Hamiltonian:

Hsc (t) = ’i ge’iδt σ+ + i g— eiδt σ’

= „¦L e’iδt σ+ + „¦— eiδt σ’ , (11.156)

L

where

d · EL

„¦L = ’ig0 ± = ’ , (11.157)

and E L is the classical ¬eld amplitude corresponding to ±. With the conventions

adopted in Section 11.1.1, the atomic state is described by

Ψ2

|Ψ ’ , (11.158)

Ψ1

where Ψ2 (Ψ1 ) is the amplitude for the excited (ground) state. In this basis the

Schr¨dinger equation becomes

o

„¦L e’iδt

d Ψ2 0 Ψ2

i = . (11.159)

— iδt

Ψ1 Ψ1

„¦L e 0

dt

The transformation Ψ2 = exp (’iδt/2) C2 and Ψ1 = exp (iδt/2) C1 produces an equa-

tion with constant coe¬cients,

’2

δ

d C2 „¦L C2

i = . (11.160)

„¦— δ

C1 C1

dt L 2

The eigenvalues of the 2 — 2 matrix on the right are ±„¦R , where

δ2 2

+ |„¦L |

„¦R = (11.161)

4

is the Rabi frequency. The general solution is

C2 (t)

= C+ ξ+ exp (’i„¦R t) + C’ ξ’ exp (i„¦R t) , (11.162)

C2 (t)

where ξ+ and ξ’ are the eigenvectors corresponding to ±„¦R and the constants C± are

determined by the initial conditions. For exact resonance (δ = 0) and an atom initially

in the ground state, the occupation probabilities are

2

|Ψ1 (t)| = cos2 („¦R t) , (11.163)

|Ψ2 (t)|2 = sin2 („¦R t) . (11.164)

The oscillation between the ground and excited states is also known as Rabi ¬‚opping.

¿

The semiclassical limit

11.3.3 The Bloch equation

The pure-state description of an atom employed in the previous section is not usually

valid, so the Schr¨dinger equation must be replaced by the quantum Liouville equation

o

introduced in Section 2.3.2-A. In the interaction picture, eqn (2.119) becomes

‚

i ρ (t) = [Hint (t) , ρ (t)] , (11.165)

‚t

where ρ (t) is the density operator for the system under study. We now consider a two-

level atom interacting with a monochromatic classical ¬eld de¬ned by the positive-

frequency part,

E (+) (r, t) = E (r, t) e’iω0 t , (11.166)

where ω0 is the carrier frequency and E (r, t) is the slowly-varying envelope. The RWA

interaction Hamiltonian is then

Hrwa (t) = ’d · E (+) (t) σ+ (t) ’ d— · E (’) (t) σ’ (t)

—

= ’d · E (t) e’iδt σ+ ’ d— · E (t) eiδt σ’ , (11.167)

where E (t) = E (R, t) is the slowly-varying envelope evaluated at the position R of

the atom. The explicit time dependence of the atomic operators has been displayed by

using eqn (11.12). In this special case, the quantum Liouville equation has the form

d

ρ (t) = ’„¦ (t) e’iδt [σ+ , ρ (t)] ’ „¦— (t) eiδt [σ’ , ρ (t)] ,

i (11.168)

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