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