Keeling Notes

8/23/2019 Keeling Notes

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Light-Matter Interactions

and Quantum Optics.

Jonathan Keeling

http://www.st-andrews.ac.uk/~jmjk/teaching/qo/
http://www.st-andrews.ac.uk/~jmjk/teaching/qo/http://www.st-andrews.ac.uk/~jmjk/teaching/qo/http://www.st-andrews.ac.uk/~jmjk/teaching/qo/http://www.st-andrews.ac.uk/~jmjk/teaching/qo/


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Contents

Contents iii

Introduction vii

1 Quantisation of electromagnetism 1

1.1 Revision: Lagrangian for electromagnetism . . . . . . . . . 2

1.2 Eliminating redundant variables . . . . . . . . . . . . . . . . 3

1.3 Canonical quantisation; photon modes . . . . . . . . . . . . 5

1.4 Dipole approximation and coupling strength . . . . . . . . . 61.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Jaynes Cummings model 9

2.1 Derivation of Jaynes-Cummings model . . . . . . . . . . . . 92.2 Semiclassical limit . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Quantum behaviour . . . . . . . . . . . . . . . . . . . . . . 11

3 Decay of a two-level system 15

3.1 Many mode quantum model irreversible decay . . . . . . 15

3.2 Density matrix equation for relaxation of two-level system . 16

3.3 Effective decay rate . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.A WignerWeisskopf approach . . . . . . . . . . . . . . . . . . 20

4 Power broadening 21

4.1 Equations of motion with coherent driving . . . . . . . . . . 21

4.2 Dephasing in addition to relaxation . . . . . . . . . . . . . . 22

4.3 Power broadening of absorption . . . . . . . . . . . . . . . . 25

4.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Purcell Effect 29

5.1 The Purcell effect in a 1D model cavity . . . . . . . . . . . 29

5.2 Weak to strong coupling via density matrices . . . . . . . . 335.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Cavity Quantum Electrodynamics 37

6.1 Optical transitions of atoms . . . . . . . . . . . . . . . . . . 386.2 Microwave transitions of atoms . . . . . . . . . . . . . . . . 42

6.3 Superconducting qubits in microwave resonators . . . . . . 44

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iv CONTENTS

6.4 Quantum-dot excitons in semiconductor microstructures . . 46

6.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.A Transfer matrix approach to calculating reflectivity of a Bragg

mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Resonance Fluorescence 51

7.1 Spectrum of emission into a reservoir . . . . . . . . . . . . . 51

7.2 Quantum regression theorem . . . . . . . . . . . . . . . . 52

7.3 Resonance fluorescence spectrum . . . . . . . . . . . . . . . 54

7.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 56

8 Quantum stochastic methods 59

8.1 Quantum jump formalism . . . . . . . . . . . . . . . . . . . 59

8.2 Heisenberg-Langevin equations . . . . . . . . . . . . . . . . 61

8.3 Fluctuation dissipation theorem . . . . . . . . . . . . . . . . 638.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 65

9 Quantum theory of the laser 67

9.1 Density matrix equation for a model laser system . . . . . . 67

9.2 Maxwell Bloch equations . . . . . . . . . . . . . . . . . . . . 69

9.3 Reduction to photon density matrix equation . . . . . . . . 70

9.4 Properties of the laser rate equations . . . . . . . . . . . . . 73

9.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 74

10 Laser fluctuations: linewidth and correlations 75

10.1 Laser linewidth . . . . . . . . . . . . . . . . . . . . . . . . . 7510.2 Spontaneous emission and the parameter . . . . . . . . . 78

10.3 Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

10.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 80

10.A Derivation of off diagonal field density matrix equations . . 81

11 Stronger coupling, micromasers, single atom lasers 83

11.1 Micromaser gain . . . . . . . . . . . . . . . . . . . . . . . . 83

11.2 Micromaser and noise . . . . . . . . . . . . . . . . . . . . . 85

11.3 Single atom lasers . . . . . . . . . . . . . . . . . . . . . . . 87

11.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 89

12 Superradiance 91

12.1 Simple density matrix equation for collective emission . . . 91

12.2 Beyond the simple model . . . . . . . . . . . . . . . . . . . 96

12.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 100

13 The Dicke model 101

13.1 Phase transitions, spontaneous superradiance . . . . . . . . 101

13.2 No-go theorem: no vacuum instability . . . . . . . . . . . . 103

13.3 Radiation in a box; restoring the phase transition . . . . . . 104

13.4 Dynamic superradiance . . . . . . . . . . . . . . . . . . . . 104

13.5 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 106


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CONTENTS v

14 Three levels, and coherent control 10914.1 Semiclassical introduction . . . . . . . . . . . . . . . . . . . 10914.2 Coherent evolution alone; why does EIT occur . . . . . . . 113

14.3 Dark state polaritons . . . . . . . . . . . . . . . . . . . . . . 11314.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 116

A Problems for lectures 14 117

B Problems for lectures 59 121

C Problems for lectures 1014 123

Bibliography 127


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Introduction

The title quantum optics covers a large range of possible courses, and sothis introduction intends to explain what this course does and does not aimto provide. Regarding the negatives, there are several things this coursedeliberately avoids:

It is not a course on quantum information theory. Some basic notionsof coherent states and entanglement will be assumed, but will not bethe focus.

It is not a course on relativistic gauge field theories; the majorityof solid state physics does not require covariant descriptions, and soit is generally not worth paying the price in complexity of using amanifestly covariant formulation.

As far as possible, it is not a course on semiclassical electromagnetism.While at times radiation will be treated classically, this will generallybe for comparison to a full quantum treatment, or where such anapproximation is valid (for at least part of the radiation).

Regarding the positive aims of this course, they are: to discuss how tomodel the quantum behaviour of coupled light and matter; to introducesome simple models that can be used to describe such systems; to dis-cuss methods for open quantum systems that arise naturally in the contextof coupled light and matter; and to discuss some of the more interestingphenomena which may arise for matter coupled to light. Semiclassical be-haviour will be discussed in some sections, both because an understanding

of semiclassical behaviour (i.e. classical radiation coupled to quantum me-chanical matter) is useful to motivate what phenomena might be expected;and also as comparison to the semiclassical case is important to see whatnew physics arises from quantised radiation.

The kind of quantum optical systems discussed will generally consistof one or many few-level atoms coupled to one quantised radiation fields.Realisations of such systems need not involve excitations of real atoms, butcan instead be artificial atoms, i.e. well defined quantum systems with dis-crete level spectra which couple to the electromagnetic field. Such conceptstherefore apply to a wide variety of systems, and a variety of character-istic energies of electromagnetic radiation. Systems currently studied ex-

perimentally include: real atomic transitions coupled to optical cavities[1];

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viii INTRODUCTION

Josephson junctions in microwave cavities (waveguides terminated by re-flecting boundaries)[2, 3]; Rydberg atoms (atoms with very high princi-ple quantum numbers, hence small differences of energy levels) in GHz

cavities[4]; and solid state excitations, i.e. excitons or trions localised inquantum dots, coupled to a variety of optical frequency cavities, includ-ing simple dielectric contrast cavities, photonic band gap materials, andwhispering gallery modes in disks[5].

These different systems provide different opportunities for control andmeasurement; in some cases one can probe the atomic state, in some casesthe radiation state. To describe experimental behaviour, one is in gen-eral interested in calculating a response function, relating the expectedoutcome to the applied input. However, to understand the predicted be-haviour, it is often clearer to consider the evolution of quantum mechanicalstate; thus, both response functions and wavefunctions will be discussed.

As such, the lectures will switch between Heisenberg and Schrodinger pic-tures frequently according to which is most appropriate. When consideringopen quantum systems, a variety of different approaches; density matrixequations, Heisenberg-Langevin equations and their semiclassical approxi-mations, again corresponding to both Schrodinger and Heisenberg pictures.

The main part of this course will start with the simplest case of a singletwo-level atom, and discuss this in the context of one or many quantisedradiation modes. The techniques developed in this will then be appliedto the problem of many two-level atoms, leading to collective effects. Thetechniques of open quantum systems will also be applied to describing las-ing, focussing on the more quantum examples of micromasers and single

atom lasers. The end of the course will consider atoms beyond the two-levelapproximation, illustrating what new physics may arise. Separate to thismain discussion, the first two lectures stand alone in discusing where thesimple models of coupled light and matter used in the rest of the coursecome from, in terms of the quantised theory of electromagnetism.


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Lecture 1

Quantisation ofelectromagnetism in the

Coulomb gauge

Our aim is to write a theory of quantised radiation interacting with quan-tised matter fields. Such a theory, e.g. the Jaynes-Cummings model (seenext lecture) has an intuitive form:

HJ.C. =

kka

kak +i,k

izi + gi,k

+i ak + H.c.

. (1.1)

The operator ak creates a photon in the mode with wavevector k, and sothis Hamiltonian describes a process where a two-level system can changeits state with the associated emission or absorption of a photon. The termka

kak then gives the total energy associated with occupation of the mode

with energy k. While the rest of the course is dedicated to studying suchmodels of coupled light-matter system, this lecture will show the relationbetween such models and the classical electromagnetism of Maxwells equa-tions.

To reach this destination, we will follow the route of canonical quan-tisation; our first aim is therefore to write a Lagrangian in terms of onlyrelevant variables. Relevant variables are those where both the variable andits time derivative appear in the Lagrangian; if the time derivative does notappear, then we cannot define the canonically conjugate momentum, andso cannot enforce canonical commutation relations. The simplest way ofwriting the Lagrangian for electromagnetism contains irrelevant variables i.e. the electric scalar potential and gauge of the vector potential A;that irrelevant variables exist is due to the gauge invariance of the theory.Since we are not worried about preserving manifest Lorentz covariance,we are free to solve this problem in the simplest way eliminating the

unnecessary variables.

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2 LECTURE 1. QUANTISATION OF ELECTROMAGNETISM

1.1 Revision: Lagrangian for electromagnetism

To describe matter interacting with radiation, we wish to write a La-

grangian whose equations of motion will reproduce Maxwells and Lorentzsequations:

B = 0J + 00E E = /0 (1.2) B = 0 E = B (1.3)mr = q[E(r) + r B(r)]. (1.4)

Equations (1.3) determine the structure of the fields, not their dynamics,and are immediately satisfied by defining B = A and E = A. Letus suggest the form of Lagrangian L that leads to Eq. (1.2) and Eq. (1.4):

L =

1

2 mr2

+

02

dV

E2

c2

B2

+

q [r A(r) (r)] .(1.5)

Here, the fields E and B should be regarded as functionals of and A.Note also that in order to be able to extract the Lorentz force acting onindividual charges, the currents and charge densities have been written as:

(r) =

q(r r), J(r) =

qr(r r). (1.6)

The identification of the Lorentz equation is simple:

d

dt

Lr

=d

dt[mr + qA(r)] = mr + q(r

)A(r) + q

tA(r)

=Lr

= q [r A(r) (r)]= q [(r ) A(r) + r ( A(r)) (r)] ,

thus one recovers the Lorentz equation,

mr = q

r [ A(r)] (r)

tA(r)

. (1.7)

Similarly, the equation that results from can be easily extracted; since

L/ = 0, this becomes

L

= 0 E

q(r r) = 0. (1.8)

Finally, the equations for A are more complicated, requiring the identity

A( A)2 = 2 ( A), (1.9)

which then gives:

d

dt

LA

= ddt

0E =LA

= 0c2 ( A) +

qr(r r),

(1.10)


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1.2. ELIMINATING REDUNDANT VARIABLES 3

which recovers the required Maxwell equation

0

d

dt

E =

1

0 B +

qr(r

r). (1.11)

Thus, the Lagrangian in Eq. (1.5), along with the definitions of E and Bin terms of A and produce the required equations.

1.2 Eliminating redundant variables

As mentioned in the introduction, we must remove any variable whosetime derivative does not appear in the Lagrangian, as one cannot writethe required canonical commutation relations for such a variable. It isclear from Eq. (1.5) that the electric scalar potential is such a variable.

Since does not appear, it is also possible to eliminate directly fromthe equation L/; using Eq. (1.8) and the definition of E, this equationgives:

0 A 02 (r) = 0. (1.12)Rewriting this in Fourier space, one has:

(k) =1

k2

(k)

0+ ik A(k)

. (1.13)

We can now try to insert this definition into the Lagrangian, to eliminate. To do this, we wish to write E2 and B2 in terms of and A; it is therefore

useful to start by writing

Ej(k) = ikj(k) + Aj(k) = i kj0k2

(k) +

jk kjkk

k2

Ak(k). (1.14)

This means that the electric field depends on the charge density, and onthe transverse part of the vector potential, which will be written:

Aj k =

jk kjkk

k2

Ak(k). (1.15)

The transverse1 part of the vector potential is by definition orthogonal tothe wavevector k, and so the electric field is the sum of two orthogonalvectors, and so:

|E(k)|2 = |A(k)|2 + 120k

2|(k)|2. (1.18)

1The combination:

jk(k) =

jk

kjkk

k2

, (1.16)

is the reciprocal space representation of the transverse delta function; with appropriateregularisation[6, Complement AI], it can be written in real space as:

jk(r) =

2

3(r)jk +

1

4r3

3rjrk

r2 jk

. (1.17)


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Similarly, the squared magnetic field in reciprocal space is given by:

|B(k)|2 = (k A(k)) (k A(k))

= k2

jk kjkkk2

Aj(k)Ak(k) = k2|A(k)|2. (1.19)

Thus, the field part of the Lagrangian becomes:

02

dV

E2 c2B2 = 1

0

d3k1

k2|(k)|2

+ 0

d3k|A(k)|2 c2k2|A(k)|2

. (1.20)

The notation

has been introduced to mean integration over reciprocalhalf-space; since A(r) is real, A(k) = A(k), thus the two half spacesare equivalent. This rewriting is important to avoid introducing redundant

fields in the Lagrangian; the field is either specified by one real variable atall points in real space, or by two real variables at all points in reciprocalhalf-space. Similar substitution into the coupling between fields and matter,written in momentum space gives:

Lemmatter = 2

d3k [J(k) A(k) (k)(k)]

= 2

d3k

J(k) A(k) (k)

(k)

0k2 ik

k2 A(k)

.

(1.21)

This can be simplified by adding a total time derivative,

L L+ dF/dt;

such transformations do not affect the equations of motion, since they addonly boundary terms to the action. If:

F = 2

d3k(k)

ik A(k)k2

, (1.22)

then

Lemmatter + dFdt

= 2

d3k

J(k) (k) ik

k2

A(k) |(k)|

2

0k2

.

(1.23)Then, using conservation of current, (k) + ik J(k) = 0, one finally has:

Lemmatter + dFdt

= 2d3k J(k) A(k) |(k)|20k2

. (1.24)

Note that this set of manipulations, adding dF/dt has eliminated the longi-tudinal part of the vector potential from the Lagrangian. The form chosenfor F is such that this procedure is equivalent to a gauge transformation;the chosen gauge is the Coulomb gauge. Putting everything together, onehas:

Lcoulomb =

1

2mr

2 10

d3k1

k2|(k)|2

+ 0d3k |A(k)|2 c2k2|A(k)|2 + 2 J(k) A(k) . (1.25)


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1.3. CANONICAL QUANTISATION; PHOTON MODES 5

Thus, the final form has divided the interaction into a part mediatedby transverse fields, described by A, and a static (and non-retarded)Coulomb interaction. Importantly, there are no irrelevant variables left

in Eq. (1.25) The Coulomb term can also be rewritten:

Vcoul =1

0

d3k1

k2|(k)|2 =

,

qq80|r r| . (1.26)

Note that since the Coulomb interaction is non-retarded, both the Coulomband transverse parts of interaction must be included to have retarded in-teractions between separated charges.

1.3 Canonical quantisation; photon modes

We have in Eq. (1.25) a Lagrangian which can now be treated via canon-ical quantisation. Since only the transverse part of the field A appears inEq. (1.25), we can drop the superscript label in A from here on. To pro-ceed, we should first identify the canonical momenta and the Hamiltonian,and then impose canonical commutation relations. Thus,

p =Lcoulomb

r= mr + qA(r) (1.27)

(k) =Lcoulomb

A(k)= 0A(k). (1.28)

Then, constructing the Hamiltonian by H = i PiRi L, one finds:H =

1

2m[p qA(r)]2 + Vcoul

+ 0

d3k

|(k)|220

+ c2k2|A(k)|2

. (1.29)

In order to quantise, it remains only to introduce commutation relationsfor the canonically conjugate operators. Noting that A(r) has only twoindependent components, because it is transverse, it is easiest to writeits commutation relations in reciprocal space, introducing directions ek,northogonal to k with n = 1, 2 ; then:

[ri,, pj,] = iij (1.30)Aek,n(k), ek,n (k

)

= i(k k)nn . (1.31)

This concludes the quantisation of matter with electromagnetic inter-actions in the Coulomb gauge. It is however instructive to rewrite thetransverse part of the fields in terms of their normal modes. The secondline of Eq. (1.29) has a clear similarity to the harmonic oscillator, witha separate oscillator for each polarisation and momentum. Rewriting innormal modes thus means introducing the ladder operators:

ak,n =0

2ck

ckAek,n(k) +

i

0 ek,n(k)

, (1.32)


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or, inverted one has:

A(r) =k

n=1,2

20kV en

ak,ne

ikr

+ a

k,neikr

. (1.33)

Inserting this into Eq. (1.29) gives the final form:

H =

1

2m[p qA(r)]2 +

k

n=1,2

kakak + Vcoul. (1.34)

1.4 Dipole approximation and coupling strength

Equation (1.34) applies to any set of point charges interacting with the

electromagnetic field. In many common cases, one is interested in dipoles,with pairs of opposite charges closely spaced, and much larger distancesbetween the dipoles. In this case it is possible to make the dipole approx-imation, by assuming that the electromagnetic field does not vary acrossthe scale of one dipole.

We consider a system with just two charges: charge +q mass m1 atR + r/2 and q mass m2 at R r/2. If r where is a characteristicwavelength of light, then one may assume that A(R+r/2) A(Rr/2) A(R), and so the remaining coupling between radiation and matter is ofthe form

HAp = q p1m1

p2

m2 A(R).

Then, in the case that one can write H0, a Hamiltonian for the dipolewithout its coupling to radiation one can use p/m = x = i[H0, x]/, thusgiving:

HAp = i q

[H0, r] A(R) (1.35)

If one considers a given basis of eigenstates |n for the free Hamil-tonian H0 obeying H0|n = n|n one may then rewrite the above in theform:

HAp = n,m

gk,nnm2

|nm|ak,neikR + ak,ne

ikR

(1.36)where we have introduced the coupling strengths gk,nnm. These describe thetransitions between bare levels that are driven by coupling to the photonfield, and are given by the expression

gk,nnm2

=q

20kVek,n n|[H0, r]|m = nmek,n dnm

20kV, (1.37)

where dnm = qn|r|m is the dipole matrix element and nm = n m.We have assumed H0 commutes with the parity operator P : Px = x,and so the coupling to radiation appears only in the off-diagonal terms in

the two-level basis.


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1.4. DIPOLE APPROXIMATION AND COUPLING STRENGTH 7

Oscillator strength

The expression in Eq. (1.37) shows that the coupling strength between

light and matter depends both on properties of the atom and the radiationmodes. The relevant property of the radiation mode is the field strength,describing the electric field (or vector potential) associated with a singlephoton. The relevant propert of the atom is a combination of its energyand dipole matrix element.

There is another characterisation of an atoms coupling to light, whichis given by the atomic oscillator strength. We will here restrict attentionto transitions involving coupling to electrons, hence q = e, m = m0.

fnm =2m02

|dnm|2e2

nm. (1.38)

This dimensionless combination is important for several reasons, which arediscussed partly here, and further elaborated in the problems. The first isthat the oscillator strength describes the relative contribution of a givenatomic transition to the atomic susceptibility, as compared to an harmonicoscillator. The second is that there is a sum rule which states

n fnm = 1

(at least for transitions involving only a single electron changing state),implying that the dimensionless combination fnm states how large the cou-pling is as compared to the maximum value it could possibly take.

The susceptibility of a Lorentz oscillator is calculated by finding P() =()0E() = ex(), where x is the displacement of the oscillator, and theoscillator obeys:

mx =

m20

x

2mx

eE(t) (1.39)

It is easy to then decompose this into frequency components and thus find

LO() =e2

m00

1

2 + 2i 20. (1.40)

To see the significance of the oscillator strength fnm, this susceptibilityshould be compared to that corresponding to Eq. (1.36). This means cal-culating:

P(t) =

dP()eit = |

mn

dmn|mn| (1.41)

in linear response theory, with H0 = m

m|m

m

|and the perturbation

H1 = i

nm

A(t)nmdnm|nm|, A(t) = i

dE()

e(it). (1.42)

Here, A(t) is a classical field corresponding to the probe field E(t), and isan infinitessimal shift to ensure the field vanishes at t . Exercise A1shows that the linear response found here evaluates to:

() =e2

m00

nm

fnm1

( + i)2 (nm/)2 (1.43)

indicating that each transition contributes like a harmonic oscillator, but

with a reduced strength given by fnm.


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1.5 Further reading

The discussion of quantisation in the Coulomb gauge in this chapter draws

heavily on the book by Cohen-Tannoudji et al. [6].


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Lecture 2

The Jaynes-Cummingsmodel

The aim of this lecture is to discuss the interaction between a single two-level system and various different configurations of photon fields. We startby introducing the Jaynes-Cummings model, and then consider variousresults of this model. These will include the interaction with a semiclassicallight field, and the behaviour of a single quantised mode of light in aninitially coherent state.

2.1 Derivation of Jaynes-Cummings model

At the end of the previous lecture, we began to discuss approximations thatresult in simplified descriptions of coupled lightmatter systems. In thissection, we continue this programme of simplifications in order to reducethe model of coupled matter and radiation modes to the Jaynes-Cummingsmodel, of a two-level atom coupled to a single photon model. To effect thisreduction, three further simplifications are required in addition to the dipoleapproximation discussed previously: neglect of the A2 terms, projection totwo-level systems, and the rotating wave approximation.

Neglect of the A2 terms in expanding [p qA(r)]2 can be justifiedin the limit of low density of atoms; considering only a single radiationmode, the contribution of the A2 term can be rewritten using Eq. (1.33)

as:HA2 =

N

V

q2

4m0

a + a

2. (2.1)

Thus, this term adds a self energy to the photon field, which scales like thedensity of atoms. The relative importance of this term can be estimatedby comparing it to the other term in the Hamiltonian which is quadraticin the photon operators, ka

kak. Their ratio is given by:

N

V

q2

4m0(k)2 N

Va3B

Rydk

2(2.2)

thus, assuming particles are more dilute than their Bohr radius, neglect

of HA2 is valid for frequencies of the order of the Rydberg for the given

9


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10 LECTURE 2. JAYNES CUMMINGS MODEL

bound system of charges.To further simplify, one can now reduce the number of states of the

atom that are considered; currently, there will be a spectrum of eigen-

states of H0, and transitions are induced between these states accordingto f|r|i. Then, restricting to only the two lowest atomic levels and toa single radiation mode, one has a model of two-level systems (describingmatter) coupled to bosonic modes (describing radiation).

H =1

2

g(a + a)

g(a + a)

+ 0aa. (2.3)

The matrix notation represents the two levels of the atom. We have intro-duced the energy splitting between the lowest two atomic levels, and wehave used the notation g for the effective coupling strength gab as definedin the previous lecture:

g2

= ek,n dba20kV

, (2.4)

The final approximation to be discussed here, the rotating wave approx-imation, is appropriate when g 0. Considering g as a perturbation,one can identify two terms:

co =g

2

0 a

a 0

, counter =

g

2

0 a

a 0

, (2.5)

where co, the co-rotating terms, conserve energy particle number; andcounter do not. More formally, the effects of counter give second orderperturbation terms like g2/(

0+), while

cogive the much larger g2/(

0). Including this approximation, one then derives the Jaynes-Cummingsmodel:

H =1

2z +

g

2(a + a+) + 0a

a. (2.6)

where represent Pauli matrices.

2.2 Semiclassical limit

On applying a classical light field to a two-level system, we can adaptthe Jaynes-Cummings model by neglecting the dynamics of the photon

field, and replacing the photon creation and annihilation operators withthe amplitude of the time-dependent classical field. This gives the effectiveHamiltonian:

H =1

2

gei0t

gei0t

. (2.7)

We have made the rotating wave approximation, and hence only includedthe positive/negative frequency components of the classical field in the rais-ing/lowering operators for the two-level system. To solve this problem, andfind the time-dependent state of the two-level atom, it is convenient tomake a transformation to a rotating frame. In general, if

U = eif(t) 0

0 eif(t)

, (2.8)


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2.3. QUANTUM BEHAVIOUR 11

then

H H =

H11 + f(t) H12e2if(t)

H21e2if(t) H22

f(t)

, (2.9)

thus, in this case we use f = t/2, giving:

H =1

2

g

g ( )

. (2.10)

Since this transformation affects only the phases of the wavefunction, we canthen find the absorption probability by considering the time averaged prob-ability of being in the upper state. The eigenvalues of Eq. (2.8) are /2where =

( )2 + g22. Writing the eigenstates as (cos , sin )T,

and ( sin , cos )T corresponding respectively to the roots (assumingtan() > 0). Substituting these forms, one finds the condition: tan(2) =

g/( ). This form is sufficient to find the probability of excitation. Ifthe two-level system begins in its ground state, then the time-dependentstate can be written as:

= sin

cos sin

eit/2 + cos

sin cos

eit/2 (2.11)

Pex = ||2 = |2sin cos sin(t/2)|2 =

1 cos(t)2

(g)2

(g)2 + ( )2 .(2.12)

Thus, the probability of excitation oscillates at the Rabi frequency =( )

2

+ (g)2

, and the amplitude of oscillation depends on detuning.On resonance ( = ), one can engineer a state with || = || = 1/2,or a state with || = 1 by applying a pulse with a duration (g)t = /2or respectively.

2.3 Quantum behaviour

Let us now repeat this analysis in the case of a quantised mode of radiation,starting with the Jaynes-Cummings model in the rotating wave approxima-tion.

H =

1

2 ga

ga + 0aa. (2.13)Rabi oscillations in the Jaynes-Cummings model

In the rotating wave approximation the total number of excitations Nex =z + aa is a conserved quantity, thus if we start in a number state of thelight field:

|n, = (a)nn!

|0, , (2.14)

then the general state at later times can always be written:

|(t) = (t) |n 1, + (t) |n, . (2.15)


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Inserting this ansatz in the equation of motion gives

it (t)(t)

= n 1

2 1 + 1

2 ( ) g

n

gn ( ) (t)

(t) .

(2.16)After removing the common phase variation exp[i(n1/2)t[, the generalsolution can be written in terms of the eigenvectors and values of the matrixin Eq. (2.16), and reusing the results for the semiclassical case, we have thestate

(t)(t)

= ei(n1/2)t

sin eit/2

cos sin

+ cos eit/2

sin cos

,

(2.17)and the excitation probability is thus:

Pex =

1 cos(t)2

g2

ng2n + ( )2 (2.18)

where we now have tan(2) = g

n/( ) and = ( )2 + g2n.Collapse and revival of Rabi oscillations

Restricting to the resonant case = , let us discuss what happens if theinitial state does not have a well defined excitation number. Since the oscil-lation frequency depends on n, the various components of the initial statewith different numbers of excitations will oscillate at different frequencies.We can therefore expect interference, washing out the signal. This is indeed

seen, but in addition one sees revivals; the signal reappears at much latertimes, when the phase difference between the contributions of succesivenumber state components is an integer multiple of 2.

Let us consider explicitly how the probability of being in the excitedstate evolves for an initially coherent state

| = e||2/2 exp(a) |0, = e||2/2n

nn!

|n, . (2.19)

If resonant, the general result in Eq. (2.17) simplifies, as = /4. Followingthe previous analysis, the state at any subsequent time is given by:

| = e(||2+it)/2

n

(eit)nn!

cos

g

nt

2

|n, + sin

g

nt

2

|n 1,

. (2.20)

Thus, the probability of being in the excited atomic state, traced over allpossible photon states, is given by

Pex =n

|n, ||2 = e||2n

||2nn!

sin2

g

nt

2

=1

2 1

2

e||2 n

||2n

n!

cos g

nt . (2.21)


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2.3. QUANTUM BEHAVIOUR 13

We consider the case 1. In this case, one may see that the coefficients||2n/n! are sharply peaked around n ||2 by writing:

||2n

n! =1

2n exp[f(n)], f(n) = n ln(n) n n ln(||2), (2.22)By differentiating f twice, one finds f(n) = ln

n/||2, f = 1/n, and

hence one may expand f(n) near its minimum at n = ||2 to give:

f(n = ||2 + x) ||2 + x2

2||2 (2.23)

Using this in Eq. (2.21) gives the approximate result:

Pex 12

122||

2m

exp

m

2

2||2 + igt

||2 + m

, (2.24)

where it is assumed that ||2 is large enough that the limits of the sum maybe taken to .

Three different timescales affect the behaviour of this sum. Concentrat-ing on the peak, near m = 0, there is a fast oscillation frequency g, de-scribing Rabi oscillations. Then, considering whether the terms in the sumadd in phase or out of phase, there are two time scales: collapse of oscilla-tions occurs when there is a phase difference of 2 across the range of terms|m| < m contributing to this sum, i.e. when gTcollapse(

||2 + m||) =. Taking m = || from the Gaussian factor, this condition becomesgTcollapse 2. On the other hand, if the phase difference between each suc-cessive term in the sum is 2, then they will rephase, and a revival occurs,

with the condition gTrevival(||2 + 1 ||) = 2, giving gTrevival = 4||.

Thus, the characteristic timescales are given by:

Toscillation 1g|| , Tdecay

1

g, Trevival ||

g. (2.25)

and the associated behaviour is illustrated in Fig. 2.1.Let us now consider how this behaviour may be approximated ana-

lytically. The simplest approach might be to replace the sum over numberstates by integration, however this would inevitably lose the revivals, whichare associated with the discreteness of the sum allowing rephasing. Thus,it is necessary to take account of the discreteness of the sum, and hence the

difference between a sum and an integral. For this, we make use of Poissonsummation which is based on the result:m

(x m) =r

e2irm m

f(m) =r

dxe2irxf(x). (2.26)

This allows us to replace the summation by integration, at the cost ofhaving a sum of final results, however in the current context this is veryhelpful, as the final sum turns out to sum over different revivals. Applyingthis formula, we may write Pex 12 [1

r {(r, ||, t)}] where

(r, ||, t) = dx

2||2

exp2irx x2

2

|

|2

+ igt|| +x

2

|

|

x2

8

|

|3

(2.27)


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0 20 40 60 80 100 120 140 160 180 200

Time g

0

0.2

0.4

0.6

0.8

1

Excitation

probability

0 2 4 60

0.5

1

Figure 2.1: Collapse and revival of Rabi oscillations, plotted for =

200.

Completing the square yields:

(r, ||, t) = eigt||2||2

dx exp

1

2||2

1 +igt

4||

x2 + i

2r +

gt

2||

x

=

eigt||2||2 exp||

2

2

(2r + gt/2

|

|)2

1 + igt/4||

dx exp

1

2||2

1 +igt

4||

x i||2 2r + gt/2||1 + igt/4||

2

=eigt||

1 + igt/4|| exp||

2

2

(2r + gt/2||)21 + igt/4||

. (2.28)

From this final expression one may directly read off: the oscillation fre-quency g||; the characteristic time for the first collapse 22/g (by takingr = 0 and assuming gt/|| is small); the delay between successive revivals4||/g; and finally one may also see that the revival at gt = 4r|| hasan increased width and decreased height compared to the initial collapse,given by a factor

1 + 2r2 as is visible in figure 2.1.

Further reading

The contents of most of this chapter is discussed in most quantum opticsbooks. For example, it is discussed in Scully and Zubairy [7], or Yamamotoand Imamoglu [8]. Further properties of collapse and revival of Jaynes-Cummings oscillations are discussed by Gea-Banacloche [9].


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Lecture 3

Two level systems coupledto many modes: Decay and

density matrix equations

The aim of this lecture is to introduce the behaviour of a two-level systemcoupled to many photon modes, such that irreversible decay now occurs.We will use the density matrix description of the two-level system, andshow how coupling to a reservoir of bath modes induces irreversible decay.The formalism we develop here will be useful also when we later considerthe laser, and its more quantum mechanical variants of single atom lasersand micromasers.

3.1 Many mode quantum model irreversibledecay

In the previous lecture we discussed a two level system coupled to a singlemode, considering both classical and quantum light. One important dif-ference between these situations is that for classical light, if the light fieldvanishes, then there can be no transitions; in contrast, for the quantumdescription, spontaneous emission also existed, in that the state |0, hy-bridises with the state

|1,

, due to the +1 in the

n + 1 matrix elements.

We will now consider the effects of this +1 factor when we couple a singletwo level system to a continuum of radiation modes.

Our Hamiltonian will remain in the rotating wave approximation, butno longer restricted to a single mode of light, and so we have:

H =

2z +

n,k

kak,nak,n +

k

gk,n2

ak,n + H.c.

(3.1)

where k = ck and

gk,n

2=

ek,n dba20kV

. (3.2)

15


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16 LECTURE 3. DECAY OF A TWO-LEVEL SYSTEM

3.2 Density matrix equation for relaxation oftwo-level system

We now discuss how the irreversible decay arising from the coupling tomany photon modes may be described in the density matrix approach. Weproceed by eliminating the the behaviour of the continuum of radiationmodes in order to write a closed equation for the density matrix of thesystem. If we define the system density matrix as the result of tracing overall reservoir degrees of freedom, we can write an equation of motion forthis quantity as:

d

dtS = iTrR {[HSR , SR]} . (3.3)

in which S means state of the system, R the reservoir (in this case, thecontinuum of photon modes), and TrR traces over the state of the reservoir.

After removing the free time evolution of the reservoir and system fields, theremaining system-reservoir Hamiltonian can be written in the interactionpicture as:

HSR =k

gk2

ake

i(k)t + +akei(k)t

. (3.4)

In general Eq. (3.3) will be complicated to solve, as interactions betweenthe system and reservoir lead to entanglement, and so the full density ma-trix develops correlations. This means that the time evolution of the systemwould depend on such correlations, and thus on the history of system reser-

voir interactions. However this will simplify is we assume a smooth densityof states for the bath, in which case it has a short correlation (memory)time, and memory effects can be neglected. This will allow us to make aMarkov approximation, meaning that the evolution of the system dependsonly on its current state i.e. neglecting memory effects of the interaction.

For a general interaction, there are a number of approximations whichare required in order to reduce the above equation to a form that is straight-forwardly soluble.

We assume the interaction is weak, so SR = S R + SR , whereSR O(gk). Since S = TrR(SR), it is clear that TrR(SR) = 0,

however the result of TrR(HSRSR) may be non-zero. Such corre-lations will be assumed to be small, but must be non-zero in orderfor there to be any influence of the bath on the system. In orderto take account of these small correlation terms, generated by thecoupling, we use the formal solution of the density matrix equationSR(t) = i

tdt[HSR(t

), SR(t)] to write:

d

dtS(t) = TrR

t0

dt

HSR(t),

HSR(t), SR(t

)

. (3.5)

We may now make the assumption of small correlations by assumingthat SR(t

) = S(t)

R(t

), and that only the correlations generated

at linear order need be considered.


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3.2. DENSITY MATRIX EQUATION FOR RELAXATION OF

TWO-LEVEL SYSTEM 17

We further assume the reservoir to be large compared to the system,and so unaffected by the evolution, thus R(t

) = R(0). Along withthe previous step, this is the Born approximation.

Finally, if the spectrum of the bath is dense (i.e. the spacing of energylevels is small), then the trace over bath modes of the factors eikt

will lead to delta functions in times. This in effect means that wemay replace S(t

) = S(t). This is the Markov approximation.

One thus has the final equation:

d

dtS(t) = TrR

t0

dt

HSR(t),

HSR(t), S(t) R(0)

. (3.6)

To evaluate the trace over the reservoir states, we need the trace ofthe various possible combinations of reservoir operators coming from HSR .

These will involve pairs of operators ak, ak. Using cyclic permutations,these trace terms can be written as thermal average X = TrR(RX). Forbulk photon modes, the relevant expectations are:

akak

=

akak

= 0,

akak

= kknk,

aka

k

= kk(nk + 1).

(3.7)In these expressions we have assumed nk = 0, i.e. we have allowed fora thermal occupation of the photon modes. The results of the previouslecture assumed an initial vacuum state, and thus nk = 0.

With these expectations, we can then write the equation of motion forthe density matrix explicitly, writing k = exp[i(k

)(t

t)], to give:

d

dtS(t) =

k

gk2

2 t0

dt

+nkk + +(nk + 1)

k

s

+(nk + 1) + +nk (k + k)+s

+nk

k +

+(nk + 1)k

. (3.8)

In this expression, the first term in parentheses comes from the expres-sion TrR [HSR(t)HSR(t

)SR ], the second comes from the conjugate pairTrR [HSR(t)SRHSR(t

)] and t t [hence (k + k)], and the final termcomes from TrR [SRHSR(t

)HSR(t)].At this point, we can now simplify the above expression by restricting

to the case where there is a smooth density of states. We define

(t t) =k

gk2

2 eik(tt) () = 2k

gk2

2 ( k). (3.9)Then, the restriction that is a smooth function means that () does notsignificantly vary over the range , i.e. d/d 1. In this case (whichcorresponds to the Markov approximation, meaning a flat effective densityof states) the integral over in Eq. (3.23) becomes a delta function, sothat:

kgk

22

k= (t

t),

kgk

22 n

kk

= n(t

t). (3.10)


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Here we have assumed that not only is the density of states flat, but thatalso nk is sufficiently flat, that we might write an averaged occupation n.Such an approximation is only valid at high enough temperatures we

will discuss this in some detail in a few lectures time.

d

dtS(t) =

2n

+S 2+S + S+

2

(n + 1)

+S 2S+ + S+

(3.11)

The factor of 1/2 here comes from the regularised half integral of the deltafunction. The two lines in this expression correspond to stimulated absorp-tion (which exists only if n > 0), and leads to excitation of the two-levelsystem, and to emission (both stimulated and spontaneous). If n = 0, onlyemission survives, and the results of the previous lecture are recovered for

the relaxation to equilibrium.One can see the behaviour from this equation most clearly by writing

the equations for the diagonal components:

t = 2

n (2) 2

(n + 1) (2) = t (3.12)

Then, using = 1 , one may find the steady state probability ofexcitation is given by:

0 = [n(1 ) (n + 1)] = n2n + 1

. (3.13)

This formula is as expected, such that if one uses the Bose-Einstein occupa-tion function for the photons, n = [exp()1]1, the excitation probabilityof the two-level system is the thermal equilibrium expression.

3.3 Effective decay rate

We have not yet calculated an actual expression for the effective decay rateappearing in the previous expression. Let us now use the form of gk inEq. (3.2) to evaluate (). Writing the wavevector in polar coordinates,with respect to the dipole matrix element dab pointing along the z axis (seeFig. 3.1), we then have:

() = 2V

(2)3

dd sin k2dk(ck )

2|dab|220ckV

n

(z ekn)2.(3.14)

From Fig. 3.1 it is clear that z ek2 = 0 and one may see that z ek1 = sin . For concreteness, the k direction and polarisations can beparametrised by the orthogonal set:

k =

sin cos sin sin

cos

, ek1 =

cos cos cos sin

sin

, ek2 =

sin

cos 0

.

(3.15)


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3.4. FURTHER READING 19

kz

e

e

2

1

Figure 3.1: Directions of polarisation vectors in polar coordinates

Putting all the terms together, one finds:

() =2|dab|2

(2)220

d

d sin3

ckcdk

3c3(ck ) = 1

40

4|dab|2333c3

(3.16)This formula therefore gives the decay rate of an atom in free space, asso-ciated with the strength of its dipole moment and its characteristic energy.

It is useful to rewrite this decay rate in terms of the oscillator strengthfab of the transition involved. Recalling the definition offab, and continuingto use the shorthand = ab we have

=

4

3

1

403c3

2e2

2m0fab =

1

3

e2

40c

2

m0c2fab (3.17)

Then, using the additional results:

=e2

40c, Ryd = 1

22m0c

2 (3.18)

where Ryd is the Rydberg energy and the fine structure constant, onecan write the decay rate in the easily interpretable form:

=

1

33

Ryd fab. (3.19)

The smallness of is thus crucial in the fact that atomic lines are welldefined (i.e. have linewidths smaller than their separation).

3.4 Further reading

Once again, the contents of this chapter can be found in most standard

quantum optics books, e.g.[7, 10].


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3.A WignerWeisskopf approach

An alternative method to show the decay rate in simple situations is to

consider solving the Schrodinger equation directly, to find the probabilitythat the two-level system remains excited. To study spontaneous emission,we start from the photon state |0, , in which there are no photons. Thesubsequent state can be written as:

| = (t)|0 +k

k,n(t)|1k,n , (3.20)

where |1k,n contains a single photon in state k, n. Then, writing theSchrodinger equations as:

it =

2

+k,n

gk,n

2

k,n, itk,n =

2

+ k k,n + gk,n2

, (3.21)

one may solve these equations by first writing = eit/2, k = kei(/2k)t,

and then formally solving the equation for k, with the initial conditionk(0) = 0, to give:

k,n = i gk,n2

tdtei(k)t

(t). (3.22)

Substituting this into the equation for gives:

t = t

dtk,n

gk,n22

ei(k)(t

t)

(t

)

= t

dt

d

2()ei()(t

t)(t) (3.23)

where we have used the same definition of as previously:

() = 2k,n

gk,n2

2 ( k,n) (3.24)We may then make the same Markov approximation as before to yield

t = t

dt()(t t)(t) = ()2

(t) (3.25)

Thus, the probability of remaining in the excited state decays exponentiallydue to spontaneous decay, with a decay rate () for the probability. Theapproach used in this section is known as the WignerWeisskopf approach.


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Lecture 4

Density matrix equationswith decay and coherent

driving

This lecture applies the framework of density matrix equations introducedin the previous lecture in order to incorporate other relaxation and decoher-ence processes, such as those arising due to noise sources, or inhomogeneousbroadening in an ensemble of atoms. We will then illustrate the use of thisapproach by discussing the behaviour of a two-level system illuminated bya coherent state of light, but able to radiate into a continuum of modes. Inthis lecture we will study the steady state of this system, and in subsequent

lectures we will look at the spectrum of emitted light, which will requireknowing more about the full time-evolution of the system.

4.1 Equations of motion with coherent driving

Before introducing extra contributions to dephasing, we begin by applyingthe method developed in the previous lecture to consider the combination ofcoherent driving and decay. We will assume from hereon that the reservoirphoton modes are empty.1 The evolution of the density matrix is thencontrolled by:

t = i [H, ] 2

+S 2S+ + S+

(4.1)

where H describes the coherent, Hamiltonian dynamics, which we will taketo be given by:

H =

2z +

g

2eit+ + H.c.. (4.2)

1This limit is frequently relevant in cavity quantum electrodynamics experiments,since the confined photon modes start at energies of the order of 1eV

104K, so the

thermal population of such photon modes is negligible.

21


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22 LECTURE 4. POWER BROADENING

One may then write out the equations of motion for each component ex-plicitly, giving

t = i 2 + g2 eit g2 eit 2 , (4.3)t = i

2

+g

2eit g

2eit

2

2. (4.4)

These two equations are sufficient, since the unit trace property of a densitymatrix implies = 1 , and the off diagonal elements are complexconjugate of one another by the Hermitian structure of the density matrix.To solve these equations, it is convenient to first go to a rotating frame,so that one writes = P e

it, and secondly to introduce the inversion,Z = ( )/2. In terms of these variables, one has:

tZ = i g2 (P P)

Z+ 12

(4.5)

tP = i( )P + i g2

2Z 2

P. (4.6)

To further simplify, we may separate the real and imaginary parts of P =X+ iY, and write = for the detuning of the optical pump, so thatone has the three coupled equations:

tX = Y 2

X (4.7)

tY = +X+ gZ 2 Y (4.8)

tZ = gY

Z+1

2

. (4.9)

These are Bloch equations for the Bloch vector parametrisation of the den-sity matrix (in the rotating frame):

=

12 + Z X iY

X + iY 12 Z

. (4.10)

The three terms, ,g, correspond to: rotation about the Z axis (i.e.

phase evolution); rotation about the X axis (i.e. excitation); and causesthe length of the Bloch vector to shrink. Note that the rotation rate aroundthe X axis is g, so a duration t = /g would lead to the transition fromground to inverted state.

Before solving Eq. (4.74.9), we will first consider other kinds of dephas-ing that may affect the density matrix evolution, leading to a generalisedversion of these equations, allowing for both relaxation and pure dephasing.

4.2 Dephasing in addition to relaxation

In addition to relaxation, where excitations are transfered from the sys-

tem to the modes of the environment, pure dephasing is also possible, in


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4.2. DEPHASING IN ADDITION TO RELAXATION 23

which the populations are unaffected but their coherence is reduced. Inconsidering this kind of dephasing, it is useful to also broaden our viewfrom considering a single two level system, to describing measurements on

an ensemble of many two level systems. For the moment we will assumethat even if there are many two level systems, they all act independently.As such, the expectation of any measurement performed on this ensemblecan be given by:

X =i

T r(iX) = T r(X), where =i

i. (4.11)

We thus have two types of decoherence one may consider:

Broadening within a two-level system, coming from time dependentshifts of energies (and possibly also coupling strengths). A variety of

sources of such terms exist; examples include collisional broadeningof gaseous atoms, where the rare events in which atoms approacheach other lead to shifts of atomic energies. In solid state contexts,motions of charges nearby the artificial atom can also lead to timedependent noise.

Inhomogeneous broadening, where the parameters of each two-levelsystem vary, and so off-diagonal matrix elements, involving time de-pendent coherences, are washed out in . This is particularly an issuein solid state experiments, where the two-level system energy may in-volve the dimensions of a fabricated or self-assembled artificial atom,

and these dimensions may vary between different systems.

To describe both such effects together, we consider adding noise,

+ (t). (4.12)

This noise will then lead to a decay of the off diagonal correlations. To seethis, consider Eq. (4.6), with the above replacement, and then perform agauge transform so as to remove the explicit time dependence. One finds:

P(t) P(t)expi t

0(t)dt (4.13)

We will take (t) = 0, and two distinct limits for the correlations of .

Fast noise limit

In order to extract a tractable model of time-dependent noise, the simplestlimit to consider is that (t) has a white noise spectrum, i.e. it is Gaussiancorrelated with:

(t)(t) = 20(t t). (4.14)This assumption is equivalent to assuming that the noise is fast, i.e. thatany correlation timescale it does possess is sufficiently short compared to

the dynamics of the system that it may be neglected.


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With this assumption of instantaneous Gaussian correlations, one maythen expand the exponential in Eq. (4.13) and write:

P(t) P(t)1 + it0

dt(t) 12!

t0

dtdt(t)(t) + . . .

=P(t)

1 20

2!

t0

dt +(20)

2

4!3

t0

dt2

+ . . .

= P(t)e0t

(4.15)

with the combinatoric factors coming from the number of possible pairingsin the Wick decomposition of n.

This noise thus leads to an enhanced decoherence rate for the off di-agonal terms, so that /2 should be replaced by 1/T2 = (/2) + 0 inthe density matrix equations. One may equivalently define the relaxation

time 1/T1 = . In general T2 < T1, as dephasing is faster than relaxation,however the factors of two in the above mean that the strict inequality isT2 < 2T1, with equality holding when there is only relaxation.

The Bloch equations are thus given by:

tX = Y 1T2

X (4.16)

tY = +X+ gZ 1T2

Y (4.17)

tZ = gY 1T1

Z+

1

2

. (4.18)

A more rigorous approach to deriving the effect of dephasing is discussedin question A3, in analogy to the way we derived decay in the density matrixformalism

Static limit (inhomogeneous broadening)

To describe inhomogeneous broadening, should represent the system-dependent variation of energies, and should have no time dependence. Thesame behaviour is also relevant if time-dependent noise terms vary slowlyon the timescale of an experiment. (For example, slow noise may arisefrom a nearby impurity have multiple stable charging states, so that the

charge environment can take multiple different values; if the charging anddischarging of this impurity is slow compared to the experiment, this energyshift will effectively give inhomogeneous broadening when averaged overmultiple experimental shots.)

For this case is static, but randomly distributed, and Eq. (4.13) be-comes:

P(t) P(t)

dp()eit. (4.19)

If we consider a Lorentzian probability distribution for with width 1,i.e. p() = 21/(

21 +

2), then one finds P(t) P(t) exp(1|t|). Alter-natively, a Gaussian probability distribution leads to a Gaussian decay of

correlations.


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4.3. POWER BROADENING OF ABSORPTION 25

If one has both noise terms as in the previous section, and inhomoge-neous broadening, one may then distinguish T2 as defined above, and T

2 ,

where 1/T2 = (/2) + 0 + 1.

Distinguishing T2 and T2

Since both 0 and 1 lead to decay of the coherence functions, a simpleexperiment measuring coherence will see the lifetime T2 . However, thedynamics of each individual two level system remains coherent for the longertimescale T2. Such extra coherence can easily be seen in any sufficientlynon-linear measurement. An example of how the difference of T2 and T

2

can be measured is given by photon echo, illustrated in Fig. 4.1.

X

Z

Y

g t= /2

X

Z

Y

X

Z

Y

g t=

X

Z

Y

Figure 4.1: Cartoon of photon-echo experiment, which removesdephasing due to inhomogeneous broadening, leaving only truedephasing rate to reduce intensity of revival.

From the Bloch equations, Eq. (4.164.18), it is clear that a resonantpulse with duration gt = /2 brings the Bloch vector to the equator.From there, inhomogeneous broadening means that (in a rotating frame atthe mean frequency) the Bloch vectors for each individual two-level systemstart to spread out. However, by applying a second pulse with gt = ,a rotation about the X axis means that whichever Bloch vector spreadat the fastest rate is now furthest from the Y axis, and the subsequentevolution sees the vectors re-converge. At the final time, the coherence ofthe resulting pulse will have been reduced by 0 and , but 1 will havehad no effect.

4.3 Power broadening of absorption

Having introduced the idea of extra dephasing terms, we may now includethese in a re-examination of the question we started an earlier lecture with what is the effect of shining classical light on a two-level atom? However,we may now account for decay and dephasing. Whereas previously theabsence of decay meant that the result was Rabi oscillations, with decay asteady state is eventually found.

Looking for steady state solutions of Eq. (4.164.18) one finds from theequations for X and Z that:

Y = 1

gT1

Z+

1

2

, X = T2Y =T2gT1

Z+

1

2

. (4.20)


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Substituting these into the equation for Y, and recalling Pex = Z + 1/2,gives:

0 = Pex T2

gT1+

1

T2

1

gT1+ g

g

2(4.21)

After re-arranging, one then finds:

Pex =1

2

T1T2(g)2

(T2)2 + 1 + T1T2(g)2=

1

2

(T1/T2)(g)2

2 + [1 + T1T2(g)2] /T22(4.22)

The addition of damping to the equations of motion has thus had anumber of important consequences:

It is responsible for a steady state existing at all (but note that thissteady state is in a frame rotating at the pump frequency , so the

physical polarisation is time dependent.)

It gives a finite width to the resonance, even at weak pump powers in the Hamiltonian case, as g 0, the width of the resonancepeak vanished.

It modifies the overall amplitude at resonance, by a factor dependingon the power, such that at g 0, there is no response.

An important feature of this excitation probability is that the linewidthdepends on the intensity of radiation. This is a consequence of the nonlin-earity implicit in a two-level system. Such dependence is not so surprising

given that in the Hamiltonian dynamics, the resonance width already de-pended on the field strength. One may note that if one considered an har-monic atomic spectrum, rather than a two-level atom, no such broadeningwould be seen (see question A4).

0

0.1

0.2

0.3

0.4

0.5

-30 -20 -10 0 10 20 30

Pex

Figure 4.2: Power broadening of absorption probability ofexcitation vs atom-photon detuning for increasing field strength.

This problem, of a single atom coherently pumped, and incoherently de-caying also shows interesting behaviour if we look at its incoherent emission

spectrum. This is however something we will postpone until lecture 7.


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4.4 Further reading

The discussion of the resonance fluorescence problem in particular can be

found in chapter 4 of Meystre and Sargent III [10].


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Lecture 5

Two-level atom in a cavity:Purcell effect and Cavity

QED

In the previous few lectures we have discussed a single two-level atom cou-pled either to a continuum of radiation modes, or considering a cavity,which picks out a particular mode. In this lecture we will consider morecarefully the situations in which a cavity will pick out a single mode, bydescribing how the system changes as one goes from no cavity, via a badcavity (which modifies the field strength but does not pick out a singlemode) to a good cavity, which can pick out a single mode.

To study this problem throughout this crossover, we will consider a twolevel system coupled both to a cavity pseudo-mode (which itself decays),and also coupled to non-cavity modes, providing incoherent decay. We willshow how in this model, a bad cavity can lead to either enhanced decay, anda good cavity can lead to periodic exchange of energy between the cavityand the two-level system, i.e. the Rabi oscillations we previously discussedfor a perfect cavity.

Before discussing the crossover between bad and good cavities, we firstinvestigate a toy model of a 1D cavity, to put into context the meaning ofthe pseudo-mode description of the cavity mode.

5.1 The Purcell effect in a 1D model cavity

The aim of this section is to describe how a cavity modifies the rate ofdecay of a two-level system, coupled to one-dimensional radiation modes.Following the earlier discussion of system-reservoir coupling, the decay ischaracterised by the combination of reservoirs density of states and thecoupling of each reservoir mode to the atom. This gave an expression forthe decay rate of the form:

() = 2k

(

ck) gk22 . (5.1)

29


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30 LECTURE 5. PURCELL EFFECT

Recalling that the coupling strength gk depends on the mode structure viagk Ek, where Ek is the field strength associated with a single photon, wecan find the effect of the cavity on () by determining the spatial profiles

of the modes of the full system, and inserting these into the above sum.

a/2

L/2

Figure 5.1: Toy model of cavity, length a, in quantisation volumeof length L.

We will consider a cavity of size a, embedded in a quantisation volumeof length L, where we will take L later on. (See figure 5.1). Our aimis to find the spatial profile of the electric field modes, and thus to calculate

() k

( ck)|Ek(x = 0)|2. (5.2)

The electric field obeys the wave equation:

d2

dx2Ek(x) = k2r(x)Ek(x), (5.3)

where r(x) is the (spatially varying) dielectric constant, and k is the

wavevector in vacuum. Both in the cavity, and outside, the modes willbe appropriate combinations of plane waves. Restricting to symmetric so-lutions (as antisymmetric solutions will vanish at x = 0), and matching theboundary at x = L/2, we may write:

Ek(x) =

f(x) Ak cos(kx) |x| < a2g(x) Bk sin

k|x| L2 |x| > a2 . (5.4)

As L , the normalisation condition will approach the simple resultBk =

2/L, since the normalisation integral will be dominated by the parts

outside the cavity. We now need to find the effect of the imperfect barriers

on this mode in order to relate Ak, Bk, and thus find the quantisationcondition specifying k, and the mode amplitude, Ek(x = 0) = Ak.

As a simple model, let us consider a varying dielectric index, = 0[1 +(|x| a/2)], giving the equation:

d2

dx2Ek(x) = k2

1 +

|x| a

2

Ek. (5.5)

Let us focus on x > 0; then, given the form of Ek(x) = (a/2 x)f(x) +(x a/2)g(x) with f = k2f and g = k2g, we may write:

d2

dx2Ek = k2Ek + 2x

a

2 g(x) f(x) + x a

2 [g(x) f(x)](5.6)


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5.1. THE PURCELL EFFECT IN A 1D MODEL CAVITY 31

Thus, to solve Eq. (5.5) we require that:

fa

2 = g

a

2 , g

a

2

f a

2 =

k2

4f

a

2+ g

a

2 (5.7)

Substituting the forms of f(x), g(x) into this equation, one may eliminateAk, Bk to give the eigenvalue condition:

cos

kL

2

+

k

2cos

ka

2

sin

ka

2 kL

2

= 0. (5.8)

For finite L, this provides a restriction on the permissible values of k. AsL , the permissible values of k become more dense, such that, in thislimit, one may instead consider this equation as depending on two separatevalues kL and ka; ifL a, it is possible to significantly change kL withoutmodifying ka. Thus, we will instead rewrite this equation as a condition

that specifies the value of kL given a fixed ka:

cos

kL

2

1 +

k

2sin

ka

2

cos

ka

2

= sin

kL

2

k

2cos2

ka

2

.

(5.9)Using this condition, one may then eliminate kL, and write Ak in terms ofBk =

2/L and functions of ka, i.e.:

Ak =

2

L

11 + k sin(ka/2) cos(ka/2) + (k/2)2 cos2(ka/2)

(5.10)

If we define = k/2, then we may use this formula to give the effectivedecay rate:

k =

c

=

0

1 + sin(ka) + 12 2(1 + cos(ka))

=0

1 + 2

2

+

1 +

2

4 cos(ka 0), tan 0 =

2

.

(5.11)

The form of this function in general is shown in Fig. 5.2. For small k, theeffect of the cavity is weak (i.e. k 1), and so there is little modificationcompared to the result without the cavity. For larger k, there are sharppeaks, which can be described by an approximately Lorentzian form (see

the inset of Fig. 5.2). This Lorentzian form can be found by expandingEq. (5.11) near its peaks, which are at:

k0a = 0 + (2n + 1). (5.12)

We can then expand () near 0 = ck0. If 1, and if we may neglectits k dependence, then we find:

0 1

1 + 2

2

22

1 + 2

2 2

4

1 12

a(0)

c

2

1

12 +

2a2

4c2 ( 0)2. (5.13)


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0

5

10

15

20

25

0 1 2 3 4 5 6 7 8

()/0

/0

0.01

0.1

1

10

100

7.2 7.6 8

Figure 5.2: Effective decay rate vs frequency for the toy model ofa 1D imperfect cavity. Inset compares the exact result, Eq. (5.11)

to the Lorentzian approximation of Eq. (5.14)

Let us introduce the full width half maximum, , such that this expressionbecomes:

0=

P(/2)2

( 0)2 + (/2)2 (5.14)

which leads to the definitions:

2=

2c

2a, P = 2 =

4c

a. (5.15)

Here 0 is the result for a 1D system in the absence of a cavity. Thefactor P describes the maximum value of /0 at resonance. At resonance,the presence of the cavity enhances the decay rate, and P is the Purcellenhancement factor. It is helpful for comparison to later results to rewritethis as:

P = 40

c

0a=

2

Q

a

, Q 0

(5.16)

where Q is the quality factor of the cavity mode. The enhancement onresonance therefore depends both on the quality factor, and on the modevolume.

Away from resonance, the decay rate decreases instead of increasing,

since there are no modes with significant weight inside the cavity; theminimum value of /0 occurs when k0a = 0 + 2n, and is given by/0 = 1/

2 = 1/P, hence for this 1D case, P describes both the enhance-ment on resonance, and the reduction off resonance.

As increases further, the Lorentzian density of states becomes sharplypeaked, and so the Markovian approximation for decay of the atom nolonger holds, instead one can use:

d

dt =

tdt(t)

d

2

0P(/2)2ei()(t

t)

( 0)2 + (/2)2

=

0P

4t dt(t)ei()(tt)(tt)/2. (5.17)


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5.2. WEAK TO STRONG COUPLING VIA DENSITY MATRICES 33

This form describes the behaviour one would see if the two-level systemwere coupled to another degree of freedom, with frequency and decayrate /2. Hence, we are led in this limit to introduce the cavity pseudo-

mode as an extra dynamical degree of freedom. The density of states impliesthat the cavity mode cannot be treated as a Markovian system, it has anon-negligible memory time 1/, and so we must consider its dynamics.

5.2 Weak to strong coupling via density matrices

This section considers the case in which a single cavity mode must betreated beyond a Markovian approximation, by considering the Jaynes-Cummings model, along with relaxation of both the two-level system andthe cavity mode. The model is thus given by:

ddt

= i [H, ] 2

aa 2aa + aa

2

+ 2+ + + (5.18)

H =g

2

+a + a

+ z + 0a

a. (5.19)

One should note that in this equation, a is the creation operator for apseudo-mode of the system, i.e. it does not describe a true eigenmode; thetrue eigenmodes are instead superpositions of modes inside and outside thecavity, just as in the previous section. Because the pseudo-mode overlapswith a range of true eigenmodes, the probability of remaining in the pseudo-mode will decay (see question B1 for the 1D case). This coupling betweencavity modes and modes outside the cavity can be described in a Markovianapproximation, leading to the decay rate in the above density matrixequation.

In addition to the decay of the pseudo-mode, we also include a rate

describing decay of the two level system into modes other than the cavitymode. In the one dimensional description considered previously, such achannel must correspond to non radiative decay. However, in three dimen-sions, if the cavity is not spherical, then decay into non-cavity directionsis possible and is described by . In the three dimensional case, one mayconsider = (1

/4), depending on solid angle the cavity encloses.

For the experimental systems discussed below, this is generally the case,due to the need to have access to insert atoms into the cavity. More gener-ally describes the possibility of relaxation into any mode other than thecavity mode; in solid state contexts, other possible reservoirs often exist(e.g phonons, other quasiparticle excitations etc).

Let us solve the equation of motion, Eq. (5.18), starting in an initiallyexcited state | , 0. From this initial state, the only possible other sub-sequent states are | , 1 and | , 0. As the last of these states cannotevolve into anything else it may be ignored, and we can write a closed setof equations for three elements of the density matrix:

PA = 0,0, PB = 1,1, CAB = 0,1 (5.20)


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By taking appropriate matrix elements, one finds:

d

dt

PA =

i

g

2

(CAB

CAB)

PA (5.21)

d

dtPB = i g

2(CAB CAB) PB (5.22)

d

dtCAB = i( 0)CAB + i g

2(PA PB) +

2CAB (5.23)

One may further simplify these equations by noting that PAPB|CAB|2 = 0is conserved, and thus writing PA = ||2, PB = ||2, CAB = . Substi-tuting this leads to the simpler equations:

d

dt

=

/2 ig/2

ig/2 i

/2

(5.24)

where we have written = 0 . In general this leads to two frequenciesfor decaying oscillations of the excitation probability:

i

2

i+ i

2

+

g2

4= 0. (5.25)

Bad cavity limit Purcell effect in 3D

In the limit of a bad cavity, i.e. g, , the two frequencies correspondingto the above equation correspond separately to decay of the excited two-level system and decay of photons. Because these are on very different

timescales, one may treat the coupling perturbatively writing = i/2 +, where at O() the determinant equation becomes:

i

i

2+

2

+

g2

4= 0, (5.26)

and since this bad cavity limit gives:

= i2

+

g2

2 + 42

+

g2

2 + 42. (5.27)

This describes a cavity-enhanced decay rate, eff = +cav, where the cav-

ity decay rate describes the Purcell effect, as discussed above. On resonancewe have:

cav = 4g

2

2 1

(5.28)

and recalling the earlier 3D results:

=30|dab|230c3

,g

2

2 = 0|dab|220V

(5.29)

(where we have assumed resonance, so 0 = ), one may write:

cav = P = 4

3c3

2V 20

1

=

3

42 Q3

V

. (5.30)


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This closely resembles the one dimensional result before; the Purcell en-hancement depends both on the quality factor, and also on the mode vol-ume. An estimate of the minimum decay rate off resonance in this case

yields cav,min = cav,max

20

2 = 3

1621Q

3V

. However, if the back-

ground decay rate also exists, this off-resonant reduction is not as rele-vant as the on-resonant enhancement.

Strong coupling Rabi oscillations

If the cavity is sufficiently good, then rather than decay, the excitationprobability oscillates. In the resonant case of = 0, one can write thegeneral solution for the oscillation frequency as:

= i( + )

4 g2

4 4

2

. (5.31)

In order that Rabi oscillations exist, it is thus necessary that g > ()/2.In order that they should be visible before decay has significantly reducedthe amplitude, it is necessary that one has g > , . In this limit, one hasstrong coupling, and decay is indeed very strongly modified.

5.3 Further reading

For a discussion of the Purcell effect, and the crossover between weak and

strong coupling, see e.g. Meystre and Sargent III [10].


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Lecture 6

Features of various of CavityQED systems

Having introduced all of the concepts required to understand a single atomcoupled to radiation in a cavity, this lecture now discusses a number ofsystems which can realise this situation. The aim here is to understandwhat parameter ranges can physically be achieved in different systems, tounderstand the advantages and disadvantages of the different systems, andto understand the physical origin of the various coupling strengths anddecay rates.

A summary of the characteristic values is given in table 6.1. The discus-sion below explains the origin and significance of these numbers. In eachcase there are several stages to consider: the nature of the two-level sys-tem, and the energies, decay rates, and coupling strengths of the transitioninvolved. In addition, one must also consider what designs of cavities arecompatible with the transition and nature of the two-level system.

System Atom[1, 11] Atom[4] SC qubit[3] Exciton[5]Optical Microwave Microwave Optical

/2 1 MHz 1 kHz a 30 MHz 30 GHz/2 6 MHz 30 Hz 3 MHz b 0.1 GHzg/2 10 MHz 50 kHz 100 MHz 100 GHz

/2 390 THz 50 GHz 10 GHz 400 THz3/V 105 101 /a = 1 101

Q 108 108 104 104

Other tmax tflight 100s 1/T2 3MHz 1/T2 3GHzaRecent work reaches 10Hz[12]bNot well known, dominated by cavity induced decay

Table 6.1: Characteristic energy scales for different cavity QEDrealisations, as discussed in the text.

37


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38 LECTURE 6. CAVITY QUANTUM ELECTRODYNAMICS

6.1 Optical transitions of atoms

We start by considering the conceptually simplest case, in which the two

level system is a transition between two low lying levels of a real atom.Although real atoms have many levels, the level spacing is anharmonic(E Ryd/n2), hence it is possible for the cavity to be near to resonanceonly for a single transition between the ground and excited states. This isonly possible because the decay rate / f 3(/Ryd), so that the naturallinewidth is much smaller than the transition. Such transitions involve lightat optical frequencies, and so require cavities designed to produce highreflectivity at these frequencies.

Atomic energy levels; fine structure, hyperfine structureand linewidths

Considering optically allowed transitions m = 0, 1, l = 1, the sim-plest spectra will correspond to alkali atoms, involving s p transitions.However, to properly understand these transitions, one must consider boththe fine structure, and the hyperfine structure.

Fine structure arises due to the spin-orbit coupling:

ESO =1

2m2c2

1

r

dV

dr

S L (6.1)

The order of magnitude of this expression is:

ESO 1mc2

2

2ma2B

e2

40aB Ryd2

mc2 2Ryd. (6.2)

The power of 2 indicates that this is much smaller than the principalelectronic level splitting, but still bigger than the linewidth, hence it canbe resolved. The spin orbit term increases with atomic number due tothe larger potential gradient that exists at small radii within the screeningcloud. Since the term produces an LS coupling, its eigenstates are given bythe quantum numbers of |L|, |S|, |J| and one other commuting observable,e.g. Jz. Thus, for alkali atoms, the s state is not split, but the p state issplit into J = 1/2, 3/2 states.

Hyperfine splitting arises due to coupling between the electronic angularmomentum J and the nuclear angular momentum I. The coupling I Jmeans that rather than using Jz, Iz to label eigenstates, one should considereigenstates of |F|, Fz with F = J + I. A level scheme for 87Rb is shownin Fig. 6.1, based on values in Steck [13]. 87Rb is a boson, as it has anodd number of electrons and an odd number of nucleons. Its ground statenuclear spin is |I| = 3/2, hence the |J| = 3/2 state splits into four levels|F| = 0 . . . 3. The notation F = 3 indicates an excited electronic state withhyperfine state F = 3, whereas F = 2 indicates a ground electronic statewith hyperfine state F = 2.

The transition between the fine structure level S1/2

P3/2 is known as

the D2 line. The dominant transition within this line is the F = 3 F =


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6.1. OPTICAL TRANSITIONS OF ATOMS 39

F=1

0.5GHz

F=0

F=1

F=2

F=3

F=2

F=1

F=21/2S

P1/2

3/2P

1.6eV=390THz=780nm

7GHz

7THz

266MHz

Figure 6.1: Level scheme for 87Rb (bosonic), showing fine andhyperfine structure. For further details, see[13]

2. This has a dipole matrix element dD2 = 1.3eA. Using the transitionenergy ED2 = 1.6eV, one finds fD2 = 0.7, so this transition is close to

saturating the oscillator sum rule. Using this transition strength, the aboveestimate of decay rate gives:

13

0.7 11373

1.6eV13.6eV

350THz 3MHz (6.3)

This estimate is smaller than the actual width, = 38MHz because it doesnot properly include summation over possible spin states. However, thislinewidth is rather small, and so it is possible to resolve even hyperfinestructure, given the scales in Fig. 6.1. To have a true two-level system, onehas two choices: either to avoid any stray magnetic fields, and rely on thefact that the cavity mode polarisation will couple to a particular superpo-

sition of the 2F + 1 = 7 levels labelled by mF = 3 . . . 3. Alternatively,this remaining degeneracy could be split by a magnetic field.

Optical cavities

From the above considerations, in order to have a cavity couple to a an indi-vidual hyperfine level transition, the cavity linewidth must also be smallerthan the hyperfine splitting, i.e 10MHz. Given the transition frequencyof E = 390THz, this implies a cavity quality factor of Q 108 is required.

One may also consider the quality factor required for Purcell enhance-ment. The minimum size of the cavity is constrained by the need to be able

to controllably insert atoms, and a typical cavity length[1, 11] is L 200m,


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and beam waist w 20m. This gives a ratio of wavelength to mode vol-ume of 3/V 105. Recalling the condition that controls the Purcelleffect, this means again that a very high Q factor, Q

108 is needed to

reach strong coupling.In the following we consider two possible approaches; metallic and di-

electric mirrors, and see which is capable of producing such high qualityfactors. In relating the quality factor the reflectivity, it is important to notethat they depend differently on the mode index involved. In particular, ifone introduces the finesse F, then:

Q =

=

n

, F=

R

1 R (6.4)

where is the mode spacing (which depends on the length of the cav-

ity), and R is the intensity reflection coefficient of the mirrors. The aboveapproximation holds for good mirrors. For very good mirrors, i.e. T =1 R 1, one has Q = nF n/(1 R), and the cases discussed belowwill be in this range. For a cavity with length L 200m, the mode indexwill be n 250, and so F 4 105 would be sufficient.

Metallic mirrors

As a brief reminder, the reflectivity of a metallic mirror can be found byconsidering a plane wave incident on its surface, and finding the reflectionand transmission coefficients associated with the effective (frequency de-

pendent) dielectric constant arising from its conductivity. For a plane wavewith frequency , one has:

B = 0( + ir0)E (6.5)

Thus the refractive index n =

r i/0 and for a good conductor atlow enough frequencies n = (1 i)/20. Then, matching electric andmagnetic fields at the interface:

Ei + Er = Et 1 + r = tBi + Br = Bt

1

r = nt

(6.6)

so r = (n 1)/(n + 1). Thus, for a good conductor one finds:

R = 1 2

20

(6.7)

For a typical metal, with 1071m1, and optical frequencies =400THz, this gives R = 1 0.05, hence F= 60. In order that the qualityfactor reach the required value, one would have to use a very high index(i.e. a very large cavity), with n 107. For such a cavity, the mode volumewould be huge, and so strong coupling unattainable. Clearly metal cavities

are not relevant.


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6.1. OPTICAL TRANSITIONS OF ATOMS 41

Dielectric mirrors: Bragg reflectors

Much higher quality mirrors for optical frequencies can be made using di-

electric Bragg mirrors. These consist of alternating layers of materials withdifferent dielectric constant, with spatial period /4, so that even if thedielectric contrast between adjacent layers is not sufficient to cause strongreflection, the interference of multiple reflections will lead to strong overallreflection. This is illustrated in Fig. 6.2.

Cavity

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

R=

|r|2

/0

na=1.5, nb=2.5

26 layers

Figure 6.2: Left: Cartoon of Bragg mirror cavity. The twocolours indicate material of two differing refractive indices. Right:reflectivity spectrum of a Bragg mirror

The reflectivity of a Bragg mirror can be found using the transfer matrixapproach, discussed in the appendix to this lecture. At resonance (i.e. ata frequency such that wana = wbnb = (c/)(/2), where wa,b are layerthicknesses and na,b the refractive indices), the reflection coefficient is givenby:

r =1 (nb/na)N1 + (nb/na)N

. (6.8)

where N is the number of pairs of layers. For the case illustrated in Fig. 6.2,with a dielectric contrast nb/na = 5/3, N = 13 layers is sufficient to reach afinesses ofF= 4105. Because the overall reflection depends on destructiveinterference between the partial reflections at each interface, the reflectanceis high only over a range of frequencies, known as the stop band. It is

therefore important that the frequency of all normal modes of the coupledcavity QED system lie within this stop band if one wants to ignore thefrequency dependence of the mirrors.

Other considerations The background atomic decay rate is deter-mined by the intrinsic properties of the atomic transition, reduced by thegeometry of the cavity, and so this sets the scale that must be overcomefor strong coupling. Typical values of the corresponding g, , are givenin Table 6.1. In addition to the timescales included within Eq. (5.18), insome experimental systems, there is another timescale, that of the time foran atom to leave the cavity, however this is typically larger than all other

timescales.


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6.2 Microwave transiti

Documents

Keeling Notes