In the Schr¨dinger picture, the reservoir and sample operators act in di¬erent spaces,

o

so [bJν , O] = 0 for any sample operator, O. Since the Schr¨dinger and Heisenberg

o

pictures are connected by a time-dependent unitary transformation, the equal-time

commutators vanish at all times,

[O (t) , bJν (t)] = O (t) , b† (t) = 0 . (14.36)

Jν

With this fact in mind, it is straightforward to use the explicit form of HW to ¬nd the

Heisenberg equations for the reservoir operators:

‚bJν (t)

= ’i„¦ν bJν (t) ’ vJ („¦ν ) OJ (t) . (14.37)

‚t

¾ Quantum noise and dissipation

Each of these equations has the formal solution

t

’i„¦ν (t’t0 )

dt e’i„¦ν (t’t ) OJ (t ) ,

’ vJ („¦ν )

bJν (t) = bJν (t0 ) e (14.38)

t0

where t0 is the initial time at which the Schr¨dinger and Heisenberg pictures coincide.

o

This convention allows the identi¬cation of bJν (t0 ) with the Schr¨dinger-picture op-

o

erator bJν . The ¬rst term on the right side of this equation describes free evolution of

the reservoir, and the second term represents radiation reaction, i.e. the emission and

absorption of reservoir excitations by the sample.

The Heisenberg equation for a sample operator OK is

‚OK (t) 1 1 1

= [OK (t) , HW (t)] = [OK (t) , HS (t)] + [OK (t) , HSE (t)] . (14.39)

‚t i i i

The explicit form (14.14) for HSE (t), together with the equal-time commutation rela-

tions, allow us to express the ¬nal term in eqn (14.39) as

1

OK (t) , OJ (t) bJν (t) ’ b† (t) [OK (t) , OJ (t)] .

†

[OK (t) , HSE (t)] = vJ („¦ν ) Jν

i ν

(14.40)

The equal-time commutation relations (14.36) guarantee that the products of sam-

ple and reservoir operators in this equation can be written in any order without chang-

ing the result, but the individual terms in the formal solution (14.38) for the reservoir

operators do not commute with the sample operators. Consequently, it is essential to

decide on a de¬nite ordering before substituting the formal solution for the reservoir

operators into eqn (14.40), and this ordering must be strictly enforced throughout the

subsequent calculation. The ¬nal physical predictions are independent of the original

order chosen, but the interpretation of intermediate results may vary. This is another

example of ordering ambiguities like those that allow one to have the zero-point en-

ergy by choosing symmetrical ordering, or to eliminate it by using normal ordering.

We have chosen to write eqn (14.40) in normal order with respect to the reservoir

operators.

Substituting the formal solution (14.38) into eqn (14.40) yields two kinds of terms.

One depends explicitly on the initial reservoir operators bJν (t0 ) and the other arises

from the radiation-reaction term. We can now proceed to eliminate the reservoir

degrees of freedom”in parallel with the elimination of the radiation ¬eld in the

Weisskopf“Wigner model”but the necessary calculations depend on the details of

the sample“environment interaction. Consequently, we will carry out the adiabatic

elimination process in several illustrative examples.

14.2 Photons in a lossy cavity

In this example, the sample consists of the discrete modes of the radiation ¬eld in an

ideal physical cavity, and the environment consists of one or more reservoirs which

schematically describe the mechanism for the loss of electromagnetic energy. For an

enclosed cavity”such as the microcavities discussed in Chapter 12”a single reservoir

¾

Photons in a lossy cavity

representing the exchange of energy between the radiation ¬eld and the cavity walls

will su¬ce. For the commonly encountered four-port devices”e.g. a resonant cavity

capped by mirrors”it is necessary to invoke two reservoirs representing the vacuum

modes entering and leaving the cavity through each port. In the present section we

will concentrate on the simpler case of the enclosed cavity; the four-port devices will

be discussed in Section 14.3.

In order for the discrete cavity modes to retain their identity, the characteristic

interaction energy, „¦S , between the sample and the reservoir must be small compared

to the minimum energy di¬erence, ∆ω, between adjacent modes, i.e.

„¦S ∆ω . (14.41)

For example, a rectangular cavity with dimensions L1 , L2 , and L3 satisfying L1

L2 L3 has ∆ω = 2πc/L3 . When eqn (14.41) is satis¬ed the radiation modes are

weakly coupled through their interaction with the reservoir modes, and”to a good

approximation”we may treat each radiation mode separately.

We may, therefore, consider a reduced sample consisting of a single mode of the

¬eld, with frequency ω0 , and drop the mode index. The unperturbed sample Hamil-

tonian is then

HS0 = ω0 a† a , (14.42)

and we will initially allow for the presence of an interaction term HS1 (t). In this case

there is only one sample operator and one reservoir, so the general expression (14.14)

reduces to

v („¦ν ) a† bν ’ b† a .

HSE = i (14.43)

ν

ν

The coupling constant v („¦ν ) is proportional to the RWA cut-o¬ function de¬ned by

eqn (11.22):

v („¦ν ) = v0 („¦ν ) K („¦ν ’ ω0 ) . (14.44)

This is an explicit realization of the assumption that the sample is coupled to a broad

spectrum of reservoir excitations.

In this connection, we note that the interaction Hamiltonian HSE is similar to

the RWA interaction Hamiltonian Hrwa , in eqn (11.46), that describes spontaneous

emission by a two-level atom. In the present case, the annihilation operator a for the

discrete cavity mode plays the role of the atomic lowering operator σ’ and the modes

of the radiation ¬eld are replaced by the reservoir excitation modes. The mathematical

similarity between HSE and Hrwa allows similar physical conclusions to be drawn. In

particular, a reservoir excitation”which carries positive energy”will never be reab-

sorbed once it is emitted. The implication that the interaction between the sample

and a physically realistic reservoir is inherently dissipative is supported by the explicit

calculations shown below.

This argument apparently rules out any description of an amplifying medium in

terms of coupling to a reservoir. There is a formal way around this di¬culty, but it

requires the introduction of an inverted-oscillator reservoir which has distinctly

unphysical properties. In this model, all reservoir excitations have negative energy;

therefore, emitting a reservoir excitation would increase the energy of the sample.

¿¼ Quantum noise and dissipation

Since the emission is irreversible, the result would be an ampli¬cation of the cavity

mode. For more details, see Gardiner (1991, Chap. 7.2.1) and Exercise 14.5.

14.2.1 The Langevin equation for the ¬eld

The Heisenberg equation for a (t) is

d 1

a (t) = ’iω0 a (t) + v („¦ν ) bν (t) + [a (t) , HS1 (t)] , (14.45)

dt i

ν

while the formal solution (14.38) for this case is

t

’i„¦ν (t’t0 )

dt e’i„¦ν (t’t ) a (t ) .

’ v („¦ν )

bν (t) = bν (t0 ) e (14.46)

t0

|v („¦ν )|, and we will also assume that HS1 is

The general rule (14.2) requires ω0

weak compared to HS0 . Thus the ¬rst term on the right side of eqn (14.45) describes

oscillations that are much faster than those due to the remaining terms. This suggests

the introduction of slowly-varying envelope operators,

a (t) = a (t) eiω0 t , bν (t) = bν (t) eiω0 t , (14.47)

that satisfy

d 1

a (t) = v („¦ν ) bν (t) + [a (t) , HS1 (t)] , (14.48)

dt i

ν

and

t

’i(„¦ν ’ω0 )(t’t0 )

dt e’i(„¦ν ’ω0 )(t’t ) a (t ) .

’ v („¦ν )

bν (t) = bν (t0 ) e (14.49)

t0

The envelope operator a (t) varies on the time scale TS = 1/„¦S , so it is the operator

version of the slowly-varying classical envelope introduced in Section 3.3.1.

We are now ready to carry out the elimination of the reservoir degrees of freedom,

by substituting eqn (14.49) into eqn (14.48). The HS1 -term plays no role in this argu-

ment, so we will simplify the intermediate calculation by omitting it. The simpli¬ed

equation for a (t) is

t

d

a (t) = ’ dt K (t ’ t ) a (t ) + ξ (t) , (14.50)

dt t0

where

|v („¦ν )|2 e’i(„¦ν ’ω0 )(t’t ) ,

K (t ’ t ) = (14.51)

ν

and

v („¦ν ) bν (t0 ) e’i(„¦ν ’ω0 )(t’t0 ) .

ξ (t) = (14.52)

ν

At this stage, the passage to the continuum limit is essential; therefore, we change

the sum over the discrete modes to an integral according to the rule

¿½

Photons in a lossy cavity

∞

fν ’ d„¦D („¦) f („¦) , (14.53)

0

ν

where D („¦) is the density of states for the reservoir modes. The exact form of D („¦)

depends on the particular model chosen for the reservoir. For example, if the reservoir

is de¬ned by modes of the radiation ¬eld, then D („¦) is given by eqn (4.158). In practice

these details are not important, since they will be absorbed into a phenomenological

decay constant. Applying the rule (14.53) to K (t ’ t ) and using eqn (14.44) leads to

the useful representation

∞

d„¦D („¦) |v0 („¦)|2 |K („¦ ’ ω0 )|2 e’i(„¦’ω0 )(t’t ) .

K (t ’ t ) = (14.54)

0

The frequency width of the Fourier transform K („¦) of K (t ’ t ) is well approxi-

mated by the width ∆K of the cut-o¬ function. According to the uncertainty principle

for Fourier transforms, the temporal width of K (t ’ t ) is therefore of the order of

1/∆K . Since K (t ’ t ) decays to zero for |t ’ t | > 1/∆K , we use the terminology in-

troduced in Section 11.1.2 to call Tmem = 1/∆K the memory interval for the reservoir.

The general rule (14.2) for cut-o¬ functions, which in the present case is

„¦S = max |v („¦)| ∆K ω0 , (14.55)

imposes the relation Tmem TS . In other words, the assumption of a broad spectral

range for the sample“reservoir interaction is equivalent to the statement that the

reservoir has a short memory. This assumption e¬ectively restricts the integral in

eqn (14.50) to the interval t ’ Tmem < t < t, in which a (t ) is essentially constant.

The short memory of the reservoir justi¬es the Markov approximation, a (t ) ≈ a (t),

and this allows us to replace the integro-di¬erential equation (14.50) by the ordinary

di¬erential equation

d Λ (t)

a (t) = ’ a (t) + ξ (t) , (14.56)

dt 2

where

t

dt K (t ’ t ) .

Λ (t) = 2 (14.57)

t0

Substituting the explicit form for K (t ’ t ) gives

∞

t’t0

d„¦D („¦) |v0 („¦)| |K („¦ ’ ω0 )| e’i(„¦’ω0 )„ .

2 2

Λ (t) = 2 d„ (14.58)

0 0

2

We can assume that the cut-o¬ function |K („¦ ’ ω0 )| is sharply peaked with respect

2

to the prefactor in the „¦-integrand, so that D („¦) |v0 („¦)| can be removed from the

„¦-integral to get

∞

t’t0

d„¦ |K („¦)|2 e’i„¦„ .

2

Λ (t) = 2D (ω0 ) |v0 (ω0 )| d„ (14.59)

’ω0

0

The width ∆K of the cut-o¬ function satis¬es ∆K ω0 , so the lower limit of the

„¦-integral can be replaced by ’∞ with negligible error. This approximation ensures

¿¾ Quantum noise and dissipation

2

that Λ (t) is real. After interchanging the „¦- and „ -integrals and noting that |K („¦)|

is an even function of „¦, one ¬nds that

∞ t’t0

1

d„ e’i„¦„ .

2 2

Λ (t) = 2D (ω0 ) |v0 (ω0 )| d„¦ |K („¦)| (14.60)

2

’∞ ’(t’t0 )

The de¬nition (14.57) shows that Λ (t0 ) = 0, but we are only concerned with much

later times such that t ’ t0 > TS Tmem , where TS = 1/„¦S is the response time

for the slowly-varying envelope operator. In this limit, i.e. after several memory times

have passed, eqn (14.56) can be replaced by

d κ

a (t) = ’ a (t) + ξ (t) , (14.61)

dt 2

where

lim Λ (t) = 2πD (ω0 ) |v (ω0 )|2 . (14.62)

κ=

t0 ’’∞

If we had not extended the lower integration limit in eqn (14.59) to ’∞, the constant

κ would have a small imaginary part. This is reminiscent of the Weisskopf“Wigner

model, in which the decay constant for the upper level of an atom is found to have a

small imaginary part nominally related to the Lamb shift. In Section 11.2.2, we showed

that a consistent application of the resonant wave approximation requires one to drop

the imaginary part. Applying this idea to the present case implies that extending the

lower limit to ’∞ is required for consistency with the resonant wave approximation.

The Fermi-golden-rule result, eqn (14.62), demonstrates that κ is positive for every

initial state of the reservoir. This agrees with the expectation”expressed at the be-

ginning of Section 14.2”that the interaction of the cavity mode and the reservoir

is necessarily dissipative. From now on we will call κ the decay rate for the cavity

mode. One can easily verify that the HS1 -contribution could have been carried along

throughout this calculation, to get the complete equation

d κ 1

a (t) = ’ a (t) + [a (t) , HS1 (t)] + ξ (t) . (14.63)

dt 2 i

The last vestiges of the reservoir degrees of freedom are in the operator ξ (t). This is

conventionally called a noise operator, since eqn (14.61) is the operator analogue of

the Langevin equations describing the evolution of a classical oscillator subjected to a

random driving force. The most famous application for these equations is the analysis

of Brownian motion (Chandler, 1987, Sec. 8.8). This formal similarity has led to the

name operator Langevin equation for eqn (14.61). This language is extended to

eqn (14.63), even when an internal interaction HSS contributes nonlinear terms.

According to eqn (14.52), ξ (t) is a linear function of the initial reservoir operators

bν (t0 ) alone; it does not depend on the ¬eld operators. Noise operators of this kind are

said to be additive, but not all noise operators have this property. In Section 14.4 we

will see that the noise operators for atoms involve products of reservoir operators and

atomic operators. Noise operators of this kind are said to represent multiplicative

noise. An example of multiplicative noise for the radiation ¬eld is given in Exercise

14.2.

¿¿

Photons in a lossy cavity

The additivity property of the noise operator ξ (t) implies that the initial sample

operators, a (t0 ) and a† (t0 ), commute with ξ (t) for any t. On the other hand, the

sample operators at later times depend on the operators bν (t0 ) and b† (t0 ); therefore,

ν

they will not in general commute with ξ (t) or ξ † (t). This is an example of the general

ordering problem discussed in Section 14.1.2; it is solved by strictly adhering to the

original ordering of factors.

At ¬rst glance, the noise operator ξ (t) may appear to be merely another nuisance”

like the zero-point energy”but this is not true. To illustrate the importance of ξ (t),

let us drop the noise operator from eqn (14.61). The solution is then a (t) =

e’κ(t’t0 )/2 a (t0 ), which in turn gives the equal-time commutator

a (t) , a† (t) = a (t) , a† (t) = e’κ(t’t0 ) a, a† = e’κ(t’t0 ) . (14.64)

This is disastrously wrong! Unitary time evolution preserves the commutation re-

lations, so we should ¬nd a (t) , a† (t) = 1 at all times. This contradiction shows

that the noise operator is essential for preserving the canonical commutation relations

and, consequently, the uncertainty principle. In this example”with no HS1 -term”the

Langevin equation is so simple that one can immediately write down the solution

t

’κ(t’t0 )/2

dt e’κ(t’t )/2 ξ (t ) ,

a (t) = e a+ (14.65)

t0

and then calculate the equal-time commutator explicitly:

t t

† ’κ(t’t0 )

dt e’κ(t’t )/2 e’κ(t’t ξ (t ) , ξ † (t ) .

)/2

a (t) , a (t) = e + dt

t0 t0

(14.66)

The de¬nition (14.52) leads to

ξ (t ) , ξ † (t ) = |v („¦ν )| e’i(„¦ν ’ω0 )(t ’t

2 )

. (14.67)

ν

In the continuum limit, the arguments used to get from eqn (14.58) to eqn (14.60) can

be applied to get

ξ (t ) , ξ † (t ) = κδ (t ’ t ) . (14.68)

It should be understood that this result is valid only when applied to functions that

vary slowly on the time scale Tmem of the reservoir. Substituting eqn (14.68) into eqn

(14.66) shows that indeed a (t) , a† (t) = 1 at all times t.

14.2.2 Noise correlation functions

We next apply the general results in Section 14.1.1-C to study the properties of the

noise operator. According to the de¬nition (14.52) of ξ (t) and the convention (14.23),

the average of ξ (t) vanishes, i.e.

ξ (t) = TrE [ρE ξ (t)] = 0 . (14.69)

¿ Quantum noise and dissipation

This is of course what one should expect of a sensible noise source. Turning next to the

correlation function, we know”from previous experience with vacuum ¬‚uctuations”

that we should proceed cautiously by evaluating ξ † (t) ξ (t ) for t = t. Since ξ † (t) ξ (t )

only acts on the reservoir degrees of freedom, an application of eqn (14.19) gives

ξ † (t) ξ (t ) = TrE ρE ξ † (t) ξ (t ) . (14.70)

Substituting the explicit de¬nition (14.52) of the noise operator yields

†

ξ † (t ) ξ (t) = v („¦ν ) v („¦µ ) bν (t0 ) bµ (t0 )

E

ν µ

— ei(„¦ν ’ω0 )(t ’t0 ) e’i(„¦ν ’ω0 )(t’t0 ) , (14.71)

and the assumption of uncorrelated reservoir modes simpli¬es this to

ξ † (t ) ξ (t) = |v („¦ν )|2 nν ei(„¦ν ’ω0 )(t ’t) , (14.72)

ν

where

nν = b † b ν (14.73)

ν

is the average occupation number of the νth mode of the reservoir. Taking the con-

tinuum limit and applying the Markov approximation yields the normal-ordered cor-

relation function,

ξ † (t ) ξ (t) = d„¦D („¦) |v („¦)| |K („¦ ’ ω0 )| n („¦) ei(„¦’ω0 )(t ’t)

2 2

≈ n0 κδ (t ’ t ) , (14.74)

where n0 = n (ω0 ). A similar calculation yields the antinormal-ordered correlation

function

ξ (t) ξ † (t ) = (n0 + 1) κδ (t ’ t ) . (14.75)

The noise operator is said to be delta correlated, because of the factor δ (t ’ t ).

Since this is an e¬ect of the short memory of the reservoir, the delta function only

makes sense when applied to functions that vary slowly on the time scale Tmem . The

noise strength is given by the power spectrum, i.e. the Fourier transform of the corre-

lation function. For delta-correlated noise operators the spectrum is said to be white

noise, because the power spectrum has the same value, n0 κ (or (n0 + 1) κ), for all fre-

quencies. This relation between the noise strength and the dissipation rate is another

example of the ¬‚uctuation dissipation theorem.

The delta correlation of the noise operator is the source of other useful properties

of the solutions of the linear Langevin equation (14.61). By using the formal solution

(14.65), one ¬nds that

ξ † (t + „ ) a (t) = ξ † (t + „ ) a (t0 ) e’κ(t’t0 )/2

t

dt1 e’κ(t’t1 )/2 ξ † (t + „ ) ξ (t1 ) .

+ (14.76)