dt

where the complex, time-dependent Rabi frequency is de¬ned by

d · E (t)

„¦ (t) = . (11.169)

Combining the notation ρqp (t) = µq |ρ (t)| µp with the hermiticity condition

ρ12 (t) = ρ— (t) allows eqn (11.168) to be written out explicitly as

21

d

ρ11 (t) = ’„¦ (t) e’iδt ρ21 (t) + „¦— (t) eiδt ρ12 (t) ,

i (11.170)

dt

d

i ρ22 (t) = „¦ (t) e’iδt ρ21 (t) ’ „¦— (t) eiδt ρ12 (t) , (11.171)

dt

d

i ρ12 (t) = ’„¦ (t) e’iδt [ρ22 (t) ’ ρ11 (t)] , (11.172)

dt

where ρ11 and ρ22 are the occupation probabilities for the two levels and the o¬-

diagonal term ρ12 is called the atomic coherence. For most applications, it is better

to eliminate the explicit exponentials by setting

ρ12 (t) = e’iδt ρ12 (t) , ρ22 (t) = ρ22 (t) , ρ11 (t) = ρ11 (t) , (11.173)

to get

¿ Coherent interaction of light with atoms

d

ρ22 (t) = i [„¦ (t) ρ12 (t) ’ „¦— (t) ρ21 (t)] , (11.174)

dt

d

ρ11 (t) = ’i [„¦ (t) ρ12 (t) ’ „¦— (t) ρ21 (t)] , (11.175)

dt

d

ρ (t) = iδρ21 (t) + i„¦ (t) (ρ11 (t) ’ ρ22 (t)) . (11.176)

dt 21

The sum of eqns (11.174) and (11.175) conveys the reassuring news that the total

occupation probability, ρ11 (t) + ρ22 (t), is conserved.

For a strictly monochromatic ¬eld, „¦ (t) = „¦, these equations can be solved to

obtain a generalized description of Rabi ¬‚opping, but there is a more pressing question

to be addressed. This is the neglect of the decay of the upper level by spontaneous

emission. We have seen in Section 11.2.2 that the upper-level amplitude C1 (t) ∼

exp (’“t/2), so in the absence of the external ¬eld the occupation probability ρ11 of

the upper level and the coherence ρ12 (t) should behave as

ρ22 (t) ∼ C2 (t) C2 (t) ∼ e’w21 t ,

—

(11.177)

ρ21 (t) ∼ C2 (t) C1 (t) ∼ e’w21 t/2 .

—

An equivalent statement is that the terms ’w21 ρ22 (t) and ’w21 ρ21 (t) /2 should ap-

pear on the right sides of eqns (11.174) and (11.175) respectively. This would be the

end of the story if spontaneous emission were the only thing that has been left out,

but there are other e¬ects to consider. In atomic vapors, elastic scattering from other

atoms will disturb the coherence ρ12 (t) and cause an additional decay rate, and in

crystals similar e¬ects arise due to lattice vibrations and local ¬eld ¬‚uctuations.

The general description of dissipative e¬ects will be studied Chapter 14, but for the

present we will adopt a phenomenological approach in which eqns (11.174)“(11.176)

are replaced by the Bloch equations:

d

ρ22 (t) = ’w21 ρ22 (t) + i [„¦ (t) ρ12 (t) ’ „¦— (t) ρ21 (t)] , (11.178)

dt

d

ρ (t) = w21 ρ22 (t) ’ i [„¦ (t) ρ12 (t) ’ „¦— (t) ρ21 (t)] , (11.179)

dt 11

d

ρ (t) = (iδ ’ “21 ) ρ21 (t) + i„¦ (t) (ρ11 (t) ’ ρ22 (t)) , (11.180)

dt 21

where the decay rate w21 and the dephasing rate “21 are parameters to be deter-

mined from experiment. In this simple two-level model the lower level is the ground

state, so the term w21 ρ22 in eqn (11.179) is required in order to guarantee conserva-

tion of the total occupation probability. This allows eqns (11.179) and (11.180) to be

replaced by

ρ11 (t) + ρ22 (t) = 1 , (11.181)

d

[ρ (t) ’ ρ11 (t)] = ’w21 ’ w21 [ρ22 (t) ’ ρ11 (t)] + 2i [„¦ (t) ρ12 (t) ’ „¦— (t) ρ21 (t)] ,

dt 22

(11.182)

where ρ22 (t) ’ ρ11 (t) is the population inversion. In the literature, the parameters

w21 and “21 are often represented as

¿

The semiclassical limit

1 1

w21 = , “21 = , (11.183)

T1 T2

where T1 and T2 are respectively called the longitudinal and transverse relaxation

times. This terminology is another allusion to the analogy with a spin-1/2 system

precessing in an external magnetic ¬eld. Another common usage is to call T1 and T2

respectively the on-diagonal and o¬-diagonal relaxation times.

In the frequency domain, the slow time variation of the ¬eld envelope E (t) is

ω0 , where ∆ω0 is the spectral width of E (ω).

represented by the condition ∆ω0

The detuning and the dephasing rate are also small compared to the carrier frequency,

but either or both can be large compared to ∆ω0 . This limit can be investigated by

means of the formal solution,

t

’i dt „¦ (t ) [ρ22 (t ) ’ ρ11 (t )] e(iδ’“21 )(t’t ) ,

(iδ’“21 )(t’t0 )

ρ21 (t) = ρ21 (t0 ) e

t0

(11.184)

w21 /2 > 0, the formal solution has the t0 ’ ’∞ limit

of eqn (11.180). Since “21

t

ρ21 (t) = ’i dt „¦ (t ) [ρ22 (t ) ’ ρ11 (t )] e(iδ’“21 )(t’t ) . (11.185)

’∞

The exponential factor exp [’“21 (t ’ t )] implies that the main contribution to the

integral comes from the interval t ’ 1/“21 < t < t, while the rapidly oscillating

exponential exp [iδ (t ’ t )] similarly restricts contributions to the interval t ’ 1/ |δ| <

max (∆ω0 , w21 ) or |δ|

t < t. Thus if either of the conditions “21 max (∆ω0 , w21 )

is satis¬ed, the main contribution to the integral comes from a small interval t ’ ∆t <

t < t. In this interval, the remaining terms in the integrand are e¬ectively constant;

consequently, they can be evaluated at the upper limit to ¬nd:

„¦ (t) [ρ22 (t) ’ ρ11 (t)]

ρ21 (t) = . (11.186)

δ + i“21

The approximation of the atomic coherence by this limiting form is called adia-

batic elimination, by analogy to the behavior of thermodynamic systems. A ther-

modynamic parameter, such as the pressure of a gas, will change in step with slow

changes in a control parameter, e.g. the temperature. The analogous behavior is seen in

eqn (11.186) which shows that the atomic coherence ρ21 (t) follows the slower changes

in the populations. For a large dephasing rate, exponential decay drives ρ21 (t) to the

equilibrium value given by eqn (11.186). In the case of large detuning, the deviation

from the equilibrium value oscillates so rapidly that its contribution averages to zero.

Once the mechanism of adiabatic elimination is understood, its application reduces

to the following simple rule.

(a) If |“qp + i∆qp | is large, set dρqp /dt = 0.

(b) Use the resulting algebraic relations to eliminate as many ρqp s as possible.

(11.187)

¿ Coherent interaction of light with atoms

Substituting ρ21 (t) from eqn (11.186) into eqn (11.182) leads to

4 |„¦ (t)|2 “21

d

[ρ (t) ’ ρ11 (t)] = ’w21 ’ [ρ22 (t) ’ ρ11 (t)] ,

w21 + (11.188)

dt 22 δ 2 + “2

21

which shows that the adiabatic elimination of the atomic coherence does not neces-

sarily imply the adiabatic elimination of the population inversion. The solution of this

di¬erential equation also shows that no pumping scheme for a strictly two-level atom

can change the population inversion from negative to positive. Since laser ampli¬ca-

tion requires a positive inversion, this implies that laser action can only be described

by atoms with at least three active levels.

If w21 = O (∆ω0 ) the population inversion and the external ¬eld change on the

same time scale. Adiabatic elimination of the population inversion will only occur for

w21 ∆ω0 . In this limit the adiabatic elimination rule yields

w21

ρ22 (t) ’ ρ11 (t) = ’ < 0. (11.189)

4|„¦(t)|2 “21

w21 + δ 2 +“2

21

When adiabatic elimination is possible for both the atomic coherence and the popula-

tion inversion, the atomic density matrix appears to react instantaneously to changes

in the external ¬eld. What this really means is that transient e¬ects are either sup-

pressed by rapid damping (w21 ∆ω0 , and “21 ∆ω0 ) or average to zero due to

rapid oscillations (|δ| ∆ω0 ). The apparently instantaneous response of the two-level

atom is also displayed by multilevel atoms when the corresponding conditions are

satis¬ed.

For later applications it is more useful to substitute the adiabatic form (11.186)

into the original equations (11.179) and (11.178) to get a pair of equations for the

occupation probabilities Pq = ρqq . In the strictly monochromatic case, one ¬nds

dP2

= W12 P1 ’ (w21 + W12 ) P2 ,

dt (11.190)

dP1

= ’W12 P1 + (w21 + W12 ) P2 ,

dt

where

2 |„¦|2 “21

W12 =2 (11.191)

δ + “2 21

is the rate of 1 ’ 2 transitions (absorptions) driven by the ¬eld. By virtue of the

equality B1’2 = B2’1 , explained in Section 1.2.2, this is equal to the rate of 2 ’ 1

transitions (stimulated emissions) driven by the ¬eld. Equations (11.190) are called

rate equations and their use is called the rate equation approximation. The

occupation probability of |µ2 is increased by absorption from |µ1 and decreased by the

combination of spontaneous and stimulated emission to |µ1 . The inverse transitions

determine the rate of change of P1 , in such a way that probability is conserved. The

rate equations can be generalized to atoms with three or more levels by adding up all

of the (incoherent) processes feeding and depleting the occupation probability of each

level.

¿

Exercises

11.4 Exercises

11.1 The antiresonant Hamiltonian

(ar)

Apply the de¬nition (11.17) of the running average to Hint (t) to ¬nd:

ωk d— · eks ’i(ω21 +ωk )t

(ar)

H int (t) = ’i e K (ω21 + ωk ) aks σ’ + HC .

2 0V

ks

Use the properties of the cut-o¬ function and the conventions ω21 > 0 and ωk > 0 to

(ar)

explain why dropping H int (t) is a good approximation.

11.2 The Weisskopf“Wigner method

(1) Fill in the steps needed to go from eqn (11.80) to eqn (11.84).

2

(2) Assume that |K (∆)| is an even function of ∆ and show that

∞ ∞

δω21 3 1

2 2

d∆ |K (∆)| + d∆ ∆2 |K (∆)| .

= 3

w21 2πω21 2πω21

’∞ ’∞

Use this to derive the estimate δω21 /w21 = O (wK /ω21 ) 1.

11.3 Atomic radiation ¬eld

(1) Use the eqns (11.26) and (B.48) to show that

ωk K (∆k ) — (d— · ∇) ∇ ik·r

ωk —

g eks e ik·r

= d+ e .

2 0 V ks k2

2 0V

s

(2) With the aid of this result, convert the k-sum in eqn (11.54) to an integral. Show

that

4π sin (kr)

d„¦k eik·r = ,

kr

and then derive eqn (11.56).

11.4 Slowly-varying envelope operators

De¬ne envelope operators σ ’ (t) = exp (iω21 t) σ’ (t), σ z (t) = σz (t), and aks (t) =

exp (iωk t) aks (t).

(1) Use eqns (11.47)“(11.49) to derive the equations satis¬ed by the envelope opera-

tors.

(2) From these equations argue that the envelope operators are slowly varying, i.e.

essentially constant over an optical period.

Two-photon cascade—

11.5

(1) Substitute the ansatz (11.106) into the Schr¨dinger equation for the Hamiltonian

o

(11.103) and obtain the di¬erential equations for the coe¬cients.

(2) Use the given initial conditions to derive eqns (11.107)“(11.109).

¿¼ Coherent interaction of light with atoms

(3) Carry out the steps needed to arrive at eqn (11.113).

(4) Starting with the normalization |K (0)| = 1 and the fact that |K (∆ )|2 is an even

function, use an argument similar to the derivation of eqn (11.89) to show that

Dk ≈ w21 /2.

(5) Evaluate the residue for the poles of Xks,k s (ζ) to ¬nd the coe¬cients G1 , G2 ,

and G3 , and then derive eqn (11.122).

12

Cavity quantum electrodynamics

In Section 4.9 we studied spontaneous emission in free space and also in the modi¬ed

geometry of a planar cavity. The large dimensions in both cases”three for free space

and two for the planar cavity”provide the densely packed energy levels that are

essential for the validity of the Fermi golden rule calculation of the emission rate.

Cavity quantum electrodynamics is concerned with the very di¬erent situation of an

atom trapped in a cavity with all three dimensions comparable to the wavelength of the

emitted radiation. In this case the radiation modes are discrete, and the Fermi golden

rule cannot be used. Instead of disappearing into the blackness of in¬nite space, the

emitted radiation is re¬‚ected from the nearby cavity walls, and soon absorbed again by

the atom. The re-excitation of the atom results in a cycle of emissions and absorptions,

rather than irreversible decay. In the limit of strong ¬elds, i.e. many photons in a single

mode, this cyclic behavior is described in Section 11.3.2 as Rabi ¬‚opping. The exact

periodicity of Rabi ¬‚opping is, however, an artifact of the semiclassical approximation,

in which the discrete nature of photons is ignored. In the limit of weak ¬elds, the grainy

nature of light makes itself felt in the nonclassical features of collapse and revival of

the probability for atomic excitation.

There are several possible experimental realizations of cavity quantum electrody-

namics, but the essential physical features of all of them are included in the Jaynes“

Cummings model discussed in Section 12.1. In Section 12.2 we will use this model to

describe the intrinsically quantum phenomena of collapse and revival of the radiation

¬eld in the cavity. A particular experimental realization is presented in Section 12.3.

12.1 The Jaynes“Cummings model

12.1.1 De¬nition of the model

In its simplest form, the Jaynes“Cummings model consists of a single two-level atom

located in an ideal cavity. For the two-level atom we will use the treatment given in

Section 11.1.1, in which the two atomic eigenstates are | 1 and | 2 with 1 < 2 . The

Hamiltonian is then

ω0

Hat = σz , (12.1)

2

where we have chosen the zero of energy so that 2 + 1 = 0, and set ω0 ≡ ( 2 ’ 1) / .

For the electromagnetic ¬eld, we use the formulation in Section 2.1, so that

ω κ a† aκ

Hem = (12.2)

κ

κ

¿¾ Cavity quantum electrodynamics

is the Hamiltonian, and

ωκ

aκ E κ (r)

E(+) (r) = i (12.3)

20

κ

is the positive-frequency part of the electric ¬eld (in the Schr¨dinger picture).

o

Adapting the general result (11.27) to the cavity problem gives the RWA interaction

Hamiltonian

Hrwa = ’d · E(+) σ+ ’ d— · E(’) σ’

gκ a† σ’ ;

—

= ’i gκ aκ σ+ + i (12.4)

κ

κ κ

where d = d is the dipole matrix element; the coupling frequencies are

1 2

ωκ d · E κ (R)

gκ = K (ω0 ’ ωκ ) ; (12.5)

20

K (ω0 ’ ωκ ) is the RWA cut-o¬ function; and R is the position of the atom.

We will now drastically simplify this model in two ways. The ¬rst is to assume

that the center-of-mass motion of the atom can be treated classically. This means

that ω0 should be interpreted as the Doppler-shifted resonance frequency. In many

cases the Doppler e¬ect is not important; for example, for microwave transitions in

Rydberg atoms passing through a resonant cavity, or single atoms con¬ned in a trap.

The second simpli¬cation is enforced by choosing the cavity parameters so that the

lowest (fundamental) mode frequency is nearly resonant with the atomic transition,

while all higher frequency modes are well out of resonance. This guarantees that only

the lowest mode contributes to the resonant Hamiltonian; consequently, the family of

annihilation operators aκ can be reduced to the single operator a for the fundamental

mode. From now on, we will call the fundamental frequency the cavity frequency

ωC and the corresponding mode function E C (R) the cavity mode.

The total Hamiltonian for the Jaynes“Cummings model is therefore HJC = H0 +

Hint , where

H0 = ωC a† a + ( ω0 /2) σz , (12.6)

Hint = ’i gaσ+ + i ga† σ’ , (12.7)

and

ωC d · E C (R)

g= . (12.8)

20

By appropriate choice of the phases in the atomic eigenstates | and | , we can

1 2

always arrange that g is real.

12.1.2 Dressed states

The interaction Hamiltonian in eqn (12.7) has the same general form as the interac-

tion Hamiltonian (11.25) for the Weisskopf“Wigner model of Section 11.2.2, but it is

greatly simpli¬ed by the fact that only one mode of the radiation ¬eld is active. In

¿¿

The Jaynes“Cummings model

the Weisskopf“Wigner case, the in¬nite-dimensional subspaces Hse are left invariant

(mapped into themselves) under the action of the Hamiltonian. Since the Hamiltonians

have the same structure, a similar behavior is expected in the present case.

The product states,

| j , n (0) = | |n (n = 0, 1, . . .) , (12.9)

j

where the | j s (j = 1, 2) are the atomic eigenstates and the |n s are number states

for the cavity mode, provide a natural basis for the Hilbert space HJC of the Jaynes“

Cummings model. The | j , n (0) s are called bare states, since they are eigenstates of

the non-interacting Hamiltonian H0 :

H0 | j , n (0) = ( + n ωC ) | j , n (0) . (12.10)

j

Turning next to Hint , a straightforward calculation shows that

Hint | 1 , 0 (0) = 0 , (12.11)

which means that spontaneous absorption from the bare vacuum is forbidden in the

resonant wave approximation. Consequently, the ground-state energy and state vector

for the atom“¬eld system are, respectively,

ω0

=’ and |G = | 1 , 0 (0) .

µG = (12.12)

1

2

Furthermore, for each photon number n the pairs of bare states | 2 , n (0) and

| 1 , n + 1 (0) satisfy

√

Hint | 2 , n (0) = i g n + 1 | 1 , n + 1 (0) ,

√ (12.13)

Hint | 1 , n + 1 (0) = ’i g n + 1 | 2 , n (0) .

Consequently, each two-dimensional subspace

Hn = span | 2 , n (0) , | 1 , n + 1 (0) (n = 0, 1, . . .) (12.14)

is left invariant by the total Hamiltonian. This leads to the natural decomposition of

HJC as

HJC = HG • H0 • H1 • · · · , (12.15)

where HG = span | 1 , 0 (0) is the one-dimensional space spanned by the ground state.

In the subspace Hn the Hamiltonian is represented by a 2 — 2 matrix

√

’2ig n + 1

1 10 δ

√

+

HJC,n = n + ωC , (12.16)

’δ

01 2 2ig n + 1

2

where δ = ω0 ’ ωC is the detuning. This construction allows us to reduce the solution

of the full Schr¨dinger equation, HJC |¦ = µ |¦ , to the diagonalization of the 2 — 2-

o

matrix HJC,n for each n. The details are worked out in Exercise 12.1. For each subspace

¿ Cavity quantum electrodynamics

Hn , the exact eigenvalues and eigenvectors, which will be denoted by µj,n and |j, n

(j = 1, 2), respectively, are

1 „¦n

µ1,n = n+ ωC + , (12.17)

2 2

|1, n = sin θn | 2 , n (0) + cos θn | 1 , n + 1 (0) , (12.18)

1 „¦n

ωC ’

µ2,n = n+ , (12.19)

2 2

|2, n = cos θn | 2 , n (0) ’ sin θn | 1 , n + 1 (0) , (12.20)

where

δ 2 + 4g 2 (n + 1)

„¦n = (12.21)

is the Rabi frequency for oscillations between the two bare states in Hn . The probability

amplitudes for the bare states are given by

„¦n ’ δ

cos θn = ,

2

(„¦n ’ δ) + 4g 2 (n + 1)

√ (12.22)

2g n + 1

sin θn = .

2

(„¦n ’ δ) + 4g 2 (n + 1)

The bare (g = 0) eigenvalues

(0)

µ1,n = (n + 1/2) ωC + δ/2 ,

(12.23)

(0)

µ2,n = (n + 1/2) ωC ’ δ/2

are degenerate at resonance (δ = 0), but the exact eigenvalues satisfy

√

µ1,n ’ µ2,n = „¦n 2 g n + 1 . (12.24)

This is an example of the ubiquitous phenomenon of avoided crossing (or level

repulsion) which occurs whenever two states are coupled by a perturbation.

The eigenstates |1, n and |2, n of the full Jaynes“Cummings Hamiltonian HJC are

called dressed states, since the interaction between the atom and the ¬eld is treated

exactly. By virtue of this interaction, the dressed states are entangled states of the

atom and the ¬eld.

12.2 Collapses and revivals

With the dressed eigenstates of HJC in hand, we can write the general solution of the

time-dependent Schr¨dinger equation as

o

∞ 2

’iµG t/

Cj,n e’iµj,n t/ |j, n ,

|Ψ (t) = e CG |G + (12.25)

n=0 j=1

where the expansion coe¬cients are determined by the initial state vector according

to CG = G |Ψ (0) and Cj,n = j, n |Ψ (0) (j = 1, 2) (n = 0, 1, . . .). If the atom

¿

Collapses and revivals

is initially in the excited state | 2 and exactly m cavity photons are present, i.e.

|Ψ (0) = | 2 , m (0) , the general solution (12.25) specializes to |Ψ (t) = | 2 , m; t , where

„¦n t „¦n t

| 2 , n; t ≡ e’i(n+1/2)ωC t cos | 2 , n (0)

+ i cos (2θn ) sin

2 2

„¦n t

’ ie’i(n+1/2)ωC t sin (2θn ) sin | 1 , n + 1 (0) . (12.26)

2

At resonance, the probabilities for the states | 2 , m (0) and | 1 , m + 1 (0) are

√

2

2 , m |Ψ (t)

(0)

= cos2 g m + 1t ,

P2,m (t) =

(12.27)

√

2

+ 1 |Ψ (t)

(0) 2

P1,m+1 (t) = 1, m = sin g m + 1t ,

so”as expected”the system oscillates between the two atomic states by emission and

absorption of a single photon. The exact periodicity displayed here is a consequence

of the special choice of an initial state with a de¬nite number of photons. For m > 0,

this is analogous to the semiclassical problem of Rabi ¬‚opping driven by a ¬eld with

de¬nite amplitude and phase. The analogy to the classical case fails for m = 0, i.e. an

excited atom with no photons present. The classical analogue of this case would be

a vanishing ¬eld, so that no Rabi ¬‚opping would occur. The occupation probabilities

P2,0 (t) = cos2 (gt) and P1,1 (t) = sin2 (gt) describe vacuum Rabi ¬‚opping, which is

a consequence of the purely quantum phenomenon of spontaneous emission, followed

by absorption, etc.

For initial states that are superpositions of several photon number states, exact

periodicity is replaced by more complex behavior which we will now study. A super-

position,

∞

|Ψ (0) = Kn | 2 , n (0) , (12.28)

n=0

of the initial states | 2 , n (0) that individually lead to Rabi ¬‚opping evolves into

∞

|Ψ (t) = Kn | 2 , n; t , (12.29)

n=0

so the probability to ¬nd the atom in the upper state, without regard to the number

of photons, is

∞ ∞

2 2

2

2 , n |Ψ (t) |Kn | 2 , n | 2 , n; t

(0) (0)

P2 (t) = = . (12.30)

n=0 n=0

At resonance, eqn (12.27) allows this to be written as

∞

√

11 2

|Kn | cos 2 n + 1gt .

P2 (t) = + (12.31)

2 2 n=0

If more than one of the coe¬cients Kn is nonvanishing, this function is a sum of

oscillatory terms with incommensurate frequencies. Thus true periodicity is only found

¿ Cavity quantum electrodynamics

for the special case |Kn | = δnm for some ¬xed value of m. For any choice of the Kn s

the time average of the upper-level population is P2 (t) = 1/2.

In order to study the behavior of P2 (t), we need to make an explicit choice for the

Kn s. Let us suppose, for example, that the initial state is |Ψ (0) = | 2 |± , where |± is

a coherent state for the cavity mode. The coe¬cients are then |Kn | = e’|±| |±| /n!,

2

2 2n

and 2∞

√

1 e’|±|

2n

|±|

P2 (t) = + cos 2 n + 1gt . (12.32)

2 2 n=0 n!

Photon numbers for the coherent state follow a Poisson distribution, so the main

contribution to the sum over n will come from the range (n ’ ∆n, n + ∆n), where

2

n = |±| is the mean photon number and ∆n = |±| is the variance. For large n,

the corresponding spread in Rabi frequencies is ∆„¦ ∼ 2g. At very early times, t

1/g, the arguments of the cosines are essentially in phase, and P2 (t) will execute an

almost coherent oscillation. At later times, the variation of the Rabi frequencies with

photon number will lead to an e¬ectively random distribution of phases and destructive

interference. This e¬ect can be estimated analytically by replacing the sum over n

with an integral and evaluating the integral in the stationary-phase approximation.

The result,

1 e’|gt|

2

cos (2 |±| gt) for gt 1 ,

P2 (t) = + (12.33)

2 2

describes the collapse of the upper-level population to the time-averaged value of

1/2. This decay in the oscillations is neither surprising nor particularly quantal in

character. A superposition of Rabi oscillations due to classical ¬elds with random ¬eld

strengths would produce a similar decay.

What is surprising is the behavior of the upper-level population at still later times.

A numerical evaluation of eqn (12.32) reveals that the oscillations reappear after a

rephasing time trp ∼ 4π |±| /g. This revival”with P2 (t) = O (1)”is a speci¬cally

quantum e¬ect, explained by photon indivisibility. The revival is in turn followed by

another collapse. The ¬rst collapse and revival are shown in Fig. 12.1.

The classical nature of the collapse is illustrated by the dashed curve in the same

¬gure, which is calculated by replacing the discrete sum in eqn (12.32) by an integral.

The two curves are indistinguishable in the initial collapse phase, but the classical

(dashed) curve remains ¬‚at at the value 1/2 during the quantum revival. Thus the

experimental observation of a revival provides further evidence for the indivisibility

of photons. After a few collapse“revival cycles, the revivals begin to overlap and”as

shown in Exercise 12.2”P2 (t) becomes irregular.

¿

The micromaser

22

1

0.8

0.6

0.4

0.2

CJ

5 10 15 20 25 30

Fig. 12.1 The solid curve shows the probability P2 (t) versus gt, where the upper-level

population P2 (t) is given by eqn (12.32), and the average photon number is n = |±|2 = 10.

The dashed curve is the corresponding classical result obtained by replacing the discrete sum

over photon number by an integral.

12.3 The micromaser

The interaction of a Rydberg atom with the fundamental mode of a microwave cavity

provides an excellent realization of the Jaynes“Cummings model. The con¬guration

sketched in Fig. 12.2 is called a micromaser (Walther, 2003). It is designed so that”

with high probability”at most one atom is present in the cavity at any given time. A

velocity-selected beam of alkali atoms from an oven is sent into a laser excitation region,

where the atoms are promoted to highly excited Rydberg states. The size of a Rydberg

atom is characterized by the radius, aRyd = n2 2 /me2 , of its Bohr orbit, where np is

p

Atomic beam oven

Maser cavity

Field ionization

Velocity selector

Laser excitation

Channeltron

detectors Atomic beam

Fig. 12.2 Rubidium Rydberg atoms from an oven pass successively through a velocity selec-

tor, a laser excitation region, and a superconducting microwave cavity. After emerging from

the cavity, they are detected”in a state-selective manner”by ¬eld ionization, followed by

channeltron detectors. (Reproduced from Rempe et al. (1990).)

¿ Cavity quantum electrodynamics

the principal quantum number, and 2 /me2 is the Bohr radius for the ground state of

the hydrogen atom. These atoms are truly macroscopic in size; for example, the radius

of a Rydberg atom with np 100 is on the order of microns, instead of nanometers.

The dipole matrix element d = np |er| np + 1 for a transition between two adjacent

Rydberg states np + 1 ’ np is proportional to the diameter of the atom, so it scales

as n2 . On the other hand, for transitions between high angular momentum (circular)

p

states the frequency scales as ω ∝ 1/n3 , which is in the microwave range. According to

p

2

eqn (4.162) the Einstein A coe¬cient scales like A ∝ |d| ω 3 ∝ 1/n5 . Thus the lifetime

p

„ = 1/A ∝ n5 of the upper level is very long, and the neglect of spontaneous emission

p

is a very good approximation.

The opposite conclusion follows for absorption and stimulated emission, since the

relation (4.166) between the A and B coe¬cients shows that B ∝ n4 . For the same

p

applied ¬eld, the absorption rate for a Rydberg atom with np 100 is typically 108

times larger than the absorption rate at the Lyman transition between the 2p and

1s states of the hydrogen atom. Since stimulated emission is also described by the

Einstein B coe¬cient, stimulated emission from the Rydberg atom can occur when

there are only a few photons inside a microwave cavity.

As indicated in Fig. 12.2, a single Rydberg atom enters and leaves a supercon-

ducting microwave cavity through small holes drilled on opposite sides. During the

transit time of the atom across the cavity the photons already present can stimulate

emission of a single photon into the fundamental cavity mode; conversely, the atom

can sometimes reabsorb a single photon. The interaction of the atom with a single

mode of the cavity is described by the Jaynes“Cummings Hamiltonian in eqn (12.7).

By monitoring whether or not the Rydberg atom has made a transition, np + 1 ’ np ,

between the adjacent Rydberg states, one can infer indirectly whether or not a single

microwave photon has been deposited in the cavity. This is possible because of the

entangled nature of the dressed states in eqns (12.18) and (12.20). A measurement of

the state of the atom, with the outcome | 2 , forces a reduction of the total state vector

of the atom“radiation system, with the result that the radiation ¬eld is de¬nitely in

the state |n . In other words, the number of photons in the cavity has not changed.

Conversely, a measurement with the outcome | 1 guarantees that the ¬eld is in the

state |n + 1 , i.e. a photon has been added to the cavity.

The discrimination between the two Rydberg states is easily accomplished, since

the ionization of the Rydberg atom by a DC electric ¬eld depends very sensitively

on its principal quantum number np . The higher number np + 1 corresponds to a

larger, more easily ionized atom, and the lower number np corresponds to a smaller,

less easily ionized atom. The electric ¬eld in the ¬rst ionization region”shown in Fig.

12.2”is strong enough to ionize all (np + 1)-atoms, but too weak to ionize any np -

atoms. Thus an atom that remains in the excited state is detected in the ¬rst region.

If the atom has made a transition to the lower state, then it will be ionized by the

stronger ¬eld in the second region. In this way, it is possible sensitively to identify the

state of the Rydberg atom. If the atom is in the appropriate state, it will be ionized and

release a single electron into the corresponding ionizing ¬eld region. The free electron

is accelerated by the ionizing ¬eld and enters into an electron-multiplication region

of a channeltron detector. As explained in Section 9.2.1, the channeltron detector

¿

The micromaser

can enormously multiply the single electron released by the Rydberg atom, and this

provides an indirect method for continuously monitoring the photon-number state of

the cavity.

A frequency-doubled dye laser (» = 297 nm) is used to excite rubidium (85 Rb)

atoms to the np = 63, P3/2 state from the np = 5, S1/2 (F = 3) state. The cavity is

tuned to the 21.456 GHz transition from the upper maser level in the np = 63, P3/2

state to the np = 61, D5/2 lower maser state. For this experiment a superconducting

cavity with a Q-value of 3—108 was used, corresponding to a photon lifetime inside the

cavity of 2 ms. The transit time of the Rydberg atom through the cavity is controlled

by changing the atomic velocity with the velocity selector. On the average, only a

small fraction of an atom is inside the cavity at any given time. In order to reduce

the number of thermally excited photons in the cavity, a liquid helium environment

reduces the temperature of the superconducting niobium microwave cavity to 2.5 K,

corresponding to the average photon number n ≈ 2.

If the transit time of the atom is larger than the collapse time but smaller than

the time of the ¬rst revival, then the solution (12.32) tells us that the atom will come

into equilibrium with the cavity ¬eld, as seen in Fig. 12.1. In this situation the atom

leaving the chamber is found in the upper or lower state with equal probability, i.e.

P2 = 1/2. When the transit time is increased to a value comparable to the ¬rst revival

time, the probability for the excited state becomes larger than 1/2. The data in Fig.

12.3 show a quantum revival of the population of atoms in the upper maser state that

occurs after a transit time of around 150 µs. Such a revival would be impossible in any

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

is the spatial cut-o¬ function for the crystal. The three-wave coupling strength is

related to the second-order susceptibility by

(2)

0 F0 F1 F2

(3)

gs0 s1 s2 (ω1 , ω2 ) = (µk0 s0 )i (µk1 s1 )j (µk2 s2 )k χijk (ω1 , ω2 ) , (13.21)

where ωp = ωkp sp and Fp = Fkp sp (p = 0, 1, 2).

In the limit of a large crystal, i.e. when all dimensions are large compared to optical

wavelengths,

3

C (k) ∼ Vc δk,0 ’ (2π) δ (k) . (13.22)

(3)

This tells us that for large crystals the only terms that contribute to Uem are those

satisfying the complete phase-matching conditions

k0 = k1 + k2 , ω0 = ω1 + ω2 . (13.23)

(4)

The same kind of analysis for Uem reveals two possible phase-matching conditions:

k0 = k1 + k2 + k3 , ω0 = ω1 + ω2 + ω3 , (13.24)

corresponding to terms of the form ±— ±1 ±2 ±3 + CC, and

0

k0 + k1 = k2 + k3 , ω0 + ω1 = ω2 + ω3 , (13.25)

corresponding to terms like ±— ±— ±2 ±3 + CC. As shown in Exercise 13.1, the coupling

01

constants associated with these processes are related to the third-order susceptibility,

χ(3) .

The de¬nition (13.21) relates the nonlinear coupling term to a fundamental prop-

erty of the medium, but this relation is not of great practical value. The ¬rst-principles

evaluation of the susceptibilities is an important problem in condensed matter physics,

but such a priori calculations typically involve other approximations. With the excep-

tion of hydrogen, the unperturbed atomic wave functions for single atoms are not

known exactly; therefore, various approximations”such as the atomic shell model”

must be used. In the important case of crystalline materials, corrections due to local

¬eld e¬ects are also di¬cult to calculate (Boyd, 1992, Sec. 3.8). In practice, approx-

imate calculations of the susceptibilities can readily incorporate the symmetry prop-

erties of the medium, but otherwise they are primarily useful as a rough guide to

the feasibility of a proposed experiment. Fortunately, the analysis of experiments does

not require the full solution of these di¬cult problems. An alternative procedure is

to use symmetry arguments to determine the form of expressions, such as (13.19),

for the energy. The coupling constants, which in principle depend on the nonlinear

susceptibilities, can then be determined by ancillary experiments.

¿ Nonlinear quantum optics

13.2.2 Quantum theory

The approximate quantization scheme for an isotropic dielectric given in Section 3.3.2

can be applied to crystals by the simple expedient of replacing the classical amplitude

±ks in eqn (13.5) by the annihilation operator aks , i.e.

i

(+) (+)

(r) = √

Eβ (r, 0) ’ Eβ Fks ±ks µks eik·r . (13.26)

V ks

In the linear approximation, the electromagnetic Hamiltonian in a crystal”which we

(0)

will now treat as the zeroth-order Hamiltonian, Hem ”is obtained from eqn (3.150) by

using the polarization-dependent frequency ωks in place of ωk :

ωks a† aks .

(0)

Hem = (13.27)

ks

ks

2

The assumption that the classical power spectrum |±ks | is peaked at the carrier

frequencies is replaced by the rule that the expressions (13.26) and (13.27) are only

valid when the operators act on a polychromatic space H ({ωβ }), as de¬ned in Section

3.3.4.

In a weakly nonlinear medium, we will employ a phenomenological approach in

which the total electromagnetic Hamiltonian is given by

(0) NL

Hem = Hem + Hem . (13.28)

NL

The higher-order terms comprising Hem can be constructed from classical energy

expressions, such as (13.19), by applying the quantization rule (13.26) and putting all

the terms into normal order. An alternative procedure is to use the correspondence

principle and symmetry arguments to determine the form of the Hamiltonian. In this

NL

approach, the weak-¬eld condition is realized by assuming that the terms in the Hem

are given by low-order polynomials in the ¬eld operators. Since the ¬eld interacts with

itself through the medium, the coupling constants must transform appropriately under

the symmetry group for the medium. The coupling constants must, therefore, have

the same symmetry properties as the classical susceptibilities. The Hamiltonian must

also be invariant with respect to time translations, and”for large crystals”spatial

translations. The general rules of quantum theory (Bransden and Joachain, 1989, Sec.

5.9) tell us that these invariances are respectively equivalent to the conservation of

energy and momentum. Applying these conservation laws to the individual terms in

the Hamiltonian yields”after dividing through by ”the classical phase-matching

conditions (13.23)“(13.25).

The expansion (13.9) for the classical energy is replaced by

Hem = Hem + Hem + · · · ,

NL (3) (4)

(13.29)

where the symmetry considerations mentioned above lead to expressions of the form

i

C (k0 ’ k1 ’ k2 ) δω0 ,ω1 +ω2

(3)

Hem =

V 3/2

k0 s0 ,k1 s1 ,k2 s2

— gs0 s1 s2 (ω1 , ω2 ) a† 1 s1 a† 2 s2 ak0 s0 ’ HC

(3)

(13.30)

k k

¿

Three-photon interactions

and

1

C (k0 ’ k1 ’ k2 ’ k3 ) δω0 ,ω1 +ω2 +ω3

(4)

Hem =

V2

k0 s0 ,...,k3 s3

— gs0 s1 s2 s3 (ω1 , ω2 , ω3 ) a† 0 s0 ak1 s1 ak2 s2 ak3 s3 + HC

(4)

k

1

C (k0 + k1 ’ k2 ’ k3 ) δω0 +ω1 ,ω2 +ω3

+

V2

k0 s0 ,...,k3 s3

— fs0 s1 s2 s3 (ω1 , ω2 , ω3 ) a† 2 s2 a† 3 s3 ak0 s0 ak1 s1 + HC .

(4)

(13.31)

k k

Another important feature follows from the observation that the susceptibilities are

necessarily proportional to the density of atoms. When combined with the assumption

that the susceptibilities are uniform over the medium, this implies that the operators

(3) (4)

Hem and Hem represent the coherent interaction of the ¬eld with the entire mate-

rial sample. First-order transition amplitudes are thus proportional to Nat , and the

2

corresponding transition rates are proportional to Nat . In contrast to this, scattering

of the light from individual atoms adds incoherently, so that the transition rate is

2

proportional to Nat rather than Nat .

The Hamiltonian obtained in this way contains many terms describing a variety of

nonlinear processes allowed by the symmetry properties of the medium. For a given

experiment, only one of these processes is usually relevant, so a model Hamiltonian is

constructed by neglecting the other terms. The relevant coupling constants must then

be determined experimentally.

13.3 Three-photon interactions

The mutual interaction of three photons corresponds to classical three-wave mixing,

which can only occur in a crystal with nonvanishing χ(2) , e.g. lithium niobate, or am-

monium dihydrogen phosphate (ADP). A familiar classical example is up-conversion

(Yariv, 1989, Sec. 17.6), which is also called sum-frequency generation (Boyd, 1992,

Sec. 2.4). In this process, waves E 1 and E 2 , with frequencies ω1 and ω2 , mix in a non-

centrosymmetric χ(2) crystal to produce a wave E 0 with frequency ω0 = ω1 +ω2 . The

traditional applications for this process involve strong ¬elds that can be treated clas-

sically, but we are interested in a quantum approach. To this end we replace classical

wave mixing by a microscopic process in which photons with energy and momentum

( k1 , ω1 ) and ( k2 , ω2 ) are absorbed and a photon with energy and momentum

( k0 , ω0 ) is emitted. The phase-matching conditions (13.23) are then interpreted as

conservation of energy and momentum in each microscopic interaction.

As a result of crystal anisotropy, phase matching can only be achieved by an ap-

propriate choice of polarizations for the three photons. The uniaxial crystals usually

employed in these experiments”which are described in Appendix B.5.3-A”have a

principal axis of symmetry, so they exhibit birefringence. This means that there are

two refractive indices for each frequency: the ordinary index no (ω) and the extraor-

dinary index ne (ω, θ). The ordinary index no (ω) is independent of the direction of

propagation, but the extraordinary index ne (ω, θ) depends on the angle θ between the

¼¼ Nonlinear quantum optics

propagation vector and the principal axis. The crystal is said to be negative (positive)

when ne < no (ne > no ). For typical crystals, the refractive indices exhibit a large

amount of dispersion between the lower frequencies of the input beams and the higher

frequency of the output beam; therefore, it is necessary to exploit the birefringence of

the crystal in order to satisfy the phase-matching conditions.

In type I phase matching, for negative uniaxial crystals, the incident beams

have parallel polarizations as ordinary rays inside the crystal, while the output beam

propagates in the crystal as an extraordinary ray. Thus the input photons obey

ω1 no (ω1 ) ω2 no (ω2 )

k1 = , k2 = , (13.32)

c c

while the output photon satis¬es the dispersion relation

ω0 ne (ω0 , θ0 )

k0 = , (13.33)

c

where θ0 is the angle between the output direction and the optic axis. In type II

phase matching, for negative uniaxial crystals, the linear polarizations of the input

beams are orthogonal, so that one is an ordinary ray, and the other an extraordinary

ray, e.g.

ω1 no (ω1 ) ω2 ne (ω2 , θ2 )

k1 = , k2 = . (13.34)

c c

In this case the output beam also propagates in the crystal as an extraordinary ray.

For positive uniaxial crystals the roles of ordinary and extraordinary rays are reversed

(Boyd, 1992).

With an appropriate choice of the angle θ0 , which can be achieved either by suitably

cutting the crystal face or by adjusting the directions of the input beams with respect

to the crystal axis, it is always possible to ¬nd a pair of input frequencies for which

all three photons have parallel propagation vectors. This is called collinear phase

matching.

From Appendix B.3.3 and Section 4.4, we know that the classical and quantum

theories of light are both invariant under time reversal; consequently, the time-reversed

process”in which an incident high-frequency ¬eld E0 generates the low-frequency out-

put ¬elds E 1 and E 2 ”must also be possible. This process is called down-conversion.

In the classical case, one of the down-converted ¬elds, say E 1 , must be initially present;

and the growth of the ¬eld E 2 is called parametric ampli¬cation (Boyd, 1992,

Sec. 2.5). The situation is quite di¬erent in quantum theory, since the initial state

need not contain either of the down-converted photons. For this reason the time-

reversed quantum process is called spontaneous down-conversion (SDC). Sponta-

neous down-conversion plays a central role in modern quantum optics. For somewhat

obscure historical reasons, this process is frequently called spontaneous parametric

down-conversion or else parametric ¬‚uorescence. In this context ˜parametric™ simply

means that the optical medium is unchanged, i.e. each atom returns to its initial state.

13.3.1 The three-photon Hamiltonian

We will simplify the notation by imposing the convention that the polarization index

is understood to accompany the wavevector. The three modes are thus represented

¼½

Three-photon interactions

by (k0 , ω0 ), (k1 , ω1 ), and (k2 , ω2 ) respectively. The fundamental interaction processes

are shown in Fig. 13.1, where the Feynman diagram (b) describes down-conversion,

while diagram (a) describes the time-reversed process of sum-frequency generation.

Strictly speaking, Feynman diagrams represent scattering amplitudes; but they are

frequently used to describe terms in the interaction Hamiltonian. The excuse is that

the ¬rst-order perturbation result for the scattering amplitude is proportional to the

matrix element of the interaction Hamiltonian between the initial and ¬nal states.

Since the nonlinear process is the main point of interest, we will simplify the prob-

lem by assuming that the entire quantization volume V is ¬lled with a medium hav-

ing the same linear index of refraction as the nonlinear crystal. This is called index

matching. The simpli¬ed version of eqn (13.30) is then

1

g (3) C (k0 ’ k1 ’ k2 ) a† 1 a† 2 ak0 + HC .

(3)

Hem = (13.35)

kk

3/2

V k0 k2 k3

This is the relevant Hamiltonian for detection in the far ¬eld of the crystal, i.e. when

the distance to the detector is large compared to the size of the crystal, since all atoms

can then contribute to the generation of the down-converted photons.

(3)

The two terms in Hem describe down-conversion and sum-frequency generation

respectively. Note that both terms must be present in order to ensure the Hermiticity

of the Hamiltonian. The down-conversion process is analogous to a radioactive decay

in which a single parent particle (the ultraviolet photon) decays into two daughter

particles, while sum-frequency generation is an analogue of particle“antiparticle anni-

hilation.

13.3.2 Spontaneous down-conversion

Spontaneous down-conversion is the preferred light source for many recent experi-

ments in quantum optics, e.g. single-photon number-state production, entanglement

phenomena (such as the Einstein“Podolsky“Rosen e¬ect and Franson two-photon in-

terference), and tunneling time measurements. One reason for the popularity of this

light source is that it is highly directional, whereas the atomic cascade sources dis-

cussed in Sections 1.4 and 11.2.3 emit light in all directions. In SDC, correlated photon

pairs are emitted into narrow cones in the form of a rainbow surrounding the pump

beam direction. The two photons of a pair are always emitted on opposite sides of the

rainbow axis. Since the photon pairs are emitted within a few degrees of the pump

(k0, ω0)

(k1, ω1) (k2, ω2)

(k1, ω1) (k2, ω2)

Fig. 13.1 Three-photon interactions (time

(k0, ω0)

¬‚ows upward in the diagrams): (a) represents

sum-frequency generation, and (b) represents

(a) (b) the time-reversed process of down-conversion.

¼¾ Nonlinear quantum optics

beam direction, detection of the output within small solid angles is relatively straight-

forward. Another practical reason for the choice of SDC is that it is much easier to

implement experimentally, since the heart of the light source is a nonlinear crystal.

This method eliminates the vacuum technology required by the use of atomic beams

in a cascade emission source.

A Generation of entangled photon pairs

In spontaneous down-conversion the incident ¬eld is called the pump beam, and the

down-converted ¬elds are traditionally called the signal and idler. To accommodate

this terminology we change the notation (E 0 , k0 , ω0 ) for the input ¬eld to (E P , p, ωP ).

There is no physical distinction between the signal and idler, so we will continue to

use the previous notation for the conjugate modes in the down-converted light. The

emission angles and frequencies of the down-converted photons vary continuously,

but they are subject to overall conservation of energy and momentum in the down-

conversion process.

The interaction Hamiltonian (13.35) is more general than is required in practice,

since it is valid for any distribution in the pump photon momenta. In typical experi-

ments, the pump photons are supplied by a continuous wave (cw) ultraviolet laser, so

the pump ¬eld is well approximated by a classical plane-wave mode with amplitude

EP . A suitable quantum model is given by a Heisenberg-picture state satisfying

ak (t) |±p = δk,p ±p e’iωP t |±p . (13.36)

In other words |±p is a coherent state built up from pump photons that are all in the

mode p. The coherent-state parameter ±p is related to the classical ¬eld amplitude

EP by

±p

EP ≡ e’ip·r ±p ep · E(+) (r) ±p = iFp √ , (13.37)

V

where the expansion (13.26) was used to get the ¬nal result. Since the number of

pump photons is large, the loss of one pump photon in each down-conversion event

can be neglected. This undepleted pump approximation allows the semiclassical

limit described in Section 11.3 to be applied. Thus we replace the Heisenberg-picture

operator ap (t) for the pump mode by ±p exp (’iωP t) + δap (t), and then neglect the

terms involving the vacuum ¬‚uctuation operators δap (t).

Since the pump mode is treated classically and the coherent state |±p is the vac-

uum for the down-converted modes, we replace the notation |±p by |0 . The classical

amplitude, ±p exp (’iωP t), is unchanged by the transformation from the Heisenberg

picture to the Schr¨dinger picture; therefore, the semiclassical Hamiltonian in the

o

Schr¨dinger picture is

o

(3)

H = H0 + Hem (t) , (13.38)

ω q a† aq ,

2

H0 = ωP |±p | + (13.39)

q

q

i

G(3) e’iωP t C (p ’ k1 ’ k2 ) a† 1 a† 2 + HC ,

Hem (t) = ’

(3)

(13.40)

kk

V

k1 ,k2

¼¿

Three-photon interactions

where the pump-enhanced coupling constant is G(3) = EP g (3) /Fp . The explicit time

dependence of the Schr¨dinger-picture Hamiltonian is a result of treating the pump

o

2

beam as an external classical ¬eld. The c-number term, ωP |±p | , in the unperturbed

Hamiltonian can be dropped, since it shifts all unperturbed energy levels by the same

amount.

We will eventually need the limit of in¬nite quantization volume, so we use the

rules (3.64) to express the (Schr¨dinger-picture) Hamiltonian as

o

(3)

H = H0 + Hem (t) , (13.41)

d3 q

ωq a† (q) a (q) ,

H0 = (13.42)