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