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-