Quantum Feedback Control

126 Los Alamos Science Number 27 2002

Salman Habib, Kurt Jacobs, and Hideo Mabuchi

How can we control quantum systems without disturbing them?

Quantum Feedback Control

The nanomechanical electrometer shown here was built in Michael Roukes’ group at Caltech.

It has a demonstrated sensitivity below a single electron charge per unit bandwidth and should

ultimately reach sensitivities of the order of parts per million. Its operation is based on the

movement of a torsional resonator that carries a detection electrode placed in an external

magnetic field. The gate electrode is seen on one side of the resonator.

Number 27 2002 Los Alamos Science 127

Ever since Niels Bohr’s firstattempt at understanding thehydrogen atom, the fundamen-

tal cautionary lesson of quantummechanics has been driven hometime after time: Processes in themicroworld transpire according tolaws and principles that directly con-tradict those governing themacroworld of human experience.This radical shift in understanding isnow almost a century old and hasbeen definitively confirmed bynumerous experiments. It might seemlikely that the strange behaviors ofquantum systems would be familiarby now and practical devices har-nessing those behaviors would becommonplace. For the most part,however, we have remained merespectators of the microphysicalrealm, where quantum mechanicsholds sway, being forced to observenaturally occurring phenomena ratherthan being able to control and manip-ulate them. In the coming decade,however, this situation may bereversed.

Recent advances in quantum andatomic optics and condensed matterphysics are providing tools to engi-neer practical quantum devices andperhaps even modestly complex networks of these devices. Quantuminformation processing, precisionmeasurement, and development ofultrasensitive sensors are driving the present development of quantumtechnologies. If quantum technologiesare ever to achieve the complexity ofclassically engineered systems suchas jet aircraft and the Internet,a quantum analog of classical feed-back control must be developed, sincefeedback control is at the heart of thestability and predictability underlyingcomplex engineered systems.

Along these lines, recent theoreti-cal results on error correction in quantum computation and on thedynamics of open quantum systemsmay be viewed as first steps in devel-oping a theoretical formalism forpractical quantum feedback control(see the articles “Introduction toQuantum Error Correction” on page 188 and “Realizing a NoiselessSubsystem” on page 260). Indeed,feedback control represents a promis-ing new approach to mitigating quan-tum noise and decoherence in bothquantum computation and precisionmeasurement. If we are to apply theconcepts and methods of feedbackcontrol theory to quantum dynamicalsystems, we must not only extendclassical control concepts to newregimes but also analyze quantummeasurement in a way that is usefulfor control systems.

The Evolution of ControlTheory

Controlling natural phenomenathrough macroscopic engineering goesback thousands of years. Consider fora moment the ingenious ways inwhich early human civilizations con-trolled irrigation. In Mesopotamia(2000 BC), where rainfall was poorand the Tigris and Euphrates Riverswere the main sources of water, engi-neers constructed an elaborate canalsystem with many diversion dams (seethe drawing to the left). In that sys-tem, the Euphrates served as a sourceand the Tigris as a drain. In a similarvein, the ancient Egyptians used waterfrom the Nile and thereby allowedtheir civilization to flourish. On asmaller scale, machines using feed-back control were developed in the

Greco-Roman period, and methods forthe automatic operation of windmillsdate back to the Middle Ages.

Perhaps the best-known exampleof feedback control in the industrialera is the Watt governor, which stabi-lizes steam engine speeds under fluc-tuating loads. James Clerk Maxwellprovided the first dynamical analysisof this system based on differentialequations. His work, which was published in 1868, founded the fieldof mathematics now known as control theory. In the early part of the 20th century, the idea of self-regulating machinery continued to bepushed in various directions, notablyin electronic amplification. Controlconcepts were further developed forindustrial, navigational, and militaryapplications.

After World War II, control sys-tems progressed to a new level ofcomplexity. Up until that time, feed-back control systems had been largelysingle loop, taking the feedback sig-nal from one point and connecting thecorrection signal to a different point.Multiloop control systems and more-sophisticated feedback techniquesemerged from progress in optimiza-tion theory and dynamical systemstheory, as well as from the advent ofdigital computers.

After 1960, there emerged what isoften referred to as “modern” (asopposed to “classical”) control theory(Brogan 1990, Zhou et al. 1996),which emphasizes optimization ofcost and performance. For the samecontrol goals, it is clear that not allcontrol strategies will be equallyeffective in terms of cost and perform-ance. Determining the best strategydefines the problem of optimal con-trol; however, optimal algorithms areoften unstable to variations in system

parameters and the external environ-ment. Theorists then turned to ensuringperformance bounds in the presence ofuncertainty. This work resulted in thetheory of “robust” control (Zhou et al.1996). Noise in the inputs, extrinsicdisturbances in the system under con-trol, measurement errors, and modelinginadequacies—all can render controlsystems less effective or, in somecases, even lead to catastrophic fail-ures. The role of robust control is tomaintain adequate stability and otherperformance margins given the uncer-tainties mentioned earlier.

Classical Control Systems

Formally speaking, a control sys-tem consists of a dynamical systeminteracting with a controller, a devicethat influences the state of the dynam-ical system toward some desired end.The objective may be to regulate theflow of an industrial process, money,energy, information, and so on. In a“closed-loop,” or feedback, controlsystem, the controller uses outputsfrom the dynamical system to monitorand influence its interaction with thatdynamical system. For a lineardynamical system, for example, sucha situation could be described by thefollowing equation:

(1)

where x is a vector describing thestate of the system, dW is a vector of Gaussian noise sources, and u isthe vector of inputs determined by the controller. The matrix A gives thesystem’s deterministic motion, and Band C describe, respectively, how thenoise and input vectors are coupledinto the system. A separate equation,namely,

(2)

describes the continuous measurement

of system outputs by the controller. In each small time interval dt, the con-troller obtains the measurement resultdy. That result is directly related to thetrue state of the system by some lineartransformation H, but it also includes a Gaussian noise process V, whichserves to represent imperfections in the measurement.

Examples of control systems canbe found in many applications. Forinstance, servomechanisms are controlsystems that use small control inputsto produce changes in large mechani-cal systems. In effect, the larger sys-tems are “slaved” to the output of theservomechanisms (for example, liquidlevels in reservoirs are controlled byfloat valves). Feedback circuits areused in ingenious ways in electronicamplification to manipulate input andoutput impedances and to improve thelinearity, distortion, and frequencybandwidth of the output signal relativeto the input signal.

In an “open-loop” control system,the controller does not monitor the output of the dynamical system. A dynamical model for the system isassumed, and control is applied withthe idea that the desired outcome willactually be achieved. Open-loop

strategies are useful in situations inwhich the system dynamics are knownprecisely and vary only slowly.Processes with long measurementdead times are sometimes better suit-ed to open-loop control methods thanto feedback methods. Open-loop con-trol strategies are applied in situationsas diverse as the maximization ofreturns from financial investments,optimal determination of aircraftflight paths, and controlled dissocia-tion of molecules.

Figure 1 shows how to implementclosed-loop control for a dynamicalsystem. One must be able to measuresome of the dynamical variables ofthe system under control (the outputs)and use them to influence some othervariables (the inputs). In other words,given the output variables, the con-troller implements a particular controlstrategy to influence the state of thedynamical system by appropriatelyvarying the inputs. Robust controllerstake into account variations in systemparameters and fluctuations from the external environment to producecontrol strategies with guaranteed stability bounds.

Control systems can involve manydifferent interacting physical systems

dy = Hxdt + RdV ,

dx = Axdt + BdW + Cu ,



Outputs

Feedback controller

Dynamical systemInputs

Measurementerrors

External perturbations,parametric drifts, and uncertainties

Figure 1. Classical Feedback ControlThe classical dynamical system to be controlled has a set of input variables, whichare processed by the system dynamics into a set of output variables. Some fractionof the set of input and output variables (possibly different for each case) is availablefor hookup to the controller. The controller has to perform in the presence of exter-nal fluctuationsthat is, uncertainties and drifts in the parameters describing thedynamical systemand measurement errors.

with a large number of sequential,parallel, and nested control loops thatare both open and closed. For exam-ple, closed- and open-loop strategiescan be combined as in the fast closed-loop systems used to stabilize theslower, inherently unstable open-loopdynamics of modern fighter aircraft.

Developing Control inQuantum Systems

The general picture of control sys-tems outlined in the previous sectionappears to be extendable to quantumsystems. Certainly, open-loop controlproblems are conceptually straightfor-ward in the quantum context. Onebegins with the time evolution opera-tor of the quantum system—theSchrödinger equation for the wavefunction, the Liouville equation for thedensity matrix, or more complicateddynamical evolution equations for the density matrix characterizing asystem coupled to an environment. A theory for time-dependent variationsin the evolution operator is then devel-oped in such a way that the wave function or the density operator atsome time is close to some target

value. This target value does not haveto be unique, nor in fact is the timeevolution to that value unique. Theapproach just outlined applies equallywell to classical probabilistic evolu-tions: Although quantum and classicalsystems are dynamically distinct, theprinciples for open-loop control are infact very similar.

Controlling chemical reactions bylaser-produced electromagnetic fieldsthat are time dependent is a well-known open-loop quantum controlproblem. In the frequency-resolvedapproach to control, the quantuminterference between different evolu-tionary paths is being manipulated; inthe time-resolved approach, thedynamics of wave packets producedby ultrafast laser pulses leads to con-trol. For some specific control of thechemical reactions, one can optimizethe temporal and spectral structure ofthose laser pulses (Shi et al. 1988).

The fundamental differencesbetween classical and quantum systemsbecome real issues, however, in thefield of closed-loop control. Quantumsystems can have two distinct types of feedback control: directly and indi-rectly coupled quantum feedback (seeFigure 2). As illustrated in Figure 2(a),

in a system with directly coupled quan-tum feedback, a quantum variable ofthe system is coupled to the quantumcontroller, and a quantum input pathfrom the controller goes directly backto the quantum system. When thequantum feedback is indirect, as shownin Figure 2(b), the quantum dynamicalsystem under control is an observedsystem. It therefore generates a classi-cal output, also known as the measure-ment record, which the controller mayanalyze to provide a best estimate ofthe original quantum state of the sys-tem. The controller then feeds back a classical signal to vary parameters in the quantum evolution operator in accord with the chosen control strategy. Hybrid couplings using bothdirect and indirect quantum feedbackchannels are easy to envisage: Thechannel from the system output to the controller input may be directlycoupled whereas the channel from thecontroller output to the system inputmay be coupled indirectly through aclassical path.

In both classical and quantum contexts, the main goal of closed-loopcontrol is to enhance system perform-ance in the presence of noise fromboth the environment and the



u

Quantumdynamicalsystem

Quantuminput

Quantumoutput

Quantumcontroller

Figure 2. Directly and Indirectly Coupled Quantum Feedback(a) Both the dynamical system and the controller are quantum systems coupled through a unitary interaction. A quantum variable is coupled to the quantum controller, and a quantum input path from the controller goes directly back to the quantumsystem. (b) A quantum dynamical system can be viewed as having two sets of inputs, one relating to the variation in the classi-cal parameters describing the Hamiltonian and the other representing fully quantum inputs. Similarly, the output channel can be divided into a quantum and a classical channel. The classical channel is, in fact, a piece of the quantum channel that has become classical after observation. The controller analyzes the classical record to form an estimate of the dynamical system’s state and uses this information to implement the appropriate control.

External perturbations orparametric drifts

pp

Classicalcontroller

Quantum input

Detector

QuantumdynamicalsystemClassical

Input

Quantum output

Classicaloutput

(a) (b)

uncertainty in the system parameters.To limit the effects of noise, the controller must perform an irre-versible operation. Noise generates alarge set of undesirable evolutions,and the controller’s task is to map thislarge set to a much smaller one ofmore desirable evolutions. Mappingfrom the larger to the smaller set is bydefinition irreversible. In other words,noise is a source of entropy for the system. To control the system, thecontroller must extract the entropyfrom the system under control and putit somewhere else. The controllermust therefore have enough degreesof freedom to respond conditionallyupon the noise realization. In indirectquantum feedback control, the meas-urement process, coupled with theconditional response of the controller,is the source of entropy reduction. In direct quantum feedback control,the evolution of the system is fullyunitary, or quantum mechanical. Thequantum controller provides a largeHilbert space of quantum mechanicalstates. That is precisely where theentropy generated by the noise maybe put (or where the history of theeffect of the noise on the system maybe stored). The quantum controllerthen reacts conditionally to this quantum record, keeping the entropyof the quantum dynamical system low, while the entropy of the storagelocation grows continually.

Inherent Noise Generation inQuantum Feedback Control

Unlike classical systems, quantumsystems may be easily disturbed wheninformation about them is extracted.Measurement disturbs a quantum system through the following intrinsicproperty of quantum mechanics:Obtaining accurate knowledge aboutone observable of a quantum systemnecessarily limits the informationabout an observable conjugate to the

first. For example, particle positionand momentum are conjugate observ-ables, and the uncertainties inherent in the knowledge of both are codifiedby the famous Heisenberg uncertaintyrelation. If the chosen feedback-control strategy involves measure-ment, one must take into account the effects of the measurement on the evolution of the quantum system.A generally applicable model forincluding those effects is that of acontinuous quantum measurement.This model was developed for quan-tum optics (Carmichael 1993), a fieldin which such measurements havebeen realized experimentally, and itwas also derived in the mathematicalphysics literature with the help ofmore abstract reasoning (Barchielli1993). In this volume, the model of acontinuous quantum measurement ispresented in the article “TheEmergence of Classical Dynamics in aQuantum World” on page 110.

Quantum measurements may introduce unwanted noise in threemore-or-less distinct ways. First, onemay measure an observable conjugateto the real variable of interest andthereby introduce more uncertainty inthe latter variable. More generally,one may attempt to obtain informationinconsistent with the state under con-trol. For example, to preserve a statethat is the superposition of two posi-tion states, position measurementsmust be avoided because they willdestroy the superposition. Thus, inquantum mechanics, the type of measurement chosen must be consis-tent with the control objectives. Thiscondition is unnecessary in classicalfeedback control. Second, if trying to control the values of observables(Doherty et al. 2000), one must con-sider that the time evolutions of differ-ent observables necessarily affect eachother over time. Observables whosevalues are uncertain at one time willcause other observables (perhaps moreaccurately known) to become uncer-

tain at a later time. For example,a very accurate measurement of theparticle position at one time intro-duces uncertainty into the value of theparticle momentum. Because the valueof momentum determines the positionof the particle at a later time,the momentum uncertainty makes thefuture position of the particle moreuncertain, hence introducing noise intothe quantity that is being measured.This mechanism for introducing noiseis usually referred to as the backaction of a quantum measurement.

The third kind of noise involves the randomness of the measurementresults. Because the state of theobserved system after a measurementdepends upon the outcome of themeasurement, the more the result fluc-tuates, the more noise there is in theevolution of the system. For classicalmeasurements, fluctuations in meas-urement results cannot be any morethan the entropy of the system beforemeasurement; that is, the measurementdoes not introduce any additionalnoise into the system. In quantummechanics, however, even if the sys-tem state is known precisely, one canstill make measurements that changethe state in a random way, therebyactually injecting noise into the sys-tem. This observation is particularlyrelevant when the overall state of thesystem, rather than a specific observ-able, is being controlled. The situationis further complicated by the fact that,for certain classes of measurements,there is actually a tradeoff between thenoise injected by the measurement andthe information gained by the observer(Doherty et al. 2001). As a result,designing measurement strategies isfar from being a trivial activity.

Strategies for QuantumFeedback Control

The differences between classicaland quantum measurements profoundly



affect the design of feedback controlalgorithms. A classical controllerextracts as much information from the system as possible. In quantumcontrol, irreducible disturbances areinherent to any measurement, andtherefore the measurement strategybecomes a significant part of the feed-back algorithm. For example, just asthe inputs to the system change withtime, the measurements too may needto be varied with time so that the bestcontrol should be achieved.

Adaptive measurement, or alteringthe measurement as it proceeds, wasfirst introduced by Howard Wiseman(1995), not for control but for accuracy. The result was a more accurate measurement of some aspectof the quantum state. Nevertheless,this approach has a unique bearing onquantum feedback control algorithms.Knowing that quantum measurementscan disturb the state being measured,one may want to start a continuousmeasurement process by measuring ina way that is not necessarily optimalbut is sufficiently weak to cause minimal disturbance to the aspect of interest. As the measurement pro-ceeds, one uses the continuouslyobtained information about the stateto make the measurement increasinglyclose to optimal.

For example, consider measuringthe oscillation amplitude of a harmonic oscillator when the phase of the oscillation is unknown but the oscillator is known to be in anamplitude-squeezed state; that is, theuncertainty in amplitude or energy ismuch smaller than the uncertainty inphase, the conjugate variable (seeFigure 3). In this case, an accuratemeasurement of amplitude is givenby a measurement of position at themoment when the particle is at itsmaximum spatial extent, or maxi-mum distance from x = 0. On theother hand, at the moment when theparticle has the most momentum (atposition x = 0), the ideal quantity to

measure is momentum. Thus, for acontinuous measurement of the oscil-lation amplitude, a linear combina-tion of position and momentumshould be measured and the relativeweighting of those two variablesshould be allowed to oscillate intime. However, without knowing the mean phase of oscillation, onecannot know which variable shouldhave the most weighting in the meas-urement at what time. Using an adaptive measurement procedure,one can start by assuming the oscilla-tor to have a particular phase andthen adjust the relative weights ofposition and momentum to moredesirable values as information aboutthe phase is obtained.

Applications of QuantumControl

Atomic optics is one field in whichit should be possible to test quantum

feedback control in the near future. It has already been demonstrated thata single atom can be trapped inside an ultralow-loss optical cavity (mirrorreflectivity is R = 0.9999984 in experiments at Caltech) in the strong-coupling quantum regime (Mabuchi etal. 1999). Figure 4 illustrates theexperimental setup used at Caltech.The strong coupling occurs betweenthe atom and the radiation field in the cavity and is proportional to theinduced atomic dipole moment andthe single-photon cavity field.Continuous measurements and real-time feedback could be used to coolsuch an atom to the “ground” state ofthe quantized mechanical potentialproduced by several photons in thecavity. The average number of pho-tons circulating inside such a cavitycan be kept very low (from 1 to10 photons) if one uses a weak driv-ing laser that barely balances the slowrate at which individual photons leakout. If the cavity mode volume is



Figure 3. “Squeezed” States for a Harmonic OscillatorSqueezing may be illustrated by considering phase-space plots of a Gaussian wavefunction. For a standard Gaussian state, the uncertainties in the x- and p-directionare equal, and the uncertainty ellipse takes the shape of a circle, provided appropri-ate position and momentum scalings are made. When states are “squeezed,” the areaof the uncertainty ellipse remains constant, but the ellipse is rotated and squeezedas shown. Squeezing momentum, for example, means reducing the uncertainty inmomentum. The constant energy surface is the dashed circle, and the position on the circle can be specified by the angle. Squeezing phase and energy again refers to changes in shape of the uncertainty ellipse for the wave function.

Momentumsqueezing

StandardGaussian

Phasesqueezing

Positionsqueezing

Number orenergy squeezing

x

p

θ

E

sufficiently small, just a few photonscan give rise to dipole (alternating-current Stark shift) forces that arestrong enough to bind an atom near alocal maximum of the optical fielddistribution. At the same time, theatomic motion can be monitored inreal time by phase-sensitive measure-ments of the light leaking out of thecavity. To a degree determined by thefidelity of these phase measurements,the information gained can be usedcontinually to adjust the strength ofthe driving laser (and hence the depthof the optical potential) in a mannerthat tends to remove kinetic energyfrom the motion of the atomic centerof mass.

In order to perform such a task inreal time, however, it is essential todevelop approximate techniques forcontinuously estimating the state ofthe atomic motion. Approximationsare needed because integrating a sto-chastic conditioned-evolution equationto obtain a continuous estimate of thedensity matrix is far too complex atask to be performed in real time.While this experiment remains to becarried out, we have developed anapproximate estimation algorithm1

and used it in combination with anexperimentally realizable feedbackalgorithm (see Figure 5).

Feedback cooling ideas can also beapplied to condensed-matter systems.Some of our recent calculations pre-dict that feedback control can be usedto cool a nanoresonator below the lim-its set by refrigeration. This methodwould reduce thermal fluctuations toapproximately the quantum energylevel spacing of the resonator. Thesefindings are important becausenanoscale devices are interesting froma more fundamental perspective thanmerely sensing and actuation applica-tions. Provided they can be cooled to



0 2 4 6 8 1060

70

80

90

100

110

120

⟨E⟩/E

eef

s

t/τ

Figure 5. Simulating a Feedback Algorithm to Cool Atomic Motion inan Optical CavityIn this simulated experiment, the light forms an effective sinusoidal potential for theatom, and the controller switches this potential between a high and a low value(separated by some ∆V) to cool the atomic motion. In this simulation, the feedbackis turned on at t = 2, and the expected value of the atomic motion energy is plottedhere as a function of time for four different values of ∆V. Although these results arestill preliminary, they indicate that the effectiveness of the feedback algorithm ishighly dependent on ∆V.

Figure 4. Quantum Feedback in a Cavity Quantum ElectrodynamicsApplicationThe dynamics between the atom and the photon field in the cavity can be modifiedby continuous measurement of the light transmitted through the cavity (which bearsinformation about the evolving system state) and by continuous adjustment of theamplitude/phase of the driving laser in a manner that depends on the measurementresults. Control objectives of fundamental interest include active cooling of themotion of an individual atom, feedback-stabilized quantum state synthesis, andactive focusing of atomic beams for applications such as direct-write lithography.

Controller

System

One atom strongly coupled to an optical cavity

IntensityPhase

DetectorLaserbeam

Modulator

1 This algorithm is described in a yet unpub-lished paper by Salman Habib, Kurt Jacobs,Hideo Mabuchi, and Daniel Steck.

sufficiently low temperatures, low-loss nanomechanical resonators wouldbe excellent candidates for the firstobservation of quantum dynamics inmechanical mesoscopic systems. Yet,as mentioned above, in order achievethis goal, we must reduce thermalfluctuations to approximately thequantum energy level spacing of theresonator, a task which requires tem-peratures in the range of millikelvins.

To cool the position coordinate ofthe nanoresonator, one needs a suit-able scheme for continuous positionmeasurement. One practical methodof performing a continuous measure-ment of a nanoresonator’s position isto use a single-electron transistor(SET)—see Figure 6. To make themeasurement, one locates the res-onator next to the central island of the SET. When the resonator ischarged and the SET is biased so thatcurrent flows through it, changes inthe resonator’s position modify theenergy of the central island, whichproduces changes in the SET current.The current therefore provides a con-tinuous measurement of the positionof the resonator, a requirement forimplementing a linear feedback cool-ing algorithm. A feedback force canbe applied to the resonator by varyingthe voltage on a “feedback electrode,”which is capacitively coupled to theresonator (see Figure 6). The appliedvoltage is adjusted so as to damp theamplitude of oscillation.

Experiments on nanomechanicaloscillators observed with SETs cur-rently start at temperatures near100 millikelvins. These oscillatorshave fundamental frequencies f0 on the order of 1 to 100 megahertz. As aconcrete example, consider a practicaloscillator with f0 = 10 megahertz,a length of 2 micrometers, and theother two dimensions on the order of100 nanometers. The effective mass of such an oscillator is roughly10–19 kilograms. An achievable qualityfactor, Q, is about 104. In order to

observe discrete quantum passagefrom one oscillator energy level toanother, the thermal energy should beon the order of the level spacing, thatis, kBT ~ hf0, which corresponds to aneffective temperature T = .24 mil-likelvin. Habib, Jacobs, Asa Hopkins,and Keith Schwab have shown thatfeedback cooling applied to this system at an initial temperature T = 100 millikelvins can yield a finaltemperature of T = 0.35 millikelvin. At this temperature, the aggregateoccupation number lies between zero,the ground state, and one, the firstexcited state of the nanomechanicalresonator. In other words, the systemis cold enough to allow observation of quantum “jumps.” Although ourcalculations are based on certain ideal-ized assumptions, those assumptionsare close enough to reality that experimentalists can hope to achievesimilar results.

Another, seemingly paradoxical,application of quantum feedback control techniques might be in sup-

pressing quantum dynamical effectssuch as tunneling. A classical memorydevice can be viewed as a two-statesystem with the two states separatedby a finite energy barrier. At low tem-peratures, there is a finite probabilityof coherent or incoherent tunnelingfrom one minimum to the other.Tunneling generates random memoryerrors, but continuous measurement,coupled with feedback, can suppressit. One such scheme is described anddemonstrated in Andrew Doherty etal. (2000). The Hamiltonian for thedouble well is taken to be

(3)

where x and p are dimensionless posi-tion and momentum. Choosing A = 2and B = 1/9 puts the minima of thewells at ±3 and gives a barrier heightof approximately 13.5. The controlleris allowed to continuously observe theposition of the particle and to apply a

H p Ax Bx= − +1

22 2 4 ,



Source Island Drain

Resonator

SET

Tunnel junctions

Feedback electrode

Vsd

Isd

Feedbackcircuitry

Vgate

. + + + + + + + + + +

Figure 6. Cooling a Nanomechanical ResonatorThis schematic diagram illustrates a concept for cooling a nanomechanical res-onator to millikelvin temperatures, at which we can possibly observe quantumdynamics. An SET measures the position of the resonator, and a feedback mecha-nism damps (cools) the resonator’s motion. The resonator, which is charged by the voltage source Vgate acts as the SET gate electrode. The resonator is alsocapacitively coupled to the SET island (red) and feedback electrode. As it movesback and forth relative to the SET island, the current Isd flowing through the SETchanges. Information about the changing current is used by the feedback circuitryto charge the feedback electrode. A force is generated that damps the resonator’soscillations.

linear force in addition to the “double-well” potential already present. Thecontinuous observation is described bythe equation

(4)

where dq is the measurement result in the time interval dt and k is a constant characterizing the accuracy, orstrength, of the measurement. The system is also driven by a thermalheat bath in the high-temperature limit.The effect of that bath is, in fact, thesame as that of a continuous quantummeasurement of position that ignores themeasurement result. When the bath isdescribed in this way, it is the strength ofthe fictitious measurement that gives therate of thermal heating, and we willdenote this constant by β.

Integrating a stochastic master equa-tion gives the observer’s state of knowl-

edge as a result of the continuous meas-urement. However, since this is a differ-ential equation for the density matrix ofthe single particle, it is numericallyexpensive to integrate. For practical pur-poses, one requires a simplified meansfor calculating a state estimate. Toachieve this goal, we note that, as aresult of the continuous observation,even though the dynamics are nonlinear,the density matrix remains approximate-ly Gaussian. When a Gaussian approxi-mation is used, the stochastic masterequation reduces to a set of five equa-tions (for all the moments of x and p upto quadratic order), and so it provides uswith a practical method for obtaining acontinuous state estimate. In practice,this Gaussian estimator can be shown towork quite well; that is, mean valuesfrom the approximate estimator agreevery well with mean values derived fromexact numerical solutions of the stochas-

tic master equation—see Figure 7(a). In addition to a state estimation

procedure, we also require a feedbackalgorithm. If the system were linear,one could apply the optimal tech-niques of modern control theory tofind a feedback algorithm. Becauseattempting an optimal control solutionfor the full nonlinear problem is com-putationally intractable, the idea is tolinearize the system dynamics aroundthe present estimate of the state with the further assumption that theprobability density, conditioned on the measurement record, remainsGaussian. As long as position meas-urements are sufficiently strong, thislast condition is satisfied. The impor-tance of this condition is twofold:Having a Gaussian approximationdoes not only mean that a small num-ber of moments (five) are needed todescribe the distribution but also thatthe quantum propagator is very closeto the classical propagator at eachtime step (for exactly Gaussian states,the two are identical), and hence techniques borrowed from classicalcontrol have an excellent chance ofworking. The control can fail if themeasurement is too weak to maintaina localized Gaussian distribution or if it is too strong. In the latter case,the state is Gaussian, but the measure-ment noise is too large.

The Gaussian state estimate is nowused to set the value of the feedbackterm in the Hamiltonian (the sign andthe magnitude of the coefficient of thelinear feedback term in the potential).By choosing appropriate strengths forthe measurement and the feedbackstrength, one can show that the feed-back scheme is effective in controllingwhether the particle is in the desiredminimum—see Figure 7(b). For thisplot, the measurement strength is k = 0.3, and the thermal heating rate isd⟨E⟩/dt = β = 0.1.

This scheme has limitations arisingfrom unwanted heating due to themeasurement. Although some of the

dq x dtdV

k= + ,

8



Figure 7. Particle in a Double Well Controlled by an Estimation-Feedback Scheme(a) Shown here, as a function of time, are the target position (blue line), the ‘true”mean position (red line) obtained with the stochastic master equation (in which themeasurement strength k equals 0.3 and the thermal heating rate β equals 0.1), andthe position obtained with the Gaussian estimator (gray line). (b) The controlstrength (size of applied force) is shown as a function of time.

heating derives directly from having to keep the state close to Gaussian, amore general limitation also con-tributes to heating: The measurementmust be sufficiently strong to provideenough information for control to beeffective. Developing new estimationand feedback schemes that can reducethe measurement-induced heating rateis an important area for future research.

Outlook for the Future

Most likely, ideas in quantum feed-back control will first be tested incondensed matter physics and inquantum and atomic optics.Experiments in atomic optics havealready furnished the cleanest testsand demonstrations of quantummechanics in the last several decades.These include violations of the Bellinequalities, quantum teleportation,quantum state tomography, quantumcryptography, and single-atom inter-ference. The ability to compare exper-imental results with precise theoreticalbenchmarks is a hallmark of thesetests. As these experiments becomeincreasingly sophisticated and com-plex, one can envisage a passage from“toy” demonstrations to real applica-tions such as feedback control. Themore strongly coupled systems ofcondensed matter physics are lessamenable to accurate theoretical pre-diction. Nevertheless, experiments arebecoming comparable in quality toearly atomic optics experiments,and the time is ripe for active interac-tion between these two fields:Theoretical development in quantumoptics, such as continuous measure-ment and quantum control, can betaken over to condensed matter con-texts, most notably in nanotechnology.As the size of the smallest structuresthat can be fabricated by lithographictechniques decreases, the need forquantum mechanics becomesinevitable. Since lithography is the

only way we know to create verycomplex systems at reasonable cost,it follows that a fundamental and predictive understanding of quantumdynamics applicable to these systems(whether coherent or incoherent) willbe required. It is also clear that, forthese systems to be designable and tofunction reliably in an engineeringsense, further development of quan-tum control theory will be necessary.

From a “more algorithmic” per-spective, the Holy Grail is the devel-opment of optimal and robust controlalgorithms that are generally applica-ble. So far, apart from the trivial casein which the system dynamics are linear and the measurement strategy isconsidered fixed (Doherty and Jacobs1999), no such optimal algorithmshave been found for quantum feed-back control. In classical control theory, optimal and robust controlalgorithms exist for linear systems,but only very few for nonlinear sys-tems despite the best effort of controltheorists in the past few decades.Nonlinear classical optimal control isa very difficult problem indeed, andprobably intractable in most cases.Systematic numerical search algo-rithms for optimal strategies exist, butthese also become intractable for sys-tems of reasonable size. Because thedynamics of noisy and measuredquantum systems is inherently nonlin-ear, the quantum control problem mayalso be intractable (Doherty et al.2000). However, in quantum dynam-ics, nonlinearity is of a restricted kind, and the possibility of obtaininggeneral analytic results providingoptimal and robust algorithms for the

feedback control of quantum systemsremains an open problem. �

Further Reading

Barchielli, A. 1993. Stochastic Differential-Equations and a Posteriori States in Quantum-Mechanics. Int. J. Theor. Phys. 32 (12): 2221.

Brogan, W. L. 1991. Modern Control Theory.Englewood Cliffs, NJ: Prentice-Hall.

Carmichael, H. J. 1993. An Open SystemsApproach to Quantum Optics, LectureNotes in Physics Monograph 18. New York:Springer-Verlag.

Doherty, A. C., and K. Jacobs. 1999. FeedbackControl of Quantum Systems UsingContinuous State Estimation. Phys. Rev. A 60 (4): 2700.

Doherty, A. C., S. Habib, K. Jacobs,H. Mabuchi, and S. M. Tan. 2000. Quantum Feedback Control and ClassicalControl Theory. Phys. Rev. A 62: 012105.

Doherty, A. C., K. Jacobs, and G. Jungman.2001. Information, Disturbance, andHamiltonian Quantum Feedback Control.Phys. Rev. A 63: 062306.

Hood, C. J., T. W. Lynn, A. C. Doherty, A. S.Parkins, and H. J. Kimble. 2000. The Atom-Cavity Microscope: Single Atoms Bound inOrbit by Single Photons. Science 287: 1447.

Mabuchi, H., J. Ye, and H. J. Kimble. 1999.Full Observation of Single-Atom Dynamicsin Cavity QED. Appl. Phys. B 68 (6): 1095.

Maxwell, J. C. 1868. On Governors. Proc. R.Soc. London 16: 270.

Shi, S., A. Woody, and H. Rabitz. 1988.Optimal Control of Selective VibrationalExcitation in Harmonic Linear ChainMolecules. J. Chem. Phys. 88: 6870.

Wiseman, H. M. 1995. Adaptive PhaseMeasurements of Optical Modes: GoingBeyond the Marginal Q Distribution. Phys.Rev. Lett. 75: 4587.

Wiseman, H. M., and G. J. Milburn. 1993.Quantum Theory of Optical Feedback viaHomodyne Detection. Phys. Rev. Lett. 70: 548.

———. 1994. Squeezing via Feedback. Phys.Rev. A 49: 1350.

Zhou, K., J. Doyle, and K. Glover. 1996.Robust and Optimal Control. EnglewoodCliffs, NJ:Prentice-Hall.



Hideo Mabuchi received an A.B. in physics from Princeton University and a Ph.D. from Caltech.Upon graduating, he became Assistant Professor of Physics at Caltech but spent hisfirst year on leave at Princeton University as a visiting fellow in chemistry. Sincereturning to Caltech, he has been named an A. P. Sloan Research Fellow, Office ofNaval Research Young Investigator, and a MacArthur Fellow. His current researchfocuses on quantum measurement, quantum feedback control, control-theoreticapproaches to the theory of multiscale phenomena, and optical measurement techniques for molecular biophysics.

For biographies of Salman Habib and Kurt Jacobs, see page 125.

Documents

Quantum Feedback Control