The master equation

In this chapter we will study the time evolution of an open system”the sample dis-

cussed in Chapter 14”by means of the quantum Liouville equation for the world

density operator. This approach, which employs the interaction-picture description of

the density operator, is complementary to the Heisenberg-picture treatment presented

in Chapter 14. The physical ideas involved in the two methods are, however, the same.

The equation of motion of the reduced sample density operator is derived by an

approximate elimination of the environment degrees of freedom that depends crucially

on the Markov approximation. This approximate equation of motion for the sample

density operator is mainly used to derive c-number equations that can be solved by

numerical methods. In this connection, we will discuss the Fokker“Planck equation in

the P -representation and the method of quantum Monte Carlo wave functions.

18.1 Reduced density operators

As explained in Section 14.1.1, the world”the composite system of the sample and

the environment”is described by a density operator ρW acting on the Hilbert space

HW = HS — HE . The application of the general de¬nition (6.21) of the reduced density

operator to ρW produces two reduced density operators: S = TrE (ρW ) and E =

TrS (ρW ), that describe the sample and the environment respectively. For example,

the rule (6.26) for partial traces shows that the average of a sample operator Q,

Q = TrW [ρW (Q — IE )] = TrS ( S Q) , (18.1)

is entirely determined by the reduced density operator for the sample.

For an open system, the reduced density operator S will always describe a mixed

state. According to Theorem 6.1 in Section 6.4.1, the reduced density operators for the

sample and the environment can only describe pure states if the world density operator

describes a separable pure state, i.e. ρW = |„¦W „¦W |, and |„¦W = |ΨS |¦E . Even

if this were initially the case, the interaction between the sample and the environment

would inevitably turn the separable pure state |„¦W into an entangled pure state. The

reduced density operators for an entangled state necessarily describe mixed states, so

the sample state will always evolve into a mixed state.

18.2 The environment picture

In the Schr¨dinger picture, the world density operator satis¬es the quantum Liouville

o

equation

¿

Averaging over the environment

‚

i ρW (t) = [HW (t) , ρW (t)]

‚t

= [HS (t) + HE + HSE , ρW (t)] , (18.2)

where the terms in HW are de¬ned in eqns (14.6)“(14.11). Since the sample“environ-

ment interaction is assumed to be weak, it is natural to regard HW 0 (t) = HS (t) + HE

as the zeroth-order part, and HSE (t) as the perturbation. This allows us to introduce

an interaction picture, through the unitary transformation,

†

|Ψenv (t) = UW 0 (t) |Ψ (t) , (18.3)

where UW 0 (t) satis¬es

‚

i UW 0 (t) = HW 0 (t) UW 0 (t) , UW 0 (0) = 1 . (18.4)

‚t

We will call this interaction picture the environment picture, since it plays a special

role in the theory.

The di¬erential equation (18.4) has the same form as eqn (4.90), but now the

ordering of the operators UW 0 (t) and HW 0 (t) is important, since the time-dependent

Hamiltonian HW 0 (t) will not in general commute with UW 0 (t). If this warning is kept

¬rmly in mind, the formal procedure described in Section 4.8 can be used again to

¬nd the Schr¨dinger equation

o

‚

|Ψenv (t) = HSE (t) |Ψenv (t) ,

env

i (18.5)

‚t

and the quantum Liouville equation

‚ env

ρW (t) = [HSE (t) , ρenv (t)]

env

i (18.6)

W

‚t

in the environment picture. The transformed operators,

†

Oenv (t) = UW 0 (t) OUW 0 (t) , (18.7)

satisfy the equations of motion

‚ env

O (t) = [Oenv (t) , HW 0 (t)] .

env

i (18.8)

‚t

env env env

One should keep in mind that HW 0 (t) = HS (t) + HE (t), and that the sample

env

Hamiltonian, HS (t), still contains all interaction terms between di¬erent degrees

of freedom in the sample. Only the sample“environment interaction is treated as a

perturbation. Thus the environment-picture sample operators obey the full Heisenberg

equations for the sample.

18.3 Averaging over the environment

In line with our general convention, we will now drop the identifying superscript ˜env™,

and replace it by the understanding that states and operators in the following discus-

sion are normally expressed in the environment picture. Exceptions to this rule will

¼ The master equation

be explicitly identi¬ed. Our immediate task is to derive an equation of motion for the

reduced density operator ρS . In pursuing this goal, we will generally follow Gardiner™s

treatment (Gardiner, 1991, Chap. 5).

The ¬rst part of the argument corresponds to the formal elimination of the reser-

voir operators in Section 14.1.2, and we begin in a similar way by incorporating the

quantum Liouville equation (18.6) and the initial density operator ρW (0) into the

equivalent integral equation,

t

i

ρW (t) = ρW (0) ’ dt1 [HSE (t1 ) , ρW (t1 )] . (18.9)

0

The assumption that HSE is weak”compared to HS and HE ”suggests solving

this equation by perturbation theory, but a perturbation expansion would only be

valid for a very short time. From Chapter 14, we know that typical sample correlation

functions decay exponentially:

Q1 (t + „ ) Q2 (t) ∼ e’γ„ , (18.10)

with a decay rate, γ ∼ g 2 , where g is the sample“environment coupling constant. This

exponential decay can not be recovered by an expansion of ρW (t) to any ¬nite order

in g.

As the ¬rst step toward ¬nding a better approach, we iterate the integral equation

(18.9) twice”this is suggested by the fact that γ is second order in g”to ¬nd

t

i

ρW (t) = ρW (0) ’ dt1 [HSE (t1 ) , ρW (0)]

0

2 t t1

i

+’ dt1 dt2 [HSE (t1 ) , [HSE (t2 ) , ρW (t2 )]] . (18.11)

0 0

Tracing over the environment then produces the exact equation

t

i

ρS (t) = ρS (0) ’ dt1 TrE ([HSE (t1 ) , ρW (0)])

0

2 t t1

i

+’ dt1 dt2 TrE ([HSE (t1 ) , [HSE (t2 ) , ρW (t2 )]]) (18.12)

0 0

for the reduced density operator. Since our objective is an equation of motion for

ρS (t), the next step is to di¬erentiate with respect to t to ¬nd

‚ i

ρS (t) = ’ TrE ([HSE (t) , ρW (0)])

‚t

t

1

’2 dt TrE ([HSE (t) , [HSE (t ) , ρW (t )]]) . (18.13)

0

This equation is exact, but it is useless as it stands, since the unknown world den-

sity operator ρW (t ) appears on the right side. Further progress depends on ¬nding

½

Averaging over the environment

approximations that will lead to a manageable equation for ρS (t) alone. The ¬rst sim-

plifying assumption is that the sample and the environment are initially uncorrelated:

ρW (0) = ρS (0) ρE (0). By combining the generic expression,

grν b† Qr ’ grν Q† brν ,

—

HSE = i (18.14)

rν r

r ν

for the sample“environment interaction with the conventional assumption, brν = 0,

E

and the initial factorization condition, it is straightforward to show that

TrE ([HSE (t) , ρW (0)]) = 0 . (18.15)

If one or more of the reservoirs has brν E = 0, one can still get this result by

writing HSE in terms of the ¬‚uctuation operators δbrν = brν ’ brν E , and absorbing

the extra terms by suitably rede¬ning HS and HE , as in Exercise 18.1.

Thus the initial factorization assumption always allows eqn (18.13) to be replaced

by the simpler form

t

‚ 1

ρS (t) = ’ 2 dt TrE {[HSE (t) , [HSE (t ) , ρW (t )]]} . (18.16)

‚t 0

Replacing ρW (t ) by ρW (0) = ρS (0) ρE (0) in eqn (18.16) would provide a perturba-

tive solution that is correct to second order, but”as we have just seen”this would

not correctly describe the asymptotic time dependence of the correlation functions.

The key to ¬nding a better approximation is to exploit the extreme asymmetry

between the sample and the environment. The environment is very much larger than

the sample; indeed, it includes the rest of the universe. It is therefore physically rea-

sonable to assume that the fractional change in the sample, caused by interaction

with the environment, is much larger than the fractional change in the environment,

caused by interaction with the sample. If this is the case, there will be no reciprocal

correlation between the sample and the environment, and the density operator ρW (t )

will be approximately factorizable at all times.

This argument suggests the ansatz

ρW (t ) ≈ ρS (t ) ρE (0) , (18.17)

and using this in eqn (18.16) produces the master equation:

t

‚ 1

ρS (t) = ’ 2 dt TrE {[HSE (t) , [HSE (t ) , ρS (t ) ρE (0)]]} . (18.18)

‚t 0

The double commutator in eqn (18.18) can be rewritten in a more convenient way

by exploiting the fact that typical interactions have the form

F (t) + F† (t) .

HSE (t) = (18.19)

This in turn allows the double commutator to be written as

¾ The master equation

1

C2 (t, t ) = [HSE (t) , [HSE (t ) , ρS (t ) ρE (0)]]

2

F† (t) , G (t ) + HC ,

= {[F (t) , G (t )] + HC} + (18.20)

where

G (t ) = [F (t ) , ρS (t ) ρE (0)] . (18.21)

18.4 Examples of the master equation

In order to go on, it is necessary to assume an explicit form for HSE . For this purpose,

we will consider the two concrete examples that were studied in Chapter 14: the single

cavity mode and the two-level atom.

18.4.1 Single cavity mode

In the environment picture, the de¬nition (14.43) of system“reservoir interaction for

a single cavity mode becomes

v („¦ν ) a† (t) bν (t) ’ HC .

HSE = i (18.22)

ν

Since a (t) and bν (t) are evaluated in the environment picture, they satisfy the Heisen-

berg equations

‚ 1

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

‚t i

1

= ’iω0 a (t) + [a (t) , HS1 (t)] , (18.23)

i

and

‚

bν (t) = ’i„¦ν bν (t) . (18.24)

‚t

By introducing the slowly-varying envelope operators a (t) = a (t) exp (iω0 t) and

bν (t) = bν (t) exp (iω0 t), we can express HSE (t) in the form (18.19), with

F (t) = ’iξ † (t) a (t) , (18.25)

where we recognize

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

ξ (t) = (18.26)

ν

as the noise operator de¬ned in eqn (14.52).

The terms in [F (t) , G (t )] contain products of ξ † (t), ξ † (t ), and ρE (0) in various

orders. When the partial trace over the environment states in eqn (18.18) is performed,

the cyclic invariance of the trace can be exploited to show that all terms are propor-

tional to

ξ † (t) ξ † (t ) E = TrE ρE (0) ξ † (t) ξ † (t ) . (18.27)

Just as in Section 14.2, we will assume that ρE (0) is diagonal in the reservoir

oscillator occupation number”this amounts to assuming that ρE (0) is a stationary

¿

Examples of the master equation

distribution”so that the correlation functions in eqn (18.27) all vanish. This assump-

tion is convenient, but it is not strictly necessary. A more general treatment”that

includes, for example, a reservoir described by a squeezed state”is given in Walls and

Milburn (1994, Sec. 6.1).

When ρE (0) is stationary, [F (t) , G (t )] and its adjoint will not contribute to the

master equation. By contrast, the commutator F† (t) , G (t ) is a sum of terms con-

taining products of ξ (t), ξ † (t ), and ρE (0) in various orders. In this case, the cyclic

invariance of the partial trace produces two kinds of terms, proportional respectively

to

ξ † (t) ξ (t ) E = ncav κδ (t ’ t ) (18.28)

and

ξ (t) ξ † (t ) = (ncav + 1) κδ (t ’ t ) , (18.29)

where ncav is the average number of reservoir oscillators at the cavity-mode frequency

ω0 .

The explicit expressions on the right sides of these equations come from eqns (14.74)

and (14.75), which were derived by using the Markov approximation. Thus the master

equation also depends on the Markov approximation, in particular on the assump-

tion that the envelope operator a (t) is essentially constant over the memory interval

(t ’ Tmem /2, t + Tmem /2).

After evaluating the partial trace of the double commutator C2 (t, t )”see Exercise

18.2”the environment-picture form of the master equation for the ¬eld is found to be

‚ κ

ρS (t) = ’ (ncav + 1) a† (t) a (t) ρS (t) + ρS (t) a† (t) a (t) ’ 2a (t) ρS (t) a† (t)

‚t 2

κ

’ ncav a (t) a† (t) ρS (t) + ρS (t) a (t) a† (t) ’ 2a† (t) ρS (t) a (t) .

2

(18.30)

The slowly-varying envelope operators a (t) and a† (t) are paired in every term;

consequently, they can be replaced by the original environment-picture operators a (t)

and a† (t) without changing the form of the equation. The right side of the equation

of motion is therefore entirely expressed in terms of environment-picture operators, so

we can easily transform back to the Schr¨dinger picture to ¬nd

o

‚

ρS (t) = LS ρS (t) + Ldis ρS (t) . (18.31)

‚t

The Liouville operators LS ”describing the free Hamiltonian evolution of the sam-

ple”and Ldis ”describing the dissipative e¬ects arising from coupling to the environ-

ment”are respectively given by

1

ω0 a† a + HS1 (t) , ρS (t)

LS ρS (t) = (18.32)

i

and

κ

Ldis ρS (t) = ’ (ncav + 1) a† , aρS (t) + ρS (t) a† , a

2 (18.33)

κ

’ ncav a, a† ρS (t) + ρS (t) a, a† .

2

The master equation

The operators we are used to, such as the Hamiltonian or the creation and annihi-

lation operators, send one Hilbert-space vector to another. By contrast, the Liouville

operators send one operator to another operator. For this reason they are sometimes

called super operators.

A Thermal equilibrium again

In Exercise 14.2 it is demonstrated that the average photon number asymptotically

approaches the Planck distribution. With the aid of the master equation, we can study

this limit in more detail. In this case, HS1 (t) = 0, so we can expect the density operator

to be diagonal in photon number. The diagonal matrix elements of eqn (18.31) in the

number-state basis yield

dpn (t)

= ’κ {(ncav + 1) n + ncav (n + 1)} pn (t)

dt (18.34)

+ κ (ncav + 1) (n + 1) pn+1 (t) + κncav n pn’1 (t) ,

where pn (t) = n |ρ (t)| n . The ¬rst term on the right represents the rate of ¬‚ow

of probability from the n-photon state to all other states, while the second and third

terms represent the ¬‚ow of probability into the n-photon state from the (n + 1)-photon

state and the (n ’ 1)-photon state respectively.

In order to study the approach to equilibrium, we write the equation as

dpn (t)

= Zn+1 (t) ’ Zn (t) , (18.35)

dt

where

Zn (t) = nκ {(ncav + 1) pn (t) ’ ncav pn’1 (t)} . (18.36)

The equilibrium condition is Zn+1 (∞) = Zn (∞), but this is the same as Zn (∞) = 0,

since Z0 (t) ≡ 0. Thus equilibrium imposes the recursion relations

(ncav + 1) pn (∞) = ncav pn’1 (∞) . (18.37)

This is an example of the principle of detailed balance; the rate of probability

¬‚ow from the n-photon state to the (n ’ 1)-photon state is the same as the rate of

probability ¬‚ow of the (n ’ 1)-photon state to the n-photon state. The solution of this

recursion relation, subject to the normalization condition

∞

pn (∞) = 1 , (18.38)

n=0

is the Bose“Einstein distribution

n

(ncav )

pn (∞) = . (18.39)

n+1

(ncav + 1)

Examples of the master equation

18.4.2 Two-level atom

For the two-level atom, the sample“reservoir interaction Hamiltonian HSE is given by

eqns (14.131)“(14.133). In this case, the operator F (t) in eqn (18.19) is the sum of two

terms: F (t) = Fsp (t) + Fpc (t), that are respectively given by

√ √

Fsp (t) = i w21 b† (t) S 12 (t) = i w21 b† (t) σ ’ (t) (18.40)

in in

and

√ √

w11 c† (t) S 11 (t) + w22 c† (t) S 11 (t)

Fpc (t) = i 1,in 2,in

√

√ 1 ’ σ z (t) 1 + σ z (t)

w11 c† (t) w22 c† (t)

=i + .

1,in 2,in

2 2

(18.41)

The envelope operators σ ’ (t) and σ z (t) are related to the environment-picture forms

by σ ’ (t) = σ’ (t) exp (iω21 t) and σ z (t) = σz (t), while the operators b† (t) and c† (t)

in q,in

are the in-¬elds de¬ned by eqns (14.146) and (14.147) respectively.

We will assume that the reservoirs are uncorrelated, i.e. ρE (0) = ρsp (0) ρpc (0),

and that the individual reservoirs are stationary. These assumptions guarantee that

most of the possible terms in the double commutator will vanish when the partial trace

over the environment is carried out.

After performing the invigorating algebra suggested in Exercise 18.4.2, the surviv-

ing terms yield the Schr¨dinger-picture master equation

o

‚

ρS (t) = LS ρS (t) + Ldis ρS (t) , (18.42)

‚t

where the Hamiltonian part,

1 ω21

LS ρS (t) = σz + HS1 (t) , ρS (t) , (18.43)