Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

Подождите немного. Документ загружается.

Quantized Field Effects 78.12 Spontaneous Emission in Free Space 1159

deﬁnition of a stochastic integral is needed to give

correct results, (ii) rules from ordinary calculus are ap-

plicable only in the Stratonovich approach, and (iii) for

Langevin equations with h(x, t) =const, as in (78.157),

the Stratonovich interpretation and the Ito interpretation

of stochastic integrals are equivalent.

78.11 Quantum Monte Carlo Formalism

The quantum Monte Carlo formalism was developed

to solve numerically master equations of the Lindblad

type [78.20, 40–44]

=−

[

,ρ

]





†

−C

†

−ρ

†



(78.169)

Here C

are arbitrary system operators.

As an illustrative example, consider

=−

[

,ρ

]



2Cρ

†

−C

†

Cρ

−ρ

†



(78.170)

where C is an arbitrary system operator. Instead of solv-

ing the master equation, one deﬁnes quantum trajectories

or stochastic wave functions as follows. Starting from

|ψ(t), there are two possibilities for the time evolution

during the interval dt:

1. The system evolves according to the non-Hermitian

Hamiltonian H

−

†

C;i.e.

|ψ(t+dt)=



1−idt



−

†





|ψ(t)

1 −ψ(t)|C

†

C|ψ(t) dt

(78.171)

2. The system makes a jump; i. e.

|ψ(t + dt)=

C|ψ(t)

ψ(t)|C

†

C|ψ(t)

(78.172)

Since both possibilities describe a nonunitary time evo-

lution, |ψ must be normalized after each step. For each

time interval dt one of these two possibilities is randomly

chosen according to the probability

P(t)dt =ψ(t)|C

†

C|ψ(t)dt (78.173)

to make a jump between t and t + dt. We can now deﬁne

a density operator

(t) =|ψ(t)ψ(t)| (78.174)

for a speciﬁc quantum trajectory |ψ(t). The density

operator

(t) = |ψ(t)ψ(t)| (78.175)

averaged over all trajectories (indicated by the bar) is

then a solution of the master equation (78.170).

This method can easily be generalized to master

equations of the form (78.169).

78.12 Spontaneous Emission in Free Space

Consider an atom which is initially in one of its excited

states and which interacts with the quantized electro-

magnetic ﬁeld of free space. Even if none of the modes

of the electromagnetic ﬁeld is excited, there are still the

vacuum ﬂuctuations which “interact” with the atom and

give rise to important effects:

1. Spontaneous emission: the atom spontaneously

emits a photon and decays from the excited state.

2. Natural linewidth: due to the ﬁnite lifetime of the

atomic levels, the radiation from an atomic transition

has a ﬁnite linewidth, called the natural linewidth.

3. Lamb shift: the energy levels of the atom are shifted.

The standard theory of spontaneous emission is the

Wigner–Weisskopf theory [78.17, 32]. Here an initially

excited atomic state | decays exponentially according



(t)|

= e

−Γ



, (78.176)

where the decay constant Γ



is given by





i

3π

, (78.177)

and the sum is over all atomic states with an en-

ergy E

lower than the energy E



of the state |.

i

= (E



−E

)/ is the transition frequency for the

transition |→|i,andd

i

=e|r|iis the correspond-

ing dipole moment.

The same decay constant Γ



is also observed as

a linewidth in the spectrum of the radiation scattered

Part F 78.12

1160 Part F Quantum Optics

by an atom when the incoming photon excites the atom

to the level |.

The energy level shift is more troublesome and needs

the concept of mass renormalization, a standard problem

in quantum electrodynamics. The theory and results are

discussed in Chapt. 27 and Chapt. 28.

Recent calculations of Pachucki [78.45] based on

fully relativistic quantum electrodynamics and includ-

ing two-loop corrections predict 1 057 838(6) kHz

for the energy difference between the 2s

1/2

stateandthe2p

1/2

-state which is in excel-

lent agreement with the experimental result of

1 057 839(12) kHz [78.46].

For a discussion of energy levels and transition fre-

quencies in hydrogen and deuterium atoms see also

Sect. 28.3.

78.13 Resonance Fluorescence

Consider a two-level atom driven by a continuous

monochromatic wave which is treated classically. The

excited state of the atom can decay by spontaneous emis-

sion into vacuum modes of the electromagnetic ﬁeld.

This emission is called resonance ﬂuorescence. Of par-

ticular interest are the properties of the emitted light. For

a detailed discussion of resonance ﬂuorescence, see for

example [78.20, 31–33].

The far ﬁeld at position R emitted by an atom at the

origin is proportional to its dipole moment and can be

expressed in terms of the dipole operators σ

and σ

−

according to the relation [78.20]

(R, t) =−

(

d× R

)

× R

4π

−

(t −r/c),

−

(R, t) =−

(

∗

× R

)

× R

4π

(t −r/c),

(78.178)

where

d = eg|r|e

(78.179)

is the atomic dipole matrix element and the ﬁeld op-

erators E

(R, t) and E

−

(R, t) as well as the dipole

operators σ

(t) and σ

−

(t) are in the Heisenberg picture.

Knowledge of the operators σ

(t) and σ

−

(t) is therefore

sufﬁcient to study the properties of the emitted light in

the far ﬁeld.

78.13.1 Equations of Motion

The total Hamiltonian for the system reads





†







−

†

∗



−



d E

∗

−

iω

∗

E σ

−iω



(78.180)

where a resonant driving term has been added to the

Hamiltonian (78.123). Here d is the projection of the

dipole matrix element eg|r|e onto the polarization

vector of the driving ﬁeld with an amplitude E.Thecor-

responding master equation in the interaction picture is

ρ =−i

Ω



+σ

−

,ρ



(

2σ

−

ρσ

−σ

−

ρ−ρσ

−

)

(78.181)

where Ω

=−E d

∗

/ is the Rabi frequency associated

with the driving ﬁeld. The vacuum modes of the ﬁeld

are described by a thermal reservoir at zero temperature.

The equations of motion for the expectation val-

ues σ

, σ

−

,andσ

 are the optical Bloch equations

with radiative damping (Chapt. 68)andare

σ

=−

σ

−i

Ω

σ

 ,

σ

−

=−

σ

−

+i

Ω

σ

 ,

σ

=−γ

(

σ

+1

)

−iΩ

(

σ

−σ

−



)

(78.182)

These expectation values determine the density opera-

tor (78.142) of the two-level atom. Because (78.182)

are a system of linear differential equations for σ

,

σ

−

,andσ

, they can be solved analytically. Further-

more, the quantum regression hypothesis allows one

to calculate two-time correlation functions as shown

in Sect. 78.9.

78.13.2 Intensity of Emitted Light

According to (78.52)) and (78.178), the intensity of the

ﬂuorescence light at position R is given by

I =E

−

(R, t)E

(R, t)∝σ

−

 , (78.183)

and can be decomposed into two parts: the coherent

intensity

coh

∝σ

σ

−

 (78.184)

Part F 78.13

Quantized Field Effects 78.13 Resonance Fluorescence 1161

originating from the mean motion of the dipole, and the

incoherent intensity

inc

∝σ

−

−σ

σ

−

 , (78.185)

which is due to ﬂuctuations of the dipole motion around

its average value. The steady state intensities are

coh

∝

Ω



+2Ω



, (78.186)

and

inc

∝

2Ω



+2Ω



. (78.187)

For weak laser intensities (Ω

small) the intensity of

the ﬂuorescence light is dominated by the coherent part

whereas for high intensities (Ω

large) it is dominated

by the incoherent part.

78.13.3 Spectrum of the Fluorescence Light

The Wiener–Khintchine theorem (78.54) allows one to

express the steady state spectrum of the ﬂuorescence

light as the Fourier transform of the correlation function

σ

(τ)σ

−

(0)

in the form

S(ω) =

2π

∞



−iωτ

E

−

(τ)E

(0)

dτ +c.c.

∝

2π

∞



−i(ω−ω

)τ

σ

(τ)σ

−

(0)

dτ +c.c.

(78.188)

Again it consists of two contributions: a coherent part

coh

(ω), and an incoherent part S

inc

(ω). The coherent

part is

coh

(ω) ∝

Ω



+2Ω



δ(ω −ω

). (78.189)

The incoherent part of the ﬂuorescence light has two

qualitatively different spectra. For Ω

<γ/4, it has a sin-

gle peak at ω

, whereas it consists of three peaks for

Ω

>γ/4. For Ω

 γ/4itisgivenby

inc

(ω) ∝

2πγ



(γ/2)

(ω −ω

)

+(γ/2)

(3γ/4)

(ω −ω

+Ω

)

+(3γ/4)

(3γ/4)

(ω −ω

−Ω

)

+(3γ/4)



(78.190)

The central peak at ω = ω

has a width of γ/2 whereas

the width of the two side peaks at ω = ω

±Ω

is 3γ/4.

Their heights are one third of the height of the central

peak. This spectrum was predicted by Burshtein [78.47]

and Mollow [78.48] and experimentally conﬁrmed by

Schuda et al. [78.49], Wu et al. [78.50], and Hartig

et al. [78.51].

This triplet can be explained in terms of the dressed

states |1, n and |2, n introduced in Chapt. 68 (78.50).

If the driving ﬁeld is resonant with the atomic transition,

these states have the energies

1,n



n +



− R

2,n



n +



− R

, (78.191)

(78.52). The energy differences between the allowed

transitions are

2,n

−E

2,n−1

= ω

+ R

− R

n−1

≈ ω

2,n

−E

1,n−1

= ω

+ R

n−1

≈ ω

+ Ω

1,n

−E

2,n−1

= ω

− R

n−1

≈ ω

− Ω

1,n

−E

1,n−1

= ω

+ R

− R

n−1

≈ ω

(78.192)

where we have made the approximations

− R

n−1

≈ 0 ,

+ R

n+1

≈ 2g

n +1 ≈ Ω

. (78.193)

This is a good approximation for an intense driv-

ing ﬁeld which can approximated by a highly

excited coherent state with an average photon num-

ber

Figure 78.6 shows these energy levels and the al-

lowed transition. Obviously, the transitions correspond

to frequencies ω

, ω

−Ω

and ω

+Ω

. The dressed

ⱍ2, n冭

ⱍ1, n冭

ⱍ2, n–1冭

ⱍ1, n–1冭

⍀

–

⍀

Fig. 78.6 Energy level diagram of dressed states. The tran-

sition frequencies are ω

, ω

−Ω

and ω

+Ω

Part F 78.13

1162 Part F Quantum Optics

state picture also explains the 2:1 ratio for the inte-

grated intensities of the central peak and the side peak

in (78.190).

78.13.4 Photon Correlations

In addition to the spectrum which is based on the cor-

relation function E

−

(τ)E

(0)

in Sect. 78.13.3,the

second-order correlation function

(2)

(τ) =E

−

(0)E

−

(τ)E

(0)

∝σ

(0)σ

(τ)σ

−

(τ)σ

−

(0)

(78.194)

can be measured to gain more insight into the ﬂu-

orescence light, see also Sect. 78.3.2. Experimentally

this is done by measuring the joint probability for

detecting a photon at time t = 0 and a subsequent

photon at time t = τ. Again the result can be ob-

tained from the quantum regression hypothesis and

reads

(2)

(τ) =

(2)

(τ)

(1)

(0)|

= 1 − e

−3γτ/4



cos δτ +

3γ

4δ

sin δτ



(78.195)

where δ is given by

δ =

Ω

−γ

/4 . (78.196)

For τ =0, g

(2)

(0) =0, indicating a tendency of photons

to be separated. This tendency is known as photon an-

tibunching and was ﬁrst predicted by Carmichael and

Walls [78.52, 53] and experimentally veriﬁed by Kim-

ble et al. [78.54, 55]. Photon antibunching of radiation

emitted from a two-level atom has a simple explanation:

After the atom has emitted a photon it is in the ground

state and must ﬁrst be excited again before it can emit

another photon.

78.14 Recent Developments

This chapter has discussed the fundamentals of the quan-

tized electromagnetic ﬁeld and applications to the broad

area of quantum optics. However, in the last eight years,

quantum optics has blossomed in several new direc-

tions particularly in the key role it is playing in recent

investigations of the fundamentals of quantum theory

and related applications. In particular, the superposition

principle (the bedrock of quantum mechanics), entan-

glement, the quantum-classical interface, and precision

measurements have become very topical research ar-

eas, especially in respect to their relevance to quantum

information processing.

78.14.1 Literature

During the last eight years, several books on quantum

optics [78.56–62] have been published. These books

cover the topics of this chapter to some extent and take

into account recent developments. For an introduction

to the rapidly evolving ﬁelds of quantum information

processing, we refer the reader to Chapt. 81 and [78.63–

65].

78.14.2 Field States

Recently, number states of the radiation ﬁeld were ob-

served in a cavity-QED experiment [78.66].

78.14.3 Reservoir Theory

New research topics, such as quantum information

processing, rely on the superposition principle and en-

tangled quantum states. Since these states are very

sensitive to decoherence, reservoir theory has attracted

a lot of interest in recent years. Furthermore, as discussed

in [78.67–72], decoherence is the physical process by

which the classical world emerges from its quantum

underpinning.

Many investigations in this area involve the presence

of a reservoir (heat-bath/environment) and master equa-

tions are a ubiquitous tool. The familiar master equations

of quantum optics are in Lindblad form [78.73], which

guarantees that the density matrix is always positive

deﬁnite during time evolution. In the derivation of this

equation [78.74, 75], rapidly oscillating terms are omit-

ted by the method of coarse-graining in time; the high

frequencies correspond to the oscillator frequency ω

and, in the usual weak coupling limit, ω

 γ ,where

γ is a typical decay constant. This is the rotating wave

approximation (Sect. 66.3.2).

We have referred to the equations obtained prior to

coarse-graining in time as pre-master (or pre-Lindblad)

equations [78.74,76], and such equations have been used

extensively in other areas of physics [78.77, 78]; other

authors have simply referred to them as master equations

Part F 78.14

Quantized Field Effects References 1163

but, to avoid confusion, we reserve the latter term for

equations in Lindblad form. Pre-master equations, like

the master equations, describe an approach to the equi-

librium state. This equilibrium state is the same in either

case [78.76], but with pre-master (non-Lindblad) equa-

tions the approach can be through non-physical states of

negative probability. However, as recently demonstrated,

pre-master equations have other advantages vis

avis

master equations:

(a) they lead to the exact expression for the mean value

of x(t) (as obtained from the exact Langevin equation

for the problem);

(b) they lead, in the classical limit (

→0), to the famil-

iar Fokker–Planck equation of classical probability; and

times of pre-master form. However, the general expec-

tation (based on the time dependence of the coefﬁcients)

that the exact master equation preserves positivity for

all times has not been realized since Ford and O’Con-

nell have recently shown that, even in high temperature

regime, the density matrix is not necessarily posi-

tive [78.84].

In traditional quantum optics, the emphasis has been

on long-time



t γ

−1



phenomena, for which the use of

either master or pre-master equations is justiﬁed. How-

ever, they are both inadequate for dealing with short-time



t γ

−1



phenomena (as can be shown most simply by

calculating the mean-square displacement, a key ingre-

dient in decoherence calculations), which are of much

recent interest. Thus, it is desirable to use exact mas-

ter equations. In that respect, the exact master equation

of Hu et al. [78.79, 80] for an oscillator is an arbitrary

dissipative environment has proved to be a popular and

useful tool for which an exact solution has now been

obtained [78.83]. However, it should also be mentioned

that the solution of the initial value quantum Langevin

equation gives all the same information as the exact mas-

ter equation, and in fact, the solutions of the former were

used to obtain the solutions of the latter [78.83].

The familiar two-level atom master equation is, of

course, similar in form to the usual Lindblad-type mas-

ter equation for the oscillator. However, motivated in

particular by the desire to study decoherence and other

short time phenomena, an exact master equation was de-

rived [78.85] to study the non-Markovian dynamics of

a two-level atom interacting with the electromagnetic

ﬁeld.

In addition, motivated by the desire to study a driven

oscillator, the usual two-level atom master equation was

generalized to include the case of an external force

ﬁeld [78.86, 87]. This generalized equation was then

used not only to obtain the familiar zero-temperature

Burshtein–Mollow spectrum, but also the corresponding

high temperature results. For strong resonant driving at

high temperature, the same three-peaked structure was

observed in the zero temperature case, but a much larger

width was found. The analysis, following other investi-

gations, used the Lax formula for calculating two-time

correlation functions. This formula is not a “quan-

tum regression theorem” as it is often designated (see

also Sect. 78.9), but simply an approximation (which

more resembles an Onsager classical regression theo-

rem [78.88]) which works very well in the case of weak

coupling and for frequencies near a resonant frequency,

but not otherwise [78.86,87].

In Sect. 78.6, we stressed the usefulness of quasi-

probability distributions instead of the density matrix,

with particular attention to the Wigner distribution. In

particular, for simple Hamiltonians, we pointed out that

the equation for the corresponding Wigner function has

the form of a Fokker–Planck equation and we con-

sidered the explicit form describing the usual master

equation. The more general equations associated with

an exact master equation and their solution was the sub-

ject of [78.83] and interesting limits of that equation,

including the pre-master equation for both momentum

coupling and coordinate coupling were discussed at

length in [78.89, 90]. In the case of two-level systems,

it is not convenient to use quasi-probability distribu-

tions; instead, it is found that the preferred tool is the

polarization vector [78.91]. Surprisingly, it has not been

generally adopted by the quantum optics community

although its usefulness in that context has been demon-

strated recently in [78.86, 87].

References

78.1 C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg:

Photons, and Atoms. An Introduction to Quantum

Electrodynamics (Wiley, New York 1989)

78.2 L. Mandel: Opt. Lett. 4, 205 (1979)

78.3 W. Schleich, M. Pernigo, Fam Le Kien: Phys. Rev. A

44, 2172 (1991)

78.4 H. J. Kimble, D. F. Walls (Eds.): J. Opt. Soc. Am. B

4(10), 1453–1737 (1987)

Part F 78

1164 Part F Quantum Optics

78.5 P. Knight, R. London (Eds.): J. Mod. Opt. 34(6) (1987)

78.6 E. Giacobino, C. Fabre (Eds.): Appl. Phys. B 55(3)

(1992)

78.7 W. Schleich, J. A. Wheeler: J. Opt. Soc. Am. B 4,1715

(1987)

78.8 H. P. Yuen: Phys. Rev. A 13, 2226 (1976)

78.9 R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz,

J. F. Valley: Phys. Rev. Lett. 55, 2409 (1985)

78.10 L. A. Wu, M. Xiao, H. J. Kimble: J. Opt. Soc. Am. B 4,

1465 (1987)

78.11 A. Heidmann, R. J. Horowicz, S. Reynaud, E. Gia-

cobino, C. Fabre: Phys. Rev. Lett. 59, 2555 (1987)

78.12 W. P. Schleich, S. M. Burnett (Eds.): Special issue on

Quantum Phase and Phase Dependent Measure-

ments, Phys. Scr.T. T48 (1993)

78.13 P. Carruthers, M. M. Nieto: Rev. Mod. Phys. 40,411

(1968)

78.14 D. T. Pegg, S. M. Barnett: Phys. Rev. A 39, 1665

(1989)

78.15 J. W. Noh, A. Fougères, L. Mandel: Phys. Rev. A 45,

424 (1992)

78.16 R. J. Glauber: Phys. Rev. 130, 2529 (1963)

78.17 W. H. Louisell: Quantum Statistical Properties of

Radiation (Wiley, New York 1973)

78.18 R. Hanbury Brown, R. Q. Twiss: Nature 177, 27 (1956)

78.19 P. L. Kelley, W. H. Kleiner: Phys. Rev. 136,316(1964)

78.20 H. Carmichael: An Open Systems Approach to Quan-

tum Optics (Springer, Berlin, Heidelberg 1993)

78.21 H. Yuen, H. P. Shapiro: IEEE Trans. Inf. Theory 26,

78 (1980)

78.22 J. H. Shapiro, S. S. Wagner: IEEE J. Quantum Elec-

tron. 20,803(1984)

78.23 K. E. Cahill, R. J. Glauber: Phys. Rev. A 177, 1882

(1969)

78.24 M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner:

Phys. Rep. 106,121(1984)

78.25 R. J. Glauber: Phys. Rev. 131 , 2766 (1963)

78.26 E. C. G. Sudarshan: Phys. Rev. Lett. 10, 277 (1963)

78.27 C. W. Gardiner: Handbook of Stochastic Methods

(Springer, Berlin, Heidelberg 1985)

78.28 C. W. Gardiner: Quantum Noise (Springer, Berlin,

Heidelberg 1991)

78.29 E. Wigner: Phys. Rev. 40,749(1932)

78.30 R. L. Hudson: Rep. Math. Phys. 6, 249 (1974)

78.31 P. Meystre, M. Sargent III: Elements of Quantum

Optics (Springer, Berlin, Heidelberg 1991)

78.32 C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg:

Atom-Photon Interactions (Wiley, New York 1992)

78.33 D. F. Walls, G. J. Milburn: Quantum Optics (Springer,

Berlin, Heidelberg 1994)

78.34 C. W. Gardiner: Phys. Rev. Lett. 56, 1917 (1986)

78.35 G. W. Ford, J. T. Lewis, R. F. O’Connell: Phys. Rev. A

37, 4419 (1988)

78.36 R. F. O’Connell: Dissipation in a squeezed-state

environment, Proceedings of the Second In-

ternational Workshop on Squeezed States and

Uncertainty Relations, Moscow, Russia 1992, ed. by

D. Han, Y.-S. Kim, V. I. Man’ko (NASA, Maryland,

USA 1993)

78.37 H. Risken: The Fokker-Planck Equation (Springer,

Berlin, Heidelberg 1989)

78.38 M. Lax: Fluctuations and Coherence Phenomena

in Classical and Quantum Physics. In: Brandeis

University Summer Institute Lectures,Vol.2,ed.

by M. Chretin, E. P. Gross, S. Deser (Gordon and

Breach, New York 1966)

78.39 G. W. Ford, R. F. O’Connell: Phys. Lett. A 157,217

(1991)

78.40 J. Dalibard, Y. Castin, K. Mølmer: Phys. Rev. Lett.

68, 580 (1992)

78.41 K. Mølmer, Y. Castin, J. Dalibard: J. Opt. Soc. Am B

10, 524 (1993)

78.42 R. Dum, P. Zoller, H. Ritsch: Phys. Rev. A 45,4879

(1992)

78.43 C. W. Gardiner, A. S. Parkins, P. Zoller: Phys. Rev. A

46, 4363 (1992)

78.44 R. Dum, A. S. Parkins, P. Zoller, C. W. Gardiner:

Phys. Rev. A 46, 4382 (1992)

78.45 K. Pachucki: Phys. Rev. Lett. 72, 3154 (1994)

78.46 E. W. Hagley, F. M. Pipkin: Phys. Rev. Lett. 72, 1172

(1994)

78.47 A. I. Burshtein: Sov. Phys. JETP 22, 939 (1966)

78.48 B. R. Mollow: Phys. Rev. 188, 1969 (1969)

78.49 F. Schuda, C. R. Straud, Jr., M. Hercher: J. Phys. B

7, L198 (1974)

78.50 F. Y. Wu, R. E. Grove, S. Ezekiel: Phys. Rev. Lett. 35,

1426 (1975)

78.51 W. Hartig, W. Rasmussen, R. Schieder, H. Walther:

Z. Phys. A 278,205(1976)

78.52 H. J. Carmichael, D. F. Walls: J. Phys. B 9,L43(1976)

78.53 H. J Carmichael, D. F. Walls: J. B. Phys 9, 1199 (1976)

78.54 H. J. Kimble, M. Dagenais, L. Mandel: Phys. Rev.

Lett. 39, 691 (1977)

78.55 M. Dagenais, L. Mandel: Phys. Rev. A 18,201(1978)

78.56 M. O. Scully, M. S. Zubairy: Quantum Optics (Cam-

bridge Univ. Press, Cambridge 1996)

78.57 H.-A. Bachor: A Guide to Experiments in Quantum

Optics (Wiley-VCH, Weinheim 1998)

78.58 H. J. Carmichael: Statistical Methods in Quan-

tum Optics 1: Master Equations and Fokker-Planck

Equations (Springer, Berlin, Heidelberg 1999)

78.59 P. Meystre, M. Sargent III: Elements of Quantum

Optics (Springer, Berlin, Heidelberg 1999)

78.60 C. W. Gardiner, P. Zoller: Quantum Noise (Springer,

Berlin, Heidelberg 2000)

78.61 M. Orszag: Quantum Optics (Springer, Berlin, Hei-

delberg 2000)

78.62 W. P. Schleich: Quantum Optics in Phase Space

(VCH-Wiley, Weinheim 2001)

78.63 D. Bouwmeester, A. Ekert, A. Zeilinger (Eds.):

The Physics of Quantum Information: Quan-

tum Cryptography, Quantum Teleportation, Quan-

tum Information (Springer, Berlin, Heidelberg

2000)

Part F 78

Quantized Field Effects References 1165

78.64 M. A. Nielsen, I. L. Chuang: Quantum Computation

and Quantum Information (Cambridge Univ. Press,

Cambridge 2000)

78.65 G. Alber, T. Beth, M. Horodecki, P. Horodecki,

R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner,

A. Zeilinger: Quantum Information: An Introduc-

tion to Basic Theoretical Concepts and Experiments,

Springer Tracts in Modern Physics, Vol. 173 (Springer,

Berlin, Heidelberg 2001)

78.66 B. T. H. Varcoe, S. Brattke, M. Weidinger, H. Walther:

Nature 403, 743 (2000)

78.67 H. D. Zeh: Found. Phys. 1, 69 (1970)

78.68 A. J. Leggett: Suppl. Prog. Theor. Phys. 69, 80 (1980)

78.69 E. P. Wigner: In: Quantum Optics, Experimen-

tal Gravity and Measurement Theory,ed.by

P. Meystre, M. O. Scully (Plenum, New York 1983)

p. 43

78.70 D. F. Walls, G. J. Milburn: Phys. Rev. A 31, 2403 (1985)

78.71 E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch,

I.-O. Stamatescu: Decoherence and the Appearence

of a Classical World in Quantum Theory, 2nd edn.

(Springer, Berlin, Heidelberg 2003)

78.72 W. H. Zurek: Rev. Mod. Phys. 75, 715 (2003)

78.73 G. Lindblad: Commun. Math. Phys. 48, 119 (1976)

78.74 G. W. Ford, J. T. Lewis, R. F. O’Connell: Ann. Phys.

(NY) 252, 362 (1996)

78.75 J. T. Lewis, R. F. O’Connell: Ann. Phys. (NY) 269,51

(1998)

78.76 G. W. Ford, R. F. O’Connell: Phys. Rev. Lett. 82, 3376

(1999)

78.77 A. O. Caldeira, A. J. Leggett: Physica A 121,587(1983)

78.78 R. Dekker: Phys. Rep. 80,1(1981)

78.79 B. L. Hu, J. P. Paz, Y. Z. Zhang: Phys. Rev. D 45, 2843

(1992)

78.80 J. J. Halliwell, T. Yu: Phys. Rev. D 53, 2012 (1996)

78.81 F. Haake, R. Reibold: Phys. Rev. A 32, 2462 (1985)

78.82 R. Karrlein, H. Grabert: Phys. Rev. E 55, 153 (1997)

78.83 G. W. Ford, R. F. O’Connell: Phys. Rev. D 64, 105020

(2001)

78.84 G. W. Ford, R. F. O’Connell: Limitations on the utility

of exact master equations, Ann. Phys (NY) 319,in

press

78.85 C. Anastopoulos, B. L. Hu: Phys. Rev. A 62, 033821

(2000)

78.86 R. F. O’Connell: Optics Comm. 179, 451 (2000)

Reprinted in Ode to a Quantum Physicist edited

by W. Schleich, H. Walther, W. E. Lamb (Else-

vier,Amsterdam, 2000)

78.87 R. F. O’Connell: Optics Comm. 179, 477 (2000)

78.88 G. W. Ford, R. F. O’Connell: Phys. Rev. Lett. 77,798

(1996)

78.89 G. W. Ford, R. F. O’Connell: Acta Phys. Hung. B 20,

91 (2004)

78.90 R. F. O’Connell: J. Optics B 5, S349 (2003)

78.91 E. Merzbacher: Quantum Mechanics, 3rd edn. (Wi-

ley, New York 1998) p. 394

Part F 78

1167

Entangled Ato

79. Entangled Atoms and Fields: Cavity QED

Although the concept of a “free atom” is of use as

a ﬁrst approximation, a full quantum description

of the interaction of atoms with an omnipresent

electromagnetic radiation ﬁeld is necessary for

a proper account of spontaneous emission and

radiative level shifts such as the Lamb shift

(Chapt. 27). This chapter is concerned with the

changes in the atom-ﬁeld interaction that take

place when the radiation ﬁeld is modiﬁed by the

presence of a cavity. An atom in the vicinity of

a plane perfect mirror serves as an example of

cavity quantum electrodynamics [79.1–5].

The primary focus in this chapter is the

two extreme cases of weak coupling and strong

coupling, as exempliﬁed by spontaneous emission.

79.1 Atoms and Fields .................................1167

79.1.1 Atoms ......................................1167

79.1.2 Electromagnetic Fields ...............1168

79.2 Weak Coupling in Cavity QED .................1169

79.2.1 Radiating Atoms in Waveguides .. 1169

79.2.2 Trapped Radiating Atoms

and Their Mirror Images .............1170

79.2.3 Radiating Atoms in Resonators ...1170

79.2.4 Radiative Shifts and Forces.........1171

79.2.5 Experiments on Weak Coupling ...1172

79.2.6 Cavity QED and Dielectrics...........1173

79.3 Strong Coupling in Cavity QED ...............1173

79.4 Strong Coupling in Experiments ............1174

79.4.1 Rydberg Atoms

and Microwave Cavities..............1174

79.4.2 Strong Coupling

in Open Optical Cavities .............1174

79.5 Microscopic Masers and Lasers ..............1175

79.5.1 The Jaynes–Cummings Model .....1175

79.5.2 Fock States, Coherent States

and Thermal States....................1175

79.5.3 Vacuum Splitting.......................1177

79.6 Micromasers........................................ 1178

79.6.1 Maser Threshold........................1178

79.6.2 Nonclassical Features of the Field 1179

79.6.3 Trapping States ......................... 1179

79.6.4 Atom Counting Statistics.............1180

79.7 Quantum Theory of Measurement .........1180

79.8 Applications of Cavity QED .................... 1181

79.8.1 Detecting and Trapping Atoms

through Strong Coupling ............1181

79.8.2 Generation of Entanglement ......1181

79.8.3 Single Photon Sources................1182

References .................................................. 1182

In the weak coupling regime, the coupling of an ex-

cited atom to a broad continuum of radiation modes

leads to exponential decay (Fig. 79.1a), as ﬁrst described

by Weisskopf and Wigner [79.6]. Spontaneous emis-

sion may be enhanced or suppressed in structures such

as waveguides or “bad” cavities. Cavities also intro-

duce van der Waals forces and the subtle Casimir level

shifts [79.7].

In the strong coupling regime, the excited atom is

strongly coupled to an isolated resonant cavity mode. In

the absence of damping, an oscillatory exchange of en-

ergy between the atom and the ﬁeld replaces exponential

decay (Fig. 79.1b) with a coherent evolution in time. Ex-

perimental investigations of these effects began [79.8]

with the development of suitable resonators and tech-

niques for producing atoms with long lived excited states

and strong dipole transition moments.

79.1 Atoms and Fields

79.1.1 Atoms

The essential features of cavity QED are elucidated

by the two-level model atom discussed in Chapts. 68,

69, 70,and77 (see also [79.9]). A ground state |g

and an excited state |e are coupled to the radiation

ﬁeld by a dipole interaction. Using the formal equiv-

alence to a spin-1/2 system, the Pauli spin operators

Part F 79

1168 Part F Quantum Optics

are

= σ

†

+σ,

=−i



†

−σ



, (79.1)

= σ

†

σ −σσ

†



†

,σ



with σ

†

=|eg| and σ =|ge|. The quadratures (out

of phase components) of the atomic polarization

are given by σ

and σ

, while σ

is the occupa-

tion number difference. The free atom Hamiltonian

atom

, (79.2)

where ω

= E

−E

is the transition energy.

79.1.2 Electromagnetic Fields

Classical Fields

Classical electromagnetic ﬁelds have longitudinal and

transverse components:

E(r, t) = E

(r, t) + E

(r, t). (79.3)

In the Coulomb gauge, the longitudinal part is the instan-

taneous electric ﬁeld. The transverse part is the radiation

0.5

0123 0123

0.5

a) b)

t/τ



c) d)

(ω – ω

) ω

(ω – ω

)



Fig. 79.1a–d Upper row: Excitation probability of an ex-

cited atom.

(a) Exponential decay in free space or bad

cavities in the weak coupling limit.

(b) Oscillatory evo-

lution in good cavities or in the strong coupling case.

Lower row: The spectral signature of exponential decay

is a Lorentzian line shape

Rabi splitting

(d) is observed in the strong coupling case

ﬁeld which obeys the wave equation



∇

−

∂

∂t



(r, t) =



∂

∂t

j(r, t).

(79.4)

In empty space, the driving current density j(r, t) van-

ishes, and the ﬁeld may be expanded in a set of

orthogonal modes as

(r, t) =



(t)e

−iω

(r) +c.c. , (79.5)

with slowly varying amplitudes E

(t). The spatial dis-

tributions u

(r) obey the vector Helmholtz equation



∇







(r, t) = 0 , (79.6)

depending on geometric boundary conditions as im-

posed by conductive or dielectric mirrors, waveguides,

and resonators. In free space, plane wave solutions

(r, t) = u



ik·r

have a continuous index µ = (k,)

with wave vector k and an index  for the two indepen-

dent polarizations. The orthogonality relation



· u

∗

r = δ

µν

(79.7)

applies. For a closed cavity, V is the resonator volume.

In waveguides and free space, an artiﬁcial boundary is

introduced and then increased to inﬁnity at the end of

a calculation, such that the ﬁnal results do not depend

on V.

Quantum Fields

The quantum analog of the classical transverse ﬁeld

in (79.4) is obtained through a quantization of its har-

monic modes leading to a number state expansion.

Field operators obey standard commutation relations

, a

†

]=δ

µ,ν

, and for a single mode with index µ,the

amplitude E

in (79.5) is replaced by the corresponding

operator

(t) = E

−iωt

, E

†

(t) = E

∗

†

iωt

(79.8)

The normalization factor E

is chosen such that the en-

ergy difference between number states |n

and |n+1

in the volume V is ω

,giving

†

2

(79.9)

The Hamiltonian of the free ﬁeld is

Field





†



(79.10)

Part F 79.1

Entangled Atoms and Fields: Cavity QED 79.2 Weak Coupling in Cavity QED 1169

In the Coulomb gauge, the vector potential A(r) is

related to the electric ﬁeld E =−∂ A/∂t by

(r, t) =−



−iωt

†

iωt



(r).

(79.11)

The ground state |0

is called the vacuum state. While

the expectation value n|E|n=0 for a number state, the

variance is not zero, since n|EE

∗

|n > 0, giving rise to

nonvanishing “ﬂuctuations” of the free electromagnetic

ﬁeld.

Dipole Coupling of Fields and Atoms

The combined system of atoms and ﬁelds can be de-

scribed by the product quantum states |a, n of atom

states |a and ﬁeld states |n. The interaction Hamilto-

nian H

of the atom and the radiation ﬁeld is given by

(The A

-term plays an important role in energy shifts

and can only be neglected when radiative processes

involving energy exchange are considered.)

=−

p· A(r) +

(r). (79.12)

This interaction causes the atom to exchange energy

with the radiation ﬁeld. In the dipole approximation, the

coupling strength is proportional to the component of

the atomic dipole moment d

= qe|r

|g along the

electric ﬁeld, with coupling constant

(r) =|d

· u

(r)E

|/ . (79.13)

In the rotating wave approximation (RWA)(Chapts.68,

69,and70),

RWA





†

∗

†



, (79.14)

where we have used the atomic operators of (79.1).

In a continuous electromagnetic spectrum, the atom

interacts with a large number of modes having quantum

numbers µ, yielding exponential decay of an excited

atomic level at the rate [79.6]

2π







−ω



(79.15)

Here we have separated the discrete (

µ) and the contin-

uous part (wave vector k) of the mode index µ.Ifg

(79.13) does not vary much across a narrow resonance,

then

 2π



(ω

)





(ω

). (79.16)

The density of states corresponding to the continuous

mode index k of dimension ν can be evaluated on a ν-

dimensional ﬁctitious volume V

(ν)



δ(ω

µ,k

−ω) →

(ν)

(2π)

∞



kδ(ω

µ,k

−ω) ,

(79.17)

provided ω(k) is known, and by converting the sum

(This is formally accomplished by taking the limit of

∆k = 2π/l for large l,wherel is a linear dimension

of an artiﬁcial resonator, and the resonator volume is

V = l

. If the relation between mode spacing ∆k and

geometric dimension is nonlinear in a more complex

geometry, this analysis can be very complicated.) over

plane wave vectors k into an integral.

The Rate of Spontaneous Emission

In free space [ω(k)

= (ck)

], the sum in (79.16) con-

tributes a factor of two, due to polarization, to the total

density of states in free space, ρ

free

(ω) = Vω

/π

When the vector coupling of atom and ﬁeld (79.13)is

replaced by its average in isotropic free space, that is,

by 1/3, the result

= A

3π

(79.18)

is obtained for the decay rate A

as measured by the

natural linewidth Γ

79.2 Weak Coupling in Cavity QED

The regime of weak cavity QED generally applies when

an atom is coupled to a continuum of radiation modes.

This is always the case with mirrors, waveguides, or bad

cavities. The signatures of weak cavity QED are modiﬁ-

cations of the rate of spontaneous emission, as well as the

existence of van der Waals and Casimir forces. Formally,

this regime is well described by perturbation theory.

79.2.1 Radiating Atoms in Waveguides

Within the continuous spectrum of a waveguide, ra-

diative decay of an excited atomic level remains

exponential, and Γ

may be determined as in the pre-

ceding section. We now consider the modiﬁcations of

spontaneous decay in a parallel plate waveguide. Ac-

Part F 79.2