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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)
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

’∞
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,„¦

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,„¦

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„¦

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)

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)

’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)

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)

’∞

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)
„¦

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„¦

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)

0
and

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

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)

0

and ∞
d„¦
cq„¦ (t0 ) e’i„¦(t’t0 ) .
cq,in (t) = (14.147)

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 )} .

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