semiclassical picture of the atom“¬eld interaction; it is prima-facie evidence for the

quantized nature of the electromagnetic ¬eld.

0.7 30

T = 2.5 K N = 3000 s’1

0.6 40

Signal depth [%]

Probability Pe(t)

0.5 50

0.4 60

63, P3/2 61, D5/2

85Rb

0.3 70

0 50 100 150

Time of flight through cavity [µs]

Fig. 12.3 Probability of ¬nding the atom in the upper maser level as a function of the time

of ¬‚ight of a Rydberg atom through a superconducting cavity. The ¬‚ux of atoms was around

3000 atoms per second. Note the revival of upper state atoms which occurs at around 150 µs.

(Reproduced from Rempe et al. (1987).)

¿¼ Cavity quantum electrodynamics

12.4 Exercises

12.1 Dressed states

(1) Verify eqns (12.10)“(12.16).

(2) Solve the eigenvalue problem for eqn (12.16) and thus derive eqns (12.17)“(12.22).

(0)

(3) Display level repulsion by plotting the (normalized) bare eigenvalues µ1,n / ωC

(0)

and µ2,n / ωC , and dressed eigenvalues µ1,n / ωC and µ2,n / ωC as functions of the

detuning δ/ωC .

12.2 Collapse and revival for pure initial states

(1) For the initial state |Ψ (0) = | 2 , m (0) , verify the solution (12.26).

(2) Carry out the steps required to derive eqn (12.31).

(3) Write a program to evaluate eqn (12.32), and use it to study the behavior of P2 (t)

at times following the ¬rst revival.

Collapse and revival for a mixed initial state—

12.3

Replace the pure initial state of the previous problem with the mixed state

∞

pn | 2 , n (0) (0)

ρ= 2 , n| .

n=0

(1) Show that this state evolves into

∞

pn | 2 , n; t (0) (0)

ρ= 2 , n; t| .

n=0

(2) Derive the expression for P2 (t).

(3) Assume that pn is the thermal distribution for a given average photon number n.

Evaluate and plot P2 (t) numerically for the value of n used in Fig. 12.1. Comment

on the comparison between the two plots.

13

Nonlinear quantum optics

The interaction of light beams with linear optical devices is adequately described by

the quantum theory of light propagation explained in Section 3.3, Chapter 7, and

Chapter 8, but some of the most important applications involve modi¬cation of the

incident light by interactions with nonlinear media, e.g. by frequency doubling, spon-

taneous down-conversion, four-wave mixing, etc. These phenomena are the province

of nonlinear optics. Classical nonlinear optics deals with ¬elds that are strong enough

to cause appreciable change in the optical properties of the medium, so that the weak-

¬eld condition of Section 3.3.1 is violated. A Bloch equation that includes dissipative

e¬ects, such as scattering from other atoms and spontaneous emission, describes the

response of the atomic density operator to the classical ¬eld.

For the present, we do not need the details of the Bloch equation. All we need to

know is that there is a characteristic response time, Tmed, for the medium. The classical

envelope ¬eld evolves on the time scale T¬‚d ∼ 1/„¦, where „¦ is the characteristic Rabi

frequency. If Tmed ≈ T¬‚d the coupled equations for the atoms and the ¬eld must be

solved together. This situation arises, for example, in the phenomenon of self-induced

transparency and in the theory of free-electron lasers (Yariv, 1989, Chaps 13, 15).

In many applications of interest for nonlinear optics, the incident radiation is de-

tuned from the atomic resonances in order to avoid absorption. As shown in Section

11.3.3, this justi¬es the evaluation of the atomic density matrix by adiabatic elimi-

nation. In this approximation, the atoms appear to follow the envelope ¬eld instan-

taneously; they are said to be slaved to the ¬eld. Even with this simpli¬cation, the

Bloch equation cannot be solved exactly, so the atomic density operator is evaluated

by using time-dependent perturbation theory in the atom“¬eld coupling. In this calcu-

lation, excited states of an atom only appear as virtual intermediate states; the atom

is always returned to its original state. This means that both spontaneous emission

and absorption are neglected.

13.1 The atomic polarization

Substituting the perturbative expression for the atomic density matrix into the source

terms for Maxwell™s equations results in the apparent disappearance, via adiabatic

elimination, of the atomic degrees of freedom. This in turn produces an expansion of

the medium polarization in powers of the ¬eld, which is schematically represented by

(1) (2) (3)

Pi = χij Ej + χijk Ej Ek + χijkl Ej Ek El + · · · , (13.1)

0

¿¾ Nonlinear quantum optics

where the χ(n) s are the tensor nonlinear susceptibilities required for dealing with

(2)

anisotropic materials and E is the classical electric ¬eld. The term χijk Ej Ek describes

the combination of two waves to provide the source for a third, so it is said to describe

(3)

three-wave mixing. In the same way χijkl Ej Ek El is associated with four-wave mix-

ing.

A substance is called weakly nonlinear if the dielectric response is accurately

represented by a small number of terms in the expansion (13.1). This approximation

is the basis for most of nonlinear optics,1 but there are nonlinear optical e¬ects that

cannot be described in this way, e.g. saturation in lasers (Yariv, 1989, Sec. 8.7). The

higher-order terms in the polarization lead to nonlinear terms in Maxwell™s equations

that represent self-coupling of individual modes as well as coupling between di¬er-

ent modes. These terms describe self-actions of the electromagnetic ¬eld that are

mediated by the interaction of the ¬eld with the medium.

Quantum nonlinear optics is concerned with situations in which there are a small

number of photons in some or all of the ¬eld modes. In this case the quantized ¬eld

theory is required, but the correspondence principle assures us that the e¬ects arising in

classical nonlinear optics must also be present in the quantum theory. Thus the classical

three- and four-wave mixing terms correspond to three- and four-photon interactions.

Since the quantum ¬elds are typically weak, these nonlinear phenomena are often

unobservably small. There are, however, at least two situations in which this is not

the case. According to eqn (2.188), the vacuum ¬‚uctuation ¬eld strength in a physical

cavity of volume V is ef = ωf /2 0 V . This shows that substantial ¬eld strengths can

be achieved, even for a single photon, in a small enough cavity. A second exception

depends on the fact that the frequency-dependent nonlinear susceptibilities display

resonant behavior. If the detuning from resonance is made as small as possible”

i.e. without violating the conditions required for adiabatic elimination”the nonlinear

couplings are said to be resonantly enhanced.

When both of these conditions are met, the interaction between the medium and

the ¬eld can be so strong that the electromagnetic ¬eld will interact with itself, even

when there are only a few quanta present. This happens, for example, when microwave

photons inside a cavity interact with each other via a medium composed of Rydberg

atoms excited near resonance. In this case the interacting microwave photons can even

form a photon ¬‚uid.

In addition to these practical issues, there are situations in which the use of quan-

tum theory is mandatory. In the phenomenon of spontaneous down-conversion, a non-

linear optical process couples vacuum ¬‚uctuations of the electromagnetic ¬eld to an

incident beam of ultraviolet light so that an ultraviolet photon decays into a pair of

lower-energy photons. E¬ects of this kind cannot be described by the semiclassical

theory.

In Section 13.2 we will brie¬‚y review some features of classical nonlinear optics and

introduce the corresponding quantum description. In the following two sections we will

discuss examples of three- and four-photon coupling. In each case the quantum theory

1 For

a selection of recent texts on nonlinear optics, see Shen (1984), Schubert and Wilhelmi (1986),

Butcher and Cotter (1990), Boyd (1992), and Newell and Moloney (1992).

¿¿

Weakly nonlinear media

will be developed in a phenomenological way, i.e. it will be based on a conjectured form

for the Hamiltonian. This is in fact the standard way of formulating a quantum theory.

The choice of the Hamiltonian must ultimately be justi¬ed by comparing the results of

calculations with experiment, as there will always be ambiguities”such as in operator

ordering, coordinate choices (e.g. Cartesian versus spherical), etc.”which cannot be

settled by theoretical arguments alone. Quantum theory is richer than classical theory;

consequently, there is no unique way of deriving the quantum Hamiltonian from the

classical energy.

13.2 Weakly nonlinear media

13.2.1 Classical theory

A Plane waves in crystals

Many applications of nonlinear optics involve the interaction of light with crystals, so

we brie¬‚y review the form of the fundamental plane waves in a crystal. As explained

in Appendix B.5.3, the ¬eld can be expressed as

1

E (+) (r, t) = i √ Fks ±ks µks ei(k·r’ωks t) , (13.2)

V ks

where µks is a crystal eigenpolarization, the polarization-dependent frequency ωks is

a solution of the dispersion relation

c2 k 2 = ω 2 n2 (ω) , (13.3)

s

and ns (ω) is the index of refraction associated with the eigenpolarization µks . The

normalization constant,

ωks vg (ωks )

Fks = , (13.4)

2 0 ns (ωks ) c

has been chosen to smooth the path toward quantization, and vg (ωks ) = dωks /dk is

the group velocity. For a polychromatic ¬eld, the expression (3.116) for the envelope

(+)

Eβ is replaced by

1

(+)

(r, t) = √

Eβ Fks ±ks µks ei(k·r’∆βk t) , (13.5)

V ks

where the prime on the k-sum indicates that it is restricted to k-values such that the

detuning, ∆βks = ωks ’ωβ , is small compared to the minimum spacing between carrier

frequencies, i.e. |∆βks | min {|ω± ’ ωβ | , ± = β}.

B Nonlinear susceptibilities

Symmetry, or lack of symmetry, with respect to spatial inversion is a fundamental

distinction between di¬erent materials. A medium is said to have a center of sym-

metry, or to be centrosymmetric, if there is a spatial point (which is conventionally

¿ Nonlinear quantum optics

chosen as the origin of coordinates) with the property that the inversion transforma-

tion r ’ ’ r leaves the medium invariant. When this is true, the polarization must

behave as a polar vector, i.e. P ’ ’P. The electric ¬eld is also a polar vector, so

(2)

eqn (13.1) implies that all even-order susceptibilities”in particular χijk ”vanish for

centrosymmetric media. Vapors, liquids, amorphous solids, and some crystals are cen-

trosymmetric. The absence of a center of symmetry de¬nes a non-centrosymmetric

(2)

crystal. This is the only case in which it is possible to obtain a nonvanishing χijk .

(3)

There is no such general restriction on χijkl ”or any odd-order susceptibility”since

(3) (3)

the third-order polarization, Pi = χijkl Ej Ek El , is odd under E ’ ’E.

The schematic expansion (13.1) does not explicitly account for dispersion, so we

now turn to the exact constitutive relation

(n) (n)

Pi dt1 · · · dtn χij1 j2 ···jn (t ’ t1 , t ’ t2 , . . . , t ’ tn )

(r, t) = 0

— Ej1 (r, t1 ) · · · Ejn (r, tn ) (13.6)

for the nth-order polarization, which is treated in greater detail in Appendix B.5.4.

This time-domain form explicitly displays the history dependence of the polarization”

previously encountered in Section 3.3.1-B”but the equivalent frequency-domain form

n

dν1 dνn

(n) (n)

Pi ··· 2πδ ν ’

(r, ν) = νp χij1 j2 ···jn (ν1 , . . . , νn )

0

2π 2π p=1

— Ej1 (r, ν1 ) · · · E jn (r, νn ) (13.7)

is more useful in practice.

C E¬ective electromagnetic energy

The derivation in Section 3.3.1-B of the e¬ective electromagnetic energy for a linear,

dispersive dielectric can be restated in the following simpli¬ed form.

(1) Start with the expression for the energy in a static ¬eld.

(2) Replace the static ¬eld by a time-dependent ¬eld.

(3) Perform a running time-average”as in eqn (3.136)”on the resulting expression.

For a nonlinear dielectric, we carry out step (1) by using the result

D

E (r) · d (D (r))

Ues = 3

dr

Vc 0

P

0

E (r) · d (P (r))

d3 rE 2 (r) + d3 r

= (13.8)

2 Vc Vc 0

for the energy of a static ¬eld in a dielectric occupying the volume Vc (Jackson, 1999,

Sec. 4.7). Substituting eqn (13.1) into this expression leads to an expansion of the

energy in powers of the ¬eld amplitude:

Ues = Ues + Ues + Ues + · · · .

(2) (3) (4)

(13.9)

¿

Weakly nonlinear media

The ¬rst term on the right is discussed in Section 3.3.1-B, so we can concentrate on

the higher-order (n 3) terms:

1 (n’1)

Ues = d3 rEi (r) Pi

(n)

(r) .

n Vc

In steps (2) and (3), we replace the static energy by the e¬ective energy,

1 (n’1)

Ues ’ Uem (t) = d3 r Ei (r, t) Pi

(n) (n)

(r, t) for n 3, (13.10)

n Vc

and use eqn (13.6) to evaluate the nth-order polarization. Our experience with the

(2)

quadratic term, Uem (t), tells us that eqn (13.10) will only be useful for polychromatic

¬elds; therefore, we impose the condition 1/ωmin T 1/∆ωmax on the averaging

time, where ωmin is the smallest carrier frequency and ∆ωmax is the largest spectral

width for the polychromatic ¬eld. This time-averaging eliminates all rapidly-varying

terms, while leaving the slowly-varying envelope ¬elds unchanged.

The lowest-order energy associated with the nonlinear polarizations is

1 (2)

Uem (t) = d3 r Ei (r, t) Pi

(3)

(r, t) , (13.11)

3 Vc

(2)

so the next task is to evaluate Pi (r, t) for a polychromatic ¬eld. This is done by

applying the exact relation (13.7) for n = 2, and using the expansion (3.119) for a

polychromatic ¬eld to ¬nd:

dν1 dν2

(2) (2)

Pi 2πδ (ν ’ ν1 ’ ν2 ) χijk (ν1 , ν2 )

(r, ν) = 0

2π 2π

β,γ σ ,σ =±

(σ ) (σ )

— E βj (r, ν1 ’ σ ωβ ) E γk (r, ν2 ’ σ ωγ ) . (13.12)

Weak dispersion means that the susceptibility is essentially constant across the spectral

(±) (2)

width of each sharply-peaked envelope function, E βj (r, ν); therefore, Pi (r, ν) can

be approximated by

dν1 dν2

(2) (2)

Pi 2πδ (ν ’ ν1 ’ ν2 ) χijk (σ ωβ , σ ωγ )

(r, ν) = 0

2π 2π

β,γ σ ,σ =±

(σ ) (σ )

— E βj (r, ν1 ’ σ ωβ ) E γk (r, ν2 ’ σ ωγ ) . (13.13)

Carrying out an inverse Fourier transform yields the time-domain relation,

(2) (2)

Pi (r, t) = χijk (σ ωβ , σ ωγ )

0

β,γ σ ,σ =±

(σ ) (σ )

— E βj (r, t) E γk (r, t) e’i(σ ωβ +σ ωγ )t

, (13.14)

which shows that the time-averaging has eliminated the history dependence of the

polarization.

¿ Nonlinear quantum optics

(3)

Using eqn (13.14) to evaluate the expression (13.11) for Uem (t) is simpli¬ed by

the observation that the slowly-varying envelope ¬elds can be taken outside the time

average, so that

1 (σ) (σ )

(2)

Uem (t) = χijk (σ ωβ , σ ωγ ) E ±i (r, t) E βj (r, t)

(3)

d3 r

3 Vc ±,β,γ σ,σ ,σ

(σ )

— E γk (r, t) e’i(σω± +σ ωβ +σ ωγ )t

. (13.15)

The frequencies in the exponential all satisfy ωT 1, so the remaining time-average,

T /2

1

’i(σω± +σ ωβ +σ ωγ )t

d„ e’i(σω± +σ ωβ +σ ωγ )(t+„ )

e = ,

T ’T /2

vanishes unless

σω± + σ ωβ + σ ωγ = 0 . (13.16)

This is called phase matching. By convention, the carrier frequencies are positive;

consequently, phase matching in eqn (13.15) always imposes conditions of the form

ω ± = ωβ + ωγ . (13.17)

(+) (+) (’) (’) (’) (+)

This in turn means that only terms of the form E E E or E E E will

contribute. By making use of the symmetry properties of the susceptibility, reviewed

in Appendix B.5.4, one ¬nds the explicit result

(2)

Uem (t) =

(3)

χijk (ωβ , ωγ ) δω± ,ωβ +ωγ

0

±,β,γ

(’) (+) (+)

— d3 r E ±i (r, t) E βj (r, t) E γk (r, t) + CC . (13.18)

Vc

In many applications, the envelope ¬elds will be expressed by an expansion in some

appropriate set of basis functions. For example, if the nonlinear medium is placed in a

resonant cavity, then the carrier frequencies can be identi¬ed with the frequencies of

the cavity modes, and each envelope ¬eld is proportional to the corresponding mode

function. More generally, the ¬eld can be represented by the plane-wave expansion

2

(13.2), provided that the power spectrum |±ks | exhibits well-resolved peaks at ωks =

ω± , where ω± ranges over the distinct monochromatic carrier frequencies. With this

restriction held ¬rmly in mind, the explicit sums over the distinct monochromatic

waves can be replaced by sums over the plane-wave modes, so that

i

gs0 s1 s2 (ω1 , ω2 ) [±0 ±— ±— ’ CC]

Uem =

(3) (3)

12

3/2

V

k0 s0 ,k1 s1 ,k2 s2

— C (k0 ’ k1 ’ k2 ) δω0 ,ω1 +ω2 , (13.19)

where ±0 = ±k0 s0 , etc., and

¿

Weakly nonlinear media

C (k) = d3 reik·r (13.20)

Vc