be dipole-coupled to the electromagnetic ¬eld. We therefore reserve θ1 ’ θ2 to deal

with any such coupling, and use eqn (14.166) as the de¬nition of θ2 ’ θ3 . For the sake

™ ™

of simplicity, we will assume that ∆21 = θ2 ’ θ1 is a constant; this assumption is valid

in most applications.

The central idea of this approach is that the envelope operators are e¬ectively in-

dependent of the randomly ¬‚uctuating pump phase ‘P (t). This means that S qp P

S qp , where · · · P denotes averaging over the distribution of pump phases. This al-

lows the rapid ¬‚uctuations in the phase to be exploited by a variant of the adiabatic

elimination argument. As an illustration of this approach, we start with the Langevin

equation,

dS 23 (t) ™ ™

= ’i θ2 ’ θ3 S 23 (t) ’ i„¦P S 21 (t) ’ “23 S 23 (t) + ξ23 (t) , (14.167)

dt

Incoherent pumping

for the atomic coherence operator S 23 (t), and impose the phase choice (14.166). Writ-

ing out the formal solution and averaging it over the phase distribution of the pump

then leads to

S 23 (t) = S 23 (t0 ) e(i∆P ’i∆21 ’“23 )(t’t0 ) CP (t, t0 )

t

dt e(i∆P ’i∆21 ’“23 )(t’t ) CP (t, t ) S 21 (t )

’ i„¦P

t0

t

dt e(i∆P ’i∆21 ’“23 )(t’t ) CP (t, t ) ξ23 (t )

+ , (14.168)

P

t0

where

CP (t, t ) ≡ e’i‘P (t) ei‘P (t ) . (14.169)

P

For a time-stationary distribution of pump phase, CP (t, t ) only depends on the time

di¬erence t’t ; and it decays rapidly for |t ’ t | larger than the pump correlation time.

For the function CP (t, t0 ), this means that transient e¬ects, associated with turning

on the pump, will fade away for t ’ t0 larger than the pump correlation time. This

is mathematically equivalent to the limit t0 ’ ’∞, so that CP (t, t0 ) ’ 0. In the

remaining terms, the rapid decay of CP (t, t ) justi¬es evaluating the other functions

in the t -integrals at t = t. The result is

S 23 (t) = ’i„¦P TP S 21 (t) + TP ξ23 (t) , (14.170)

P

where

t

e’i[‘P (t)’‘P (t )]

TP = lim dt (14.171)

t0 ’’∞ P

t0

is a measure of the correlation time for the incoherent pump. The same procedure

applied to S 13 (t) yields

S 13 (t) = ’i„¦P TP S 11 (t) ’ S 33 (t) + TP ξ13 (t) . (14.172)

P

The strengths of the noise operators ξqp (t) P are determined by the atomic tran-

sition rates, which we can assume are small compared to „¦P . This justi¬es neglecting

the noise terms in eqns (14.170) and (14.172) to get

S 23 (t) = ’i„¦P TP S 21 (t) ,

(14.173)

S 13 (t) = ’i„¦P TP S 11 (t) ’ S 33 (t) .

Substituting these results in the remaining Langevin equations and restoring the con-

tributions from HS1 produces the reduced equations:

dS 11 (t) 1

= ’RP S 11 (t) + w21 S 22 (t) + (w31 + RP ) S 33 (t) + S 11 (t) , HS1 + ξ11 (t) ,

dt i

(14.174)

dS 22 (t) 1

= w32 S 33 (t) ’ w21 S 22 (t) + S 22 (t) , HS1 + ξ22 (t) , (14.175)

dt i

¼ Quantum noise and dissipation

dS 33 (t) 1

= RP S 11 (t)’(w31 + RP + w32 ) S 33 (t)+ S 33 (t) , HS1 +ξ33 (t) , (14.176)

dt i

dS 12 (t) 1 1

= i∆21 ’ (w21 + RP ) S 12 (t) + S 12 (t) , HS1 + ξ12 (t) , (14.177)

dt 2 i

where RP = 2„¦2 TP is the incoherent pumping rate. The more familiar c-number Bloch

P

equations describing incoherent pumping are derived in Exercise 14.7 by averaging

these equations with the initial density operator ρ. The correlation functions for the

remaining noise operators can be calculated by means of the Einstein relation discussed

in Section 14.6.2 and Exercise 14.8.

In eqn (14.177) we have explicitly exhibited the dephasing rate (w21 + RP ) /2, in

order to show that the pumping rate, RP , contributes to the dephasing rate in exactly

the same way as the decay rate w21 . This suggests that we modify the general de¬nition

(14.162) for “pq to include the e¬ects of any pumping transitions that may be present.

This is done by replacing the decay rates wqp with wqp + Rqp , where Rqp = Rpq is the

rate for an incoherent pump driving q ” p.

The ¬‚uctuation dissipation theorem—

14.6

Now that we have seen several examples of the ¬‚uctuation dissipation theorem, it is

time to seek a more general result. In the examples considered above, the OJ s satisfy

commutation relations of the general form

ΛI OI

[OJ , OK ] = (14.178)

JK

I

(e.g. the operators 1, a, a† or S qp ), and in some cases product relations

¦I OI

OJ OK = (14.179)

JK

I

(e.g. the transition operators S qp ), where the ΛI s and ¦I s are c-number coe¬-

JK JK

cients. The OJ s in the previous examples also satisfy

[OJ , HS0 ] = ωJ OJ . (14.180)

The last property permits the de¬nition of slowly-varying envelope operators O J (t)

by

O J (t) = OJ (t) exp (iωJ t) . (14.181)

In practice these features are quite typical; they are not restricted to the speci¬c

examples in Sections 14.2 and 14.4. For a given sample, it is usually easy to pick out

these operators by inspection.

A potentially signi¬cant weakness of the discussions in Sections 14.2 and 14.4 is

their neglect of the e¬ects of internal sample interactions or interactions with external

classical ¬elds. In particular, the proof of the important nonanticipating property in

eqn (14.77) uses the explicit solution (14.65) of the linear Langevin equation (14.61),

The ¬‚uctuation dissipation theorem— ½

which is only correct for HS1 = 0. This is an example of the following general feature

of the theory of noise and dissipation. If the Heisenberg equations for the sample

operators are linear, then results that are needed for subsequent applications”such

as the nonanticipating property”can be proved by fairly simple arguments. Since the

internal interaction HSS describes coupling between di¬erent degrees of freedom of the

sample, it will necessarily produce nonlinear terms in the Heisenberg equations for

the sample operators. In order to avoid these complications as much as possible, we

will make two assumptions. The ¬rst is that the internal interactions can be neglected

when considering dissipative e¬ects, i.e. HSS ∼ 0. The second is that any external

interactions produce linear terms in the Heisenberg equation, i.e.

1

OJ (t) , VS (t) = i „¦JK (t) OK (t) , (14.182)

i

K

where the „¦JK (t)s are c-number functions. The plausibility of these assumptions de-

pends on the following points.

(1) The e¬ect of HSS and VS (t) is to cause additional unitary”and thus non-dissipa-

tive”evolution of the sample.

(2) By convention, HSS is weak compared to HS0 .

(3) In typical cases”e.g. atoms interacting with a laser or ¬eld modes excited by a

classical current”VS (t) is linear in the sample operators, and they satisfy the

commutation relations (14.178).

With these facts in mind, it is quite plausible that ignoring HSS and imposing eqn

(14.182) on VS (t) will not cause any serious errors in the treatment of dissipation and

noise. A more sophisticated argument that dispenses with these simplifying assump-

tions is brie¬‚y sketched in Exercise 14.9.

14.6.1 Generic Langevin equations

The argument just given allows us to replace the general Heisenberg equation (14.39)

for the OJ s by the equation of motion

d

i O J (t) = OJ (t) , HSE (t) + O J (t) , VS (t) (14.183)

dt

for the slowly-varying envelope operators. We can then substitute the formal solutions

(14.38) for the reservoir operators into this equation, and impose the Markov approx-

imation, i.e. the assumption that the reservoir memory Tmem is much shorter than

any dynamical time scale for the sample. The resulting Langevin equations take the

general form

d

OJ (t) = DJ (t) + ξJ (t) , (14.184)

dt

where

DJ (t) = ZJK (t) OK (t) (14.185)

K

is the (generalized) drift term, and the noise operators are de¬ned so that

¾ Quantum noise and dissipation

ξJ (t) = 0 . (14.186)

The complex coe¬cients ZJK (t) are given by

ZJK (t) = ’“JK + i„¦JK (t) , (14.187)

where the real, positive constants “JK arise from the elimination of the reservoir

variables”combined with the Markov approximation”and the real functions „¦JK (t)

are de¬ned by eqn (14.182). The decay constants “JK can be expressed as functions

of the coupling strengths vJ („¦ν ), but in practice they are treated as phenomenolog-

ical parameters. The Markov approximation includes the assumption that the noise

operators ξJ (t) are delta correlated,

†

ξJ (t) ξK (t ) = CJK δ (t ’ t ) . (14.188)

The coe¬cients CJK de¬ne the correlation matrix for the noise operators, and

CJK /2 is also known as the di¬usion matrix. The names ˜drift term™ and ˜di¬usion

matrix™ arise in connection with the master equation approach, which will be discussed

in Chapter 18.

14.6.2 The Einstein relations

The direct calculation of the correlation matrix CJK is very di¬cult, except in the case

of additive noise. Fortunately, yet another consequence of the Markov approximation

can be used to express the CJK s in terms of sample correlation functions. We ¬rst show

that the sample operators are nonanticipating with respect to the noise operators. For

this purpose we can use eqns (14.184) and (14.188) to ¬nd the equations of motion for

†

the correlation functions ξK (t ) OJ (t) :

‚ † †

ξK (t ) O J (t) = ξK (t ) DJ (t) + CKJ δ (t ’ t ) . (14.189)

‚t

For t > t the delta function term vanishes, and we ¬nd a set of linear, homogeneous

di¬erential equations

‚ † †

ξK (t ) O J (t) = ZJI ξK (t ) O I (t) (14.190)

‚t

I

†

for the set of correlation functions ξK (t ) OJ (t) . The assumption that the sample

and the reservoirs are uncorrelated at t = t0 ensures that all the correlation functions

vanish at t = t0 ,

†

ξK (t ) O I (t0 ) = 0 ; (14.191)

therefore, we can conclude that

†

ξK (t ) O J (t) = 0 for t > t . (14.192)

†

Similar arguments show that O J (t) ξK (t ) = 0 for t > t, etc.

The ¬‚uctuation dissipation theorem— ¿

To use this fact, we start with the identity (Meystre and Sargent, 1990, Sec. 14-4)

t

dOJ (t )

O J (t) = O J (t ’ ∆t) + dt

dt

t’∆t

t

= O J (t ’ ∆t) + dt {DJ (t ) + ξJ (t )} , (14.193)

t’∆t

which in turn implies

t

† † †

(t) = O J (t ’

OJ (t) ξK ∆t) ξK (t) + dt DJ (t ) ξK (t)

t’∆t

t

†

+ dt ξJ (t ) ξK (t) . (14.194)

t’∆t

The nonanticipating property guarantees that the ¬rst term vanishes and that the

integrand of the second term also vanishes, except possibly at the end point t = t. Thus

†

the integral must vanish unless the correlation function DJ (t ) ξK (t) is proportional

to δ (t ’ t ). This cannot be the case, since the drift term is slowly varying compared

to the noise term. Thus only the third term contributes, and

t

† †

OJ (t) ξK (t) = dt ξJ (t ) ξK (t)

t’∆t

t

1

dt CJK δ (t ’ t ) =

= CJK . (14.195)

2

t’∆t

A similar calculation shows that

1

†

ξJ (t) O K (t) = CJK . (14.196)

2

We will now use these results to investigate the equation of motion of the equal-

†

time correlation function OJ (t) O K (t) . The Langevin equation (14.184) combines

with eqns (14.195) and (14.196) to yield

d † † † †

OJ (t) O K (t) = {DJ (t) + ξJ (t)} OK (t) + OJ (t) DK (t) + ξK (t)

dt

† †

= DJ (t) O K (t) + O J (t) DK (t)

†

†

+ O J (t) ξK (t) + ξJ (t) O K (t)

† †

= DJ (t) O K (t) + O J (t) DK (t) + CJK . (14.197)

We turn this around to obtain the Einstein relation,

d † † †

O J (t) OK (t) ’ DJ (t) O K (t) ’ O J (t) DK (t) ,

CJK = (14.198)

dt

that expresses the noise correlation matrix in terms of equal-time sample correlation

functions. The sample correlation functions depend on the decay constants, so this is

Quantum noise and dissipation

the general form of the ¬‚uctuation dissipation theorem. The calculation of the noise

correlation matrix is thereby reduced to obtaining the values of the equal-time corre-

†

lation functions OI (t) OK (t) . In general the sample correlation functions must be

independently calculated”e.g. by means of the master equation discussed in Chapter

18”but approximate estimates are often su¬cient.

For an illustration of the use of eqn (14.198), we turn to the incoherently pumped

three-level atom of Section 14.5. The index J now runs over the nine pairs (q, p), with

q, p = 1, 2, 3. Let us, for example, calculate the correlation coe¬cient C12,12 appearing

in

†

ξ12 (t) ξ12 (t ) = C12,12 δ (t ’ t ) . (14.199)

For the case of pure pumping, i.e. HS1 = 0, the Langevin equation (14.177) tells us

that the drift term D12 = ’“12 S 12 . Applying eqn (14.198) yields

d † † †

S 12 S 12 ’ D12 S 12 ’ S 12 D12

C12,12 =

dt

d †

= S 11 + 2“12 S 12 S 12

dt

= ’RP N1 (t) + w21 N2 (t) + (w31 + RP ) N3 (t) + 2“12 N1 (t) , (14.200)

where Nq (t) = S qq (t) . At long times (i.e. for t0 ’ ’∞) the populations are given

by the steady-state solution of the c-number Bloch equations obtained by averaging

eqns (14.174)“(14.177). One then ¬nds

2“12 w21 (RP + w31 + w32 )

C12,12 = . (14.201)

RP (2w21 + w32 ) + w21 (w31 + w32 )

Note that C12,12 , which represents the strength of the noise operator ξ12 , vanishes for

w21 = 0. This justi¬es the interpretation of ξ12 as the noise due to the spontaneous

emission 2 ’ 1. A similar calculation yields

2“12 RP w32

C21,21 = , (14.202)

RP (2w21 + w32 ) + w21 (w31 + w32 )

which implies

† †

ξ12 (t) ξ12 (t ) = ξ21 (t) ξ21 (t ) = C21,21 δ (t ’ t ) . (14.203)

Quantum regression—

14.7

All experimentally relevant numerical information is contained in the expectation val-

ues of functions of the sample operators, so we begin by observing that the expectation

values O J (t) obey the averaged form of the Langevin equations (14.184):

d

O J (t) = ZJK (t) OK (t) . (14.204)

dt

K

Photon bunching—

A standard method for solving sets of linear ¬rst-order equations like (14.204) is to

de¬ne a Green function GJK (t, t ) by

d

GJK (t, t ) = ZJI (t) GIK (t, t ) ,

dt (14.205)

I

GJK (t , t ) = δJK ,

which allows the solution of eqn (14.204) to be written as

OJ (t) = GJK (t, t ) OK (t ) . (14.206)

K

In classical statistics, the relation (14.206) between the averages of the stochast-

ically-dependent variables OJ (t) and O K (t ) is called a linear regression. This so-

lution for the time dependence of the averages of the sample operators is moderately

useful, but the correlation functions O J (t) O K (t ) are of much greater interest, since

their Fourier transforms describe the spectral response functions measured in experi-

ments. Using the Langevin equation for OJ (t) to evaluate the time derivative of the

correlation function leads to

d

O J (t) O K (t ) = ’ ZJI (t) O I (t) O K (t ) + ξJ (t) O K (t ) . (14.207)

dt

I

For t < t the nonanticipating property (14.192) imposes ξJ (t) OK (t ) = 0, and the

correlation function satis¬es

d

OJ (t) O K (t ) = ’ ZJI (t) OI (t) OK (t ) . (14.208)

dt

I

Since this has the same form as eqn (14.204), the solution is obtained by using the

same Green function:

O J (t) OK (t ) = GJI (t, t ) OI (t ) O K (t ) . (14.209)

I

In other words, the two-time correlation function O J (t) O K (t ) is related to the

equal-time correlation functions OI (t ) OK (t ) by the same regression law that re-

lates the single-time averages O J (t) at time t to the averages O I (t ) at the earlier

time t . A little thought shows that a similar derivation gives the more general result