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)

t0

The ¬rst term on the right side vanishes, by virtue of the assumption that the ¬eld

and the reservoir are initially uncorrelated. The second term also vanishes, because the

¿

The input“output method

delta function from eqn (14.74) vanishes for 0 t1 t and „ > 0. Thus the operator

a (t) satis¬es

ξ † (t + „ ) a (t) = 0 for „ > 0 , (14.77)

and is consequently said to be nonanticipating with respect to the noise operator

(Gardiner, 1985, Sec. 4.2.4). In anthropomorphic language, the ¬eld at time t cannot

know what the randomly ¬‚uctuating noise term will do in the future.

14.3 The input“output method

In Section 14.2 our attention was focussed on the interaction of cavity modes with a

noise reservoir, but there are important applications in which the excitation of reservoir

modes themselves is the experimentally observable signal. In these situations some of

the reservoirs are not noise reservoirs; consequently, averages like bν need not vanish.

Consider”as shown in Fig. 14.1”an open-sided cavity formed by two mirrors M1 and

M2 that match the curvature of a particular Gaussian mode. Analysis of this classical

wave problem shows that the mode is e¬ectively con¬ned to the resonator (Yariv, 1989,

Chap. 7), so that the main loss mechanism is transmission through the end mirrors.

The geometry of the cavity might lead one to believe that it is a two-port device,

but this would be a mistake. The reason is that radiation can both enter and leave

through each of the mirrors. We have indicated this feature by drawing the input and

output ports separately in Fig. 14.1. The labeling conventions are modeled after the

beam splitter in Fig. 8.2, but in this case the radiation is normally incident to the

partially transmitting mirror. Thus the resonant cavity is a four-port device.

If we only consider the fundamental cavity mode with frequency ω0 , the sample

Hamiltonian is

HS = HS0 + HS1 (t) , (14.78)

where

HS0 = ω0 a† a , (14.79)

2 1

1 2

M1 M2

Fig. 14.1 A Gaussian mode in a resonant cavity. The upper and lower dashed curves repre-

sent lines of constant intensity for a Gaussian solution given by eqn (7.50), and the left and

right dashed curves represent the local curvature of the wavefront. The curvature of mirrors

M1 and M2 are chosen to match the wavefront curvature at their locations. Under these

conditions the Gaussian mode is con¬ned to the cavity. Ports 1 and 2 are input ports for

photons entering from the left and right respectively. Ports 1 and 2 are output ports for

photons exiting to the right and left respectively. The cavity is therefore a four-port device

like the beam splitter.

¿ Quantum noise and dissipation

and a is the mode annihilation operator. The internal sample interaction Hamiltonian

HS1 (t) can depend explicitly on time in the presence of external classical ¬elds, and a

model of this sort will be used later on to describe nonlinear coupling between cavity

modes induced by spontaneous down-conversion.

Losses through the end mirrors are described by two reservoirs consisting of vac-

uum modes of the ¬eld propagating in space to the left and right of the cavity. We

could treat these reservoirs by using the exact theory of vacuum propagation, but

the simpler description in terms of the generic reservoir operators bJν introduced in

Section 14.1.1 is su¬cient. For this application it is better to go to the continuum

limit from the beginning, as opposed to the end, of the analysis. For this purpose, we

construct a simpli¬ed reservoir model by imposing periodic boundary conditions on a

one-dimensional (1D) cavity of length L. The index ν then runs over the integers, and

the corresponding wavevectors are k = 2πν/L. In the limit L ’ ∞, the operators bJν

√

are replaced by new operators bJk = LbJν satisfying

bJk , b† = 2πδ (k ’ k ) , (14.80)

Jk

and the environment Hamiltonian is

∞

2

dk

„¦k b† bJ,k .

HE = (14.81)

J,k

2π

’∞

J=1

The standard approach to in- and out-¬elds (Gardiner, 1991, Sec. 5.3) employs

creation and annihilation operators for modes of de¬nite frequency „¦, rather than

de¬nite wavenumber k. In the 1D model this can be achieved by assuming that the

mode frequency „¦k is a monotone-increasing function of the continuous label k. This

assumption justi¬es the change of variables k ’ „¦ in eqn (14.81), with the result

∞ 2

d„¦

b† bJ,„¦ ,

HE = „¦ (14.82)

J,„¦

2π

0 J=1

where

1

bJ,„¦ = bJ,k . (14.83)

|d„¦k /dk|

Using this de¬nition in eqn (14.80) leads to the Heisenberg-picture, equal-time com-

mutation relations

bJ,„¦ (t) , b†

K,„¦ (t) = 2πδJK δ („¦ ’ „¦ ) , J, K = 1, 2 , (14.84)

bJ,„¦ (t) , a† (t) = 0 , J = 1, 2 . (14.85)

It should be kept in mind that „¦ simply replaces the mode label k; it is not a Fourier

transform variable.

We should also mention that the usual presentation of this theory extends the „¦-

integral in eqn (14.82) to ’∞, and thus introduces unphysical negative-energy modes.

In expert hands, this formal device simpli¬es the mathematics without really violating

¿

The input“output method

any physical principles, but it clearly de¬es Einstein™s rule. Furthermore, the restriction

to the physically allowed, positive-energy modes clari¬es the physical signi¬cance of

the approximations to be imposed below.

In our approach, the generic sample“environment Hamiltonian, given by eqn

(14.43), is

∞

2

d„¦ dk

vJ („¦) a† bJ,„¦ ’ b† a .

HSE = i L (14.86)

J,„¦

2π d„¦

0

J=1

√

The looming disaster of the uncompensated factor L is an illusion. In the ¬nite

cavity, the unit cell for wavenumbers is 2π/L; therefore, the density of states D („¦)

satis¬es

dk

D („¦) d„¦ = . (14.87)

2π/L

This observation allows the dangerous-looking result for HSE to be replaced by

∞

2

D („¦)

vJ („¦) a† bJ,„¦ ’ b† a .

HSE = i d„¦ (14.88)

J,„¦

2π

0

J=1

The terms in eqn (14.88) have simple interpretations; for example, b† a represents

2,„¦

the disappearance of a cavity photon balanced by the emission of a photon into the

environment through the mirror M2.

The slowly-varying envelope operators”a (t) = a (t) exp (iω0 t) and bJ,„¦ (t) =

bJ,„¦ (t) exp (iω0 t) (J = 1, 2)”obey the Heisenberg equations of motion:

d

bJ,„¦ (t) = ’i („¦ ’ ω0 ) bJ,„¦ (t) ’ 2πD („¦)vJ („¦) a (t) (J = 1, 2) , (14.89)

dt

∞

2

D („¦)

d 1

a (t) = [a (t) , HS1 (t)] + d„¦ vJ („¦) bJ,„¦ (t) . (14.90)

dt i 2π

0

J=1

14.3.1 In-¬elds

We begin by choosing a time t0 earlier than any time at which interactions occur. A

formal solution of eqn (14.89) is given by

t

’i(„¦’ω0 )(t’t0 )

dt e’i(„¦’ω0 )(t’t ) a (t ) ,

’

bJ,„¦ (t) = bJ,„¦ (t0 ) e 2πD („¦)vJ („¦)

t0

(14.91)

and substituting this into eqn (14.90) yields

d 1

a (t) = [a (t) , HS1 (t)]

dt i

∞

2

D („¦)

vJ („¦) bJ,„¦ (t0 ) e’i(„¦’ω0 )(t’t0 )

+ d„¦

2π

0

J=1

∞

2 t

dt e’i„¦(t’t ) a (t ) ,

2

’ d„¦D („¦ + ω0 ) |vJ („¦ + ω0 )|

’ω0 t0

J=1

(14.92)

¿ Quantum noise and dissipation

where the integration variable „¦ has been shifted by „¦ ’ „¦ + ω0 in the ¬nal term.

Since the operator a (t ) is slowly varying, the t -integral in this term de¬nes a function

of „¦ that is sharply peaked at „¦ = 0; in particular, the width of this function is small

compared to ω0 . This implies that the lower limit of the „¦-integral can be extended to

’∞ with negligible error. In addition, we impose the Markov approximation by the

ansatz :

2πD („¦) |vJ („¦)|2 κJ = 2πD (ω0 ) |vJ (ω0 )|2 (J = 1, 2) , (14.93)

representing the assumption that the sample interacts with a broad spectrum of reser-

voir excitations. Note that this replaces eqn (14.91) by

√ t

’i(„¦’ω0 )(t’t0 )

dt e’i(„¦’ω0 )(t’t ) a (t ) .

’ κJ

bJ,„¦ (t) = bJ,„¦ (t0 ) e (14.94)

t0

When the approximation (14.93) is used in eqn (14.92), the extended „¦-integral in

the third term produces 2πδ (t ’ t ). Evaluating the t -integral, with the aid of the

end-point rule (A.98), then leads to the Langevin equation,

d 1 κC

[a (t) , HS1 (t)] ’

a (t) = a (t) + ξC (t) , (14.95)

dt i 2

where

κC = κ 1 + κ 2 (14.96)

is the total cavity damping rate. The cavity noise-operator,

2

√

ξC (t) = κJ bJ,in (t) , (14.97)

J=1

is expressed in terms of the in-¬elds

∞

d„¦

bJ,„¦ (t0 ) e’i(„¦’ω0 )(t’t0 ) .

bJ,in (t) = (14.98)

2π

0

For later use it is convenient to write out the Langevin equation as

√ √

d 1 κC

[a (t) , HS1 (t)] + κ1 b1,in (t) + κ2 b2,in (t) ’

a (t) = a (t) . (14.99)

dt i 2

The operator a (t) depends on the initial reservoir operators through the in-¬elds,

so eqn (14.99) is called the retarded Langevin equation. Since t0 precedes any

interactions, the reservoir ¬elds and the sample ¬elds are uncorrelated at t = t0 .

The in-¬elds have an unexpected algebraic property. Combining the equal-time

commutation relations (14.84) with the de¬nition (14.98) leads to

∞

d„¦ ’i„¦(t’t )

†

bJ,in (t) , bK,in (t ) = δJK e . (14.100)

2π

’ω0

The correct interpretation of the ambiguous expression on the right side involves both

mathematics and physics. The mathematical part of the argument is to interpret the

¿

The input“output method

„¦-integral as a generalized function of t’ t . According to Appendix A.6.2, this is done

by applying the generalized function to a good function f (t ) to ¬nd:

∞ ∞ ∞

d„¦ ’i„¦(t’t ) d„¦ ’i„¦t

dt e f (t ) = e f („¦) . (14.101)

2π 2π

’∞ ’ω0 ’ω0

The physical part of the argument is that only slowly-varying good functions are

relevant. In the frequency domain, this means that f („¦) is peaked at „¦ = 0 and has

a width that is small compared to ω0 . Thus, just as in the argument following eqn

(14.92), the lower limit can be extended to ’∞ with negligible error. This last step

replaces the right side of eqn (14.101) by f (t), and this in turn implies the unequal-

time commutation relations:

«

†

bJ,in (t) , bK,in (t ) = δJK δ (t ’ t )¬

(J, K = 1, 2) , (14.102)

⎭

b (t) , b (t ) = 0

J,in K,in

for the in-¬elds.

If the environment density operator represents the vacuum, i.e.

†

bJ,„¦ (t0 ) ρE = ρE bJ,„¦ (t0 ) = 0 (J = 1, 2) , (14.103)

and [a, Hss ] = 0, then one can show that

d† κC †

a (t) a (t) = ’ (κ1 + κ2 ) a† (t) a (t) .

a (t) a (t) = ’2 (14.104)

dt 2

This justi¬es the interpretation of κ1 and κ2 as the rate of loss of cavity photons

through mirrors M1 and M2 respectively.

14.3.2 Out-¬elds

In most applications, only the emitted ¬elds are experimentally accessible; thus we

will be interested in the reservoir ¬elds at late times, after all interactions inside the

cavity have occurred. For this purpose, we choose a late time t1 and write a formal

solution of eqn (14.89) as

√ t1

’i(„¦’ω0 )(t’t1 )

dt e’i(„¦’ω0 )(t’t ) a (t ) (J = 1, 2) .

bJ,„¦ (t) = bJ,„¦ (t1 ) e + κJ

t

(14.105)

After substituting this into eqn (14.90), we ¬nd the advanced Langevin equation

√ √

d 1 κC

a (t) = [a (t) , HS1 (t)] + κ2 b2,out (t) + κ1 b1,out (t) + a (t) , (14.106)

dt i 2

where the out-¬elds bJ,out (t) are de¬ned by

∞

d„¦

bJ,„¦ (t1 ) e’i(„¦’ω0 )(t’t1 ) .

bJ,out (t) = (14.107)

2π

0

The sign di¬erence between the ¬nal terms of eqns (14.106) and (14.99) can be traced

back to the minus sign in the second term of eqn (14.105). This in turn re¬‚ects the

¼ Quantum noise and dissipation

free evolution of b1,out (t) and b2,out (t) toward the future values b1,„¦ (t1 ) and b2,„¦ (t1 ).

Another important di¬erence from the retarded case is that the operators b1,„¦ (t1 ) and

b2,„¦ (t1 ) are necessarily correlated with the sample operator a (t1 ), since the time t1

follows all interactions inside the cavity.

A relation between the in- and out-¬elds”similar to the scattering relations dis-

cussed in Section 8.2”follows from equating the alternate expressions (14.94) and

(14.105) for bJ,„¦ (t) to get

√ t1

bJ,„¦ (t1 ) e’i(„¦’ω0 )(t’t1 ) = bJ,„¦ (t0 ) e’i(„¦’ω0 )(t’t0 ) ’ dt e’i(„¦’ω0 )(t’t ) a (t ) .

κJ

t0

(14.108)

The left side of this equation is the integrand of the expression (14.107) de¬ning

bJ,out (t), so we take the hint and integrate over „¦ to ¬nd the input“output equation:

√

bJ,out (t) = bJ,in (t) ’ κJ a (t) . (14.109)

14.3.3 The empty cavity

In order to get some insight into the meaning of all this formalism, we consider the

case of an empty cavity, i.e. HS1 = 0. In this case, the equation of motion (14.99) for

the intracavity ¬eld is a linear di¬erential equation with constant coe¬cients,

√ √

d κC

a (t) = κ2 b2,in (t) + κ1 b1,in (t) ’ a (t) . (14.110)

dt 2

Equations of this type are commonly solved by introducing the Fourier transform pairs

∞

dteiωt F (t) ,

F (ω) = (14.111)

’∞

∞

dω ’iωt

F (t) = e F (ω) . (14.112)

2π

’∞

In the present case, F (t) stands for any of the envelope operators a (t), b1,in (t), and

b2,in (t). Since these operators are not hermitian, a convention regarding adjoints is

needed. We choose to use the same convention in the time and frequency domains:

† † † †

F (t) = F (t) , F (ω) = F (ω) . (14.113)

With this convention in force, the adjoint of eqn (14.112) yields

∞ ∞

dω iωt † dω ’iωt †

†

F (t) = e F (ω) = e F (’ω) . (14.114)

2π 2π

’∞ ’∞

Substituting the expansions (14.112) and (14.114) into eqn (14.102) produces the

frequency-domain commutation relations

†

bJ,in (ω) , bK,in (ω ) = 2πδJK δ (ω ’ ω ) ,

(14.115)

bJ,in (ω) , bK,in (ω ) = 0 .

In general, it is not correct to think of eqn (14.112) as a mode expansion for

F (t). For example, a (t) is the Heisenberg-picture annihilation operator associated

½

The input“output method

with a particular cavity mode; this is as far as mode expansions go. Consequently the

application of eqn (14.112) to a (t) cannot be regarded as a further mode expansion.

The in-¬elds are a special case in this regard, since Fourier transforming the de¬nition

(14.98) yields

bJ,in (ω) = bJ,ω+ω0 (t0 ) eiωt0 (J = 1, 2) . (14.116)

This close relation between the Fourier transform and the mode expansion is a result of

the explicit de¬nition of the in-¬eld as a superposition of freely propagated annihilation

operators for the individual modes.

We can now proceed by Fourier transforming the di¬erential equation (14.110) for

a (t) , to get the algebraic equation

√ √ κC

’iωa (t) = κ2 b2,in (ω) + κ1 b1,in (ω) ’ a (ω) , (14.117)

2

with the solution √ √

κ2 b2,in (ω) + κ1 b1,in (ω)

a (ω) = . (14.118)

κC /2 ’ iω

In the frequency domain, the unmodi¬ed operators and the slowly-varying envelope

operators are related by the translation rule

F (ω) = F (ω + ω0 ) . (14.119)

This kind of rule is often expressed by saying that ω is replaced by ω + ω0 , but this is

a bit misleading. The translation rule really means that the argument of the function

is translated; for example, F (’ω) is replaced by F (’ω + ω0 ), not F (’ω ’ ω0 ). Thus

the argument in F (ω) represents the displacement, either positive or negative, from

the carrier frequency ω0 . Applying the translation rule to eqn (14.116) and to the

expression for a (ω) yields

bJ,in (ω) = bJ,ω (t0 ) eiωt0 (J = 1, 2) , (14.120)

√ √

and

κ2 b2,in (ω) + κ1 b1,in (ω)

a (ω) = . (14.121)

κC /2 ’ i (ω ’ ω0 )

The frequency-domain version of the scattering equation (14.109) for b1,out (ω),

where b1,out (ω) = b1,out (ω ’ ω0 ), combines with the explicit solution (14.121) to yield

the input“output equation

√

[(κ2 ’ κ1 ) /2 ’ i (ω ’ ω0 )] b1,in (ω) ’ κ1 κ2 b2,in (ω)

b1,out (ω) = . (14.122)

κC /2 ’ i (ω ’ ω0 )

For far o¬-resonance radiation, i.e. |ω ’ ω0 | κC /2, this relation reduces to

b1,out (ω) ≈ b1,in (ω) , (14.123)

which corresponds to complete re¬‚ection of the radiation incident on M1. For a sym-

metrical resonator, i.e. κ1 = κ2 = κC /2, the input“output relation simpli¬es to

¾ Quantum noise and dissipation

i (ω ’ ω0 ) b1,in (ω) + (κC /2) b2,in (ω)

b1,out (ω) = , (14.124)

i (ω ’ ω0 ) ’ κC /2

and for nearly resonant radiation, ω ≈ ω0 , this becomes

(κC /2) b2,in (ω)

b1,out (ω) = . (14.125)

i (ω ’ ω0 ) ’ κC /2

In this limit, the output ¬eld from mirror M1 is simply proportional to the input ¬eld

at mirror M2, i.e. there is essentially no re¬‚ection of radiation incident on mirror M2.

In this situation the cavity is called a Lorentzian ¬lter, since the output intensity,

2

(κC /2)

b† b† (t0 ) b2,ω (t0 ) ,

1,out (ω) b1,out (ω) = (14.126)

2,ω

2 2

(κC /2) + (ω ’ ω0 )

has a typical Lorentzian line shape.

14.4 Noise and dissipation for atoms

In Section 11.3.3 we obtained a dissipative form of the Bloch equation for a two-level

atom by adding phenomenological damping terms to the quantum Liouville equation

for the atomic density operator. The Liouville equation is de¬ned in the Schr¨dinger

o

picture, or sometimes in the interaction picture; consequently, the Bloch equation does

not immediately ¬t into the Heisenberg-picture formulation of the sample“reservoir

model employed above. In order to make the connection, we ¬rst recall that an N -

level atom is completely described by the transition operators, Sqp = |µq µp |, de¬ned

in Section 11.1.4. In particular, the matrix elements ρpq (t) = µp |ρ (t)| µq of the

density operator are given by

ρpq (t) = Tr ρ (t) Sqp . (14.127)

The trace is invariant under unitary transformations, so this result can equally well

be written as

ρpq (t) = Tr ρSqp (t) , (14.128)

where ρ and Sqp (t) are both expressed in the Heisenberg picture. Since the Heisenberg-

picture density operator ρ is time independent, the Bloch equation for the matrix

elements of the density operator is an immediate consequence of the Heisenberg equa-

tions of motion for the transition operators. For this reason, we will sometimes use the

name operator Bloch equation for these particular Heisenberg equations.

14.4.1 Two-level atoms

In order to avoid unnecessary complications, we will restrict the detailed discussion to

the simplest case of two-level atoms. With these results in hand, the generalization to

¿

Noise and dissipation for atoms

N -level atoms is straightforward. For a sample consisting of a single two-level atom,

the sample Hamiltonian is HS = HS0 + HS1 (t), where

ω21

[S22 ’ S11 ] .

HS0 = (14.129)

2

The terms in the Heisenberg equations of motion contributed by HS1 (t) play no role

in the following discussion, so we will omit them from the intermediate calculations

and restore them at the end to get the ¬nal form of the Langevin equations.

A Noise reservoirs

There are now two forms of dissipation to be considered: spontaneous emission (sp)

and phase-changing perturbations (pc). We already have the complete theory for

spontaneous emission, but in the present context it is more instructive to use the

schematic approach of Section 14.1.2. The creation and annihilation operators for the

reservoir excitations (photons) that are emitted and absorbed in the 2 ” 1 transition

are denoted by b† and b„¦ . The second form of dissipation is associated with the decay

„¦

of the atomic dipole, due to perturbations that do not cause real transitions between

the two levels. In the simplest case, the atom is excited from an initial state to a virtual

intermediate state and then returned to the original state. In a vapor, this e¬ect arises

primarily from collisions with other atoms. In a solid, phase-changing perturbations

are often caused by local ¬eld ¬‚uctuations. The phase-changing perturbations of the

two levels may arise from di¬erent mechanisms, so we need a reservoir for each level,

with creation and annihilation operators c† and cq„¦ (q = 1, 2).

q„¦

The environment Hamiltonian is therefore

∞ ∞

2

d„¦ d„¦

„¦b† b„¦ + „¦c† cq„¦ ,

HE = (14.130)

„¦ q„¦

2π 2π

0 0

q=1

and the sample“environment interaction Hamiltonian HSE is

HSE = Hsp + Hpc , (14.131)

where Hsp and Hpc are responsible for spontaneous emission and phase-changing per-

turbations respectively. The spontaneous emission Hamiltonian,

∞

D („¦)

v („¦) b† S12 ’ S21 b„¦ ,

Hsp = i d„¦ (14.132)

„¦

2π

0

is modeled directly on the RWA Hamiltonian of eqn (11.25), with the coupling constant

v („¦) playing the role of the dipole matrix element.

The simplest phase-changing perturbation is a second-order process in which the

atom starts and ends in the same state. The transition from an initial state |µq to

an intermediate state |µp is represented by the operator Spq , and the return to the

original state is described by Sqp ; consequently, the complete transition is described

by the product Sqp Spq = Sqq . Since there is no overall change in energy, the resonance

Quantum noise and dissipation

for this transition occurs at zero frequency. We model the phase-changing mechanism

by coupling the atom to two reservoirs according to

∞

2

D („¦)

uq („¦) c† Sqq ’ Sqq cq„¦ .

Hpc = i d„¦ (14.133)

q„¦

2π

0

q=1

Coupling to the zero-frequency resonance is enforced by assuming that the coupling

constant uq („¦) is proportional to the cut-o¬ function centered at zero frequency.

B Langevin equations

Since the sample and environment operators commute at equal times, the terms in the

total Hamiltonian can be written in any desired order. We chose to put them in normal

order with respect to the environment operators, so that the Heisenberg equations

d 1

Sqp (t) = [Sqp (t) , HE + HS + Hsp + Hpc ] (14.134)

dt i

are also normally ordered. The resonance frequencies for the interaction of the sample

with the spontaneous-emission and phase-changing reservoirs are ω = ω21 and ω = 0

respectively; therefore, we express eqn (14.134) in terms of the envelope operators

S 12 (t) = S12 (t) eiω21 t , S qq (t) = Sqq (t) ,

(14.135)

b„¦ (t) = b„¦ (t) eiω21 t , cq„¦ (t) = cq„¦ (t) ,

to ¬nd

d † †

S 12 (t) = S 22 (t) ’ S 11 (t) β (t) + γ2 (t) ’ γ1 (t) S 12 (t)

dt

’ S 12 (t) {γ2 (t) ’ γ1 (t)} , (14.136)

d

S 22 (t) = ’β † (t) S 12 (t) ’ S 21 (t) β (t) , (14.137)

dt

d

b„¦ (t) = ’i („¦ ’ ω21 ) b„¦ (t) + 2πD („¦) v („¦) S 12 (t) , (14.138)

dt

d

cq„¦ (t) = ’i„¦cq„¦ (t) + 2πD („¦) uq („¦) S qq (t) , (14.139)

dt

where

∞

D („¦)

β (t) = d„¦ v („¦) b„¦ (t) (14.140)

2π

0

and

∞

D („¦)

γq (t) = d„¦ uq („¦) cq„¦ (t) (q = 1, 2) . (14.141)

2π

0

The equation for S 11 (t) has been omitted, by virtue of the identity S 11 (t)+S 22 (t) = 1.

The Langevin equations for the atomic transition operators are derived by an ar-

gument similar to the one employed in Section 14.2.1. The formal solutions of eqns

Noise and dissipation for atoms

(14.138) and (14.139) for the reservoir operators are combined with the Markov con-

ditions

2πD („¦) |v („¦)|2 w21 = 2πD (ω21 ) |v (ω21 )|2 (14.142)

and

2 2

2πD („¦) |uq („¦)| wqq = 2πD (0) |uq (0)| , (14.143)

to get

√ w21

β (t) = w21 bin (t) + S 12 (t) (14.144)

2

and

√ wqq

γq (t) = wqq cq,in (t) + S qq (t) (q = 1, 2) . (14.145)

2

The in-¬elds for the reservoirs are given by

∞

d„¦

b„¦ (t0 ) e’i(„¦’ω21 )(t’t0 )

bin (t) = (14.146)

2π

0

and ∞

d„¦

cq„¦ (t0 ) e’i„¦(t’t0 ) .

cq,in (t) = (14.147)

2π

0

Substituting these results into eqns (14.136) and (14.137) yields the Langevin equations

for the transition operators:

d 1

S12 (t) = [iω12 ’ “12 ] S12 (t) + [S12 (t) , HS1 (t)] + ξ12 (t) , (14.148)

dt i

d 1

S22 (t) = ’w21 S22 (t) + [S22 (t) , HS1 (t)] + ξ22 (t) , (14.149)

dt i

d 1

S11 (t) = w21 S22 (t) + [S11 (t) , HS1 (t)] + ξ11 (t) , (14.150)

dt i

where w21 is the spontaneous decay rate for the 2 ’ 1 transition, w11 and w22 are the

rates of the phase-changing perturbations, and

1

“12 = (w21 + w22 + w11 ) (14.151)

2

is the dephasing rate for the atomic dipole. We have restored the HS1 (t)-terms and

also imposed ξ11 (t) = ’ξ22 (t) in accord with the conservation of population.

The operators ξ12 (t) and ξ22 (t) represent multiplicative noise, since they involve

products of sample and reservoir operators. This raises a new di¬culty, because there is

no general argument proving that multiplicative noise operators are delta correlated.

Even in the special cases for which a proof can be given”e.g. those considered in

Exercise 14.2”the calculations are quite involved. In this situation, the only general

procedure available is to include the delta-correlation assumption as part of the Markov

approximation. For the problem at hand the ansatz is

†

δ (t ’ t ) .

ξqp (t) ξq p (t ) = Cqp,q (14.152)

p

The coe¬cients Cqp,q p can be evaluated, at least partially, by the general methods

described in Section 14.6.

Quantum noise and dissipation

We will see, in the following section, that the use of atomic transition operators

is a great advantage for the generalization from two-level to N -level atoms, but for

applications to two-level atoms themselves, it is often easier to work in terms of the

familiar Pauli matrices. The relations

1 1

(1 + σz ) , S11 = (1 ’ σz ) , S12 = σ’ , S21 = σ+

S22 = (14.153)

2 2

lead to the equivalent Langevin equations

d 1

σ ’ (t) = ’“12 σ ’ (t) + [σ ’ (t) , HS1 (t)] + ξ’ (t) , (14.154)

dt i

d 1

σ z (t) = ’w21 [1 + σ z (t)] + [σ z (t) , HS1 (t)] + ξz (t) , (14.155)

dt i

where ξ’ (t) = ξ12 (t) and ξz (t) = 2ξ22 (t).

N -level atoms

14.4.2

The derivation of the Langevin equations for atoms with N levels could be carried

out by applying the approach followed for the two-level atom, but this would require

assigning a reservoir for every real decay and another reservoir for each level subjected

to phase-changing perturbations. One can escape burial under this avalanche of reser-

voirs by paying careful attention to the structure of eqns (14.148)“(14.151) for the

two-level atom. If we assume that the dissipative e¬ects involve transitions between

pairs of atomic levels or phase-changing perturbations of single levels, then a little

thought shows that the N -level Langevin equations must have the general form

d 1

Sqp (t) = (iωqp ’ “qp ) Sqp (t) + [Sqp (t) , HS1 (t)] + ξqp (t) for q = p , (14.156)

dt i

d 1

wpq Spp (t) ’

Sqq (t) = wqp Sqq (t) + [Sqq (t) , HS1 (t)] + ξqq (t) . (14.157)

dt i

p p

The envelope operators are de¬ned by generalizing eqn (14.135) to

Sqp (t) = S qp (t) eiωqp t ei[θq (t)’θp (t)] , (14.158)

where each θq (t) is a real function. The reason for including the θq s in this de¬nition is

that”in favorable cases”they can be chosen to eliminate explicit time dependencies

due to S qp (t) , HS1 (t) . Substituting eqn (14.158) into eqns (14.156) and (14.157)

leads to the envelope equations

d 1

™ ™

S qp (t) = ’i θq ’ θp ’ “qp S qp (t) + S qp (t) , HS1 (t) + ξqp (t) for q = p ,

dt i

(14.159)

d 1

wpq S pp (t) ’

S qq (t) = wqp S qq (t) + S qq (t) , HS1 (t) + ξqq (t) , (14.160)

dt i

p p

where

Incoherent pumping

§

⎪ transition rate for p ’ q if µp > µq ,

⎨

= 0 if µp < µq ,

wpq (14.161)

⎪

©

the phase-changing rate for the qth level when q = p .

1

For q = p, “qp = (wqr + wpr ) is the dephasing rate for S qp (t) . (14.162)

2 r

Strictly speaking, one should also de¬ne envelope noise operators,

ξ qp (t) = e’iωqp t e’i[θq (t)’θp (t)] ξqp (t) , (14.163)

but the assumption that the original operators ξqp (t) are delta correlated implies that

the envelope noise operators would have the same correlation functions. Since the

correlation functions are all that matters for noise operators, it is safe to ignore the

distinction between ξqp (t) and ξ qp (t).

14.5 Incoherent pumping

Incoherent pumping processes”which raise rather than lower the energy of an atom”

are used to produce population inversion; consequently, they play a central role in

laser physics. As we have seen in Section 14.4, the interaction of an atom with a

short-memory reservoir is necessarily dissipative. This raises the following question:

Can incoherent pumping be described by a reservoir model? This feat has been ac-

complished, but only at the cost of introducing an unphysical reservoir (Gardiner,

1991, Sec. 7.2.1). The idea is to describe pumping by coupling the atom to a reservoir

composed of oscillators with an inverted energy spectrum, µ„¦ = ’ „¦, as in Exercise

14.5. Emitting an excitation into this reservoir lowers the reservoir energy and there-

fore raises the energy of the atom. We have previously mentioned the formal use of

unphysical negative-energy modes in the discussion of the input“output method in

Section 14.3, but in that situation the probability for exciting the unphysical modes

is negligible. This cannot be the case for the inverted-oscillator reservoir; otherwise,

there would be no pumping. Since this model violates Einstein™s rule, we must accept

some added complexity.

The interaction between an atom and a classical ¬eld, with rapid ¬‚uctuations in

phase, provides a physically acceptable model for incoherent pumping. Unfortunately,

building such a model for the simplest case of a two-level atom is pointless, since the

discussion in Section 11.3.3 shows that no pumping scheme for a two-level atom can

produce an inverted population. We will, therefore, grudgingly admit that real atoms

have more than two levels and add a third. The added complexity will be o¬set by

ignoring phase-changing perturbations.

The sample is a collection of three-level atoms, with the energy-level diagram shown

in Fig. 14.2. The 3 ” 1 transition is driven by a strong, classical pump ¬eld

E P (t) = eP EP 0 ei‘P (t) e’iωP t , (14.164)

where ωP ≈ ω31 and ‘P (t) is a rapidly ¬‚uctuating phase. Since there is no coupling

between the atoms, we can restrict our attention to a single atom located at r = 0.

For this reduced sample, the interaction Hamiltonian is HS1 = VP (t) + HS1 , where

Quantum noise and dissipation

!

Fig. 14.2 A three-level atom with dipole al-

M!

lowed transitions 1 ” 3 and 1 ” 2. The spon-

taneous emission rates are w31 and w21 respec-

„¦2

tively. The 1 ” 3 transition is also driven by

M!

M

a classical ¬eld with Rabi frequency „¦P . A

non-radiative decay 3 ’ 2, with rate w32 , is

indicated by the dashed arrow. The wavy ar-

rows denote the spontaneous emissions.

„¦P ei‘P (t) e’iωP t eiω31 t ei[θ3 (t)’θ1 (t)] S 31 + HC ,

VP (t) = (14.165)

S 31 is the envelope operator de¬ned by eqn (14.158), „¦P is the Rabi frequency associ-

ated with the constant amplitude EP 0 , and HS1 includes any other interactions with

external ¬elds as well as any sample“sample interactions. The remaining interaction

term HS1 in¬‚uences some of the choices to be made, but the terms contributed by

HS1 to the equations of motion play no direct role in the following argument. We

will therefore omit them from the intermediate steps and restore them at the end. In

addition to the spontaneous emissions, 2 ’ 1 and 3 ’ 1, we assume that there is a

non-radiative decay channel: 3 ’ 2.

The Langevin equations for this problem are derived in Exercise 14.6 by dropping

the phase-changing terms from the (N = 3)-case of eqns (14.159) and (14.160). It is

also useful to impose θ3 (t) ’ θ1 (t) = ∆P t ’ ‘P (t)”where ∆P = ωP ’ ω31 is the pump

detuning”in order to eliminate the explicit time dependence of VP (t). The remaining

phase di¬erences θ1 ’ θ2 and θ2 ’ θ3 are related by

θ2 ’ θ3 = (θ1 ’ θ3 ) ’ (θ1 ’ θ2 )

= ∆P t ’ ‘P (t) ’ (θ1 ’ θ2 ) , (14.166)

so we can only impose one more condition on the phases. The choice of this condition

depends on HS1 . In the problem at hand, we have assumed that the transition 2 ” 1

is dipole allowed, but the transition 3 ” 2 is not. Thus only the transition 2 ” 1 can

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

X (t ) O J (t) Y (t ) = GJK (t, t ) X (t ) O K (t ) Y (t ) , (14.210)

K

where X (t ) and Y (t ) are sample operators that depend on O J (t ) for t < t <

t. Equations (14.209) and (14.210) are special cases of the quantum regression

theorem ¬rst proved by Lax (1963). We will study the general version in Chapter 18.

Quantum noise and dissipation

Photon bunching—

14.8

We mentioned in Section 10.1.1 that the Hanbury Brown“Twiss e¬ect can be measured

by coincidence counting. As explained in Section 9.2.4, the coincidence-count rate is

proportional to the second-order correlation function

G(2) (r , t , r, t; r , t , r, t) = E (’) (r , t ) E (’) (r, t) E (+) (r, t) E (+) (r , t ) , (14.211)

where r and r are the locations of the detectors, t = t + „ , and the ¬elds are all

projected on a common polarization vector. By placing suitable ¬lters in front of the

detectors, we can con¬ne our attention to a single mode, so that G(2) is proportional

to the correlation function

C („ ) = a† (t + „ ) a† (t) a (t) a (t + „ ) = a† (t) N (t + „ ) a (t) , (14.212)

where N (t) = a† (t) a (t) is the mode number operator in the Heisenberg picture.

The quantum regression theorem can be applied to the evaluation of C („ ) by using

the Langevin equation for a (t) to derive the di¬erential equation

d N (t)

= ’κ N (t) + ξ † (t) a (t) + a† (t) ξ (t) (14.213)

dt

for the average photon number. It is shown in Exercise 14.2 that

ξ † (t) a (t) + a† (t) ξ (t) = n0 κ , (14.214)

so that the equation for N (t) can be rewritten as

d δN (t)

= ’κ δN (t) , (14.215)

dt

where δN (t) = N (t) ’ n0 . The solution,

δN (t) = e’κ(t’t0 ) δN (t0 ) , (14.216)

of this equation is a special case of the linear regression equation (14.206), with the

Green function G („ ) = exp (’κ„ ). According to the quantum regression theorem

(14.210), the correlation function a† (t) δN (t + „ ) a (t) obeys the same regression

law, so

a† (t) δN (t + „ ) a (t) = e’κ„ a† (t) δN (t) a (t) , (14.217)

and

C („ ) = e’κ„ a†2 (t) a2 (t) + 1 ’ e’κ„ n0 N (t) . (14.218)

For large times, κ (t ’ t0 ) 1, eqn (14.216) shows that N (t) ≈ n0 . The remaining

†2 2

expectation value a (t) a (t) can be calculated by using the solution (14.65) for

Resonance ¬‚uorescence—

a (t). In the same large-time limit, the initial-value term in eqn (14.65) can be dropped

to get the asymptotic result

⎡ ¤

4

t t

κ

dt4 exp ⎣’ (t ’ tj )¦ ξ † (t1 ) ξ † (t2 ) ξ (t3 ) ξ (t4 ) .

a†2 (t) a2 (t) = dt1 · · ·

2 j=1

t0 t0

(14.219)

For a thermal noise distribution,

ρE = exp ’β „¦ν N ν , (14.220)

ν

the discussion in Section 14.2.2 shows that

ξ † (t1 ) ξ † (t2 ) ξ (t3 ) ξ (t4 ) = (n0 κ) {δ (t1 ’ t3 ) δ (t2 ’ t4 ) + δ (t1 ’ t4 ) δ (t2 ’ t3 )} .