includes HS1 (t). The dissipative part is

w21

Ldis ρS (t) = ’ (nsp + 1) {[σ+ , σ’ ρ] + [ρσ+ , σ’ ]}

2

w21

’ nsp {[σ’ , σ+ ρ] + [ρσ’ , σ+ ]}

2

wpc

’ [ρσz , σz ] , (18.44)

2

where nsp is the average number of reservoir excitations (photons) at the transition

frequency ω21 . The phase-changing rate in the last term is

2

1

wpc = (2npc,q + 1) wqq , (18.45)

2 q=1

where npc,q is the average number of excitations in the phase-perturbing reservoir

coupled to the atomic state |µq .

The master equation

18.5 Phase space methods

In Section 5.6.3 we have seen that the density operator for a single cavity mode can be

described in the Glauber“Sudarshan P (±) representation (5.165). As we will show be-

low, this representation provides a natural way to express the operator master equation

(18.31) as a di¬erential equation for the P (±)-function. In single-mode applications”

and also in some more complex situations”this equation is mathematically identical

to the Fokker“Planck equation studied in classical statistics (Risken, 1989, Sec. 4.7).

By de¬ning an atomic version of the P -function”as the Fourier transform of a

properly chosen quantum characteristic function”it is possible to apply the same

techniques to the master equation for atoms, but we will restrict ourselves to the

simpler case of a single mode of the radiation ¬eld. The application to atomic master

equations can be found, for example, in Haken (1984, Sec. IX.2) or Walls and Milburn

(1994, Chap. 13).

For the discussion of the master equation in terms of P (±), it is better to use

the alternate convention in which P (±) is regarded as a function, P (±, ±— ), of the

independent variables ± and ±— . In this notation the P -representation is

d2 ±

|± P (±, ±— ; t) ±| .

ρS (t) = (18.46)

π

The function P (±, ±— ; t) is real and satis¬es the normalization condition

d2 ±

P (±, ±— ; t) = 1 , (18.47)

π

but it cannot always be interpreted as a probability distribution. The trouble is that,

for nonclassical states, P (±, ±— ; t) must take on negative values in some region of the

±-plane.

18.5.1 The Fokker“Planck equation

In order to use the P -representation in the master equation, we must translate the

products of Fock space operators, e.g. a, a† , and ρ, in the master equation into the

action of di¬erential operators on the c-number function P (±, ±— ; t). For this purpose

it is useful to write the coherent state |± as

2

|± = e’|±| |±; B ,

/2

(18.48)

where the Bargmann state |±; B is

∞

±n

√ |n .

|±; B = (18.49)

n!

n=0

The virtue of the Bargmann states is that they are analytic functions of ±. More

precisely, for any ¬xed state |Ψ the c-number function

∞

±n

√

Ψ |±; B = Ψ |n (18.50)

n!

n=0

is analytic in ±. In the same sense, ±; B| is an analytic function of ±— , so it is inde-

pendent of ±.

Phase space methods

Since |±; B is proportional to |± , the action of a on the Bargmann states is just

a |±; B = ± |±; B . (18.51)

The action of a† is found by using eqn (18.49) to get

∞

±n √

†

√

a |±; B = n + 1 |n + 1

n!

n=0

‚

|±; B .

= (18.52)

‚±

The adjoint of this rule is

‚

±; B| a =

±; B| . (18.53)

‚±—

In the Bargmann notation, the P -representation of the density operator is

d2 ±

P (±, ±— ; t) e’|±| |±; B ±; B| .

2

ρS (t) = (18.54)

π

The rule (18.51) then gives

d2 ±

P (±, ±— ; t) e’|±| ± |±; B

2

aρS (t) = ±; B|

π

d2 ±

±P (±, ±— ; t) |± ±| .

= (18.55)

π

Applying the rule (18.52) yields

d2 ± ‚

†

P (±, ±— ; t) e’|±|

2

|±; B

a ρS (t) = ±; B| , (18.56)

π ‚±

but this is not expressed in terms of a di¬erential operator acting on P (±, ±— ; t).

Integrating by parts on ± leads to the desired form:

d2 ± ‚ —

a† ρS (t) = ’ P (±, ±— ; t) e’±± |±; B ±; B|

π ‚±

2

d± ‚

±— ’ P (±, ±— ; t) |± ±| .

= (18.57)

π ‚±

This result depends on the fact that the normalization condition requires P (±, ±— ; t)

to vanish as |±| ’ ∞.

Combining eqns (18.55) and (18.57) with their adjoints gives us the translation

table

aρS (t) ” ±P (±, ±— ; t) ρS (t) a ” (± ’ ‚/‚±— ) P (±, ±— ; t)

(18.58)

a† ρS (t) ” (±— ’ ‚/‚±) P (±, ±— ; t) ρS (t) a† ” ±— P (±, ±— ; t) .

The master equation

Applying the rules in eqn (18.58) to eqn (18.32)”for the simple case with HS1 =

0”and to eqn (18.33) yields the translations

1 ‚ ‚—

ω0 a† a, ρS (t) ” iω0 ± P (±, ±— ; t)

LS ρS (t) = ±’ (18.59)

—

i ‚± ‚±

and

“ ‚ ‚

[±P (±, ±— ; t)] + [±— P (±, ±— ; t)]

Ldis ρS (t) ” —

2 ‚± ‚±

‚2

P (±, ±— ; t) ,

+ “ncav (18.60)

—

‚±‚±

for the Hamiltonian and dissipative Liouville operators respectively.

In the course of carrying out these calculations, it is easy to get confused about

the correct order of operations. The reason is that products like a† aρ”with operators

standing on the left of ρ”and products like ρa† a”with operators standing to the

right of ρ”are both translated into di¬erential operators acting from the left on the

function P (±, ±— ; t).

Studying a simple example, e.g. carrying out a direct derivation of both a† aρ and

ρa† a, shows that the order of the di¬erential operators is reversed from the order of the

Fock space operators when the Fock space operators stand to the right of ρ. Another

way of saying this is that one should work from the inside to the outside; the ¬rst

di¬erential operator acting on P corresponds to the Hilbert space operator closest to

ρ. This rule gives the correct result for Fock space operators to the left or to the right

of ρ.

The master equation for an otherwise unperturbed cavity mode is, therefore, rep-

resented by

‚ ‚ ‚

P (±, ±— ; t) = [Z (±) P (±, ±— ; t)] + [Z — (±) P (±, ±— ; t)]

—

‚t ‚± ‚±

2

‚

P (±, ±— ; t) ,

+ “ncav (18.61)

—

‚±‚±

where

“

+ iω0 ± . (18.62)

Z (±) =

2

We can achieve a ¬rmer grip on the meaning of this equation by changing variables

from (±, ±— ) to u = (u1 , u2 ), where u1 = Re ± and u2 = Im ±. In these variables,

P (±, ±— ; t) = P (u; t), and the ±-derivative is

‚ 1 ‚ ‚

’i

= . (18.63)

‚± 2 ‚u1 ‚u2

In this notation, the master equation takes the form of a classical Fokker“Planck

equation in two dimensions:

‚ D0 2

P (u; t) = ’∇ · [F (u) P (u; t)] + ∇ P (u; t) , (18.64)

‚t 2

Phase space methods

where

D0 = “ncav /2 (18.65)

is the di¬usion constant, and we have introduced the following shorthand notation:

‚X1 ‚X2

∇·X = + ,

‚u1 ‚u2

“ “

F (u) = (’ Re Z, ’ Im Z) = ω0 u 2 ’ u1 , ’ω0 u1 ’ u2 , (18.66)

2 2

2 2

‚ ‚

∇=2

+ .

‚u1 ‚u2

The ¬rst- and second-order di¬erential operators in eqn (18.64) are respectively called

the drift term and the di¬usion term.

A Classical Langevin equations

The Fokker“Planck equation (18.64) is a special case of a general family of equations

of the form

N N

‚ 1 ‚ ‚

P (u; t) = ’∇ · [F (u, t) P (u; t)] + Dmn (u, t) P (u; t) , (18.67)

‚t 2 m=1 n=1 ‚um ‚un

where u = (u1 , . . . , uN ), F = (F1 , . . . , FN ), Dmn is the di¬usion matrix, and

X · Y = X1 Y1 + · · · + XN YN . (18.68)

For the two-component case, given by eqn (18.64), the di¬usion matrix is diagonal,

Dmn = D0 δmn , so it has a single eigenvalue D0 > 0. The corresponding condition in

the general N -component case is that all eigenvalues of the di¬usion matrix D are

positive, i.e. D is a positive-de¬nite matrix. In this case D has a square root matrix

B that satis¬es D = BB T .

When D is positive de¬nite, then eqn (18.67) is exactly equivalent to the set of

classical Langevin equations (Gardiner, 1985, Sec. 4.3.5)

N

dun (t)

= Cn (u, t) + Bnm (u, t) wm (t) , (18.69)

dt m=1

where the un s are stochastic variables and the wm s are independent white noise sources

of unit strength, i.e. wm (t) = 0 and

wm (t) wn (t ) = δmn δ (t ’ t ) . (18.70)

In particular, the Langevin equations corresponding to eqn (18.64) are

du (t)

= F (u) + D0 w (t) . (18.71)

dt

¼ The master equation

These real Langevin equations are essential for numerical simulations, but for ana-

lytical work it is useful to write them in complex form. This is done by combining

± = u1 + iu2 with eqns (18.62) and (18.66) to get

d± (t)

= ’Z (± (t)) + 2D0 · (t) , (18.72)

dt

where ± (t) is a complex stochastic variable, and

1

· (t) = √ [w1 (t) + iw2 (t)] (18.73)

2

is a complex white noise source satisfying

· — (t) · (t ) = δ (t ’ t ) .

· (t) = 0 , · (t) · (t ) = 0 , (18.74)

The equivalence of the Fokker“Planck equation and the classical Langevin equa-

tions for a positive-de¬nite di¬usion matrix is important in practice, since the nu-

merical simulation of the Langevin equations is usually much easier than the direct

numerical solution of the Fokker“Planck equation itself.

For some problems”e.g. when the appropriate reservoir is described by a squeezed

state”the di¬usion matrix derived from the Glauber“Sudarshan P -function is not

positive de¬nite, so the Fokker“Planck equation is not equivalent to a set of classical

Langevin equations. In such cases, another representation of the density operator may

be more useful (Walls and Milburn, 1994, Sec. 6.3.1).

18.5.2 Applications of the Fokker“Planck equation

A Coherent states are robust

Let us begin with a simple example in which ncav = 0, so that the di¬usion term in eqn

(18.64) vanishes. If we interpret the reservoir oscillators as phonons in the cavity walls,

then this model describes the idealized situation of material walls at absolute zero.

Alternatively, the reservoir could be de¬ned by other modes of the electromagnetic

¬eld, into which the particular mode of interest is scattered by a gas of nonresonant

atoms. In this case, it is natural to assume that the initial reservoir state is the vacuum.

In other words, the universe is big and dark and cold.

The terms remaining after setting ncav = 0 can be rearranged to produce

‚

P (u; t) + F (u) · ∇P (u; t) = “P (u; t) . (18.75)

‚t

Let us study the evolution of a ¬eld state initially de¬ned by P (u; 0) = P0 (u). The

general technique for solving linear, ¬rst-order, partial di¬erential equations like eqn

(18.75) is the method of characteristics (Zauderer, 1983, Sec. 2.2), but we will employ

an equivalent method that is well suited to the problem at hand.

The ¬rst step is to introduce an integrating factor, by setting

P (u; t) = P (u; t) e“t , (18.76)

½

Phase space methods

so that

‚

P (u; t) + F (u) · ∇P (u; t) = 0 . (18.77)

‚t

The second step is to transform to new variables (u , t ) by

u = V (u, t) , t = t , (18.78)

where we require u = u at t = 0, and also assume that the function V (u, t) is linear

in u, i.e.

2

Vn (u, t) = Gnm (t) um . (18.79)

m=1

The reason for trying a linear transformation is that the coe¬cient vector,

’“/2 ω0

, (18.80)

Fj (u) = Wjl ul , where W =

’ω0 ’“/2

l

is itself linear in u.

The chain rule calculation in Exercise 18.5 yields expressions for the operators ‚/‚t

and ‚/‚ul in terms of the new variables, so that eqn (18.77) becomes

‚ dG (t) ‚

P (u ; t ) + G (t) W + ul P (u ; t) = 0 . (18.81)

‚t dt ‚un

nl

n l

Choosing the matrix G (t) to satisfy

dG (t)

+ G (t) W = 0 (18.82)

dt

ensures that the coe¬cient of ‚/‚un vanishes identically in u, and this in turn simpli¬es

the equation for P (u ; t ) to

‚

P (u ; t ) = 0 . (18.83)

‚t

Thus P (u ; t ) = P (u ; 0), but t = 0 is the same as t = 0, so P (u ; t ) = P0 (u ).

In this way the solution to the original problem is found to be

P (u, t) = e“t P0 (V (u, t)) , (18.84)

and the only remaining problem is to evaluate V (u, t). This is most easily done by

writing G (t) as

b (t) b2 (t)

G (t) = 1 , (18.85)

c1 (t) c2 (t)

and substituting this form into eqn (18.82). This yields simple di¬erential equations

for the vectors b (t) and c (t), with initial conditions b (0) = (1, 0) and c (0) = (0, 1).

The solution of these auxiliary equations gives

¾ The master equation

cos (ω0 t) ’ sin (ω0 t)

G (t) = e“t/2 R (t) = e“t/2 , (18.86)

sin (ω0 t) cos (ω0 t)

so that

P (u; t) = P0 e“t/2 R (t) u e“t . (18.87)

Thus, in the absence of the di¬usive term, the shape of the distribution is un-

changed; the argument u is simply scaled by exp (“t/2) and rotated by the angle ω0 t.

In the complex-± description the solution is given by

P (±, ±— ; t) = P0 e(“/2+iω0 )t ±, e(“/2’iω0 )t ±— e“t . (18.88)

This result is particularly interesting if the ¬eld is initially in a coherent state |±0 ,

i.e. the initial P -function is P0 (u) = δ2 (u ’ u0 ). In this case, the standard properties

of the delta function lead to

P (u; t) = e“t δ2 e“t/2 R (t) u ’ u0 = δ2 (u ’ u (t)) , (18.89)

where

’1

u (t) = e’“t/2 [R (t)] u0 . (18.90)

The conclusion is that a coherent state interacting with a zero-temperature reservoir

will remain a coherent state, with a decaying amplitude

± (t) = ±0 e’“t/2 e’iω0 t . (18.91)

Consequently, the time-dependent joint variance of a† and a vanishes at all times:

V a† , a ; t = ± (t) a† a ± (t) ’ ± (t) a† ± (t) ± (t) |a| ± (t) = 0 . (18.92)

In other words, coherent states are robust: scattering and absorption will not destroy

the coherence properties, as long as the environment is at zero temperature.

This apparently satisfactory result raises several puzzling questions. The ¬rst is that

the initially pure state remains pure, even after interaction with the environment. This

seems to contradict the general conclusion, established in Section 18.1, that interaction

with a reservoir inevitably produces a mixed state for the sample.

The resolution of this discrepancy is that the general argument is true for the exact

theory, while the master equation is derived with the aid of the approximation”see

eqn (18.17)”that back-action of the sample on the reservoir can be neglected. This

means that the robustness property of the coherent states is only as strong as the

approximations leading to the master equation. Furthermore, we will see in Section