocity of the oscillator. The equation of motion for theoscillator is readily obtained from

Q = C0[−V (x = 0+, t)− L0I(x = 0+, t)]. (C18)

Using Eq. (C17) we obtain a harmonic oscillator dampedby the transmission line and driven by the incomingwaves

Q = −Ω20Q− γQ−

2L0Vin(t), (C19)

where the resonant frequency is Ω20 ≡ 1/

√L0C0. Note

that the term ZcI(x = 0, t) in Eq. (C17) results in thelinear viscous damping rate γ ≡ Zc/L0.

If we solve the equation of motion of the oscillator, wecan predict the outgoing field. In the present instance ofa simple oscillator we have a particular example of thegeneral case where the system responds linearly to theinput field. We can characterize any such system by a

10

complex, frequency dependent impedance Z[ω] definedby

Z[ω] = −V (x = 0, ω)I(x = 0, ω)

. (C20)

Note the peculiar minus sign which results from our def-inition of positive current flowing to the right (out of thesystem and into the transmission line). Using Eqs. (C12,C13) and Eq. (C20) we have

Vout[ω] = r[ω]Vin[ω], (C21)

where the reflection coefficient r is determined by theimpedance mismatch between the system and the lineand is given by the well known result

r[ω] =Z[ω]− Zc

Z[ω] + Zc. (C22)

If the system is constructed from purely reactive (i.e.lossless) components, then Z[ω] is purely imaginary andthe reflection coefficient obeys |r| = 1 which is consistentwith Eq. (C16) and the energy conservation requirementof no net power flow into the lossless system. For exam-ple, for the series LC oscillator we have been considering,we have

Z[ω] =1

jωC0+ jωL0, (C23)

where, to make contact with the usual electrical engi-neering sign conventions, we have used j = −i. If thedamping γ of the oscillator induced by coupling it to thetransmission line is small, the quality factor of the reso-nance will be high and we need only consider frequenciesnear the resonance frequency Ω0 ≡ 1/

√L0C0 where the

impedance has a zero. In this case we may approximate

Z[ω] ≈ 2jC0Ω2

0

[Ω0 − ω] = 2jL0(ω − Ω0) (C24)

which yields for the reflection coefficient

r[ω] =ω − Ω0 + jγ/2ω − Ω0 − jγ/2

(C25)

showing that indeed |r| = 1 and that the phase of thereflected signal winds by 2π upon passing through theresonance. 4

Turning to the more general case where the systemalso contains lossy elements, one finds that Z[ω] is nolonger purely imaginary, but has a real part satisfyingRe Z[ω] > 0. This in turn implies via Eq. (C22) that|r| < 1. In the special case of impedance matchingZ[ω] = Zc, all the incident power is dissipated in the

4 For the case of resonant transmission through a symmetric cav-ity, the phase shift only winds by π.

system and none is reflected. The other two limits of in-terest are open circuit termination with Z =∞ for whichr = +1 and short circuit termination Z = 0 for whichr = −1.

Finally, if the system also contains an active devicewhich has energy being pumped into it from a separateexternal source, it may under the right conditions be de-scribed by an effective negative resistance ReZ[ω] < 0over a certain frequency range. Eq. (C22) then gives|r| ≥ 1, implying |Vout| > |Vin|. Our system will thus actlike the one-port amplifier discussed in Sec. V.D: it am-plifies signals incident upon it. We will discuss this ideaof negative resistance further in Sec. C.4; a physical real-ization is provided by the two-port reflection parametricamplifier discussed in Appendix V.C.

2. Lagrangian, Hamiltonian, and wave modes for atransmission line

Prior to moving on to the case of quantum noise itis useful to review the classical statistical mechanics oftransmission lines. To do this we need to write downthe Lagrangian and then determine the canonical mo-menta and the Hamiltonian. Very conveniently, the sys-tem is simply a large collection of harmonic oscillators(the normal modes) and hence can be readily quantized.This representation of a physical resistor is essentially theone used by Caldeira and Leggett (Caldeira and Leggett,1983) in their seminal studies of the effects of dissipationon tunneling. The only difference between this modeland the vacuum fluctuations in free space is that the rel-ativistic bosons travel in one dimension and do not carrya polarization label. This changes the density of states asa function of frequency, but has no other essential effect.

It is convenient to define a flux variable (Devoret, 1997)

ϕ(x, t) ≡∫ t

−∞dτ V (x, τ), (C26)

where V (x, t) = ∂tϕ(x, t) is the local voltage on the trans-mission line at position x and time t. Each segment ofthe line of length dx has inductance ` dx and the voltagedrop along it is −dx ∂x∂tϕ(x, t). The flux through thisinductance is thus −dx ∂xϕ(x, t) and the local value ofthe current is given by the constitutive equation

I(x, t) = −1`∂xϕ(x, t). (C27)

The Lagrangian for the system is

Lg ≡∫ ∞

0

dxL(x, t) =∫ ∞

0

dx

(c

2(∂tϕ)2 − 1

2`(∂xϕ)2

),

(C28)The Euler-Lagrange equation for this Lagrangian is sim-ply the wave equation

v2p∂

2xϕ− ∂2

t ϕ = 0. (C29)

11

The momentum conjugate to ϕ(x) is simply the chargedensity

q(x, t) ≡ δLδ∂tϕ

= c∂tϕ = cV (x, t) (C30)

and so the Hamiltonian is given by

H =∫dx

12cq2 +

12`

(∂xϕ)2

. (C31)

We know from our previous results that the chargedensity consists of left and right moving solutions of ar-bitrary fixed shape. For example we might have for theright moving case

q(t−x/vp) = αk cos[k(x−vpt)]+βk sin[k(x−vpt)]. (C32)

A confusing point is that since q is real valued, we seethat it necessarily contains both eikx and e−ikx termseven if it is only right moving. Note however that fork > 0 and a right mover, the eikx is associated with thepositive frequency term e−iωkt while the e−ikx term isassociated with the negative frequency term e+iωkt whereωk ≡ vp|k|. For left movers the opposite holds. We canappreciate this better if we define

Ak ≡1√L

∫dx e−ikx

1√2cq(x, t)− i

√k2

2`ϕ(x, t)

(C33)

where for simplicity we have taken the fields to obey pe-riodic boundary conditions on a length L. Thus we have(in a form which anticipates the full quantum theory)

H =12

∑k

(A∗kAk +AkA∗k) . (C34)

The classical equation of motion (C29) yields the simpleresult

∂tAk = −iωkAk. (C35)

Thus

q(x, t)

=√

c

2L

∑k

eikx[Ak(0)e−iωkt +A∗−k(0)e+iωkt

](C36)

=√

c

2L

∑k

[Ak(0)e+i(kx−ωkt) +A∗k(0)e−i(kx−ωkt)

].

(C37)

We see that for k > 0 (k < 0) the wave is right (left)moving, and that for right movers the eikx term is asso-ciated with positive frequency and the e−ikx term is as-sociated with negative frequency. We will return to thisin the quantum case where positive (negative) frequencywill refer to the destruction (creation) of a photon. Notethat the right and left moving voltages are given by

V→ =

√1

2Lc

∑k>0


](C38)

V← =

√1

2Lc

∑k<0


](C39)

The voltage spectral density for the right moving waves is thus

S→V V [ω] =2π2Lc

∑k>0

〈AkA∗k〉δ(ω − ωk) + 〈A∗kAk〉δ(ω + ωk) (C40)

The left moving spectral density has the same expression but k < 0.Using Eq. (C16), the above results lead to a net power flow (averaged over one cycle) within a frequency band

defined by a pass filter G[ω] of

P = P→ − P← =vp

2L

∑k

sgn(k) [G[ωk]〈AkA∗k〉+G[−ωk]〈A∗kAk〉] . (C41)

3. Classical statistical mechanics of a transmission line

Now that we have the Hamiltonian, we can considerthe classical statistical mechanics of a transmission line in

thermal equilibrium at temperature T . Since each modek is a simple harmonic oscillator we have from Eq. (C34)and the equipartition theorem

〈A∗kAk〉 = kBT. (C42)

12

Using this, we see from Eq. (C40) that the right movingvoltage signal has a simple white noise power spectrum.Using Eq. (C41) we have for the right moving power ina bandwidth B (in Hz rather than radians/sec) the verysimple result

P→ =vp

2L

∑k>0

〈G[ωk]A∗kAk +G[−ωk]AkA∗k〉

=kBT

2

∫ +∞

−∞

dω

2πG[ω]

= kBTB. (C43)

where we have used the fact mentioned in connectionwith Eq. (A26) and the discussion of square law detec-tors that all passive filter functions are symmetric in fre-quency.

One of the basic laws of statistical mechanics is Kirch-hoff’s law stating that the ability of a hot object to emitradiation is proportional to its ability to absorb. Thisfollows from very general thermodynamic arguments con-cerning the thermal equilibrium of an object with its ra-diation environment and it means that the best possibleemitter is the black body. In electrical circuits this princi-ple is simply a form of the fluctuation dissipation theoremwhich states that the electrical thermal noise producedby a circuit element is proportional to the dissipation itintroduces into the circuit. Consider the example of a ter-minating resistor at the end of a transmission line. If theresistance R is matched to the characteristic impedanceZc of a transmission line, the terminating resistor actsas a black body because it absorbs 100% of the powerincident upon it. If the resistor is held at temperature Tit will bring the transmission line modes into equilibriumat the same temperature (at least for the case where thetransmission line has finite length). The rate at which theequilibrium is established will depend on the impedancemismatch between the resistor and the line, but the finaltemperature will not.

A good way to understand the fluctuation-dissipationtheorem is to represent the resistor R which is terminat-ing the Zc line in terms of a second semi-infinite trans-mission line of impedance R as shown in Fig. (2). Firstconsider the case when the R line is not yet connected tothe Zc line. Then according to Eq. (C22), the open termi-nation at the end of the Zc line has reflectivity |r|2 = 1so that it does not dissipate any energy. Additionallyof course, this termination does not transmit any sig-nals from the R line into the Zc. However when thetwo lines are connected the reflectivity becomes less thanunity meaning that incoming signals on the Zc line seea source of dissipation R which partially absorbs them.The absorbed signals are not turned into heat as in atrue resistor but are partially transmitted into the R linewhich is entirely equivalent. Having opened up this portfor energy to escape from the Zc system, we have alsoallowed noise energy (thermal or quantum) from the Rline to be transmitted into the Zc line. This is com-pletely equivalent to the effective circuit shown in Fig. (3

a) in which a real resistor has in parallel a random cur-rent generator representing thermal noise fluctuations ofthe electrons in the resistor. This is the essence of thefluctuation dissipation theorem.

In order to make a quantitative analysis in terms ofthe power flowing in the two lines, voltage is not the bestvariable to use since we are dealing with more than onevalue of line impedance. Rather we define incoming andoutgoing fields via

Ain =1√Zc

V←c (C44)

Aout =1√Zc

V→c (C45)

Bin =1√RV→R (C46)

Bout =1√RV←R (C47)

Normalizing by the square root of the impedance allowsus to write the power flowing to the right in each line inthe simple form

Pc = (Aout)2 − (Ain)2 (C48)PR = (Bin)2 − (Bout)2 (C49)

The out fields are related to the in fields by the s matrix(Aout

Bout

)= s

(Ain

Bin

)(C50)

Requiring continuity of the voltage and current at theinterface between the two transmission lines, we can solvefor the scattering matrix s:

s =(

+r tt −r

)(C51)

where

r =R− Zc

R+ Zc(C52)

t =2√RZc

R+ Zc. (C53)

Note that |r|2 + |t|2 = 1 as required by energy conserva-tion and that s is unitary with det (s) = −1. By movingthe point at which the phase of the Bin and Bout fieldsare determined one-quarter wavelength to the left, wecan put s into different standard form

s′ =(

+r itit +r

)(C54)

which has det (s′) = +1.As mentioned above, the energy absorbed from the Zc

line by the resistor R is not turned into heat as in atrue resistor but is is simply transmitted into the R line,which is entirely equivalent. Kirchhoff’s law is now easy

13

VR

VcVR

VcR Zc

FIG. 2 (Color online) Semi-infinite transmission line ofimpedance Zc terminated by a resistor R which is representedas a second semi-infinite transmission line.

to understand. The energy absorbed from the Zc line byR, and the energy transmitted into it by thermal fluctua-tions in theR line are both proportional to the absorptioncoefficient

A = 1− |r|2 = |t|2 =4RZc

(R+ Zc)2. (C55)

R

IN

SII SVV

R

VN

a) b)FIG. 3 Equivalent circuits for noisy resistors.

The requirement that the transmission line Zc come toequilibrium with the resistor allows us to readily computethe spectral density of current fluctuations of the randomcurrent source shown in Fig. (3 a). The power dissipatedin Zc by the current source attached to R is

P =∫ +∞

−∞

dω

2πSII [ω]

R2Zc

(R+ Zc)2

(C56)

For the special case R = Zc we can equate this to theright moving power P→ in Eq. (C43) because left movingwaves in the Zc line are not reflected and hence cannotcontribute to the right moving power. Requiring P =P→ yields the classical Nyquist result for the currentnoise of a resistor

SII [ω] =2RkBT (C57)

or in the electrical engineering convention

SII [ω] + SII [−ω] =4RkBT. (C58)

We can derive the equivalent expression for the volt-age noise of a resistor (see Fig. 3 b) by considering the

voltage noise at the open termination of a semi-infinitetransmission line with Zc = R. For an open terminationV→ = V← so that the voltage at the end is given by

V = 2V← = 2V→ (C59)

and thus using Eqs. (C40) and (C42) we find

SV V = 4S→V V = 2RkBT (C60)

which is equivalent to Eq. (C57).

4. Amplification with a transmission line and a negativeresistance

We close our discussion of transmission lines by fur-ther expanding upon the idea mentioned at the end ofApp. C.1 that one can view a one-port amplifier as atransmission line terminated by an effective negative re-sistance. The discussion here will be very general: we willexplore what can be learned about amplification by sim-ply extending the results we have obtained on transmis-sion lines to the case of an effective negative resistance.Our general discussion will not address the important is-sues of how one achieves an effective negative resistanceover some appreciable frequency range: for such ques-tions, one must focus on a specific physical realization,such as the parametric amplifier discussed in Sec. V.C.

We start by noting that for the case −Zc < R < 0 thepower gain G is given by

G = |r|2 > 1, (C61)

and the s′ matrix introduced in Eq. (C54) becomes

s′ = −( √

G ±√G− 1

±√G− 1

√G

)(C62)

where the sign choice depends on the branch cut chosenin the analytic continuation of the off-diagonal elements.This transformation is clearly no longer unitary (becausethere is no energy conservation since we are ignoring thework done by the amplifier power supply). Note howeverthat we still have det (s′) = +1. It turns out that thisnaive analytic continuation of the results from positive tonegative resistance is not strictly correct. As we will showin the following, we must be more careful than we havebeen so far in order to insure that the transformationfrom the in fields to the out fields must be canonical.

In order to understand the canonical nature of thetransformation between input and output modes, it isnecessary to delve more deeply into the fact that thetwo quadrature amplitudes of a mode are canonicallyconjugate. Following the complex amplitudes definedin Eqs. (C44-C47), let us define a vector of real-valuedquadrature amplitudes for the incoming and outgoingfields

~q in =

X inA

X inB

Y inB

Y inA

, ~q out =

XoutA

XoutB

Y outBY outA

.

(C63)

14

The Poisson brackets amongst the different quadratureamplitudes is given by

qini , q

inj ∝ Jij , (C64)

or equivalently the quantum commutators are

[qini , q

inj ] = iX2

ZPFJij , (C65)

where

J ≡

0 0 0 +10 0 +1 00 −1 0 0−1 0 0 0

. (C66)

In order for the transformation to be canonical, the samePoisson bracket or commutator relations must hold forthe outgoing field amplitudes

[qouti , qout

j ] = iX2ZPFJij . (C67)

In the case of a non-linear device these relations wouldapply to the small fluctuations in the input and outputfields around the steady state solution. Assuming a lineardevice (or linearization around the steady state solution)we can define a 4 × 4 real-valued scattering matrix s inanalogy to the 2× 2 complex-valued scattering matrix sin Eq. (C51) which relates the output fields to the inputfields

qouti = sijq

inj . (C68)

Eq. (C67) puts a powerful constraint on on the s matrix,namely that it must be symplectic. That is, s and itstranspose must obey

sJ sT = J. (C69)

From this it follows that

det s = ±1. (C70)

This in turn immediately implies Liouville’s theoremthat Hamiltonian evolution preserves phase space volume(since det s is the Jacobian of the transformation whichpropagates the amplitudes forward in time).

Let us further assume that the device is phase preserv-ing, that is that the gain or attenuation is the same forboth quadratures. One form for the s matrix consistentwith all of the above requirements is

s =

+ cos θ sin θ 0 0sin θ − cos θ 0 0

0 0 − cos θ sin θ0 0 sin θ + cos θ

. (C71)

This simply corresponds to a beam splitter and is theequivalent of Eq. (C51) with r = cos θ. As mentioned inconnection with Eq. (C51), the precise form of the scat-tering matrix depends on the choice of planes at which

the phases of the various input and output waves aremeasured.

Another allowed form of the scattering matrix is:

s′ = −

+ cosh θ + sinh θ 0 0+ sinh θ + cosh θ 0 0

0 0 + cosh θ − sinh θ0 0 − sinh θ + cosh θ

.

(C72)If one takes cosh θ =

√G, this scattering matrix is

essentially the canonically correct formulation of thenegative-resistance scattering matrix we tried to write inEq. (C62). Note that the off-diagonal terms have changedsign for the Y quadrature relative to the naive expressionin Eq. (C62) (corresponding to the other possible an-alytic continuation choice). This is necessary to satisfythe symplecticity condition and hence make the transfor-mation canonical. The scattering matrix s′ can describeamplification. Unlike the beam splitter scattering ma-trix s above, s′ is not unitary (even though det s′ = 1).Unitarity would correspond to power conservation. Here,power is not conserved, as we are not explicitly trackingthe power source supplying our active system.

The form of the negative-resistance amplifier scatteringmatrix s′ confirms many of the general statements wemade about phase-preserving amplification in Sec. V.B.First, note that the requirement of finite gain G > 1 andphase preservation makes all the diagonal elements of s′(i.e. cosh θ ) equal. We see that to amplify the A mode,it is impossible to avoid coupling to the B mode (via thesinh θ term) because of the requirement of symplecticity.We thus see that it is impossible classically or quantummechanically to build a linear phase-preserving amplifierwhose only effect is to amplify the desired signal. Thepresence of the sinh θ term above means that the outputsignal is always contaminated by amplified noise fromat least one other degree of freedom (in this case the Bmode). If the thermal or quantum noise in A and Bare equal in magnitude (and uncorrelated), then in thelimit of large gain where cosh θ ≈ sinh θ, the output noise(referred to the input) will be doubled. This is true forboth classical thermal noise and quantum vacuum noise.

The negative resistance model of an amplifier heregives us another way to think about the noise added byan amplifier: crudely speaking, we can view it as beingdirectly analogous to the fluctuation-dissipation theoremsimply continued to the case of negative dissipation. Justas dissipation can occur only when we open up a newchannel and thus we bring in new fluctuations, so ampli-fication can occur only when there is coupling to an ad-ditional channel. Without this it is impossible to satisfythe requirement that the amplifier perform a canonicaltransformation.

15

Appendix D: Quantum Modes and Noise of a TransmissionLine

1. Quantization of a transmission line

Recall from Eq. (C30) and the discussion in AppendixC that the momentum conjugate to the transmission lineflux variable ϕ(x, t) is the local charge density q(x, t).Hence in order to quantize the transmission line modeswe simply promote these two physical quantities to quan-tum operators obeying the commutation relation

[q(x), ϕ(x′)] = −i~δ(x− x′) (D1)

from which it follows that the mode amplitudes definedin Eq. (C33) become quantum operators obeying

[Ak′ , A†k] = ~ωkδkk′ (D2)

and we may identify the usual raising and lowering oper-ators by

Ak =√

~ωk bk (D3)

where bk destroys a photon in mode k. The quantumform of the Hamiltonian in Eq. (C34) is thus

H =∑k

~ωk[b†k bk +

12

]. (D4)

For the quantum case the thermal equilibrium expressionthen becomes

〈A†kAk〉 = ~ωknB(~ωk), (D5)

which reduces to Eq. (C42) in the classical limit ~ωk kBT .

We have seen previously in Eqs. (C6) that the volt-age fluctuations on a transmission line can be resolvedinto right and left moving waves which are functions of acombined space-time argument

V (x, t) = V→(t− x

vp) + V←(t+

x

vp). (D6)

Thus in an infinite transmission line, specifying V→ ev-erywhere in space at t = 0 determines its value for alltimes. Conversely specifying V→ at x = 0 for all timesfully specifies the field at all spatial points. In prepa-ration for our study of the quantum version of input-output theory in Appendix E, it is convenient to extendEqs. (C38-C39) to the quantum case (x = 0):

V→(t) =

√1

2Lc

∑k>0

√~ωk

[bke−iωkt + h.c.

]=∫ ∞

0

dω

2π

√~ωZc

2

[b→[ω]e−iωt + h.c.

](D7)

In the second line, we have defined:

b→[ω] ≡ 2π√vpL

∑k>0

bkδ(ω − ωk) (D8)

In a similar fashion, we have:

V←(t) =∫ ∞

0

dω

2π

√~ωZc

2

[b←[ω]e−iωt + h.c.

](D9)

b←[ω] ≡ 2π√vpL

∑k<0

bkδ(ω − ωk) (D10)

One can easily verify that among the b→[ω], b←[ω] opera-tors and their conjugates, the only non-zero commutatorsare given by:[b→[ω],

(b→[ω′]

)†]=[b←[ω],

(b←[ω′]

)†]= 2πδ(ω − ω′)

(D11)We have taken the continuum limit L→∞ here, allowingus to change sums on k to integrals. We have thus ob-tained the description of a quantum transmission line interms of left and right-moving frequency resolved modes,as used in our discussion of amplifiers in Sec. VI (seeEqs. 6.2). Note that if the right-moving modes are fur-ther taken to be in thermal equilibrium, one finds (again,in the continuum limit):⟨(

b→[ω])†b→[ω′]

⟩= 2πδ(ω − ω′)nB(~ω) (D12a)⟨

b→[ω](b→[ω′]

)†⟩= 2πδ(ω − ω′) [1 + nB(~ω)]

. (D12b)

We are typically interested in a relatively narrow bandof frequencies centered on some characteristic drive orresonance frequency Ω0. In this case, it is useful to workin the time-domain, in a frame rotating at Ω0. Fouriertransforming 5 Eqs. (D8) and (D10), one finds:

b→(t) =√vp

L

∑k>0

e−i(ωk−Ω0)tbk(0), (D13a)

b←(t) =√vp

L

∑k<0

e−i(ωk−Ω0)tbk(0). (D13b)

These represent temporal right and left moving modes.Note that the normalization factor in Eqs. (D13) has beenchosen so that the right moving photon flux at x = 0 andtime t is given by

〈N〉 = 〈b†→(t)b→(t)〉 (D14)

In the same rotating frame, and within the approxima-tion that all relevant frequencies are near Ω0, Eq. (D7)becomes simply:

V→(t) ≈√

~Ω0Zc

2

[b→(t) + b†→(t)

](D15)

5 As in the main text, we use in this appendix a convention whichdiffers from the one commonly used in quantum optics: a[ω] =R +∞−∞ dt e+iωta(t) and a†[ω] = [a[−ω]]† =

R +∞−∞ dt e+iωta†(t).

16

We have already seen that using classical statisticalmechanics, the voltage noise in equilibrium is white.The corresponding analysis of the temporal modes usingEqs. (D13) shows that the quantum commutator obeys

[b→(t), b†→(t′)] = δ(t− t′). (D16)

In deriving this result, we have converted summationsover mode index to integrals over frequency. Further,because (for finite time resolution at least) the integral isdominated by frequencies near +Ω0 we can, within theMarkov (Wigner Weisskopf) approximation, extend thelower limit of frequency integration to minus infinity andthus arrive at a delta function in time. If we further takethe right moving modes to be in thermal equilibrium,then we may similarly approximate:

〈b†→(t′)b→(t)〉 = nB(~Ω0)δ(t− t′) (D17a)

〈b→(t)b†→(t′)〉 = [1 + nB(~Ω0)] δ(t− t′). (D17b)

Equations (D15) to (D17b) indicate that V→(t) can betreated as the quantum operator equivalent of whitenoise; a similar line of reasoning applies mutatis mutan-dis to the left moving modes. We stress that these re-sults rely crucially on our assumption that we are dealingwith a relatively narrow band of frequencies in the vicin-ity of Ω0; the resulting approximations we have madeare known as the Markov approximation. As one can al-ready see from the form of Eqs. (D7,D9), and as will bediscussed further, the actual spectral density of vacuumnoise on a transmission line is not white, but is linear infrequency. The approximation made in Eq. (D16) treatsit as a constant within the narrow band of frequenciesof interest. If the range of frequencies of importance islarge then the Markov approximation is not applicable.

2. Modes and the windowed Fourier transform

While delta function correlations can make the quan-tum noise relatively easy to deal with in both the timeand frequency domain, it is sometimes the case that itis easier to deal with a ‘smoothed’ noise variable. Theintroduction of an ultraviolet cutoff regulates the math-ematical singularities in the noise operators evaluated atequal times and is physically sensible because every realmeasurement apparatus has finite time resolution. A sec-ond motivation is that real spectrum analyzers output atime varying signal which represents the noise power ina certain frequency interval (the ‘resolution bandwidth’)averaged over a certain time interval (the inverse ‘videobandwidth’). The mathematical tool of choice for dealingwith such situations in which time and frequency bothappear is the ‘windowed Fourier transform’. The win-dowed transform uses a kernel which is centered on somefrequency window and some time interval. By summa-tion over all frequency and time windows it is possibleto invert the transformation. The reader is directed to(Mallat, 1999) for the mathematical details.

For our present purposes where we are interested injust a single narrow frequency range centered on Ω0, aconvenient windowed transform kernel for smoothing thequantum noise is simply a box of width ∆t representingthe finite integration time of our detector. In the framerotating at Ω0 we can define

B→j =1√∆t

∫ tj+1

tj

dτ b→(τ) (D18)

where tj = j(∆t) denotes the time of arrival of the jthtemporal mode at the point x = 0. Recall that b→ hasa photon flux normalization and so B→j is dimensionless.From Eq. (D16) we see that these smoothed operatorsobey the usual bosonic commutation relations

[B→j , B†→k ] = δjk. (D19)

The state B†j |0〉 has a single photon occupying basismode j, which is centered in frequency space at Ω0 and intime space on the interval j∆t < t < (j + 1)∆t (i.e. thistemporal mode passes the point x = 0 during the jthtime interval.) This basis mode is much like a note in amusical score: it has a certain specified pitch and occursat a specified time for a specified duration. Just as wecan play notes of different frequencies simultaneously, wecan define other temporal modes on the same time in-terval and they will be mutually orthogonal provided theangular frequency spacing is a multiple of 2π/∆t. Theresult is a set of modes Bm,p labeled by both a frequencyindex m and a time index p. p labels the time interval asbefore, while m labels the angular frequency:

ωm = Ω0 +m2π∆t

(D20)

The result is, as illustrated in Fig. (4), a complete lat-tice of possible modes tiling the frequency-time phasespace, each occupying area 2π corresponding to the time-frequency uncertainty principle.

We can form other modes of arbitrary shapes centeredon frequency Ω0 by means of linear superposition of ourbasis modes (as long as they are smooth on the time scale∆t). Let us define

Ψ =∑j

ψjB→j . (D21)

This is also a canonical bosonic mode operator obeying

[Ψ,Ψ†] = 1 (D22)

provided that the coefficients obey the normalization con-dition ∑

j

|ψj |2 = 1. (D23)

We might for example want to describe a mode which iscentered at a slightly higher frequency Ω0 + δΩ (obeying

17

area = 2!

t

" #t

2!/#t

B-mp

Bmp

"=$0

m

-m

p

FIG. 4 (Color online) Schematic figure indicating how thevarious modes defined by the windowed Fourier transform tilethe time-frequency plane. Each individual cell corresponds toa different mode, and has an area 2π.

(δΩ)(∆t) << 1) and spread out over a large time intervalT centered at time T0. This could be given for exampleby

ψj = N e−(j∆t−T0)2

4T2 e−i(δΩ)(j∆t) (D24)

where N is the appropriate normalization constant.The state having n photons in the mode is simply

1√n!

(Ψ†)n |0〉. (D25)

The concept of ‘wave function of the photon’ is fraughtwith dangers. In the very special case where we re-strict attention solely to the subspace of single photonFock states, we can usefully think of the amplitudes ψjas the ‘wave function of the photon’ (Cohen-Tannoudjiet al., 1989) since it tells us about the spatial mode whichis excited. In the general case however it is essential tokeep in mind that the transmission line is a collection ofcoupled LC oscillators with an infinite number of degreesof freedom. Let us simplify the argument by consideringa single LC oscillator. We can perfectly well write a wavefunction for the system as a function of the coordinate(say the charge q on the capacitor). The ground statewave function χ0(q) is a gaussian function of the coor-dinate. The one photon state created by Ψ† has a wavefunction χ1(q) ∼ qχ0(q) proportional to the coordinatetimes the same gaussian. In the general case χ is a wavefunctional of the charge distribution q(x) over the entiretransmission line.

Using Eq. (D17a) we have

〈B†→j B→k 〉 = nB(~Ω0)δjk (D26)

independent of our choice of the coarse-graining time win-dow ∆t. This result allows us to give meaning to thephrase one often hears bandied about in descriptions of

amplifiers that ‘the noise temperature corresponds to amode occupancy of X photons’. This simply means thatthe photon flux per unit bandwidth is X. Equivalentlythe flux in bandwidth B is

N =X

∆t(B∆t) = XB. (D27)

The interpretation of this is that X photons in a tempo-ral mode of duration ∆t pass the origin in time ∆t. Eachmode has bandwidth ∼ 1

∆t and so there are B∆t inde-pendent temporal modes in bandwidth B all occupyingthe same time interval ∆t. The longer is ∆t the longer ittakes a given mode to pass the origin, but the more suchmodes fit into the frequency window.

As an illustration of these ideas, consider the followingelementary question: What is the mode occupancy of alaser beam of power P and hence photon flux N = P

~Ω0?

We cannot answer this without knowing the coherencetime or equivalently the bandwidth. The output of agood laser is like that of a radio frequency oscillator–ithas essentially no amplitude fluctuations. The frequencyis nominally set by the physical properties of the oscilla-tor, but there is nothing to pin the phase which conse-quently undergoes slow diffusion due to unavoidable noiseperturbations. This leads to a finite phase coherence timeτ and corresponding frequency spread 1/τ of the laserspectrum. (A laser beam differs from a thermal sourcethat has been filtered to have the same spectrum in thatit has smaller amplitude fluctuations.) Thus we expectthat the mode occupancy is X = Nτ . A convenient ap-proximate description in terms of temporal modes is totake the window interval to be ∆t = τ . Within the jthinterval we take the phase to be a (random) constant ϕjso that (up to an unimportant normalization constant)we have the coherent state∏

j

e√Xeiϕj B†→j |0〉 (D28)

which obeys

〈B→k 〉 =√Xeiϕk (D29)

and

〈B†→k B→k 〉 = X. (D30)

3. Quantum noise from a resistor

Let us consider the quantum equivalent to Eq. (C60),SV V = 2RkBT , for the case of a semi-infinite transmis-sion line with open termination, representing a resistor.From Eq. (C27) we see that the proper boundary con-dition for the ϕ field is ∂xϕ(0, t) = ∂xϕ(L, t) = 0. (Wehave temporarily made the transmission line have a largebut finite length L.) The normal mode expansion thatsatisfies these boundary conditions is

ϕ(x, t) =

√2L

∞∑n=1

ϕn(t) cos(knx), (D31)

18

where ϕn is the normal coordinate and kn ≡ πnL . Sub-

stitution of this form into the Lagrangian and carryingout the spatial integration yields a set of independentharmonic oscillators representing the normal modes.

Lg =∞∑n=1

(c

2ϕ2n −

12`k2nϕ

2n

). (D32)

From this we can find the momentum operator pn canon-ically conjugate to the coordinate operator ϕn and quan-tize the system to obtain an expression for the operatorrepresenting the voltage at the end of the transmissionline in terms of the mode creation and destruction oper-ators

V =∞∑n=1

√~ΩnLc

i(b†n − bn). (D33)

The spectral density of voltage fluctuations is then foundto be

SVV[ω] =2πL

∞∑n=1

~Ωnc

nB(~Ωn)δ(ω + Ωn)

+[nB(~Ωn) + 1]δ(ω − Ωn), (D34)

where nB(~ω) is the Bose occupancy factor for a photonwith energy ~ω. Taking the limit L→∞ and convertingthe summation to an integral yields

SVV(ω) = 2Zc~|ω|nB(~|ω|)Θ(−ω)+[nB(~|ω|)+1]Θ(ω)

,

(D35)where Θ is the step function. We see immediately thatat zero temperature there is no noise at negative frequen-cies because energy can not be extracted from zero-pointmotion. However there remains noise at positive frequen-cies indicating that the vacuum is capable of absorbingenergy from another quantum system. The voltage spec-tral density at both zero and non-zero temperature isplotted in Fig. (1).

Eq. (D35) for this ‘two-sided’ spectral density of a re-sistor can be rewritten in a more compact form

SVV[ω] =2Zc~ω

1− e−~ω/kBT, (D36)

which reduces to the more familiar expressions in variouslimits. For example, in the classical limit kBT ~ω thespectral density is equal to the Johnson noise result6

SVV[ω] = 2ZckBT, (D37)

in agreement with Eq. (C60). In the quantum limit itreduces to

SVV[ω] = 2Zc~ωΘ(ω). (D38)

6 Note again that in the engineering convention this would beSVV[ω] = 4ZckBT .

Again, the step function tells us that the resistor can onlyabsorb energy, not emit it, at zero temperature.

If we use the engineering convention and add the noiseat positive and negative frequencies we obtain

SVV[ω] + SVV[−ω] = 2Zc~ω coth~ω

2kBT(D39)

for the symmetric part of the noise, which appears in thequantum fluctuation-dissipation theorem (cf. Eq. (2.16)).The antisymmetric part of the noise is simply

SVV[ω]− SVV[−ω] = 2Zc~ω, (D40)

yielding

SVV[ω]− SVV[−ω]SVV[ω] + SVV[−ω]

= tanh~ω

2kBT. (D41)

This quantum treatment can also be applied to anyarbitrary dissipative network (Burkhard et al., 2004; De-voret, 1997). If we have a more complex circuit con-taining capacitors and inductors, then in all of the aboveexpressions, Zc should be replaced by ReZ[ω] where Z[ω]is the complex impedance presented by the circuit.

In the above we have explicitly quantized the stand-ing wave modes of a finite length transmission line. Wecould instead have used the running waves of an infiniteline and recognized that, as the in classical treatment inEq. (C59), the left and right movers are not independent.The open boundary condition at the termination requiresV← = V→ and hence b→ = b←. We then obtain

SV V [ω] = 4S→V V [ω] (D42)

and from the quantum analog of Eq. (C40) we have

SV V [ω] =4~|ω|2cvp

Θ(ω)(nB + 1) + Θ(−ω)nB

= 2Zc~|ω| Θ(ω)(nB + 1) + Θ(−ω)nB(D43)

in agreement with Eq. (D35).

Appendix E: Back Action and Input-Output Theory forDriven Damped Cavities

A high Q cavity whose resonance frequency can beparametrically controlled by an external source can actas a very simple quantum amplifier, encoding informa-tion about the external source in the phase and ampli-tude of the output of the driven cavity. For example,in an optical cavity, one of the mirrors could be move-able and the external source could be a force acting onthat mirror. This defines the very active field of optome-chanics, which also deals with microwave cavities cou-pled to nanomechanical systems and other related setups(Arcizet et al., 2006; Brown et al., 2007; Gigan et al.,

19

2006; Harris et al., 2007; Hohberger-Metzger and Kar-rai, 2004; Marquardt et al., 2007, 2006; Meystre et al.,1985; Schliesser et al., 2006; Teufel et al., 2008; Thomp-son et al., 2008; Wilson-Rae et al., 2007). In the case of amicrowave cavity containing a qubit, the state-dependentpolarizability of the qubit acts as a source which shiftsthe frequency of the cavity (Blais et al., 2004; Schusteret al., 2005; Wallraff et al., 2004).

The dephasing of a qubit in a microwave cavity andthe fluctuations in the radiation pressure in an opticalcavity both depend on the quantum noise in the numberof photons inside the cavity. We here use a simple equa-tion of motion method to exactly solve for this quantumnoise in the perturbative limit where the dynamics of thequbit or mirror degree of freedom has only a weak backaction effect on the cavity.

In the following, we first give a basic discussion ofthe cavity field noise spectrum, deferring the detailedmicroscopic derivation to subsequent subsections. Wethen provide a review of the input-output theory fordriven cavities, and employ this theory to analyze theimportant example of a dispersive position measurement,where we demonstrate how the standard quantum limitcan be reached. Finally, we analyze an example wherea modified dispersive scheme is used to detect only onequadrature of a harmonic oscillator’s motion, such thatthis quadrature does not feel any back-action.

1. Photon shot noise inside a cavity and back action

Consider a degree of freedom z coupled parametricallywith strength A to the cavity oscillator

Hint = ~ωc(1 +Az) [a†a− 〈a†a〉] (E1)

where following Eq. (3.12), we have taken A to be dimen-sionless, and use z to denote the dimensionless systemvariable that we wish to probe. For example, z couldrepresent the dimensionless position of a mechanical os-cillator

z ≡ x

xZPF. (E2)

We have subtracted the 〈a†a〉 term so that the meanforce on the degree of freedom is zero. To obtain the fullHamiltonian, we would have to add the cavity dampingand driving terms, as well as the Hamiltonian governingthe intrinsic dynamics of the system z. From Eq. (3.18)we know that the back action noise force acting on z isproportional to the quantum fluctuations in the numberof photons n = a†a in the cavity,

Snn(t) = 〈a†(t)a(t)a†(0)a(0)〉 − 〈a†(t)a(t)〉2. (E3)

For the case of continuous wave driving at frequencyωL = ωc + ∆ detuned by ∆ from the resonance, thecavity is in a coherent state |ψ〉 obeying

a(t) = e−iωLt[a+ d(t)] (E4)

where the first term is the ‘classical part’ of the mode am-plitude ψ(t) = ae−iωLt determined by the strength of thedrive field, the damping of the cavity and the detuning∆, and d is the quantum part. By definition,

a|ψ〉 = ψ|ψ〉 (E5)

so the coherent state is annihilated by d:

d|ψ〉 = 0. (E6)

That is, in terms of the operator d, the coherent statelooks like the undriven quantum ground state . The dis-placement transformation in Eq. (E4) is canonical since

[a, a†] = 1 ⇒ [d, d†] = 1. (E7)

Substituting the displacement transformation intoEq. (E3) and using Eq. (E6) yields

Snn(t) = n〈d(t)d†(0)〉, (E8)

where n = |a|2 is the mean cavity photon number. If weset the cavity energy damping rate to be κ, such that theamplitude damping rate is κ/2, then the undriven stateobeys

〈d(t)d†(0)〉 = e+i∆te−κ2 |t|. (E9)

This expression will be justified formally in the subse-quent subsection, after introducing input-output theory.We thus arrive at the very simple result

Snn(t) = nei∆t−κ2 |t|. (E10)

The power spectrum of the noise is, via the Wiener-Khinchin theorem (Appendix A.2), simply the Fouriertransform of the autocorrelation function given inEq. (E10)

Snn[ω] =∫ +∞

−∞dt eiωtSnn(t) = n

κ

(ω + ∆)2 + (κ/2)2.

(E11)As can be seen in Fig. 5a, for positive detuning ∆ =ωL − ωc > 0, i.e. for a drive that is blue-detuned withrespect to the cavity, the noise peaks at negative ω. Thismeans that the noise tends to pump energy into the de-gree of freedom z (i.e. it contributes negative damping).For negative detuning the noise peaks at positive ω cor-responding to the cavity absorbing energy from z. Basi-cally, the interaction with z (three wave mixing) tries toRaman scatter the drive photons into the high density ofstates at the cavity frequency. If this is uphill in energy,then z is cooled.

As discussed in Sec. B.2 (c.f. Eq. (2.8)), at each fre-quency ω, we can use detailed balance to assign the noisean effective temperature Teff [ω]:

Snn[ω]Snn[−ω]

= e~ω/kBTeff [ω] ⇔

kBTeff [ω] ≡ ~ω

log[Snn[ω]Snn[−ω]

] (E12)

20

Noise

pow

er(a)

Frequency

(b)No

ise te

mpe

ratu

re

(c)

Frequency

4

2

0-6 0 6 60-6

4

0

-4

3

2

1

5 600 1 2 3 4

-

FIG. 5 (Color online) (a) Noise spectrum of the photon num-ber in a driven cavity as a function of frequency when thecavity drive frequency is detuned from the cavity resonanceby ∆ = +3κ (left peak) and ∆ = −3κ (right peak). (b) Ef-fective temperature Teff of the low frequency noise, ω → 0,as a function of the detuning ∆ of the drive from the cavityresonance. (c) Frequency-dependence of the effective noisetemperature, for different values of the detuning.

or equivalently

Snn[ω]− Snn[−ω]Snn[ω] + Snn[−ω]

= tanh(β~ω/2). (E13)

If z is the coordinate of a harmonic oscillator of frequencyω (or some non-conserved observable of a qubit with levelsplitting ω), then that system will acquire a temperatureTeff [ω] in the absence of coupling to any other environ-ment. In particular, if the characteristic oscillation fre-quency of the system z is much smaller than κ, then wehave the simple result

1kBTeff

= limω→0+

2~ω

Snn[ω]− Snn[−ω]Snn[ω] + Snn[−ω]

= 2d lnSnn[ω]

d~ω

=1~

−4∆∆2 + (κ/2)2

. (E14)

As can be seen in Fig. 5, the asymmetry in the noisechanges sign with detuning, which causes the effectivetemperature to change sign.

First we discuss the case of a positive Teff , where thismechanism can be used to laser cool an oscillating me-chanical cantilever, provided Teff is lower than the in-trinsic equilibrium temperature of the cantilever. (Ar-cizet et al., 2006; Brown et al., 2007; Gigan et al., 2006;Harris et al., 2007; Hohberger-Metzger and Karrai, 2004;Marquardt et al., 2007; Schliesser et al., 2006; Thompsonet al., 2008; Wilson-Rae et al., 2007). A simple classicalargument helps us understand this cooling effect. Sup-pose that the moveable mirror is at the right hand endof a cavity being driven below the resonance frequency.If the mirror moves to the right, the resonance frequencywill fall and the number of photons in the cavity will rise.There will be a time delay however to fill the cavity andso the extra radiation pressure will not be fully effectivein doing work on the mirror. During the return part ofthe oscillation as the mirror moves back to the left, thetime delay in emptying the cavity will cause the mirror tohave to do extra work against the radiation pressure. Atthe end of the cycle it ends up having done net positivework on the light field and hence is cooled. The effect cantherefore be understood as being due to the introductionof some extra optomechanical damping.

The signs reverse (and Teff becomes negative) if thecavity is driven above resonance, and consequently thecantilever motion is heated up. In the absence of in-trinsic mechanical losses, negative values of the effectivetemperature indicate a dynamical instability of the can-tilever (or population inversion in the case of a qubit),where the amplitude of motion grows until it is finallystabilized by nonlinear effects. This can be interpretedas negative damping introduced by the optomechanicalcoupling and can be used to create parametric amplifica-tion of mechanical forces acting on the oscillator.

Finally, we mention that cooling towards the quantumground state of a mechanical oscillator (where phononnumbers become much less than one), is only possible(Marquardt et al., 2007; Wilson-Rae et al., 2007) in the“far-detuned regime”, where −∆ = ω κ (in contrastto the ω κ regime discussed above).

2. Input-output theory for a driven cavity

The results from the previous section can be more for-mally and rigorously derived in a full quantum theoryof a cavity driven by an external coherent source. Thetheory relating the drive, the cavity and the outgoingwaves radiated by the cavity is known as input-outputtheory and the classical description was presented in Ap-pendix C. The present quantum discussion closely fol-lows standard references on the subject (Walls and Mil-burn, 1994; Yurke, 1984; Yurke and Denker, 1984). Thecrucial feature that distinguishes such an approach frommany other treatments of quantum-dissipative systems

21

is the goal of keeping the bath modes instead of trac-ing them out. This is obviously necessary for the situa-tions we have in mind, where the output field emanatingfrom the cavity contains the information acquired duringa measurement of the system coupled to the cavity. Aswe learned from the classical treatment, we can elimi-nate the outgoing waves in favor of a damping term forthe system. However we can recover the solution for theoutgoing modes completely from the solution of the equa-tion of motion of the damped system being driven by theincoming waves.

In order to drive the cavity we must partially open oneof its ports which exposes the cavity both to the externaldrive and to the vacuum noise outside which permits en-ergy in the cavity to leak out into the surrounding bath.We will formally separate the degrees of freedom into in-ternal cavity modes and external bath modes. Strictlyspeaking, once the port is open, these modes are not dis-tinct and we only have ‘the modes of the universe’ (Gea-Banacloche et al., 1990a,b; Lang et al., 1973). Howeverfor high Q cavities, the distinction is well-defined and wecan model the decay of the cavity in terms of a spon-taneous emission process in which an internal boson isdestroyed and an external bath boson is created. Weassume a single-sided cavity. For a high Q cavity, thisphysics is accurately captured in the following Hamilto-nian

H = Hsys + Hbath + Hint. (E15)

The bath Hamiltonian is

Hbath =∑q

~ωq b†q bq (E16)

where q labels the quantum numbers of the independentharmonic oscillator bath modes obeying

[bq, b†q′ ] = δq,q′ . (E17)

Note that since the bath terminates at the system, thereis no translational invariance, the normal modes arestanding not running waves, and the quantum numbersq are not necessarily wave vectors.

The coupling Hamiltonian is (within the rotating waveapproximation)

Hint = −i~∑q

[fqa†bq − f∗q b†qa

]. (E18)

For the moment we will leave the system (cavity) Hamil-tonian to be completely general, specifying only that itconsists of a single degree of freedom (i.e. we concentrateon only a single resonance of the cavity with frequencyωc) obeying the usual bosonic commutation relation

[a, a†] = 1. (E19)

(N.B. this does not imply that it is a harmonic oscilla-tor. We will consider both linear and non-linear cavities.)

Note that the most general linear coupling to the bathmodes would include terms of the form b†qa

† and bqa butthese are neglected within the rotating wave approxima-tion because in the interaction representation they os-cillate at high frequencies and have little effect on thedynamics.

The Heisenberg equation of motion (EOM) for thebath variables is

˙bq =

i

~[H, bq] = −iωqbq + f∗q a (E20)

We see that this is simply the EOM of a harmonic oscil-lator driven by a forcing term due to the motion of thecavity degree of freedom. Since this is a linear system,the EOM can be solved exactly. Let t0 < t be a time inthe distant past before any wave packet launched at thecavity has reached it. The solution of Eq. (E20) is

bq(t) = e−iωq(t−t0)bq(t0) +∫ t

t0

dτ e−iωq(t−τ)f∗q a(τ).

(E21)The first term is simply the free evolution of the bathwhile the second represents the waves radiated by thecavity into the bath.

The EOM for the cavity mode is

˙a =i

~[Hsys, a]−

∑q

fq bq. (E22)

Substituting Eq. (E21) into the last term above yields∑q

fq bq =∑q

fqe−iωq(t−t0)bq(t0)

+∑q

|fq|2∫ t

t0

dτ e−i(ωq−ωc)(t−τ)[e+iωc(τ−t)a(τ)], (E23)

where the last term in square brackets is a slowly varyingfunction of τ . To simplify our result, we note that ifthe cavity system were a simple harmonic oscillator offrequency ωc then the decay rate from the n = 1 singlephoton excited state to the n = 0 ground state would begiven by the following Fermi Golden Rule expression

κ(ωc) = 2π∑q

|fq|2δ(ωc − ωq). (E24)

From this it follows that∫ +∞

−∞

dν

2πκ(ωc + ν)e−iν(t−τ) =

∑q

|fq|2e−i(ωq−ωc)(t−τ).

(E25)We now make the Markov approximation which assumesthat κ(ν) = κ is a constant over the range of frequenciesrelevant to the cavity so that Eq. (E25) may be repre-sented as∑

q

|fq|2e−i(ωq−ωc)(t−τ) = κδ(t− τ). (E26)

22

Using ∫ x0

−∞dx δ(x− x0) =

12

(E27)

we obtain for the cavity EOM

˙a =i

~[Hsys, a]− κ

2a−

∑q

fqe−iωq(t−t0)bq(t0). (E28)

The second term came from the part of the bath motionrepresenting the wave radiated by the cavity and, withinthe Markov approximation, has become a simple lineardamping term for the cavity mode. Note the importantfactor of 2. The amplitude decays at half the rate of theintensity (the energy decay rate κ).

Within the spirit of the Markov approximation it isfurther convenient to treat f ≡

√|fq|2 as a constant and

define the density of states (also taken to be a constant)by

ρ =∑q

δ(ωc − ωq) (E29)

so that the Golden Rule rate becomes

κ = 2πf2ρ. (E30)

We can now define the so-called ‘input mode’

bin(t) ≡ 1√2πρ

∑q

e−iωq(t−t0)bq(t0) . (E31)

For the case of a transmission line treated in AppendixD, this coincides with the field b→ moving towards thecavity [see Eq. (D13a)]. We finally have for the cavityEOM

˙a =i

~[Hsys, a]− κ

2a−√κ bin(t). (E32)

Note that when a wave packet is launched from the bathtowards the cavity, causality prevents it from knowingabout the cavity’s presence until it reaches the cavity.Hence the input mode evolves freely as if the cavity werenot present until the time of the collision at which pointit begins to drive the cavity. Since bin(t) evolves underthe free bath Hamiltonian and acts as the driving term inthe cavity EOM, we interpret it physically as the inputmode. Eq. (E32) is the quantum analog of the classi-cal equation (C19), for our previous example of an LC-oscillator driven by a transmission line. The latter wouldalso have been first order in time if as in Eq. (C35) wehad worked with the complex amplitude A instead of thecoordinate Q.

Eq. (E31) for the input mode contains a time labeljust as in the interaction representation. However it isbest interpreted as simply labeling the particular linearcombination of the bath modes which is coupled to thesystem at time t. Some authors even like to think of

the bath modes as non-propagating while the cavity fliesalong the bath (taken to be 1D) at a velocity v. Thesystem then only interacts briefly with the local modepositioned at x = vt before moving on and interactingwith the next local bath mode. We will elaborate on thisview further at the end of this subsection.

The expression for the power Pin (energy per time)impinging on the cavity depends on the normalizationchosen in our definition of bin. It can be obtained, forexample, by imagining the bath modes bq to live on a one-dimensional waveguide with propagation velocity v andlength L (using periodic boundary conditions). In thatcase we have to sum over all photons to get the averagepower flowing through a cross-section of the waveguide,Pin =

∑q ~ωq(vp/L)

⟨b†q bq

⟩. Inserting the definition for

bin, Eq. (E31), the expression for the input power carriedby a monochromatic beam at frequency ω is

Pin(t) = ~ω⟨b†in(t)bin(t)

⟩(E33)

Note that this has the correct dimensions due to ourchoice of normalization for bin (with dimensions

√ω). In

the general case, an integration over frequencies is needed(as will be discussed further below). An analogous for-mula holds for the power radiated by the cavity, to bediscussed now.

The output mode bout(t) is radiated into the bath andevolves freely after the system interacts with bin(t). Ifthe cavity did not respond at all, then the output modewould simply be the input mode reflected off the cav-ity mirror. If the mirror is partially transparent thenthe output mode will also contain waves radiated by thecavity (which is itself being driven by the input modepartially transmitted into the cavity through the mirror)and hence contains information about the internal dy-namics of the cavity. To analyze this output field, lett1 > t be a time in the distant future after the inputfield has interacted with the cavity. Then we can writean alternative solution to Eq. (E20) in terms of the finalrather than the initial condition of the bath

bq(t) = e−iωq(t−t1)bq(t1)−∫ t1

t

dτ e−iωq(t−τ)f∗q a(τ).

(E34)Note the important minus sign in the second term as-sociated with the fact that the time t is now the lowerlimit of integration rather than the upper as it was inEq. (E21).

Defining

bout(t) ≡1√2πρ

∑q

e−iωq(t−t1)bq(t1), (E35)

we see that this is simply the free evolution of the bathmodes from the distant future (after they have interactedwith the cavity) back to the present, indicating that it isindeed appropriate to interpret this as the outgoing field.

23

Proceeding as before we obtain

˙a =i

~[Hsys, a] +

κ

2a−√κ bout(t). (E36)

Subtracting Eq. (E36) from Eq. (E32) yields

bout(t) = bin(t) +√κ a(t) (E37)

which is consistent with our interpretation of the out-going field as the reflected incoming field plus the fieldradiated by the cavity out through the partially reflectingmirror.

The above results are valid for any general cavityHamiltonian. The general procedure is to solve Eq. (E32)for a(t) for a given input field, and then solve Eq. (E37)to obtain the output field. For the case of an empty cav-ity we can make further progress because the cavity modeis a harmonic oscillator

Hsys = ~ωca†a. (E38)

In this simple case, the cavity EOM becomes

˙a = −iωca−κ

2a−√κ bin(t). (E39)

Eq. (E39) can be solved by Fourier transformation, yield-ing

a[ω] = −√κ

i(ωc − ω) + κ/2bin[ω] (E40)

= −√κχc[ω − ωc]bin[ω] (E41)

and

bout[ω] =ω − ωc − iκ/2ω − ωc + iκ/2

bin[ω] (E42)

which is the result for the reflection coefficient quoted inEq. (3.13). For brevity, here and in the following, we willsometimes use the susceptibility of the cavity, defined as

χc[ω − ωc] ≡ 1−i(ω − ωc) + κ/2

(E43)

For the case of steady driving on resonance where ω = ωc,the above equations yield

bout[ω] =√κ

2a[ω]. (E44)

In steady state, the incoming power equals the outgoingpower, and both are related to the photon number insidethe single-sided cavity by

P = ~ω⟨b†out(t)bout(t)

⟩= ~ω

κ

4⟨a†(t)a(t)

⟩(E45)

Note that this does not coincide with the naive expecta-tion, which would be P = ~ωκ

⟨a†a⟩. The reason for this

discrepancy is the the interference between the part of the

incoming wave which is promptly reflected from the cav-ity and the field radiated by the cavity. The naive expres-sion becomes correct after the drive has been switched off(where ignoring the effect of the incoming vacuum noise,we would have bout =

√κa). We note in passing that for

a driven two-sided cavity with coupling constants κL andκR (where κ = κL + κR), the incoming power sent intothe left port is related to the photon number by

P = ~ωκ2/(4κL)⟨a†a⟩. (E46)

Here for κL = κR the interference effect completely elim-inates the reflected beam and we have in contrast toEq. (E45)

P = ~ωκ

2⟨a†a⟩. (E47)

Eq. (E39) can also be solved in the time domain toobtain

a(t) = e−(iωc+κ/2)(t−t0)a(t0)

−√κ

∫ t

t0

dτ e−(iωc+κ/2)(t−τ)bin(τ). (E48)

If we take the input field to be a coherent drive at fre-quency ωL = ωc + ∆ so that its amplitude has a classicaland a quantum part

bin(t) = e−iωLt[bin + ξ(t)] (E49)

and if we take the limit t0 →∞ so that the initial tran-sient in the cavity amplitude has damped out, then thesolution of Eq. (E48) has the form postulated in Eq. (E4)with

a = −√κ

−i∆ + κ/2bin (E50)

and (in the frame rotating at the drive frequency)

d(t) = −√κ

∫ t

−∞dτ e+(i∆−κ/2)(t−τ)ξ(τ). (E51)

Even in the absence of any classical drive, the inputfield delivers vacuum fluctuation noise to the cavity. No-tice that from Eqs. (E31, E49)

[bin(t), b†in(t′)] = [ξ(t), ξ†(t′)]

=1

2πρ

∑q

e−i(ωq−ωL)(t−t′)

= δ(t− t′), (E52)

which is similar to Eq. (D16) for a quantum transmissionline. This is the operator equivalent of white noise. UsingEq. (E48) in the limit t0 → −∞ in Eqs. (E4,E51) yields

[a(t), a†(t)] = [d(t), d†(t)]

= κ

∫ t

−∞dτ

∫ t

−∞dτ ′ e−(−i∆+κ/2)(t−τ)

e−(+i∆+κ/2)(t−τ ′)δ(τ − τ ′)= 1 (E53)

24

as is required for the cavity bosonic quantum degree offreedom. We can interpret this as saying that the cavityzero-point fluctuations arise from the vacuum noise thatenters through the open port. We also now have a simplephysical interpretation of the quantum noise in the num-ber of photons in the driven cavity in Eqs. (E3,E8,E11).It is due to the vacuum noise which enters the cavitythrough the same ports that bring in the classical drive.The interference between the vacuum noise and the clas-sical drive leads to the photon number fluctuations in thecavity.

In thermal equilibrium, ξ also contains thermal radi-ation. If the bath is being probed only over a narrowrange of frequencies centered on ωc (which we have as-sumed in making the Markov approximation) then wehave to a good approximation (consistent with the abovecommutation relation)

〈ξ†(t)ξ(t′)〉 = Nδ(t− t′) (E54)

〈ξ(t)ξ†(t′)〉 = (N + 1)δ(t− t′) (E55)

where N = nB(~ωc) is the thermal equilibrium occupa-tion number of the mode at the frequency of interest. Wecan gain a better understanding of Eq. (E54) by Fouriertransforming it to obtain the spectral density

S[ω] =∫ +∞

−∞dt 〈ξ†(t)ξ(t′)〉eiω(t−t′) = N. (E56)

As mentioned previously, this dimensionless quantity isthe spectral density that would be measured by a photo-multiplier: it represents the number of thermal photonspassing a given point per unit time per unit bandwidth.Equivalently the thermally radiated power in a narrowbandwidth B is

P = ~ωNB. (E57)

One often hears the confusing statement that the noiseadded by an amplifier is a certain number N of photons(N = 20, say for a good cryogenic HEMT amplifier op-erating at 5 GHz). This means that the excess outputnoise (referred back to the input by dividing by the powergain) produces a flux of N photons per second in a 1 Hzbandwidth, or 106N photons per second in 1 MHz ofbandwidth (see also Eq. (D27)).

We can gain further insight into input-output theoryby using the following picture. The operator bin(t) repre-sents the classical drive plus vacuum fluctuations whichare just about to arrive at the cavity. We will be ableto show that the output field is simply the input fielda short while later after it has interacted with the cav-ity. Let us consider the time evolution over a short timeperiod ∆t which is very long compared to the inversebandwidth of the vacuum noise (i.e., the frequency scalebeyond which the vacuum noise cannot be treated as con-stant due to some property of the environment) but veryshort compared to the cavity system’s slow dynamics. In

this circumstance it is useful to introduce the quantumWiener increment related to Eq. (D18)

dW ≡∫ t+∆t

t

dτ ξ(τ) (E58)

which obeys

[dW , dW †] = ∆t. (E59)

In the interaction picture (in a displaced frame inwhich the classical drive has been removed) the Hamilto-nian term that couples the cavity to the quantum noiseof the environment is from Eq. (E18)

V = −i~√κ(a†ξ − aξ†). (E60)

Thus the time evolution operator (in the interaction pic-ture) on the jth short time interval [tj , tj + ∆t] is

Uj = e√κ(a dcW †−a† dcW ) (E61)

Using this we can readily evolve the incoming temporalmode forward in time by a small step ∆t

dW ′ = U†dW U ≈ dW +√κ∆t a. (E62)

Recall that in input-output theory we formally definedthe outgoing field as the bath field far in the future prop-agated back (using the free field time evolution) to thepresent, which yielded

bout = bin +√κa. (E63)

Eq. (E62) is completely equivalent to this. Thus we con-firm our understanding that the incoming field is the bathtemporal mode just before it interacts with the cavity andthe outgoing field is the bath temporal mode just after itinteracts with the cavity.

This leads to the following picture which is especiallyuseful in the quantum trajectory approach to conditionalquantum evolution of a system subject to weak continu-ous measurement (Gardiner et al., 1992; Walls and Mil-burn, 1994). On top of the classical drive bin(t), the bathsupplies to the system a continuous stream of “fresh” har-monic oscillators, each in their ground state (if T = 0).Each oscillator with its quantum fluctuation dW inter-acts briefly for a period ∆t with the system and thenis disconnected to propagate freely thereafter, never in-teracting with the system again. Within this picture itis useful to think of the oscillators arrayed in an infinitestationary line and the cavity flying over them at speedvp and touching each one for a time ∆t.

3. Quantum limited position measurement using a cavitydetector

We will now apply the input-output formalism intro-duced in the previous section to the important example

25

of a dispersive position measurement, which employs acavity whose resonance frequency shifts in response tothe motion of a harmonic oscillator. This physical sys-tem was considered heuristically in Sec. III.B.3. Here wewill present a rigorous derivation using the (linearized)equations of motion for the coupled cavity and oscillatorsystem.

Let the dimensionless position operator

z =1

xZPFx = [c† + c] (E64)

be the coordinate of a harmonic oscillator whose energyis

HM = ~ωMc†c (E65)

and whose position uncertainty in the quantum groundstate is xZPF =

√〈0|x2|0〉.

This Hamiltonian could be realized for example bymounting one of the cavity mirrors on a flexible cantilever(see the discussion above).

When the mirror moves, the cavity resonance fre-quency shifts,

ωc = ωc[1 +Az(t)] (E66)

where for a cavity of length L, A = −xZPF/L.Assuming that the mirror moves slowly enough for the

cavity to adiabatically follow its motion (i.e. Ω κ), theoutgoing light field suffers a phase shift which follows thechanges in the mirror position. This phase shift can bedetected in the appropriate homodyne set up as discussedin Sec. III.B, and from this phase shift we can determinethe position of the mechanical oscillator. In addition tothe actual zero-point fluctuations of the oscillator, ourmeasurement will suffer from shot noise in the homodynesignal and from additional uncertainty due to the backaction noise of the measurement acting on the oscillator.All of these effects will appear naturally in the derivationbelow.

We begin by considering the optical cavity equationof motion based on Eq. (E32) and the optomechanicalcoupling Hamiltonian in Eq. (E1). These yield

˙a = −iωc(1 +Az)a− κ

2a−√κbin. (E67)

Let the cavity be driven by a laser at a frequency ωL =ωc +∆ detuned from the cavity by ∆. Moving to a framerotating at ωL we have

˙a = +i(∆−Aωcz)a−κ

2a−√κbin. (E68)

and we can write the incoming field as a constant pluswhite noise vacuum fluctuations (again, in the rotatingframe)

bin = bin + ξ (E69)

and similarly for the cavity field following Eq. (E4)

a = a+ d. (E70)

Substituting these expressions into the equation of mo-tion, we find that the constant classical fields obey

a = −√κ

κ/2− i∆bin (E71)

and the new quantum equation of motion is, after ne-glecting a small term dz:

˙d = +i∆d− iAωcaz −

κ

2d−√κξ. (E72)

The quantum limit for position measurement will bereached only at zero detuning, so we specialize to thecase ∆ = 0. We also choose the incoming field amplitudeand phase to obey

bin = −i√N , (E73)

so that

a = +2i

√N

κ, (E74)

where N is the incoming photon number flux. The quan-tum equation of motion for the cavity then becomes

˙d = +gz − κ

2d−√κξ, (E75)

where the opto-mechanical coupling constant is propor-tional to the laser drive amplitude

g ≡ 2Aωc

√N

κ= Aωc

√n. (E76)

and

n = |a|2 = 4N

κ(E77)

is the mean cavity photon number. Eq. (E75) is easilysolved by Fourier transformation

d[ω] =1

[κ/2− iω]

gz[ω]−

√κξ[ω]

. (E78)

Let us assume that we are in the limit of low mechan-ical frequency relative to the cavity damping, Ω κ,so that the cavity state adiabatically follows the motionof the mechanical oscillator. Then we obtain to a goodapproximation

d[ω] =2κ

gz[ω]−

√κξ[ω]

(E79)

d†[ω] =2κ

gz[ω]−

√κξ†[ω]

(E80)

26

The mechanical oscillator equation of motion which isidentical in form to that of the optical cavity

∂tc = −[γ0

2+ iΩ]c−√γ0η(t) +

i

~[Hint, c(t)], (E81)

where Hint is the Hamiltonian in Eq. (E1) and η isthe mechanical vacuum noise from the (zero tempera-ture) bath which is causing the mechanical damping atrate γ0. Using Eq. (E70) and expanding to first orderin small fluctuations yields the equation of motion lin-earized about the steady state solution

∂tc = −[γ0

2+ iΩ]c−√γ0η(t) + 2

g√κ

[ξ(t)− ξ†(t)]. (E82)

It is useful to consider an equivalent formulation inwhich we expand the Hamiltonian in Eq. (E1) to secondorder in the quantum fluctuations about the classical so-lution

Hint ≈ ~ωcd†d+ xF , (E83)

where the force (including the coupling A) is (up to asign)

F = −i ~gxZPF

[d− d†]. (E84)

Note that the radiation pressure fluctuations (photonshot noise) inside the cavity provide a forcing term. Thestate of the field inside the cavity in general depends onthe past history of the cantilever position. However forthis special case of driving the cavity on resonance, thedependence of the cavity field on the cantilever history issuch that the latter drops out of the radiation pressure.To see this explicitly, consider the equation of motion forthe force obtained from Eq. (E75)

˙F = −κ

2F + i

~gxZPF

√κ[ξ − ξ†]. (E85)

Within our linearization approximation, the position ofthe mechanical oscillator has no effect on the radiationpressure (photon number in the cavity), but of course itdoes affect the phase of the cavity field (and hence theoutgoing field) which is what we measure in the homo-dyne detection.

Thus for this special case z does not appear on theRHS of either Eq. (E85) or Eq. (E82), which means thatthere is no optical renormalization of the cantilever fre-quency (‘optical spring’) or optical damping of the can-tilever. The lack of back-action damping in turn impliesthat the effective temperature Teff of the cavity detectoris infinite (cf. Eq. (2.8)). For this special case of zerodetuning the back action force noise is controlled by asingle quadrature of the incoming vacuum noise (whichinterferes with the classical drive to produce photon num-ber fluctuations). This is illustrated in the cavity ampli-tude phasor diagram of Fig. (6). We see that the vac-uum noise quadrature ξ+ ξ† conjugate to F controls the

FIG. 6 (Color online) Phasor diagram for the cavity ampli-tude showing that (for our choice of parameters) the imag-

inary quadrature of the vacuum noise ξ interferes with theclassical drive to produce photon number fluctuations whilethe real quadrature produces phase fluctuations which lead tomeasurement imprecision. The quantum fluctuations are il-lustrated in the usual fashion, depicting the Gaussian Wignerdensity of the coherent state in terms of color intensity.

phase noise which determines the measurement impreci-sion (shot noise in the homodyne signal). This will bediscussed further below.

The solution for the cantilever position can again beobtained by Fourier transformation. For frequenciessmall on the scale of κ the solution of Eq. (E85) is

F [ω] =2i~g

xZPF√κ

ξ[ω]− ξ†[ω]

(E86)

and hence the back action force noise spectral density isat low frequencies

SFF [ω] =4~2g2

x2ZPFκ

(E87)

in agreement with Eq. (3.18).Introducing a quantity proportional to the cantilever

(mechanical) susceptibility (within the rotating wave ap-proximation we are using)

χM[ω − Ω] ≡ 1−i(ω − Ω) + γ0

2

, (E88)

we find from Eq. (E82)

z[ω] = z0[ω]− i

~xZPF χM[ω − Ω]− χM[ω + Ω] F [ω],

(E89)

27

where the equilibrium fluctuations in position are givenby

z0[ω] ≡ −√γ0

χM[ω − Ω]η[ω] + χM[ω + Ω]η†[ω]

.

(E90)We can now obtain the power spectrum Szz describing

the total position fluctuations of the cantilever driven bythe mechanical vacuum noise plus the radiation pressureshot noise. From Eqs. (E89, E90) we find

Sxx[ω]x2

ZPF

= Szz[ω]

= γ0|χ[ω − Ω]|2 (E91)

+x2

ZPF

~2|χM[ω − Ω]− χM[ω + Ω]|2 SFF .

Note that (assuming high mechanical Q, i.e. γ0 Ω) theequilibrium part has support only at positive frequencieswhile the back action induced position noise is symmetricin frequency reflecting the effective infinite temperatureof the back action noise. Symmetrizing this result withrespect to frequency (and using γ0 Ω) we have

Sxx[ω] ≈ S0xx[ω]

(1 +

S0xx[Ω]~2

SFF

), (E92)

where S0xx[ω] is the symmetrized spectral density for posi-

tion fluctuations in the ground state given by Eq. (3.54).Now that we have obtained the effect of the back action

noise on the position fluctuations, we must turn our at-tention to the imprecision of the measurement due to shotnoise in the output. The appropriate homodyne quadra-ture variable to monitor to be sensitive to the outputphase shift caused by position fluctuations is

I = bout + b†out, (E93)

which, using the input-output results above, can be writ-ten

I = −(ξ + ξ†) + λx. (E94)

We see that the cavity homodyne detector system actsas a position transducer with gain

λ =4g

xZPF√κ. (E95)

The first term in Eq. (E94) represents the vacuum noisethat mixes with the homodyne local oscillator to producethe shot noise in the output. The resulting measurementimprecision (symmetrized) spectral density referred backto the position of the oscillator is

SIxx =

1λ2. (E96)

Comparing this to Eq. (E87) we see that we reach thequantum limit relating the imprecision noise to the backaction noise

SIxxSFF =

~2

4(E97)

in agreement with Eq. (3.10).Notice also from Eq. (E94) that the quadrature of the

vacuum noise which leads to the measurement impreci-sion is conjugate to the one which produces the backaction force noise as illustrated previously in Fig. (6).Recall that the two quadratures of motion of a harmonicoscillator in its ground state have no classical (i.e., sym-metrized) correlation. Hence the symmetrized cross cor-relator

SIF [ω] = 0 (E98)

vanishes. Because there is no correlation between theoutput imprecision noise and the forces controlling theposition fluctuations, the total output noise referred backto the position of the oscillator is simply

Sxx,tot[ω] = Sxx[ω] + SIxx (E99)

= S0xx[ω]

(1 +

S0xx[Ω]~2

SFF

)+

~2

4SFF.

This expression again clearly illustrates the competitionbetween the back action noise proportional to the drivelaser intensity and the measurement imprecision noisewhich is inversely proportional. We again emphasize thatall of the above relations are particular to the case of zerodetuning of the cavity drive field from the cavity.

The total output noise at some particular frequencywill be a minimum at some optimal drive intensity. Theprecise optimal value depends on the frequency chosen.Typically this is taken to be the mechanical resonancefrequency where we find that the optimal coupling leadsto an optimal back action noise

SFF,opt =~2

2S0xx[Ω]

=~2γ0

4x2ZPF

. (E100)

This makes sense because the higher the damping the lesssusceptible the oscillator is to back action forces. At thisoptimal coupling the total output noise spectral densityat frequency Ω referred to the position is simply twicethe vacuum value

Sxx,tot[Ω] = 2S0xx[Ω], (E101)

in agreement with Eq. (3.62). Evaluation of Eq. (E100)at the optimal coupling yields the graph shown inFig. (6). The background noise floor is due to thefrequency independent imprecision noise with value12 S

0xx[Ω]. The peak value at ω = Ω rises a factor of three

above this background.We derived the gain λ in Eq. (E95) by direct solution

of the equations of motion. With the results we have de-rived above, it is straightforward to show that the Kuboformula in Eq. (4.3) yields equivalent results. We havealready seen that the classical (i.e. symmetrized) correla-tions between the output signal I and the force F whichcouples to the position vanishes. However the Kubo for-mula evaluates the quantum (i.e. antisymmetric) corre-lations for the uncoupled system (A = g = 0). Hence wehave

28

χIF (t) = − i~θ(t)

⟨[−(ξ(t− δt) + ξ†(t− δt)

),

2i~gxZPF

√κ

(ξ(0)− ξ†(0)

)]⟩0

, (E102)

where δt is a small (positive) time representing the delaybetween the time when the vacuum noise impinges onthe cavity and when the resulting outgoing wave reachesthe homodyne detector. (More precisely it also compen-sates for certain small retardation effects neglected in thelimit ω κ used in several places in the above deriva-tions.) Using the fact that the commutator between thetwo quadratures of the vacuum noise is a delta function,Fourier transformation of the above yields (in the limitω δt 1 the desired result

χIF [ω] = λ. (E103)

Similarly we readily find that the small retardationcauses the reverse gain to vanish. Hence all our resultsare consistent with the requirements needed to reach thestandard quantum limit.

Thus with this study of the specific case of an oscillatorparametrically coupled to a cavity, we have reproducedall of the key results in Sec. V.E derived from completelygeneral considerations of linear response theory.

4. Back-action free single-quadrature detection

We now provide details on the cavity single-quadraturedetection scheme discussed in Sec. V.H.2. We again con-sider a high-Q cavity whose resonance frequency is mod-ulated by a high-Q mechanical oscillator with co-ordinatex (cf. Eqs. (E1) and (E64)). To use this system for ampli-fication of a single quadrature, we will consider the typi-cal case of a fast cavity (ωc Ω), and take the “good cav-ity” limit, where Ω κ. As explained in the main text,the crucial ingredient for single-quadrature detection isto take an amplitude-modulated cavity drive describedby the classical input field bin given in Eq. (5.101). Asbefore (cf. Eq. (E4)), we may write the cavity annihila-tion operator a as the sum of a classical piece a(t) and aquantum piece d; only d is influenced by the mechanicaloscillator. a(t) is easily found from the classical (noise-free) equations of motion for the isolated cavity; makinguse of the conditions ωc Ω κ, we have

a(t) '

√Nκ

2Ωcos (Ωt+ δ) e−iωct (E104)

To proceed with our analysis, we work in an interac-tion picture with respect to the uncoupled cavity andoscillator Hamiltonians. Making standard rotating-waveapproximations, the Hamiltonian in the interaction pic-

ture takes the simple form corresponding to Eq. (5.102b):

Hint = ~A(d+ d†

) (eiδ c+ e−iδ c†

)= ~A

(d+ d†

) Xδ

xZPF, (E105)

where

A = A · ωc

√Nκ

4Ω, (E106)

and in the second line, we have made use of the definitionof the quadrature operators Xδ, Yδ given in Eqs. (5.92).The form of Hint was discussed heuristically in the maintext in terms of Raman processes where photons are re-moved from the classical drive bin and either up or downconverted to the cavity frequency via absorption or emis-sion of a mechanical phonon. Alternatively, we can thinkof the drive yielding a time-dependent cavity-oscillatorcoupling which “follows” the Xδ quadrature. Note thatwe made crucial of use of the good cavity limit (κ Ω)to drop terms in Hint which oscillate at frequencies ±2Ω.These terms represent Raman sidebands which are awayfrom the cavity resonance by a distance ±2Ω. In the goodcavity limit, the density of photon states is negligible sofar off resonance and these processes are suppressed.

Similar to Eqs. (E39) and (E81), the Heisenberg equa-tions of motion (in the rotating frame) follow directlyfrom Hint and the dissipative terms in the total Hamil-tonian:

∂td = −κ2d−√κξ(t)eiωct − iA

(eiδ c+ e−iδ c†

)(E107a)

∂tc = −γ0

2c−√γ0η(t)eiΩt − ie−iδ

[A(d+ d†

)− f(t)

](E107b)

As before, ξ(t) represents the unavoidable noise in thecavity drive, and η(t), γ0 are the noisy force and damp-ing resulting from an equilibrium bath coupled to themechanical oscillator. Note from Eq. (E107a) that as an-ticipated, the cavity is only driven by one quadrature ofthe oscillator’s motion. We have also included a drivingforce F (t) on the mechanical oscillator which has somenarrow bandwidth centered on the oscillator frequency;this force is parameterized as:

F (t) =2~xZPF

Re[f(t)e−iΩte−iδ

](E108)

where f(t) is a complex function which is slowly varyingon the scale of an oscillator period.

29

The equations of motion are easily solved upon Fouriertransformation, resulting in:

Xδ[ω] = −xZPF · χM [ω]

[i (f∗[−ω]− f [ω]) (E109a)

+√γ0

(eiδ η(ω + Ω) + e−iδ η†(ω − Ω)

) ]

Yδ[ω] = ixZPF · χM [ω]

[(−i) (f [ω] + f∗[−ω]) (E109b)

+√γ0

(eiδ η(ω + Ω)− e−iδ η†(ω − Ω)

)−2iAχc[ω]

√κ(ξ(ω + ωc) + ξ†(ω − ωc)

)]

where the cavity and mechanical susceptibilities χc, χM

are defined in Eqs. (E43) and (E88).As anticipated, the detected quadrature Xδ is com-

pletely unaffected by the measurement: Eq. (E109a) isidentical to what we would have if there were no couplingbetween the oscillator and the cavity. In contrast, theconjugate quadrature Yδ experiences an extra stochasticforce due to the cavity: this is the measurement back-action.

Turning now to the output field from the cavity bout,we use the input-output relation Eq. (E37) to find in thelab (i.e. non-rotating) frame:

bout[ω] = bout[ω] +[−i(ω − ωc)− κ/2−i(ω − ωc) + κ/2

]ξ[ω]

−i A√κ

xZPFχc[ω − ωc] · Xδ(ω − ωc)

(E110)

The first term on the RHS simply represents the outputfield from the cavity in the absence of the mechanicaloscillator and any fluctuations. It will yield sharp peaksat the two sidebands associated with the drive, ω = ωc±Ω. The second term on the RHS of Eq. (E110) representsthe reflected noise of the incident cavity drive. This noisewill play the role of the “intrinsic ouput noise” of thisamplifier.

Finally, the last term on the RHS of Eq. (E110) is theamplified signal: it is simply the amplified quadratureXδ of the oscillator. This term will result in a peak inthe output spectrum at the resonance frequency of thecavity, ωc. As there is no back-action on the measuredXδ quadrature, the added noise can be made arbitrarilysmall by simply increasing the drive strength N (andhence A).

Appendix F: Information Theory and Measurement Rate

Suppose that we are measuring the state of a qubitvia the phase shift ±θ0 from a one-sided cavity. Let I(t)

be the homodyne signal integrated up to time t as inSec. III.B. We would like to understand the relationshipbetween the signal-to-noise ratio defined in Eq. (3.23),and the rate at which information about the state of thequbit is being gained. The probability distribution for Iconditioned on the state of the qubit σ = ±1 is

p(I|σ) =1√

2πSθθtexp

[−(I − σθ0t)2

2Sθθt

]. (F1)

Based on knowledge of this conditional distribution, wenow present two distinct but equivalent approaches togiving an information theoretic basis for the definition ofthe measurement rate.

1. Method I

Suppose we start with an initial qubit density matrix

ρ0 =(

12 00 1

2

). (F2)

After measuring for a time t, the new density matrixconditioned on the results of the measurement is

ρ1 =(p+ 00 p−

)(F3)

where it will be convenient to parameterize the two prob-abilities by the polarization m ≡ Tr(σzρ1) by

p± =1±m

2. (F4)

The information gained by the measurement is the en-tropy loss7 of the qubit

I = Tr(ρ1 ln ρ1 − ρ0 ln ρ0). (F5)

We are interested in the initial rate of gain of informationat short times θ2

0t Sθθ where m will be small. In thislimit we have

I ≈ m2

2. (F6)

We must now calculate m conditioned on the measure-ment result I

mI ≡∑σ

σp(σ|I). (F7)

From Bayes theorem we can express this in terms ofp(I|σ), which is the quantity we know,

p(σ|I) =p(I|σ)p(σ)∑σ′ p(I|σ′)p(σ′)

. (F8)

7 It is important to note that we use throughout here the physi-cist’s entropy with the natural logarithm rather than the log base2 which gives the information in units of bits.

30

Using Eq. (F1) the polarization is easily evaluated

mI = tanh(Iθ0

Sθθ

). (F9)

The information gain is thus

II =12

tanh2

(Iθ0

Sθθ

)≈ I2

2

(θ0

Sθθ

)2

(F10)

where the second equality is only valid for small |m|.Ensemble averaging this over all possible measurementresults yields the mean information gain at short times

I ≈ 12θ2

0

Sθθt (F11)

which justifies the definition of the measurement rategiven in Eq. (3.24).

2. Method II

An alternative information theoretic derivation is toconsider the qubit plus measurement device to be a sig-naling channel. The two possible inputs to the channelare the two states of the qubit. The output of the chan-nel is the result of the measurement of I. By togglingthe qubit state back and forth, one can send informationthrough the signal channel to another party. The chan-nel is noisy because even for a fixed state of the qubit,the measured values of the signal I have intrinsic fluctu-ations. Shannon’s noisy channel coding theorem (Coverand Thomas, 1991) tells us the maximum rate at whichinformation can be reliably sent down the channel by tog-gling the state of the qubit and making measurements ofI. It is natural to take this rate as defining the measure-ment rate for our detector.

The reliable information gain by the receiver on a noisychannel is a quantity known as the ‘mutual information’of the communication channel (Clerk et al., 2003; Coverand Thomas, 1991)

R = −∫ +∞

−∞dI

p(I) ln p(I)−

∑σ

p(σ) [p(I|σ) ln p(I|σ)]

(F12)

The first term is the Shannon entropy in the signal Iwhen we do not know the input signal (the value of thequbit). The second term represents the entropy giventhat we do know the value of the qubit (averaged overthe two possible input values). Thus the first term issignal plus noise, the second is just the noise. Subtractingthe two gives the net information gain. Expanding thisexpression for short times yields

R =18

(〈I(t)〉+ − 〈I(t)〉−)2

Sθθt

=θ2

0

2Sθθt

= Γmeast (F13)

exactly the same result as Eq. (F11). (Here 〈I(t)〉σ is themean value of I given that the qubit is in state σ.)

Appendix G: Number Phase Uncertainty

In this appendix, we briefly review the number-phaseuncertainty relation, and from it we derive the relation-ship between the spectral densities describing the photonnumber fluctuations and the phase fluctuations. Con-sider a coherent state labeled by its classical amplitudeα

|α〉 = exp−|α|

2

2

expαa†|0〉. (G1)

This is an eigenstate of the destruction operator

a|α〉 = α|α〉. (G2)

It is convenient to make the unitary displacement trans-formation which maps the coherent state onto a new vac-uum state and the destruction operator onto

a = α+ d (G3)

where d annihilates the new vacuum. Then we have

N = 〈N〉 = 〈0|(α∗ + d†)(α+ d)|0〉 = |α|2, (G4)

and

(∆N)2 = 〈(N − N)2〉 = |α|2〈0|dd†|0〉 = N . (G5)

Now define the two quadrature amplitudes

X =1√2

(a+ a†) (G6)

Y =i√2

(a† − a). (G7)

Each of these amplitudes can be measured in a homodyneexperiment. For convenience, let us take α to be real andpositive. Then

〈X〉 =√

2α (G8)

and

〈Y 〉 = 0. (G9)

If the phase of this wave undergoes a small modulationdue for example to weak parametric coupling to a qubitthen one can estimate the phase by

〈θ〉 =〈Y 〉〈X〉

. (G10)

This result is of course only valid for small angles, θ 1.For N 1, the uncertainty will be

(∆θ)2 =〈Y 2〉

(〈X〉)2=

12 〈0|dd

†|0〉2N

=1

4N. (G11)

31

Thus using Eq. (G5) we arrive at the fundamental quan-tum uncertainty relation

∆θ∆N =12. (G12)

Using the input-output theory described in Ap-pendix E we can restate the results above in terms ofnoise spectral densities. Let the amplitude of the fieldcoming in to the homodyne detector be

bin = bin + ξ(t) (G13)

where ξ(t) is the vacuum noise obeying

[ξ(t), ξ†(t′)] = δ(t− t′). (G14)

We are using a flux normalization for the field operatorsso

N = 〈b†inbin〉 = |bin|2 (G15)

and

〈N(t)N(0)〉 − N2

= 〈0|(b∗in + ξ†(t))(bin + ξ(t))(b∗in + ξ†(0))(bin + ξ(0))|0〉 −∣∣bin∣∣4 = Nδ(t). (G16)

From this it follows that the shot noise spectral densityis

SNN = N . (G17)

Similarly the phase can be estimated from the quadra-ture operator

θ =i(b†in − bin)

〈b†in + bin〉= 〈θ〉+ i

(ξ† − ξ)2bin

(G18)

which has noise correlator

〈δθ(t)δθ(0)〉 =1

4Nδ(t) (G19)

corresponding to the phase imprecision spectral density

Sθθ =1

4N. (G20)

We thus arrive at the fundamental quantum limit relation√SθθSNN =

12. (G21)

Appendix H: Using feedback to reach the quantum limit

In Sec. (VI.B), we demonstrated that any two port am-plifier whose scattering matrix has s11 = s22 = s12 = 0will fail to reach quantum limit when used as a weaklycoupled op-amp; at best, it will miss optimizing the quan-tum noise constraint of Eq. (5.88) by a factor of two.Reaching the quantum limit thus requires at least oneof s11, s22 and s12 to be non-zero. In this subsection,we demonstrate how this may be done. We show thatby introducing a form of negative feedback to the “min-imal” amplifier of the previous subsection, one can takeadvantage of noise correlations to reduce the back-actioncurrent noise SII by a factor of two. As a result, one is

able to reach the weak-coupling (i.e. op-amp) quantumlimit. Note that quantum amplifiers with feedback arealso treated in Courty et al. (1999); Grassia (1998).

On a heuristic level, we can understand the need foreither reflections or reverse gain to reach the quantumlimit. A problem with the “minimal” amplifier of thelast subsection was that its input impedance was too lowin comparison to its noise impedance ZN ∼ Za. Fromgeneral expression for the input impedance, Eq. (6.7d),we see that having non-zero reverse gain (i.e. s12 6= 0)and/or non-zero reflections (i.e. s11 6= 0 and/or s22 6= 0)could lead to Zin Za. This is exactly what occurswhen feedback is used to reach the quantum limit. Keepin mind that having non-vanishing reverse gain is danger-ous: as we discussed earlier, an appreciable non-zero λ′Ican lead to the highly undesirable consequence that theamplifier’s input impedance depends on the impedanceof the load connected to its output (cf. Eq. (6.6)).

1. Feedback using mirrors

To introduce reverse gain and reflections into the “min-imal” two-port bosonic amplifier of the previous subsec-tion, we will insert mirrors in three of the four arms lead-ing from the circulator: the arm going to the input line,the arm going to the output line, and the arm goingto the auxiliary “cold load” (Fig. 7). Equivalently, onecould imagine that each of these lines is not perfectlyimpedance matched to the circulator. Each mirror willbe described by a 2× 2 unitary scattering matrix:(

aj,out

bj,out

)= Uj ·

(bj,inaj,in

)(H1)

Uj =(

cos θj − sin θjsin θj cos θj

)(H2)

Here, the index j can take on three values: j = z for themirror in the input line, j = y for the mirror in the arm

32

bz,out

bz,in

az,in

az,out

ideal1-portamp.

coldload


cin cout=G1/2cin+(G-1)1/2v†in

vin vout

ax,out bx,in

by,outby,in

ay,inay,out

ax,in bx,outXZ

Ymirror

FIG. 7 Schematic of a modified minimal two-port amplifier,where partially reflecting mirrors have been inserted in theinput and output transmission lines, as well as in the lineleading to the cold load. By tuning the reflection coefficientof the mirror in the cold load arm (mirror Y ), we can in-duce negative feedback which takes advantage of correlationsbetween current and voltage noise. This then allows this sys-tem to reach the quantum limit as a weakly coupled voltageop amp. See text for further description

going to the cold load, and j = x for the mirror in theoutput line. The mode aj describes the “internal” modewhich exists between the mirror and circulator, while themode bj describes the “external” mode on the other sideof the mirror. We have taken the Uj to be real for con-venience. Note that θj = 0 corresponds to the case of nomirror (i.e. perfect transmission).

It is now a straightforward though tedious exercise toconstruct the scattering matrix for the entire system.From this, one can identify the reduced scattering matrixs appearing in Eq. (6.3), as well as the noise operatorsFj . These may then in turn be used to obtain the op-ampdescription of the amplifier, as well as the commutatorsof the added noise operators. These latter commutatorsdetermine the usual noise spectral densities of the ampli-fier. Details and intermediate steps of these calculationsmay be found in Appendix I.6.

As usual, to see if our amplifier can reach the quantumlimit when used as a (weakly-coupled) op-amp , we needto see if it optimizes the quantum noise constraint ofEq. (5.88). We consider the optimal situation where boththe auxiliary modes of the amplifier (uin and v†in) are inthe vacuum state. The surprising upshot of our analysis(see Appendix I.6) is the following: if we include a small

amount of reflection in the cold load line with the correctphase, then we can reach the quantum limit, irrespectiveof the mirrors in the input and output lines. In particular,if sin θy = −1/

√G, our amplifier optimizes the quantum

noise constraint of Eq. (5.88) in the large gain (i.e. largeG) limit, independently of the values of θx and θy. Notethat tuning θy to reach the quantum limit does not havea catastrophic impact on other features of our amplifier.One can verify that this tuning only causes the voltagegain λV and power gain GP to decrease by a factor oftwo compared to their θy = 0 values (cf. Eqs. (I60) and(I64)). This choice for θy also leads to Zin Za ∼ ZN

(cf. (I62)), in keeping with our general expectations.Physically, what does this precise tuning of θy corre-

spond to? A strong hint is given by the behaviour of theamplifier’s cross-correlation noise SV I [ω] (cf. Eq. (I65c)).In general, we find that SV I [ω] is real and non-zero.However, the tuning sin θy = −1/

√G is exactly what is

needed to have SV I vanish. Also note from Eq. (I65a)that this special tuning of θy decreases the back-actioncurrent noise precisely by a factor of two compared toits value at θy = 0. A clear physical explanation nowemerges. Our original, reflection-free amplifier had cor-relations between its back-action current noise and out-put voltage noise (cf. Eq. (6.18c)). By introducing nega-tive feedback of the output voltage to the input current(i.e. via a mirror in the cold-load arm), we are able touse these correlations to decrease the overall magnitudeof the current noise (i.e. the voltage fluctuations V par-tially cancel the original current fluctuations I). For anoptimal feedback (i.e. optimal choice of θy), the currentnoise is reduced by a half, and the new current noise isnot correlated with the output voltage noise. Note thatthis is indeed negative (as opposed to positive) feedback–it results in a reduction of both the gain and the powergain. To make this explicit, in the next section we willmap the amplifier described here onto a standard op-ampwith negative voltage feedback.

2. Explicit examples

To obtain a more complete insight, it is useful to goback and consider what the reduced scattering matrix ofour system looks like when θy has been tuned to reachthe quantum limit. From Eq. (I58), it is easy to see thatat the quantum limit, the matrix s satisfies:

s11 = −s22 (H3a)

s12 =1Gs21 (H3b)

The second equation also carries over to the op-amp pic-ture; at the quantum limit, one has:

λ′I =1GλV (H4)

One particularly simple limit is the case where thereare no mirrors in the input and output line (θx = θz = 0),

33

only a mirror in the cold-load arm. When this mirror istuned to reach the quantum limit (i.e. sin θy = −1/

√G),

the scattering matrix takes the simple form:

s =(

0 1/√G√

G 0

)(H5)

In this case, the principal effect of the weak mirror inthe cold-load line is to introduce a small amount of re-verse gain. The amount of this reverse gain is exactlywhat is needed to have the input impedance diverge(cf. Eq. (6.7d)). It is also what is needed to achieve anoptimal, noise-canceling feedback in the amplifier. Tosee this last point explicitly, we can re-write the ampli-fier’s back-action current noise (I) in terms of its originalnoises I0 and V0 (i.e. what the noise operators would havebeen in the absence of the mirror). Taking the relevantlimit of small reflection (i.e. r ≡ sin θy goes to zero as|G| → ∞), we find that the modification of the currentnoise operator is given by:

I ' I0 +2√Gr

1−√Gr

V0

Za(H6)

As claimed, the presence of a small amount of reflectionr ≡ sin θy in the cold load arm “feeds-back” the originalvoltage noise of the amplifier V0 into the current. Thechoice r = −1/

√G corresponds to a negative feedback,

and optimally makes use of the fact that I0 and V0 arecorrelated to reduce the overall fluctuations in I.

While it is interesting to note that one can reach thequantum limit with no reflections in the input and outputarms, this case is not really of practical interest. Thereverse current gain in this case may be small (i.e. λ′I ∝1/√G), but it is not small enough: one finds that because

of the non-zero λ′I , the amplifier’s input impedance isstrongly reduced in the presence of a load (cf. Eq. (6.6)).

There is a second simple limit we can consider which ismore practical. This is the limit where reflections in theinput-line mirror and output-line mirror are both strong.Imagine we take θz = −θx = π/2 − δ/G1/8. If againwe set sin θy = −1/

√G to reach the quantum limit, the

scattering matrix now takes the form (neglecting termswhich are order 1/

√G):

s =(

+1 0δ2G1/4

2 −1

)(H7)

In this case, we see that at the quantum limit, the reflec-tion coefficients s11 and s22 are exactly what is neededto have the input impedance diverge, while the reversegain coefficient s12 plays no role. For this case of strongreflections in input and output arms, the voltage gain isreduced compared to its zero-reflection value:

λV →√ZbZa

(δ

2

)2

G1/4 (H8)

The power gain however is independent of θx, θz, and isstill given by G/2 when θy is tuned to be at the quantumlimit.

VaVb

R1 R2

+-Gf

FIG. 8 Schematic of a voltage op-amp with negative feedback.

3. Op-amp with negative voltage feedback

We now show that a conventional op-amp with feed-back can be mapped onto the amplifier described inthe previous subsection. We will show that tuning thestrength of the feedback in the op-amp corresponds totuning the strength of the mirrors, and that an optimallytuned feedback circuit lets one reach the quantum limit.This is in complete correspondence to the previous sub-section, where an optimal tuning of the mirrors also letsone reach the quantum limit.

More precisely, we consider a scattering description ofa non-inverting op-amp amplifier having negative voltagefeedback. The circuit for this system is shown in Fig. 8.A fraction B of the output voltage of the amplifier is fedback to the negative input terminal of the op-amp. Inpractice, B is determined by the two resistors R1 and R2

used to form a voltage divider at the op-amp output. Theop-amp with zero feedback is described by the “ideal”amplifier of Sec. VI.B: at zero feedback, it is describedby Eqs. (6.11a)- (6.11d). For simplicity, we consider therelevant case where:

Zb R1, R2 Za (H9)

In this limit, R1 and R2 only play a role through thefeedback fraction B, which is given by:

B =R2

R1 +R2(H10)

Letting Gf denote the voltage gain at zero feedback(B = 0), an analysis of the circuit equations for our op-amp system yields:

λV =Gf

1 +B ·Gf(H11a)

λ′I =B

1 +B ·Gf(H11b)

Zout =Zb

1 +B ·Gf(H11c)

Zin = (1 +B ·Gf )Za (H11d)

GP =G2f/2

B ·Gf + 2Zb/Za(H11e)

Again, Gf represents the gain of the amplifier in the ab-sence of any feedback, Za is the input impedance at zero

34

feedback, and Zb is the output impedance at zero feed-back.

Transforming this into the scattering picture yields ascattering matrix s satisfying:

s11 = −s22 = −BGf (Za − (2 +BGf )Zb)BGfZa + (2 +BGf )2Zb

(H12a)

s21 = − 2√ZaZbGf (1 +BGf )

BGfZa + (2 +BGf )2Zb(H12b)

s12 =B

Gfs21 (H12c)

Note the connection between these equations and the nec-essary form of a quantum limited s-matrix found in theprevious subsection.

Now, given a scattering matrix, one can always find aminimal representation of the noise operators Fa and Fbwhich have the necessary commutation relations. Theseare given in general by:

Fa =√

1− |s11|2 − |s12|2 + |l|2 · uin + l · v†in(H13)

Fb =√|s21|2 + |s22|2 − 1 · v†in (H14)

l =s11s

∗21 + s12s

∗22√

|s21|2 + |s22|2 − 1(H15)

Applying this to the s matrix for our op-amp, and thentaking the auxiliary modes uin and v†in to be in the vac-uum state, we can calculate the minimum allowed SV Vand SII for our non-inverting op-amp amplifier. One canthen calculate the product SV V SII and compare againstthe quantum-limited value (SIV is again real). In thecase of zero feedback (i.e. B = 0), one of course findsthat this product is twice as big as the quantum limitedvalue. However, if one takes the large Gf limit while

keeping B non-zero but finite, one obtains:

SV V SII → (~ω)2

(1− 2B

Gf+O

(1Gf

)2)

(H16)

Thus, for a fixed, non-zero feedback ratio B, it is possibleto reach the quantum limit. Note that if B does not tendto zero as Gf tends to infinity, the voltage gain of thisamplifier will be finite. The power gain however will beproportional to Gf and will be large. If one wants alarge voltage gain, one could set B to go to zero withGf i.e. B ∝ 1√

Gf. In this case, one will still reach

the quantum limit in the large Gf limit, and the voltagegain will also be large (i.e. ∝

√Gf ). Note that in all

these limits, the reflection coefficients s11 and s22 tendto −1 and 1 respectively, while the reverse gain tendsto 0. This is in complete analogy to the amplifier withmirrors considered in the previous subsection, in the casewhere we took the reflections to be strong at the inputand at the output (cf. Eq. (H7)). We thus see yet againhow the use of feedback allows the system to reach thequantum limit.Appendix I: Additional Technical Details

This appendix provides further details of calculationspresented in the main text.

1. Proof of quantum noise constraint

Note first that we may write the symmetrized I and Fnoise correlators defined in Eqs. (4.4a) and (4.4b) as sumsover transitions between detector energy eigenstates:

SFF [ω] = π~∑i,f

〈i|ρ0|i〉 · |〈f |F |i〉|2 [δ(Ef − Ei + ~ω) + δ(Ef − Ei − ~ω)] (I1)

SII [ω] = π~∑i,f

〈i|ρ0|i〉 · |〈f |I|i〉|2 [δ(Ef − Ei + ~ω) + δ(Ef − Ei − ~ω)] (I2)

Here, ρ0 is the stationary density matrix describing thestate of the detector, and |i〉 (|f〉) is a detector energyeigenstate with energy Ei (Ef ). Eq. (I1) expresses thenoise at frequency ω as a sum over transitions. Eachtransition starts with an an initial detector eigenstate|i〉, occupied with a probability 〈i|ρ0|i〉, and ends with afinal detector eigenstate |f〉, where the energy differencebetween the two states is either +~ω or −~ω . Further,each transition is weighted by an appropriate matrix el-ement.

To proceed, we fix the frequency ω > 0, and let theindex ν label each transition |i〉 → |f〉 contributing to thenoise. More specifically, ν indexes each ordered pair ofdetector energy eigenstates states |i〉, |f〉 which satisfyEf − Ei ∈ ±~[ω, ω + dω] and 〈i|ρ0|i〉 6= 0. We can nowconsider the matrix elements of I and F which contributeto SII [ω] and SFF [ω] to be complex vectors ~v and ~w.

35

Letting δ be any real number, let us define:

[~w]ν = 〈f(ν)|F |i(ν)〉 (I3)

[~v]ν =

e−iδ〈f(ν)|I|i(ν)〉 if Ef(ν) − Ei(ν) = +~ω,eiδ〈f(ν)|I|i(ν)〉 if Ef(ν) − Ei(ν) = −~ω.

(I4)

Introducing an inner product 〈·, ·〉ω via:

〈~a,~b〉ω = π∑ν

〈i(ν)|ρ0|i(ν)〉 · (aν)∗ bν , (I5)

we see that the noise correlators SII and SFF may bewritten as:

SII [ω]dω = 〈~v,~v〉ω (I6)SFF [ω]dω = 〈~w, ~w〉ω (I7)

We may now employ the Cauchy-Schwartz inequality:

〈~v,~v〉ω〈~w, ~w〉ω ≥ |〈~v, ~w〉ω|2 (I8)

A straightforward manipulation shows that the realpart of 〈~v, ~w〉ω is determined by the symmetrized cross-correlator SIF [ω] defined in Eq. (4.4c):

Re 〈~v, ~w〉ω = Re[eiδSIF [ω]

]dω (I9)

In contrast, the imaginary part of 〈~v, ~w〉ω is independentof SIF ; instead, it is directly related to the gain χIF andreverse gain χFI of the detector:

Im 〈~v, ~w〉ω =~2

Re[eiδ(χIF [ω]− [χFI [ω]]∗

)]dω (I10)

Substituting Eqs. (I10) and (I9) into Eq. (I8), one im-mediately finds the quantum noise constraint given inEq. (4.16). As in the main text, we let χIF = χIF −χ∗FI .Maximizing the RHS of this inequality with respect tothe phase δ, one finds that the maximum is achieved forδ = δ0 = − arg(χIF ) + δ0 with

tan 2δ0 = − |SIF | sin 2φ(~/2)|χIF |+ |SIF | cos 2φ

(I11)

where φ = arg(SIF χIF ∗). At δ = δ0, Eq. (4.16) becomesthe final noise constraint of Eq. (4.11).

The proof given here also allows one to see what mustbe done in order to achieve the “ideal” noise conditionof Eq. (4.17): one must achieve equality in the Cauchy-Schwartz inequality of Eq. (I8). This requires that thevectors ~v and ~w be proportional to one another; theremust exist a complex factor α (having dimensions [I]/[F ])such that:

~v = α · ~w (I12)

Equivalently, we have that

〈f |I|i〉 =

eiδα〈f |F |i〉 if Ef − Ei = +~ωe−iδα〈f |F |i〉 if Ef − Ei = −~ω.

(I13)

for each pair of initial and final states |i〉, |f〉 contributingto SFF [ω] and SII [ω] (cf. Eq. (I1)). Note that this notthe same as requiring Eq. (I13) to hold for all possiblestates |i〉 and |f〉. This proportionality condition in turnimplies a proportionality between the input and output(unsymmetrized) quantum noise spectral densities:

SII [ω] = |α|2SFF [ω] (I14)

It thus also follows that the imaginary parts of the inputand output susceptibilities are proportional:

Im χII [ω] = |α|2Im χFF [ω], (I15)

as well as the symmetrized input and output noise (i.e.Eq. (4.18)). Finally, one can also use Eq. (I13) to relatethe unsymmetrized I-F quantum noise correlator SIF [ω]to SFF [ω]: (cf. Eq. (4.7)):

SIF [ω] =

e−iδα∗SFF [ω] if ω > 0,eiδα∗SFF [ω] if ω < 0

(I16)

Note that SFF [ω] is necessarily real and positive.Finally, for a detector with quantum-ideal noise prop-

erties, the magnitude of the constant α can be found fromEq. (4.18). The phase of α can also be determined from:

−Im α

|α|=

~|χIF |/2√SII SFF

cos δ0 (I17)

For zero frequency or for a large detector effective tem-perature, this simplifies to:

−Im α

|α|=

~χIF /2√SII SFF

(I18)

Note importantly that to have a non-vanishing gainand power gain, one needs Im α 6= 0. This in turn placesa very powerful constraint on a quantum-ideal detectors:all transitions contributing to the noise must be to finalstates |f〉 which are completely unoccupied. To see this,imagine a transition taking an initial state |i〉 = |a〉 toa final state |f〉 = |b〉 makes a contribution to the noise.For a quantum-ideal detector, Eq. (I13) will be satisfied:

〈b|I|a〉 = e±iδα〈b|F |a〉 (I19)

where the plus sign corresponds to Eb > Ea, the mi-nus to Ea > Eb. If now the final state |b〉 was also oc-cupied (i.e. 〈b|ρ0|b〉 6= 0), then the reverse transition|i = b〉 → |f = a〉) would also contribute to the noise.The proportionality condition of Eq. (I13) would now re-quire:

〈a|I|b〉 = e∓iδα〈a|F |b〉 (I20)

As I and F are both Hermitian operators, and as α musthave an imaginary part in order for there to be gain, wehave a contradiction: Eq. (I19) and (I20) cannot both betrue. It thus follows that the final state of a transition

36

contributing to the noise must be unoccupied in orderfor Eq. (I13) to be satisfied and for the detector to haveideal noise properties. Note that this necessary asym-metry in the occupation of detector energy eigenstatesimmediately tells us that a detector or amplifier cannotreach the quantum limit if it is in equilibrium.

2. Proof that a noiseless detector does not amplify

With the above results in hand, we can now prove as-sertions made in Sec. IV.A.4 that detectors which evadethe quantum noise constraint of Eq. (4.11) and simplysatisfy

SFF SII = |SIF |2 (I21)

are at best transducers, as their power gain is limited tobeing at most one.

The first way to make the RHS of Eq. (4.11) vanish isto have χIF = χ∗FI . We have already seen that when-ever this relation holds, the detector power gain cannotbe any larger than one (c.f. Eq. (5.53)). Now, imaginethat the detector also has a minimal amount of noise,i.e. Eq. (I21) also holds. This latter fact implies that theproportionality condition of Eq. (I13) also must hold. Inthis situation, the detector must have a power gain ofunity, and is thus a transducer. There are two possibil-ities to consider here. First, SFF and SII could bothbe non-zero, but perfectly correlated: |SIF |2 = SFF SII .In this case, the proportionality constant α must be real(c.f. Eq. (I17)). Using this fact along with Eqs. (I16) and(4.8b), one immediately finds that SFF [ω] = SFF [−ω].This implies the back-action damping γ associated withthe detector input vanishes (c.f. Eq. (2.12)). It thus fol-lows immediately from Eq. (5.52) and Eq. (5.53) that thepower gain GP,rev (defined in a way that accounts for thereverse gain) is exactly one. The detector is thus simply atransducer. The other possibility here is that χIF = χ∗FIand one or both of SII , SFF are equal zero. Note that ifthe symmetrized noise vanishes, then so must the asym-metric part of the noise. Thus, it follows that either thedamping induced by the detector input, γ, or that in-duced by the output, γout (c.f. Eq. (5.48)) (or both) mustbe zero. Eqs. (5.52) and (5.53) then again yield a powergain GP,rev = 1. We thus have shown that any detectorwhich has χFI = χ∗IF and satisfies SFF SII = |SIF |2 mustnecessarily be a transducer, with a power gain preciselyequal to one.

A second way to make the RHS of Eq. (4.11) vanish isto have SIF /χIF be purely imaginary and larger in mag-nitude than ~/2. Suppose this is the case, and that thedetector also satisfies the minimal noise requirement ofEq. (I21). Without loss of generality, we take χIF to bereal, implying that SIF is purely imaginary. Eqs. (I10)and (I11) then imply that the phase factor eiδ appear-ing in the proportionality relation of Eq. (I16) is purelyimaginary, while the constant α is purely real. Using this

proportionality relation in Eq. (4.8b) for χIF yields:

χIF =α

~(SFF [ω]− SFF [−ω])

= 2α [−Im χFF [ω]] (I22)

Using this result and the relation between χFF and χIIin Eq. (I15), we can write the power gain in the absenceof reverse gain, GP (c.f. Eq. (5.52)), as

GP = 1/ |1− χFI/χIF |2 (I23)

If the reverse gain vanishes (i.e. χFI = 0), we immedi-ately find that GP = 1: the detector has a power gain ofone, and is thus simply a transducer. If the reverse gainis non-zero, we must take the expression for GP aboveand plug it into Eq. (5.53) for the power gain with re-verse gain, GP,rev. Some algebra again yields that thefull power gain is at most unity. We again have the con-clusion that the detector does not amplify.

3. Simplifications for a quantum-limited detector

In this appendix, we derive the additional constraintson the property of a detector that arise when it satisfiesthe quantum noise constraint of Eq. (4.17). We focus onthe ideal case where the reverse gain χFI vanishes.

To start, we substitute Eq. (I16) into Eqs. (4.8b)-(4.8a); writing SFF [ω] in terms of the detector effectivetemperature Teff (cf. Eq. (2.8)) yields:

~λ[ω]2

= −e−iδ~ [−Im χFF [ω]] (I24)[(Im α) coth

(~ω

2kBTeff

)+ i (Re α)

]SIF [ω] = e−iδ~ [−Im χFF [ω]] (I25)[

(Re α) coth(

~ω2kBTeff

)− i (Im α)

]To proceed, let us write:

e−iδ =λ

|λ|e−iδ (I26)

The condition that |λ| is real yields the condition:

tan δ =Re αImα

tanh(

~ω2kBTeff

)(I27)

We now consider the relevant limit of a large detectorpower gain GP . GP is determined by Eq. (5.56); the onlyway this can become large is if kBTeff/(~ω) → ∞ whileIm α does not tend to zero. We will thus take the largeTeff limit in the above equations while keeping both α andthe phase of λ fixed. Note that this means the parameterδ must evolve; it tends to zero in the large Teff limit. In

37

this limit, we thus find for λ and SIF :

~λ[ω]2

= −2e−iδkBTeffγ[ω] (Im α)

[1 +O

[(~ω

kBTeff

)2]]

(I28)

SIF [ω] = 2e−iδkBTeffγ[ω] (Re α)[1 +O

(~ω

kBTeff

)](I29)

Thus, in the large power-gain limit (i.e. large Teff

limit), the gain λ and the noise cross-correlator SIF havethe same phase: SIF /λ is purely real.

4. Derivation of non-equilibrium Langevin equation

In this appendix, we prove that an oscillator weaklycoupled to an arbitrary out-of-equilibrium detector is de-scribed by the Langevin equation given in Eq. (5.41), anequation which associates an effective temperature anddamping kernel to the detector. The approach taken hereis directly related to the pioneering work of Schwinger(Schwinger, 1961).

We start by defining the oscillator matrix Keldyshgreen function:

G(t) =(GK(t) GR(t)GA(t) 0

)(I30)

where GR(t− t′) = −iθ(t− t′)〈[x(t), x(t′)]〉, GA(t− t′) =iθ(t′− t)〈[x(t), x(t′)]〉, and GK(t− t′) = −i〈x(t), x(t′)〉.At zero coupling to the detector (A = 0), the oscillatoris only coupled to the equilibrium bath, and thus G0 hasthe standard equilibrium form:

G0[ω] =~m

(−2Im g0[ω] coth

(~ω

2kBTbath

)g0[ω]

g0[ω]∗ 0

)(I31)

where:

g0[ω] =1

ω2 − Ω2 + iωγ0/m(I32)

and where γ0 is the intrinsic damping coefficient, andTbath is the bath temperature.

We next treat the effects of the coupling to the detectorin perturbation theory. Letting Σ denote the correspond-ing self-energy, the Dyson equation for G has the form:[

G[ω]]−1

=[G0[ω]

]−1 −(

0 ΣA[ω]ΣR[ω] ΣK [ω]

)(I33)

To lowest order in A, Σ[ω] is given by:

Σ[ω] = A2D[ω] (I34)

≡ A2

~

∫dt eiωt (I35)(

0 iθ(−t)〈[F (t), F (0)]〉−iθ(t)〈[F (t), F (0)]〉 −i〈F (t), F (0)〉

)

Using this lowest-order self energy, Eq. (I33) yields:

GR[ω] =~

m(ω2 − Ω2)−A2Re DR[ω] + iω(γ0 + γ[ω])(I36)

GA[ω] =[GR[ω]

]∗(I37)

GK [ω] = −2iIm GR[ω]×

γ0 coth(

~ω2kBTbath

)+ γ[ω] coth

(~ω

2kBTeff

)γ0 + γ[ω]

(I38)

where γ[ω] is given by Eq. (2.12), and Teff [ω] is defined byEq. (2.8). The main effect of the real part of the retardedF Green function DR[ω] in Eq. (I36) is to renormalize theoscillator frequency Ω and massm; we simply incorporatethese shifts into the definition of Ω and m in what follows.

If Teff [ω] is frequency independent, then Eqs. (I36)- (I38) for G corresponds exactly to an oscillator cou-pled to two equilibrium baths with damping kernels γ0

and γ[ω]. The correspondence to the Langevin equa-tion Eq. (5.41) is then immediate. In the more generalcase where Teff [ω] has a frequency dependence, the cor-relators GR[ω] and GK [ω] are in exact correspondenceto what is found from the Langevin equation Eq. (5.41):GK [ω] corresponds to symmetrized noise calculated fromEq. (5.41), while GR[ω] corresponds to the response co-efficient of the oscillator calculated from Eq. (5.41). Thisagain proves the validity of using the Langevin equationEq. (5.41) to calculate the oscillator noise in the presenceof the detector to lowest order in A.

5. Linear-response formulas for a two-port bosonic amplifier

In this appendix, we use the standard linear-responseKubo formulas of Sec. V.F to derive expressions forthe voltage gain λV , reverse current gain λ′I , inputimpedance Zin and output impedance Zout of a two-portbosonic voltage amplifier (cf. Sec. VI). We recover thesame expressions for these quantities obtained in Sec. VIfrom the scattering approach. We stress throughout thisappendix the important role played by the causal struc-ture of the scattering matrix describing the amplifier.

In applying the general linear response formulas, wemust bear in mind that these expressions should be ap-plied to the uncoupled detector, i.e. nothing attached tothe detector input or output. In our two-port bosonicvoltage amplifier, this means that we should have ashort circuit at the amplifier input (i.e. no input voltage,Va = 0), and we should have open circuit at the out-put (i.e. Ib = 0, no load at the output drawing current).These two conditions define the uncoupled amplifier. Us-ing the definitions of the voltage and current operators(cf. Eqs. (6.2a) and (6.2b)), they take the form:

ain[ω] = −aout[ω] (I39a)

bin[ω] = bout[ω] (I39b)

38

The scattering matrix equation Eq. (6.3) then allows usto solve for ain and aout in terms of the added noiseoperators Fa and Fb.

ain[ω] = −1− s22

DFa[ω]− s12

DFb[ω] (I40a)

bin[ω] = −s21

DFa[ω] +

1 + s11

DFb[ω] (I40b)

where D is given in Eq. (6.8), and we have omitted writ-ing the frequency dependence of the scattering matrix.Further, as we have already remarked, the commutatorsof the added noise operators is completely determined bythe scattering matrix and the constraint that output op-erators have canonical commutation relations. The non-vanishing commutators are thus given by:[Fa[ω], F†a(ω′)

]= 2πδ(ω − ω′)

(1− |s11|2 − |s12|2

)(I41a)[

Fb [ω], F†b (ω′)]

= 2πδ(ω − ω′)(1− |s21|2 − |s22|2

)(I41b)[

Fa[ω], F†b (ω′)]

= −2πδ(ω − ω′) (s11s∗21 + s12s

∗22)

(I41c)

The above equations, used in conjunction withEqs. (6.2a) and (6.2b), provide us with all all the infor-mation needed to calculate commutators between currentand voltage operators. It is these commutators which en-ter into the linear-response Kubo formulas. As we willsee, our calculation will crucially rely on the fact thatthe scattering description obeys causality: disturbancesat the input of our system must take some time beforethey propagate to the output. Causality manifests itselfin the energy dependence of the scattering matrix: as afunction of energy, it is an analytic function in the upperhalf complex plane.

a. Input and output impedances

Eq. (5.84) is the linear response Kubo formula forthe input impedance of a voltage amplifier. Recallthat the input operator Q for a voltage amplifier is re-lated to the input current operator Ia via −dQ/dt =Ia (cf. Eq. (5.83)). The Kubo formula for the inputimpedance may thus be re-written in the more familiarform:

Yin,Kubo[ω] ≡ i

ω

(− i

~

∫ ∞0

dt〈[Ia(t), Ia(0)]〉eiωt)

(I42)

where Yin[ω] = 1/Zin[ω],Using the defining equation for Ia (Eq. (6.1b)) and

Eq. (I39a) (which describes an uncoupled amplifier), weobtain:

Yin,Kubo[ω] = (I43)2Za

∫ ∞0

dteiωt∫ ∞

0

dω′

2πω′

ωΛaa(ω′)

(e−iω

′t − eiω′t)

where we have defined the real function Λaa[ω] for ω > 0via: [

ain[ω], a†in(ω′)]

= 2πδ(ω − ω′)Λaa[ω] (I44)

It will be convenient to also define Λaa[ω] for ω < 0via Λaa[ω] = Λaa[−ω]. Eq. (I43) may then be written as:

Yin,Kubo[ω] =2Za

∫ ∞0

dt

∫ ∞−∞

dω′

2πω′

ωΛaa(ω′)ei(ω−ω

′)t

=Λaa[ω]Za

+i

πωP∫ ∞−∞

dω′ω′Λaa(ω′)/Za

ω − ω′

(I45)

Next, by making use of Eq. (I40a) and Eqs. (I41) forthe commutators of the added noise operators, we canexplicitly evaluate the commutator in Eq. (I44) to cal-culate Λaa[ω]. Comparing the result against the resultEq. (6.7d) of the scattering calculation, we find:

Λaa[ω]Za

= Re Yin,scatt[ω] (I46)

where Yin,scatt[ω] is the input admittance of the ampli-fier obtained from the scattering approach. Returning toEq. (I45), we may now use the fact that Yin,scatt[ω] isan analytic function in the upper half plane to simplifythe second term on the RHS, as this term is simply aKramers-Kronig integral:

1πωP∫ ∞−∞

dω′ω′Λaa(ω′)/Za

ω − ω′

=1πωP∫ ∞−∞

dω′ω′Re Yin,scatt(ω′)

ω − ω′

= Im Yin,scatt[ω] (I47)

It thus follows from Eq. (I45) that input impedance cal-culated from the Kubo formula is equal to what we foundpreviously using the scattering approach.

The calculation for the output impedance proceeds inthe same fashion, starting from the Kubo formula givenin Eq. (5.85). As the steps are completely analogous tothe above calculation, we do not present it here. Oneagain recovers Eq. (6.7c), as found previously within thescattering approach.

b. Voltage gain and reverse current gain

Within linear response theory, the voltage gain of theamplifier (λV ) is determined by the commutator betweenthe “input operator” Q and Vb (cf. Eq. (4.3); recall thatQ is defined by dQ/dt = −Ia. Similarly, the reversecurrent gain (λ′I) is determined by the commutator be-tween Ia and Φ, where Φ is defined via dΦ/dt = −Vb(cf. Eq. (4.6)). Similar to the calculation of the inputimpedance, to properly evaluate the Kubo formulas for

39

the gains, we must make use of the causal structure ofthe scattering matrix describing our amplifier.

Using the defining equations of the current and volt-age operators (cf. Eqs. (6.1a) and (6.1b)), as well asEqs. (I39a) and (I39b) which describe the uncoupled am-plifier, the Kubo formulas for the voltage gain and reversecurrent gain become:

λV,Kubo[ω] = 4√ZbZa

(I48)

×∫ ∞

0

dt eiωtRe[∫ ∞

0

dω′

2πΛba(ω′)e−iω

′t

]λ′I,Kubo[ω] = −4

√ZbZa

(I49)

×∫ ∞

0

dt eiωtRe[∫ ∞

0

dω′

2πΛba(ω′)eiω

′t

]where we define the complex function Λba[ω] for ω > 0via: [

bin[ω], a†in(ω′)]≡ (2πδ(ω − ω′)) Λba[ω] (I50)

We can explicitly evaluate Λba[ω] by using Eqs. (I40a)-(I41) to evaluate the commutator above. Comparing theresult against the scattering approach expressions for thegain and reverse gain (cf. Eqs. (6.7a) and (6.7b)), onefinds:

Λba[ω] = λV,scatt[ω]−[λ′I,scatt[ω]

]∗ (I51)

Note crucially that the two terms above have differentanalytic properties: the first is analytic in the upper halfplane, while the second is analytic in the lower half plane.This follows directly from the fact that the scatteringmatrix is causal.

At this stage, we can proceed much as we did in thecalculation of the input impedance. Defining Λba[ω] forω < 0 via Λba[−ω] = Λ∗ba[ω], we can re-write Eqs. (I48)and (I49) in terms of principle part integrals.

λV,Kubo[ω] =ZbZa

(Λba[ω] +

i

πP∫ ∞−∞

dω′Λba(ω′)ω − ω′

)λ′I,Kubo[ω] = −Zb

Za

(Λba[ω] +

i

πP∫ ∞−∞

dω′Λba(ω′)ω + ω′

)(I52)

Using the analytic properties of the two terms in Eq. (I51)for Λba[ω], we can evaluate the principal part integralsabove as Kramers-Kronig relations. One then finds thatthe Kubo formula expressions for the voltage and cur-rent gain coincide precisely with those obtained from thescattering approach.

While the above is completely general, it is useful togo through a simpler, more specific case where the role ofcausality is more transparent. Imagine that all the energy

dependence in the scattering in our amplifier arises fromthe fact that there are small transmission line “stubs” oflength a attached to both the input and output of theamplifier (these stubs are matched to the input and out-put lines). Because of these stubs, a wavepacket incidenton the amplifier will take a time τ = 2a/v to be eitherreflected or transmitted, where v is the characteristic ve-locity of the transmission line. This situation is describedby a scattering matrix which has the form:

s[ω] = e2iωa/v · s (I53)

where s is frequency-independent and real. To furthersimplify things, let us assume that s11 = s22 = s12 = 0.Eqs. (I51) then simplifies to

Λba[ω] = = s21[ω] = s21eiωτ (I54)

where the propagation time τ = 2a/v We then have:

λV [ω0] = 2√ZbZas21

∫ ∞0

dt eiω0tδ(t− τ) (I55)

λI [ω0] = −2√ZbZas21

∫ ∞0

dt eiω0tδ(t+ τ) (I56)

If we now do the time integrals and then take the limitτ → 0+, we recover the results of the scattering approach(cf. Eqs. (6.7a) and (6.7b)); in particular, λI = 0. Notethat if we had set τ = 0 from the outset of the calculation,we would have found that both λV and λI are non-zero!

6. Details for the two-port bosonic voltage amplifier withfeedback

In this appendix, we provide more details on the calcu-lations for the bosonic-amplifier-plus-mirrors system dis-cussed in Sec. H. Given that the scattering matrix foreach of the three mirrors is given by Eq. (H2), and thatwe know the reduced scattering matrix for the mirror-freesystem (cf. Eq. (6.10)), we can find the reduced scatteringmatrix and noise operators for the system with mirrors.One finds that the reduced scattering matrix s is nowgiven by:

s =1M× (I57)(

sin θz +√G sin θx sin θy − cos θx cos θz sin θy√

G cos θx cos θz sin θx +√G sin θy sin θz

)where the denominator M describes multiple reflectionprocesses:

M = 1 +√G sin θx sin θz sin θy (I58)

Further, the noise operators are given by:

40

(FaFb

)=

1M

(cos θy cos θz

√G− 1 cos θz sin θx sin θy

−√G cos θx cos θy sin θz

√G− 1 cos θx

)(uinv†in

)(I59)

The next step is to convert the above into the op-amprepresentation, and find the gains and impedances of theamplifier, along with the voltage and current noises. Thevoltage gain is given by:

λV =√ZBZA

2√G

1−√G sin θy

· 1 + sin θxcos θx

1− sin θzcos θz

(I60)

while the reverse gain is related to the voltage gain bythe simple relation:

λ′I = − sin θy√GλV (I61)

The input impedance is determined by the amount ofreflection in the input line and in the line going to thecold load:

Zin = Za1−√G sin θy

1 +√G sin θy

· 1 + sin θz1− sin θz

(I62)

Similarly, the output impedance only depends on theamount of reflection in the output line and the in thecold-load line:

Zout = Zb1 +√G sin θy

1−√G sin θy

· 1 + sin θx1− sin θx

(I63)

Note that as sin θy tends to −1/√G , both the input

admittance and output impedance tend to zero.Given that we now know the op-amp parameters of our

amplifier, we can use Eq. (6.5) to calculate the amplifier’spower gain GP . Amazingly, we find that the power gainis completely independent of the mirrors in the input andoutput lines:

GP =G

1 +G sin2 θy(I64)

Note that at the special value sin θy = −1√G (which

allows one to reach the quantum limit), the power gainis reduced by a factor of two compared to the reflectionfree case (i.e. θy = 0).

Turning to the noise spectral densities, we assume theoptimal situation where both the auxiliary modes uin andv†in are in the vacuum state. We then find that both ˆI

and ˆV are independent of the amount of reflection in the

output line (e.g. θx):

SII =2~ωZa

[1− sin θz1 + sin θz

]×G sin2 θy + cos(2θy)(√

G sin θy − 1)2

(I65a)

SV V = ~ωZa[

1 + sin θz1− sin θz

]×(

3 + cos(2θy)4

− 12G

)(I65b)

SV I =√G(1− 1/G) sin θy + cos2 θy

1−√G sin θy

(I65c)

As could be expected, introducing reflections in the inputline (i.e. θz 6= 0) has the opposite effect on SII versusSV V : if one is enhanced, the other is suppressed.

It thus follows that the product of noise spectraldensities appearing in the quantum noise constraint ofEq. (5.88) is given by (taking the large-G limit):

SII SV V

(~ω)2 =(2− sin2 θy

)· 1 +G sin2 θy(

1−√G sin θy

)2 (I66)

Note that somewhat amazingly, this product (and hencethe amplifier noise temperature) is completely indepen-dent of the mirrors in the input and output arms (i.e. θzand θx). This is a result of both SV V and SII havingno dependence on the output mirror (θx), and their hav-ing opposite dependencies on the input mirror (θz). Alsonote that Eq. (I66) does indeed reduce to the result ofthe last subsection: if θy = 0 (i.e. no reflections in theline going to the cold load), the product SII SV V is equalto precisely twice the quantum limit value of (~ω)2. Forsin(θy) = −1/

√G, the RHS above reduces to one, imply-

ing that we reach the quantum limit for this tuning ofthe mirror in the cold-load arm.

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Introduction to quantum noise, measurement, and …optical–quantum optics communities. This review gives a pedagogical introduction to the physics of quantum noise and its connections