o

H = H0 + Hint , (11.7)

where

ω21

ωk a† aks +

H0 = σz . (11.8)

ks

2

ks

In the interaction picture, the operators satisfy the uncoupled equations of motion

‚

aks (t) = [aks (t) , H0 ] = ωk aks (t) , (11.9)

i

‚t

‚

i

σz (t) = [σz (t) , H0 ] = 0 , (11.10)

‚t

‚ ω21

σ± (t) = [σ± (t) , H0 ] = “

i σ± (t) , (11.11)

‚t 2

with the solution

aks (t) = aks e’iωk t , σz (t) = σz , σ± (t) = e±iω21 t σ± , (11.12)

where aks , σz , and σ± are the Schr¨dinger-picture operators. Thus the time depen-

o

dence of the operators is explicitly expressed in terms of the atomic transition fre-

quency ω21 and the optical frequencies ωk . This is a great advantage for the calcula-

tions to follow.

The interaction-picture state vector |˜ (t) satis¬es the Schr¨dinger equation

o

‚

|˜ (t) = Hint (t) |˜ (t) ,

i (11.13)

‚t

where

(r) (ar)

Hint (t) = Hint (t) + Hint (t) , (11.14)

ωk d · eks i(ω21 ’ωk )t

(r)

Hint (t) = ’i e aks σ+ + HC (11.15)

2 0V

ks

and

ωk d— · eks ’i(ω21 +ωk )t

(ar)

(t) = ’i

Hint e aks σ’ + HC (11.16)

2 0V

ks

are obtained by replacing the operators in eqns (11.5) and (11.6) by the explicit solu-

tions in eqn (11.12).

¿¿

Resonant wave approximation

11.1.2 Time averaging

The slow and fast time scales can be separated explicitly by means of a temporal

¬ltering operation, like the one introduced in Section 9.1.2-C to describe narrowband

detection. We use an averaging function, (t), satisfying eqns (9.35)“(9.37), to de¬ne

running averages by

∞ ∞

f (t) ≡ (t ’ t ) f (t ) =

dt dt (t ) f (t + t ) . (11.17)

’∞ ’∞

The temporal width ∆T de¬ned by eqn (9.37) will now be renamed the memory

interval Tmem . The idea behind this new language is that the temporally coarse-

grained picture imposed by averaging over the time scale Tmem causes amnesia, i.e.

averaged operators at time t will not be correlated with averaged operators at an

earlier time, t < t ’ Tmem . The average in eqn (11.17) washes out oscillations with

periods smaller than Tmem , and the average of the derivative is the derivative of the

average:

df d

(t) = f (t) . (11.18)

dt dt

The separation of the two time scales is enforced by imposing the condition

1 1

Tmem (11.19)

ω21 „¦

on Tmem . A function g (t) that varies on the time scale 1/„¦ is essentially constant over

the averaging interval, so that

∞

g (t) ≡ (t ’ t ) g (t ) ≈ g (t) .

dt (11.20)

’∞

The combination of this feature with the normalization condition (9.36) leads to the

following rule:

(t ’ t ) ≈ δ (t ’ t ) when applied to slowly-varying functions . (11.21)

It is also instructive to describe the averaging procedure in the frequency domain.

We would normally denote the Fourier transform of (t) by (ω), but this particular

function plays such an important role in the theory that we will honor it with a special

name: ∞

(t) eiωt .

K (ω) = dt (11.22)

’∞

The properties of (t) guarantee that K (ω) is real and even, K — (ω) = K (’ω) =

K (ω), and that it has a ¬nite width, wK , related to the averaging interval by wK ∼

1/Tmem. The frequency-domain conditions corresponding to eqn (11.19) are

„¦ wK ω21 , (11.23)

and the time-domain normalization condition (9.36) implies K (0) = 1. Performing

the Fourier transform of eqn (11.17) gives the frequency-domain description of the

¿ Coherent interaction of light with atoms

averaging procedure as f (ω) = K (ω) f (ω). Thus for small frequencies, ω wK , the

original function f (ω) is essentially unchanged, but frequencies larger than the width

wK are strongly suppressed. For this reason K (ω) is called the cut-o¬ function.1

11.1.3 Time-averaged Schr¨dinger equation

o

Since |˜ (t) only varies on the slow time scale, the rule (11.21) tells us that it is

e¬ectively unchanged by the running average, i.e. |˜ (t) ≈ |˜ (t) . Consequently,

averaging the Schr¨dinger equation (11.13), with the help of eqn (11.18), yields the

o

approximate equation

‚

|˜ (t) = H int (t) |˜ (t) .

i (11.24)

‚t

(ar)

According to eqn (11.16), all terms in Hint (t) are rapidly oscillating; therefore, we

(ar)

expect that H int (t) ≈ 0. This expectation is justi¬ed by the explicit calculation in

(ar)

Exercise 11.1, which shows that the cut-o¬ function in each term of H int (t) is evalu-

ated with its argument on the optical scale. In the resonant wave approximation

(RWA), the antiresonant part is discarded, i.e. the full interaction Hamiltonian Hint (t)

(r)

is replaced by the resonant part H int (t). The traditional name, rotating wave approx-

imation, is suggested by the mathematical similarity between the two-level atom and

a spin-1/2 particle precessing in a magnetic ¬eld (Yariv, 1989, Chap. 15).

(r)

Turning next to the expression (11.15) for H int (t), we see that the exponentials

involve the detuning ∆k = ωk ’ ω21 which will be small near resonance; therefore, the

(r)

average of Hint (t) will not vanish. The explicit calculation gives

(r)

gks e’i∆k t σ+ aks + HC ,

Hrwa (t) ≡ H int (t) = ’i (11.25)

ks

where

ωk d · eks

gks = K (∆k ) , (11.26)

2 0V

and we have introduced the new notation Hrwa (t) as a reminder of the approximation

in use. The cut-o¬ function in the de¬nition of the coupling constant guarantees that

only terms satisfying the resonance condition |ω21 ’ ωk | < wK will contribute to

Hrwa .

With the resonant wave approximation in force, we can transform to the Schr¨- o

dinger picture by the simple expedient of omitting the time-dependent exponentials

in eqn (11.25). Thus the RWA Hamiltonian in the Schr¨dinger picture is

o

Hrwa = H0 ’ d · E(+) σ+ ’ d— · E(’) σ’

g— a† σ’ ,

= H0 ’ i gks aks σ+ + i (11.27)

ks ks

ks ks

where H0 is given by eqn (11.8). This observation provides the following general scheme

for de¬ning the resonant wave approximation directly in the Schr¨dinger picture.

o

1 This is physics jargon. An engineer would probably call K (ω) a low-pass ¬lter.

¿

Resonant wave approximation

(1) Discard all terms in Hint that do not conserve energy in a ¬rst-order perturbation

calculation.

(2) Multiply the coupling constants in the remaining terms by the cut-o¬ function

K (∆k ).

It is also useful to note that this rule mandates that each term in Hrwa is the prod-

uct of an energy-raising (-lowering) operator for the atom with an energy-lowering

(-raising) operator for the ¬eld. We emphasize that the discarded part, H (ar) , is not

unphysical; it simply does not contribute to the ¬rst-order transition amplitude. The

antiresonant Hamiltonian H (ar) can and does contribute in higher orders of perturba-

tion theory, but the time averaging argument shows that Hrwa is the dominant part

of the Hamiltonian for long-term evolution under the in¬‚uence of weak ¬elds.

11.1.4 Multilevel atoms

Our object is this section is to introduce a family of operators that play the role of the

Pauli matrices for an atom with more than two active levels. We will only consider the

interaction of the ¬eld with a single atom, since the generalization to the many-atom

case is straightforward. The atomic transition operators Sqp are de¬ned by

Sqp = |µq µp | , (11.28)

where |µq and |µp are eigenstates of Hat . As explained in Appendix C.1.2, this nota-

tion means that the operator Sqp projects any atomic state |Ψ onto |µq with coe¬cient

µp |Ψ , i.e.

Sqp |Ψ = |µq µp |Ψ . (11.29)

When this de¬nition is applied to the two-level case, it is easy to see that S21 =

σ+ , S12 = σ’ , and S22 ’ S11 = σz . The energy eigenvalue equation for the states,

Hat |µq = µq |µq , implies the operator eigenvalue equation [Sqp , Hat ] = ’ ωqp Sqp for

Sqp , so the transition operators are sometimes called eigenoperators.

The eigenstates |µq of Hat satisfy the completeness relation

|µq µq | = IA , (11.30)

q

where IA is the identity operator in HA ; therefore,

O ≡ IA OIA = µq |O| p Sqp . (11.31)

q p

Thus the Sqp s form a complete set for the expansion of any atomic operator, just as

every 2 — 2 matrix can be expressed as a linear combination of Pauli matrices.

The algebraic properties

†

Sqp = Spq , (11.32)

Sqp Sq p = δpq Sqp , (11.33)

[Sqp , Sq p ] = {δpq Sqp ’ δp q Sq p } (11.34)

are readily derived by using the orthogonality of the eigenstates. The special case q = p

and q = p of eqn (11.33) shows that the Sqq s are a set of orthogonal projection oper-

ators for the atom. For any atomic state |Ψ , eqn (11.29) yields Sqq |Ψ = |µq µq |Ψ ,

¿ Coherent interaction of light with atoms

i.e. Sqq projects out the |µq component of |Ψ . The Sqq s are called population oper-

ators, since the expectation value,

2

Ψ |Sqq | Ψ = | µq |Ψ | , (11.35)

is the probability for ¬nding the value µq , and the corresponding eigenstate |µq , in

a measurement of the energy of an atom prepared in the state |Ψ . Because of the

convention that q > p implies µq > µp , the operator Sqp for q > p is called a rais-

ing operator. It is analogous to the angular momentum raising operator, or to the

creation operator a† for a photon. By the same token, Spq = Sqp is a lowering op-

†

ks

erator, analogous to the lowering operator for angular momentum, or to the photon

annihilation operator aks .

In this representation the atomic Hamiltonian in the Schr¨dinger picture has the

o

simple form

Hat = µq Sqq , (11.36)

q

and the interaction Hamiltonian is given by

Hint = ’ Sqp dqp · E (0) , (11.37)

q,p

where dqp = µq d µp . Since dqq = 0, the sum over q and p splits into two parts

with q > p and p > q. Combining this with E = E(+) + E(’) leads to an expression

involving four sums. After interchanging the names of the summation indices in the

q < p sums, the result can be arranged as follows:

(r) (ar)

Hint = Hint + Hint , (11.38)

(r)

Hint = ’ Sqp dqp · E(+) (0) + HC ,

q>p

(11.39)

(ar)

=’ Sqp dqp · E (’)

Hint (0) + HC .

q>p

(r)

In Hint the raising (lowering) operator Sqp (Spq ) is associated with the annihilation

(ar)

(creation) operator E(+) E(’) , while the opposite pairing appears in Hint . It is

not necessary to carry out the explicit time averaging procedure; the results of the

two-level problem have already provided us with a general rule for writing down the

RWA Hamiltonian. Since all antiresonant terms are to be discarded, we can dispense

(ar)

with Hint and set

Hrwa = ’ Sqp dqp · E(+) (0) + HC . (11.40)

q>p

Expanding the ¬eld operator in plane waves yields the equivalent form

Hrwa = ’i gqp,ks Sqp aks + HC , (11.41)

ks q>p

¿

Spontaneous emission II

where the coupling frequencies,

dqp · eks

ωk

gqp,ks = K (ωqp ’ ωk ) , (11.42)

2 0V

include the cut-o¬ function, so that only those terms satisfying a resonance con-

dition |ωqp ’ ωk | < wK will contribute to the RWA interaction Hamiltonian. The

Schr¨dinger-picture form in eqn (11.41) becomes

o

gqp,ks ei(ωqp ’ωk )t Sqp aks + HC

Hrwa (t) = ’i (11.43)

ks q>p

in the interaction picture.

11.2 Spontaneous emission II

11.2.1 Propagation of spontaneous emission

The discussion of spontaneous emission in Section 4.9.3 is concerned with the calcu-

lation of the rate of quantum jumps associated with the emission of a photon. This

approach does not readily lend itself to answering other kinds of questions. For ex-

ample, if an atom at the origin is prepared in its excited state at t = 0, what is the

earliest time at which a detector located at a distance r can register the arrival of a

photon? Questions of this kind are best answered by using the Heisenberg picture.

Since the Heisenberg, Schr¨dinger, and interaction pictures all coincide at t = 0,

o

the interaction Hamiltonian in the Heisenberg picture can be inferred from eqn (11.25)

by setting t = 0 in the exponentials. The total Hamiltonian in the resonant wave

approximation is therefore

H = Hat + Hem + Hrwa , (11.44)

ω21

ωk a† (t) aks (t) ,

Hat = σz (t) , Hem = (11.45)

ks

2

ks

gks σ+ (t) aks (t) ’ g— σ’ (t) a† (t) ,

Hrwa = ’i (11.46)

ks ks

ks

where the operators are all evaluated in the Heisenberg picture. The Heisenberg equa-

tions of motion,

d

gks σ+ (t) aks (t) + g— σ’ (t) a† (t) ,

σz (t) = ’2 (11.47)

ks ks

dt

ks

d

σ’ (t) = ’iω21 σ’ (t) + gks aks (t) σz (t) , (11.48)

dt

ks

d

aks (t) = ’iωk aks (t) + g— σ’ (t) , (11.49)

ks

dt

show that the ¬eld operators aks (t) and the atomic operators σ (t), which are inde-

pendent at t = 0, are coupled at all later times. For this reason, it is usually impossible

to obtain closed-form solutions.

¿ Coherent interaction of light with atoms

Let us study the time dependence of the ¬eld emitted by an initially excited atom.

In the Heisenberg picture, the plane-wave expansion (3.69) for the positive-frequency

part of the ¬eld is

ωk

E(+) (r, t) = aks (t) eks eik·r ,

i (11.50)

2 0V

ks

so we begin by using the standard integrating factor method to get the formal solution,

t

’iωk t

g— dt e’iωk (t’t ) σ’ (t ) ,

aks (t) = aks (0) e + (11.51)

ks

0

of eqn (11.49). Substituting this into eqn (11.50) gives E(+) (r, t) as the sum of two

terms:

(+)

E(+) (r, t) = E(+) (r, t) + Erad (r, t) , (11.52)

vac

where

ωk

E(+) (r, t) = aks (0) eks ei(k·r’ωk t)

i (11.53)

vac

2 0V

ks

describes vacuum ¬‚uctuations and

t

ωk —

dt e’iωk (t’t ) σ’ (t )

(+)

gks eks eik·r

Erad (r, t) = i (11.54)

2 0V 0

ks

represents the ¬eld radiated by the atom. The state vector,

|in = |µ2 , 0 = |µ2 |0 , (11.55)

describes the situation with the atom in the excited state and no photons in the ¬eld.

In Section 9.1 we saw that the counting rate for a detector located at r is proportional

(+)

to in E(’) (r, t) · E(+) (r, t) in . Since |in is the vacuum for photons, Evac (r, t) will

(’) (+)

not contribute, and the counting rate is proportional to in Erad (r, t) · Erad (r, t) in .

(+)

Calculating the atomic radiation operator Erad (r, t) from eqn (11.54) requires an

evaluation of the sum over polarizations, followed by the conversion of the k-sum to an

integral, as outlined in Exercise 11.3. After carrying out the integral over the directions

of k, the result is

k 2 dk ωk K (ωk ’ ω21 ) — (d— · ∇) ∇ 4π sin (kr)

(+)

Erad (r, t) = i d+

(2π)3 k2

20 kr

t

dt e’iωk (t’t ) σ’ (t ) .

— (11.56)

0

The cut-o¬ function K (ωk ’ ω21 ) imposes k ≈ k21 = ω21 /c, so we can de¬ne the

radiation zone by kr ≈ k21 r 1. For a detector in the radiation zone,

1 4π sin (kr) 4π sin (kr) — 1

d— + (d— · ∇) ∇ = d +O , (11.57)

k2 k 2 r2

kr kr

¿

Spontaneous emission II

where

d— = d— ’ (r · d— ) r = (d— — r) — r (11.58)

is the component of d— transverse to the vector r linking the atom to the detector.

This is the same as the rule for the polarization of radiation emitted by a classical

dipole (Jackson, 1999, Sec. 9.2). After changing the integration variable from k to

ω = ωk = ck, we ¬nd

∞

d— ωr

i

(+)

dωω 2 K (ω ’ ω21 ) sin

Erad (r, t) =

4π 2 c2 0r c

0

t

dt e’iω(t’t ) σ’ (t ) .

— (11.59)

0

Approximating the slowly-varying factor ω 2 by ω21 , and unpacking sin (kr), yields the

2

expression

k 2 d—

(+)

[I (r) ’ I (’r)]

Erad (r, t) = 21 (11.60)

8π 2 0 r

for the ¬eld, where

∞

t

dωK (ω ’ ω21 ) eiωr/c e’iω(t’t ) σ’ (t )

I (r) = dt

0 0

∞

t

ik21 r ’iω21 t

dωK (ω) eiω[r/c’(t’t )] eiω21 t σ’ (t ) .

=e e dt (11.61)

’ω21

0

ω21 allows us to extend the lower limit of the ω-integral to ’∞

The condition wK

with negligible error, so

∞ ∞

dω

dωK (ω) eiω„ ≈ 2π K (ω) eiω„

’∞ 2π

’ω21

= 2π („ ) , (11.62)

where („ ) is the averaging function introduced in eqn (11.17). The results derived

in Exercise 11.4 include the fact that

σ ’ (t ) = eiω21 t σ’ (t ) (11.63)

is a slowly-varying envelope operator, i.e. it varies on the time scale set by |gks |.

Combining these observations with the approximate delta function rule (11.21) leads

to

t

I (r) = 2πeik21 r e’iω21 t dt δ (r/c ’ (t ’ t )) σ ’ (t )

0

ik21 r ’iω21 t

σ ’ (t ’ r/c) ,

= 2πe e (11.64)

and

t

’ik21 r ’iω21 t

dt δ (’r/c ’ (t ’ t )) σ ’ (t ) = 0 .

I (’r) = 2πe e (11.65)

0

¿¼ Coherent interaction of light with atoms

The ¬nal result for the radiated ¬eld is

k21 d— eik21 r ’iω21 t

2

(+)

σ ’ (t ’ r/c) .

Erad (r, t) = e (11.66)

4π 0 r

Thus the ¬eld operator behaves as an expanding spherical wave with source given by

the atomic dipole operator at the retarded time t ’ r/c. Just as in the classical theory,

the detector will not ¬re before the ¬rst arrival time t = r/c. We should emphasize

that this fundamental result does not depend on the resonant wave approximation

and the other simpli¬cations made here. A rigorous calculation leading to the same

conclusion has been given by Milonni (1994).

11.2.2 The Weisskopf“Wigner method

The perturbative calculation of the spontaneous emission rate can apparently be im-

proved by including higher-order terms from eqn (4.103). Since the initial and ¬nal

states are ¬xed, these terms must describe virtual emission and absorption of pho-

tons. In other words, the higher-order terms”called radiative corrections”involve

vacuum ¬‚uctuations. We know, from Section 2.5, that the contributions from vacuum

¬‚uctuations are in¬nite, so it will not come as a surprise to learn that all of the integrals

de¬ning the higher-order contributions are divergent.

A possible remedy would be to include the cut-o¬ function K (∆k ), in the coupling

frequencies, i.e. to replace Gks by gks . This will cure the divergent integrals, but it must

then be proved that the results do not depend on the detailed shape of K (∆k ). This

can be done, but only at the expense of importing the machinery of renormalization

theory from quantum electrodynamics (Greiner and Reinhardt, 1994).

A more important drawback of the perturbative approach is that it is only valid

1/ |gks | ≈ „sp = 1/A2’1 . Thus perturbation theory

in the limited time interval t

cannot be used to follow the evolution of the system for times comparable to the spon-

taneous decay time. We will use the RWA to pursue a nonperturbative approach (see

Cohen-Tannoudji et al. (1977b, Complement D-XIII), or the original paper Weisskopf

and Wigner (1930)) which can describe the behavior of the atom“¬eld system for long

times, t > „sp .

The key to this nonperturbative method is the following simple observation. In the

resonant wave approximation, the atom“¬eld state |µ2 ; 0 , in which the atom is in the

excited state and there are no photons, can only make transitions to one of the states

|µ1 ; 1ks , in which the atom is in the ground state and there is exactly one photon

present. Conversely, the state |µ1 ; 1ks can only make a transition into the state |µ2 ; 0 .

This is demonstrated more explicitly by using eqn (11.25) for Hrwa to ¬nd

g— ei∆k t |µ1 ; 1ks ,

Hrwa (t) |µ2 ; 0 = i (11.67)

ks

ks

and

Hrwa (t) |µ1 ; 1ks = ’i gks e’i∆k t |µ2 ; 0 . (11.68)

Consequently, the spontaneous emission subspace

Hse = span {|µ2 ; 0 , |µ1 ; 1ks for all ks} (11.69)

¿½

Spontaneous emission II

is sent into itself by the action of the RWA Hamiltonian: Hrwa (t) Hse ’ Hse . This

means that an initial state in Hse will evolve into another state in Hse . The time-

dependent state can therefore be expressed as

C1ks (t) e’i∆k t |µ1 ; 1ks ,

|˜ (t) = C2 (t) |µ2 ; 0 + (11.70)

ks

where the exponential in the second term is included to balance the explicit time depen-

dence of the interaction-picture Hamiltonian. Substituting this into the Schr¨dinger

o

equation (11.13) produces equations for the coe¬cients:

dC2 (t)

=’ gks C1ks (t) , (11.71)

dt

ks

d

+ i∆k C1ks (t) = g— C2 (t) . (11.72)

ks

dt

For the discussion of spontaneous emission, it is natural to assume that the atom is

initially in the excited state and no photons are present, i.e.

C2 (0) = 1 , C1ks (0) = 0 . (11.73)

Inserting the formal solution,

t

dt g— e’i∆k (t’t ) C2 (t ) ,

C1ks (t) = (11.74)

ks

0

of eqn (11.72) into eqn (11.71) leads to the integro-di¬erential equation

t

dC2 (t)

|gks | e’i∆k (t’t )

2

=’ dt C2 (t ) (11.75)

dt 0 ks

for C2 . This presents us with a di¬cult problem, since the evolution of C2 (t) now

depends on its past history. The way out is to argue that the function in curly brackets

decays rapidly as t ’ t increases, so that it is a good approximation to set C2 (t ) =

C2 (t). This allows us to replace eqn (11.75) by

t

dC2 (t)

|gks | e’i∆k t

2

=’ dt C2 (t) , (11.76)

dt 0 ks

which has the desirable feature that C2 (t + ∆t) only depends on C2 (t), rather than

C2 (t ) for all t < t. As we already noted in Section 9.2.1, evolutions with this property

are called Markov processes, and the transition from eqn (11.75) to eqn (11.76) is called

the Markov approximation. In the following paragraphs we will justify the assumptions

underlying the Markov approximation by a Laplace transform method that is also

useful in related problems.

¿¾ Coherent interaction of light with atoms

The di¬erential equations for C1 (t) and C2 (t) de¬ne a linear initial value problem

that can be solved by the Laplace transform method reviewed in Appendix A.5. Ap-

plying the general scheme in eqns (A.73)“(A.75) to the initial conditions (11.73) and

the di¬erential equations (11.71) and (11.72) produces the algebraic equations

ζ C2 (ζ) = 1 ’ gks C1ks (ζ) , (11.77)

ks

(ζ + i∆k ) C1ks (ζ) = g— C2 (ζ) . (11.78)

ks

Substituting the solution of the second of these equations into the ¬rst leads to

1

C2 (ζ) = , (11.79)

ζ + D (ζ)

where

2

|gks |

D (ζ) = . (11.80)

ζ + i∆k

ks

In order to carry out the limit V ’ ∞, we introduce

|gks |2 ,

g2 (k) = V (11.81)

s

which allows D (ζ) to be expressed as

g2 (k) g2 (k)

d3 k

1

’

D (ζ) = . (11.82)

(2π)3 ζ + i∆k

V ζ + i∆k

k

According to eqn (4.160),

ωk |K (∆k )|2 2

2

g (k) = |d| ’ d · k

2

, (11.83)

20

and the integral over the directions of k in eqn (11.82) can be carried out by the method

used in eqn (4.161). The relation |k| = ωk /c is then used to change the integration

variable from |k| to ∆ = ωk ’ ω21 . The lower limit of the ∆-integral is ∆ = ’ω21 ,

but the width of the cut-o¬ function is small compared to the transition frequency

ω21 ); therefore, there is negligible error in extending the integral to ∆ = ’∞

(wK

to get

3

2

|K (∆)|

∆

∞ 1+

w21 ω21

D (ζ) = d∆ , (11.84)

2π ζ + i∆

’∞

where

2

|d| ω21

3

w21 = = A2’1 (11.85)

3π 0 c3

is the spontaneous decay rate previously found in Section 4.9.3.

¿¿

Spontaneous emission II

The time dependence of C2 (t) is determined by the location of the poles in C2 (ζ),

which are in turn determined by the roots of

ζ + D (ζ) = 0 . (11.86)

A peculiar feature of this approach is that it is absolutely essential to solve this equa-

tion without knowing the function D (ζ) exactly. The reason is that an exact evaluation

2

of D (ζ) would require an explicit model for |K (∆)| , but no physically meaningful

results can depend on the detailed behavior of the cut-o¬ function. What is needed is

an approximate evaluation of D (ζ) which is as insensitive as possible to the shape of

2

|K (∆)| . The key to this approximation is found by combining eqn (11.86) with eqn

(11.84) to conclude that the relevant values of ζ are small compared to the width of

the cut-o¬ function, i.e.

ζ = O (w21 ) wK . (11.87)

This is the step that will justify the Markov approximation (11.76). In the time do-

main, the function C2 (t) varies signi¬cantly over an interval of width ∆t ∼ 1/w21 ;

consequently, the condition (11.87) is equivalent to Tmem ∆t; that is, the memory

of the averaging function is short compared to the time scale on which the function

C2 (t) varies. The physical source of this feature is the continuous phase space of ¬nal

states available to the emitted photon. Summing over this continuum of ¬nal photon

states e¬ectively erases the memory of the atomic state that led to the emission of the

photon.

For values of ζ satisfying eqn (11.87), D (ζ) can be approximated by combining

the normalization condition K (0) = 1 with the identity

1 1

= πδ (∆) ’ iP ,

lim (11.88)

ζ’0 ζ + i∆ ∆

where P denotes the Cauchy principal value”see eqn (A.93). The result is

w21

D (ζ) = + iδω21 , (11.89)

2

where the imaginary part,

∞ 3 2

|K (∆)|

w21 ∆

=’

δω21 P d∆ 1 + , (11.90)

2π ω21 ∆

’∞

is the frequency shift. It is customary to compare δω21 to the Lamb shift (Cohen-

Tannoudji et al., 1992, Sec. II-E.1), but this is somewhat misleading. The result for

Re D (ζ) is robust, in the sense that it is independent of the details of the cut-o¬

function, but the result for Im D (ζ) is not robust, since it depends on the shape

2

of |K (∆)| . In Exercise 11.2, eqn (11.90) is used to get the estimate, δω21 /w21 =

O (wK /ω21 ) 1, for the size of the frequency shift. This is comforting, since it tells

us that δω21 is at least very small, even if its exact numerical value has no physical

signi¬cance. The experimental fact that measured shifts are small compared to the line

¿ Coherent interaction of light with atoms

widths is even more comforting. A strictly consistent application of the RWA neglects

all terms of the order wK /ω21 ; therefore, we will set δω21 = 0.

Substituting D (ζ) from eqn (11.89) into eqn (11.79) gives the simple result

1

C2 (ζ) = , (11.91)

ζ + w21 /2

and evaluating the inverse transform (A.72) by the rule (A.80) produces the corre-

sponding time-domain result

C2 (t) = e’w21 t/2 . (11.92)

Thus the nonperturbative Weisskopf“Wigner method displays an irreversible decay,

|C2 (t)|2 = e’w21 t , (11.93)

of the upper-level occupation probability. This conclusion depends crucially on the

coupling of the discrete atomic states to the broad distribution of electromagnetic

modes available in the in¬nite volume limit. In the time domain, we can say that

the atom forgets the emission event before there is time for reabsorption. We will see

later on that the irreversibility of the decay does not hold for atoms in a cavity with

dimensions comparable to a wavelength.

In addition to following the decay of the upper-level occupation probability, we can

study the probability that the atom emits a photon into the mode ks. According to

eqn (11.78),

g—

ks

C1ks (ζ) = . (11.94)

(ζ + i∆k ) (ζ + w21 /2)

The probability amplitude for a photon with wavevector k and polarization eks is

C1ks (t) ei∆k t , so another application of eqn (A.80) yields

e’i∆k t ’ e’w21 t/2

C1ks (t) = ig— . (11.95)

ks

∆k + iw21 /2

After many decay times (w21 t 1), the probability for emission is

2

pks = lim C1ks (t) ei∆k t

t’∞

2

|gks |

=

w21 2

2

(∆k ) + 2

2 2

ωk |K |d · eks |

(∆k )|

= . (11.96)

w21 2

2

20 V (∆k ) + 2

The denominator of the second factor e¬ectively constrains ∆k by |∆k | < w21 , so it is

permissible to set |K (∆k )| = 1 in the following calculations.

¿

Spontaneous emission II

As explained in Section 3.1.4, physically meaningful results are found by passing

to the limit of in¬nite quantization volume. In the present case, this is done by using

3

the rule 1/V ’ d3 k/ (2π) , which yields

2

|d · eks | d3 k

ωk

dps (k) = (11.97)

w21 2 (2π)3

2

20 (∆k ) + 2

for the probability of emitting a photon with polarization eks into the momentum-

space volume element d3 k. Summing over polarizations and integrating over the angles

of k, by the methods used in Section 4.9.3, gives the probability for emission of a photon

in the frequency interval (ω, ω + dω):

w21

dω

2

dp (ω) = 2π. (11.98)

2

(ω ’ ω21 ) + w21

2

This has the form of the Lorentzian line shape

γ

L (ν) = , (11.99)

ν2 + γ2

where ν is the detuning from the resonance frequency, γ is the half-width-at-half-

maximum (HWHM), and the normalization condition is

∞

dν

L (ν) = 1 . (11.100)

π

’∞

From eqn (11.98) we see that the line width w21 is the full-width-at-half-maximum,

but also that the normalization condition is not exactly satis¬ed. The trouble is that

ω = ωk is required to be positive, so the integral over all physical frequencies is

∞ w21

dν 2

< 1. (11.101)

2

π ν 2 + w21

’ω21

2

This is not a serious problem since ω21 w21 , i.e. the optical transition frequency

is much larger than the line width. Thus the lower limit of the integral can be ex-

tended to ’∞ with small error. The spectrum of spontaneous emission is therefore

well represented by a Lorentzian line shape.

Two-photon cascade—

11.2.3

The photon indivisibility experiment of Grangier, Roget, and Aspect, discussed in

Section 1.4, used a two-photon cascade transition as the source of an entangled two-

photon state. The simplest model for this process is a three-level atom, as shown in

Fig. 11.1.

This concrete example will illustrate the use of the general techniques discussed

in the previous section. The one-photon detunings, ∆32,k = ck ’ ω32 and ∆21,k =

ck ’ ω21 , are related to the two-photon detuning, ∆31,kk = ck + ck ’ ω31 , by

∆31,kk = ∆32,k + ∆21,k = ∆32,k + ∆21,k . (11.102)

¿ Coherent interaction of light with atoms

!

kI

Fig. 11.1 Two-photon cascade emission from

kI

a three-level atom. The frequencies are as-

sumed to satisfy ω = ck ≈ ω32 , ω = ck ≈ ω21 ,

and ω32 ω21 .

According to the general result (11.43), the RWA Hamiltonian is

g32,ks e’i∆32,k t S32 aks + g21,ks e’i∆21,k t S21 aks + HC ,

Hrwa (t) = ’i (11.103)

ks

where the coupling constants are

ωk d32 · eks

g32,ks = K (∆32,k ) ,

2 0V

(11.104)

ωk d21 · eks

g21,ks = K (∆21,k ) .

2 0V

Initially the atom is in the uppermost excited state |µ3 and the ¬eld is in the

vacuum state |0 , so the combined system is described by the product state |µ3 ; 0 =

|µ3 |0 . The excited atom can decay to the intermediate state |µ2 with the emission of

a photon, and then subsequently emit a second photon while making the ¬nal transition

to the ground state |µ1 . It may seem natural to think that the 3 ’ 2 photon must

be emitted ¬rst and the 2 ’ 1 photon second, but the order could be reversed. The

reason is that we are not considering a sequence of completed spontaneous emissions,

each described by an Einstein A coe¬cient, but instead a coherent process in which

the atom emits two photons during the overall transition 3 ’ 1. Since the ¬nal states

are the same, the processes (3 ’ 2 followed by 2 ’ 1) and (2 ’ 1 followed by 3 ’ 2)

are indistinguishable. Feynman™s rules then tell us that the two amplitudes must be

coherently added before squaring to get the transition probability. If the level spacings

were nearly equal, both processes would be equally important. In the situation we

ω21 , the process (2 ’ 1 followed by 3 ’ 2) would be far o¬

are considering, ω32

resonance; therefore, we can safely neglect it. This approximation is formally justi¬ed

by the estimate

g32,ks g21,ks ≈ 0 , (11.105)

which is a consequence of the fact that the cut-o¬ functions |K (∆32,k )| and |K (∆21,k )|

do not overlap.

The states |µ2 ; 1ks = |µ2 |1ks and |µ1 ; 1ks , 1k s = |µ1 |1ks , 1k s will appear

as the state vector |˜ (t) evolves. It is straightforward to show that applying the

¿

Spontaneous emission II

Hamiltonian to each of these states results in a linear combination of the same three

states. The standard terminology for this situation is that the subspace spanned by

|µ3 ; 0 , |µ2 ; 1ks , and |µ1 ; 1ks , 1k s is invariant under the action of the Hamiltonian.

We have already met with a case like this in Section 11.2.2, and we can use the ideas

of the Weisskopf“Wigner model to analyze the present problem. To this end, we make

the following ansatz for the state vector:

|˜ (t) = Z (t) |µ3 ; 0 + Yks (t) ei∆32,k t |µ2 ; 1ks

ks

Xks,k s (t) ei∆31,kk t |µ1 ; 1ks , 1k s ,

+ (11.106)

ks k s

where the time-dependent exponentials have been introduced to cancel the time de-

pendence of Hrwa (t). Note that the coe¬cient Xks,k s is necessarily symmetric under

ks ” k s .

Substituting this expansion into the Schr¨dinger equation”see Exercise 11.5”

o

leads to a set of linear di¬erential equations for the coe¬cients. We will solve these

equations by the Laplace transform technique, just as in Section 11.2.2. The initial

conditions are Z (0) = 1 and Yks (0) = Xks,k s (0) = 0, so the di¬erential equations

are replaced by the algebraic equations

ζ Z (ζ) = 1 ’ g32,ks Yks (ζ) , (11.107)

ks

[ζ + i∆32,k ] Yks (ζ) = g—

32,ks Z (ζ) ’ 2 g21,k s Xks,k s (ζ) , (11.108)

ks

1—

Yk s (ζ) + g— s Yks (ζ) .

g

[ζ + i∆31,kk ] Xks,k s (ζ) = (11.109)

2 21,ks 21,k

Solving the ¬nal equation for Xks,k s and substituting the result into eqn (11.108)

produces

g21,k s g—

g—

21,ks

(ζ) ’

[ζ + i∆32,k + Dk (ζ)] Yks (ζ) = 32,ks Z Yk s (ζ) , (11.110)

ζ + i∆31,kk

ks

where

|g21,k s |2

Dk (ζ) = . (11.111)

ζ + i∆32,k + i∆21,k

ks

As far as the k-dependence is concerned, eqn (11.110) is an integral equation for

Yks (ζ), but there is an approximation that simpli¬es matters. The ¬rst-order term

on the right side shows that Yks ∼ g— 32,ks , but this implies that the k -sum in the

second term includes the product g21,k s g— s , which can be neglected by virtue of

32,k

¿ Coherent interaction of light with atoms

eqn (11.105). Thus the second term can be dropped, and an approximate solution to

eqn (11.110) is given by

g—

32,ks Z (ζ)

Yks (ζ) = . (11.112)

ζ + i∆32,k + Dk (ζ)

Calculations similar to those in Section 11.2.2 allow us to carry out the limit V ’ ∞

and express Dk (ζ) as

2

|K (∆ )|

w21

Dk (ζ) = d∆ , (11.113)

2π ζ + i∆32,k + i∆

where w21 , the decay rate for the 2 ’ 1 transition, is given by eqn (11.85).

The poles of Yks (ζ) are partly determined by the zeroes of ζ + i∆32,k + Dk (ζ), so

the relevant values of ζ satisfy

ζ + i∆32,k = O (w21 ) . (11.114)

Another application of the argument used in Section 11.2.2 yields Dk ≈ w21 /2, so the

expression for Yks (ζ) simpli¬es to

g—

32,ks Z (ζ)

Yks (ζ) = . (11.115)

ζ + i∆32,k + w2 21

Substituting this into eqn (11.107) gives

1

Z (ζ) = , (11.116)

ζ + F (ζ)

where

2

|K (∆)|

w32

F (ζ) = d∆ , (11.117)

ζ + w2 + i∆

2π 21

and w32 is the decay rate for the 3 ’ 2 transition. In this case ζ = O (w32 ), so ζ +w21 /2

is also small compared to the width wK of the cut-o¬ function. A third application of

the same argument yields F (ζ) = w32 /2, so the Laplace transforms of the expansion

coe¬cients are given by

1

Z (ζ) = , (11.118)

ζ + w232

g—

32,ks

Yks (ζ) = , (11.119)

w21 w32

ζ + i∆32,k + ζ+

2 2

g— —

32,ks g21,k s

1

+ (ks ” k s ) . (11.120)

Xks,k s (ζ) =

2 [ζ + i∆31,kk ] ζ + w2 w21

ζ + i∆32,k +

32

2

The rule (A.80) shows that the inverse Laplace transform of eqn (11.120) has the

form

¿

The semiclassical limit

w32 t w21 t

Xks,k s (t) = G1 exp ’ + G2 exp ’ exp [’i∆32,k t]

2 2

+ G3 exp [’i∆31,kk t] . (11.121)

In the limit of long times, i.e. w32 t 1 and w21 t 1, only the third term survives.

Evaluating the residue for the pole at ζ = ’i∆31,kk provides the explicit expression

for G3 and thus the long-time probability amplitude for the state |µ3 ; 1ks , 1k s :

g— —

32,ks g21,k s

1

∞

=’ + (ks ” k s ) .

Xks,k s (11.122)

i i

2 ∆31,kk + 2 w32 ∆21,k + 2 w21

Since the two one-photon resonances are nonoverlapping, only one of these two terms

will contribute for a given (ks, k s )-pair. In order to√

pass to the in¬nite volume limit,

√

we introduce g32,s (k) = V g32,ks and g21,s (k ) = V g21,k s and use the argument

leading to eqn (11.97) to get the di¬erential probability

|g32,s (k)|2 |g21,s (k )|2 d3 k d3 k

1

dp (ks, k s ) = . (11.123)

3 3

4 [∆13,kk ]2 + 1 w2 2

[∆21,k ] + 1 w21 (2π) (2π)

2

4 32 4

For early times, i.e. w32 t < 1, w21 t < 1, the full solution in eqn (11.121) must be

used, and the expansion (11.106) shows that the atom and the ¬eld are described by

an entangled state. At late times, the irreversible decay of the upper-level occupation

probabilities destroys the necessary coherence, and the system is described by the

product state |µ3 ; 1ks , 1k s = |µ3 |1ks , 1k s . Thus the atom is no longer entangled

with the ¬eld, but the two photons remain entangled with one another, as described by

the state |1ks , 1k s . The entanglement of the photons in the ¬nal state is the essential

feature of the design of the photon indivisibility experiment.

11.3 The semiclassical limit

Since we have a fully quantum treatment of the electromagnetic ¬eld, it should be pos-

sible to derive the semiclassical approximation”which was simply assumed in Section

4.1”and combine it with the quantized description of spontaneous emission. This is an

essential step, since there are many applications in which an e¬ectively classical ¬eld,

e.g. the single-mode output of a laser, interacts with atoms that can also undergo

spontaneous emission into other modes. Of course, the entire electromagnetic ¬eld

could be treated by the quantized theory, but this would unnecessarily complicate

the description of the interesting applications. The ¬nal result”which is eminently

plausible on physical grounds”can be stated as the following rule.

In the presence of an external classical ¬eld E (r, t) = ’‚A (r, t) /‚t, the total

Schr¨dinger-picture Hamiltonian is

o

sc

H = Hchg (t) + Hem + Hint , (11.124)

where

ω f a† af

Hem = (11.125)

f

f

¿¼ Coherent interaction of light with atoms

is the Hamiltonian for the quantized radiation ¬eld, and

Hint = ’ d3 r j (r) · A(+) (r) (11.126)

is the interaction Hamiltonian between the quantized ¬eld and the charges. The re-

maining term,

N N

p2 1 qn ql qn

’ A (rn , t) · pn ,

n

sc

Hchg (t) = + (11.127)

|rn ’ rl | n=1 Mn

2Mn 4π 0

n=1 n=l

includes the mutual Coulomb interaction between the charges and the interaction of

the charges with the external classical ¬eld.

The rule (11.124) is derived in Section 11.3.1”where some subtleties concerning

the separation of the quantized radiation ¬eld and the classical ¬eld are explained”

and applied to the treatment of Rabi oscillations and the optical Bloch equation in

the following sections.

The semiclassical Hamiltonian—

11.3.1

In the presence of a classical source current J (r, t), the complete Schr¨dinger-picture

o

Hamiltonian is the sum of the microscopic Hamiltonian, given by eqn (4.29), and the

hemiclassical interaction term given by eqn (5.36):

H = Hem + Hchg + Hint + HJ (t) , (11.128)

where Hem , Hchg , Hint , and HJ are given by eqns (5.29), (4.31), (5.27), and (5.36)

respectively. The description of the internal states of atoms, etc. is contained in this

Hamiltonian, since Hchg includes all Coulomb interactions between the charges. The

hemiclassical interaction Hamiltonian is an explicit function of time”by virtue of the

presence of the prescribed external current”which is conveniently expressed as

Gκ (t) a† + G— (t) aκ ,

HJ (t) = ’ (11.129)

κ κ

κ

where

d3 r J (r, t) · E — (r)

Gκ (t) = (11.130)

κ

2 0 ωκ

is the multimode generalization of the coe¬cients introduced in eqn (5.39).

The familiar semiclassical approximation involves a prescribed classical ¬eld, rather

than a classical current, so our immediate objective is to show how to replace the

current by the ¬eld. For this purpose, it is useful to transform to the Heisenberg

picture, i.e. to replace the time-independent, Schr¨dinger-picture operators by their

o

time-dependent, Heisenberg forms:

aκ , a† , rn , pn ’ aκ (t) , a† (t) , rn (t) , pn (t) . (11.131)

κ κ

The c-number current J (r, t) is unchanged, so the full Hamiltonian in the Heisenberg

picture is still an explicit function of time. The advantage of this transformation is that

¿½

The semiclassical limit

we can apply familiar methods for treating ¬rst-order, ordinary di¬erential equations

to the Heisenberg equations of motion for the quantum operators.

By using the equal-time commutation relations to evaluate [aκ (t) , H (t)], one ¬nds

the Heisenberg equation for the annihilation operator aκ (t):

d

aκ (t) = ωκ aκ (t) ’ Gκ (t) + [aκ (t) , Hint ] .

i (11.132)

dt

The general solution of this linear, inhomogeneous di¬erential equation for aκ (t) is the

sum of the general solution of the homogeneous equation and any special solution of

the inhomogeneous equation. The result (5.40) for the single-mode problem suggests

the choice of the special solution ±κ (t), where ±κ (t) is a c-number function satisfying

d

±κ (t) = ωκ ±κ (t) ’ Gκ (t) .

i (11.133)

dt

The ansatz

aκ (t) = ±κ (t) + arad (t) (11.134)

κ

for the general solution de¬nes a new operator, arad (t), that satis¬es the canonical,

κ

equal-time commutation relations

arad (t) , arad † (t) = δκ» . (11.135)

κ »

Substituting eqn (11.134) into eqn (11.132) produces the homogeneous di¬erential

equation

d rad

aκ (t) = ωκ arad (t) + arad (t) , Hint .

i (11.136)

κ κ

dt

In order to express Hint in terms of the new operators arad (t), we substitute eqn

κ

(11.134) into the Heisenberg-picture version of the expansion (5.28) to get

A(+) (r, t) = A(+) (r, t) + Arad(+) (r, t) . (11.137)

The operator part,

arad (t) E κ (r) ,

Arad(+) (r, t) = (11.138)

κ

2 0 ωκ

κ

is de¬ned in terms of the new annihilation operators arad (t). The c-number part,

κ

A(+) (r, t) = ±κ (t) E κ (r) = ± A(+) (r, t) ± , (11.139)

2 0 ωκ

κ

is the positive-frequency part of the classical ¬eld A de¬ned by the coherent state,

|± , that is emitted by the classical current J . Substitution of eqn (11.137) into eqn

(5.27) yields

sc rad

Hint = Hint + Hint , (11.140)

¿¾ Coherent interaction of light with atoms

where

Hint = ’ d3 r j (r, t) · A (r, t)

sc

(11.141)

and

Hint = ’ d3 r j (r, t) · Arad (r, t)

rad

(11.142)

respectively describe the interaction of the charges with the classical ¬eld, A (r, t),

and the quantized radiation ¬eld Arad (r, t). Since arad (t) commutes with Hint , the

sc

κ

Heisenberg equation for arad (t) is

κ

d rad

aκ (t) = ωκ arad (t) + arad (t) , Hint .

rad

i (11.143)

κ κ

dt

The operators rn (t) and pn (t) for the charges commute with HJ (t), so their

Heisenberg equations are

d sc rad

i rn (t) = [rn (t) , Hchg ] + [rn (t) , Hint ] + rn (t) , Hint ,

dt (11.144)

d sc rad

i pn (t) = [pn (t) , Hchg ] + [pn (t) , Hint ] + pn (t) , Hint ,

dt

where Hchg is given by eqn (4.31).

The complete Heisenberg equations, (11.143) and (11.144), follow from the new

form,

sc rad rad

H = Hchg + Hem + Hint , (11.145)

of the Hamiltonian, where

sc sc

Hchg = Hchg + Hint (11.146)

and

ωκ arad † (t) arad (t) .

rad

Hem = (11.147)

κ κ

κ

We have, therefore, succeeded in replacing the classical current J by the classical ¬eld

A.

The de¬nition (5.26) of the current operator and the explicit expression (4.31) for

Hchg yield

N N

p2 (t) 1 qn ql qn

’ A (rn (t) , t) · pn (t) , (11.148)

n

sc

Hchg = +

|rn (t) ’ rl (t)| n=1 Mn

2Mn 4π 0

n=1 n=l

which agrees with the semiclassical Hamiltonian in eqn (4.3), in the approximation

that the A2 -terms are neglected. The explicit time dependence of the Schr¨dinger-

o

picture form for the Hamiltonian”which is obtained by inverting the replacement

rule (11.131)”now comes from the appearance of the classical ¬eld A (r, t), rather

than the classical current J (r, t).

¿¿

The semiclassical limit

The replacement of aκ by arad is not quite as straightforward as it appears to be.

κ

The equal-time canonical commutation relation (11.135) guarantees the existence of a

vacuum state 0rad for the arad s, i.e.

κ

arad (t) 0rad = 0 for all modes , (11.149)

κ

but the physical interpretation of 0rad requires some care. The meaning of the new

vacuum state becomes clear if one uses eqn (11.134) to express eqn (11.149) as

aκ (t) 0rad = ±κ (t) 0rad . (11.150)

This shows that the Heisenberg-picture ˜vacuum™ for arad (t) is in fact the coherent

κ

state |± generated by the classical current. In the Schr¨dinger picture this becomes

o

aκ 0rad (t) = ±κ (t) 0rad (t) , (11.151)

which means that the modi¬ed vacuum state is even time dependent. In either picture,

the excitations created by arad † represent vacuum ¬‚uctuations relative to the coherent

κ

state |± . These subtleties are not very important in practice, since the classical ¬eld

is typically con¬ned to a single mode or a narrow band of modes. For other modes,

i.e. those modes for which ±κ (t) vanishes at all times, the modi¬ed vacuum is the

true vacuum. For this reason the superscript ˜rad™ in arad , etc. will be omitted in the

κ

applications, and we arrive at eqn (11.124).

11.3.2 Rabi oscillations

The resonant wave approximation is also useful for describing the interaction of a

two-level atom with a classical ¬eld having a long coherence time Tc , e.g. the ¬eld of

a laser. From Section 4.8.2, we know that perturbation theory cannot be used if Tc >

1/A, where A is the Einstein A coe¬cient, but the RWA provides a nonperturbative

approach. We will assume that there is only one mode, with frequency ω0 , which is

nearly resonant with the atomic transition. In this case the interaction-picture state

vector |˜ (t) satis¬es

‚

|˜ (t) = Hrwa (t) |˜ (t) ,

i (11.152)

‚t

and specializing eqn (11.25) to the single mode (k0 , s0 ) gives

Hrwa (t) = ’i g0 e’iδt σ+ a0 + i g— eiδt σ’ a† , (11.153)

0 0

where δ = ω0 ’ ω21 is the detuning.

In Chapter 12 we will study the full quantum dynamics associated with this Hamil-

tonian (also known as the Jaynes“Cummings Hamiltonian), but for our immediate pur-

poses we will assume that the combined system of ¬eld and atom is initially described

by the state

|˜ (0) = |Ψ (0) |± , (11.154)

where |Ψ (0) is the initial state vector for the atom and |± is a coherent state for a0 ,

i.e.

a0 |± = ± |± . (11.155)

This is a simple model for the output of a laser. As explained above, the operator

arad = a0 ’ ± represents vacuum ¬‚uctuations around the coherent state, so replacing

0

¿ Coherent interaction of light with atoms

a0 by ± in eqn (11.153) amounts to neglecting all vacuum ¬‚uctuations, including spon-

taneous emission from the upper level. This approximation de¬nes the semiclassical

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)] ,