for the meter.

(1) Use the results of Exercise 13.3 to show that |Ψ (t) = n1 , ±2 e’iγn1 , where

γ = g12 t.

(2) Devise a homodyne measurement scheme that can distinguish between the phase

shifts experienced by the meter beam for di¬erent values of n1 , e.g. n1 = 0 and

n1 = 1. For example, measure the quadrature X2 = a2 exp [’i•] + a† exp [i•] /2,

2

where • is the phase of the local oscillator in the homodyne apparatus.

14

Quantum noise and dissipation

In the majority of the applications considered so far”e.g. photons in an ideal cavity,

photons passing through passive linear media, atoms coupled to the radiation ¬eld,

etc.”we have neglected all dissipative e¬ects, such as absorption and scattering. In

terms of the fundamental microscopic theory, this means that all interactions between

the system under study and the external world have been ignored. When this assump-

tion is in force, the system is said to be closed. The evolution of a closed system

is completely determined by its Hamiltonian. A pure state of a closed system is de-

scribed by a state vector obeying the Schr¨dinger equation (2.108), and a mixed state

o

is represented by a density operator obeying the quantum Liouville equation (2.119).

With the possible exception of the entire universe, the assumption that a system is

closed is always an approximation. Every experimentally relevant physical system is

unavoidably coupled to other physical systems in its vicinity, and usually very little is

known about the neighboring systems or about the coupling mechanisms. If interac-

tions with the external world cannot be neglected, the system is said to be open. In

this chapter, we begin the study of open systems.

14.1 The world as sample and environment

For the discussion of open systems, we will divide the world into two parts: the

sample1 ”the physical objects of experimental interest”and the environment”

everything else. Deciding which degrees of freedom should be assigned to the sample

and which to the environment requires some care, as we will shortly see.

In fact, we have already studied three open systems in previous chapters. In the

discussion of blackbody radiation in Section 2.4.2, the radiation ¬eld is assumed to be

in thermal equilibrium with the cavity walls. In this case the sample is the radiation

¬eld in the cavity, and some coupling to the cavity walls (the environment) is required

to enforce thermal equilibrium. In line with standard practice in statistical mechanics,

we simply assume the existence of a weak coupling that imposes equilibrium, but

otherwise plays no role. In the discussion of the Weisskopf“Wigner method in Section

11.2.2 the sample is a two-level atom, and the modes of the radiation ¬eld are assigned

to the environment. In this case, an approximate treatment of the coupling to the

environment leads to a derivation of the irreversible decay of the excited atom. A

purely phenomenological treatment of other dissipative terms in the Bloch equation

for the two-level atom can be found in Section 11.3.3.

1 Overuse has leached almost all meaning from the word ˜system™, so we have replaced it with

˜sample™ for this discussion.

¾½

The world as sample and environment

As an illustration of the choices involved in separating the world into sample and

environment, we begin by revisiting the problem of transmission through a stop. In

Section 8.7 the radiation ¬eld is treated as a closed system by assuming that the

screen is a perfect re¬‚ector, and by including both the incident and the re¬‚ected

modes in the sample. Let us now look at this problem in a di¬erent way, by assigning

the re¬‚ected modes”i.e. the modes propagating from right to left in Fig. 8.5”to

the environment. The newly de¬ned sample consists of the modes propagating from

left to right. It is clearly an open system, since the right-going modes of the sample

scatter into left-going modes that belong to the environment. The loss of photons from

the sample represents dissipation, and the result (8.82) shows that this dissipation is

accompanied by an increase in ¬‚uctuations of photon number in the transmitted ¬eld.

This is a simple example of a general principle which is often called the ¬‚uctuation

dissipation theorem.

14.1.1 Reservoir model for the environment

Our next task is to work out a more systematic way of dealing with open systems.

This e¬ort would be doomed from the start if it required a detailed description of the

environment, but there are many experimentally interesting situations for which such

knowledge is not necessary. These favorable cases are characterized by generalizations

of the conditions required for the Weisskopf“Wigner (WW) treatment of spontaneous

emission.

(1) The modes of the environment (the radiation ¬eld for WW) have a continuous

spectrum.

(2) The sample (the two-level atom for WW) has”to a good approximation”the

following properties.

(a) The sample Hamiltonian has a discrete spectrum. This is guaranteed if the

sample (like the atom) has a ¬nite number of degrees of freedom. If the sample

has an in¬nite number of degrees of freedom (like the radiation ¬eld) a discrete

spectrum is guaranteed by con¬nement to a ¬nite region of space, e.g. a cavity.

(b) The sample is weakly coupled to a broad spectral range of environmental

modes.

In the Weisskopf“Wigner model these features justify the Markov approximation. Ap-

plying the general rule (11.23) of the resonant wave approximation to the WW model

provides the condition

|„¦ks | ∆K ω21 , (14.1)

where |„¦ks | is the one-photon Rabi frequency de¬ned in eqn (4.153), and ∆K is the

width of the cut-o¬ function for the RWA. This inequality de¬nes what is meant by

coupling to a broad spectral range of the radiation ¬eld.

Turning now to the general problem, we assume the environmental degrees of free-

dom that couple to the sample have continuous spectra, and that the coupling is weak.

Expressing the characteristic coupling strength as „¦S de¬nes a characteristic response

frequency „¦S , and the condition of weak coupling to a broad range of environmental

excitations is

„¦S ∆E ωS . (14.2)

¾¾ Quantum noise and dissipation

Here ∆E is the spectral width of the environmental modes that are coupled to the

sample, and ωS is a characteristic mode frequency for the unperturbed sample.

In the Weisskopf“Wigner model, the environment is the radiation ¬eld, and we

have a detailed theory for this example. This luxury is missing in the general case,

so we will instead devise a generic model that is based on the assumption of weak

interaction between the sample and the environment. An important consequence of

this assumption is that the sample can only excite low energy modes of the environ-

ment. As we have previously remarked, the low-lying modes of many systems can be

approximated by harmonic oscillators. For example, suppose that the environment

includes some solid material, e.g. the walls of a cavity, and that interaction with the

sample excites vibrations in the crystal lattice of the solid. In the quantum theory

of solids, these lattice vibrations are called phonons (Cohen-Tannoudji et al. 1977a,

Complement JV, p. 586; Kittel 1985, Chap. 2). The νth phonon mode”which is an

analogue of the ks-mode of the radiation ¬eld”is represented by a harmonic oscil-

lator with fundamental frequency „¦ν , analogous to ωks . For macroscopic bodies, the

discrete index ν becomes e¬ectively continuous, so this environment has a continuous

spectrum. Generalizing from this example suggests modeling the environment by one

or more families of harmonic oscillators with continuous spectra. Each family of oscil-

lators is called a reservoir. Weak coupling to the reservoir implies that the amplitudes

of the oscillator displacements and momenta will be small; therefore, we will make the

crucial assumption that the interaction Hamiltonian HSE is linear in the creation and

annihilation operators for the reservoir modes.

Within this schematic model of the world”the combined system of sample and

environment”the reservoirs can be grouped into two classes, according to their uses. A

reservoir which is not itself subjected to any experimental measurements will be called

a noise reservoir. In this case, the reservoir model simply serves as a useful theoretical

device for describing dissipative e¬ects. This is the most common situation, but there

are important applications in which the primary experimental signal is carried by

the modes of one of the reservoirs. In these cases, we will call the reservoir under

observation a signal reservoir. In the optical experiments discussed below, the signal

reservoir excitations are”naturally enough”photons.

For noise reservoirs, the objective is to carry out an approximate elimination of

the reservoir degrees of freedom, in order to arrive at a description of the sample as

an open system. The two principal methods used for this purpose are the quantum

Langevin equations for the ¬eld operator and atomic operator (which are formulated

in the Heisenberg picture) and the master equation for the density operator (which

is expressed in the interaction picture). The Langevin approach is, in some ways,

more intuitive and technically simpler. It is particularly useful for problems that have

simple analytical solutions or are amenable to perturbation theory, but it produces

equations of motion for sample operators that do not lend themselves to the numerical

simulations required for more complex problems. For such cases, the approach through

the master equation is essential. We will explain the Langevin method in the present

chapter, and introduce the master equation in Chapter 18.

In the case of a signal reservoir”which, after all, carries the experimental inform-

ation”it would evidently be foolish to eliminate the reservoir degrees of freedom.

¾¿

The world as sample and environment

Instead, the objective is to determine the e¬ect of the sample on the reservoir modes

to be observed. Despite this di¬erence in aim, the theoretical techniques developed

for dealing with noise reservoirs can also be applied to signal reservoirs. The principal

reason for this happy outcome is the assumption that both kinds of reservoirs are

coupled to the sample by an interaction Hamiltonian that is linear in the reservoir

operators. This approach to signal reservoirs, which is usually called the input“output

method, is described in Section 14.3.

A The world Hamiltonian

The division of the world into sample and environment implies that the Hilbert space

for the world is the tensor product,

HW = HS — HE , (14.3)

of the sample and environment spaces. For most applications, it is necessary to model

the environment by means of several independent reservoirs; therefore, the space HE

is itself a tensor product,

HE = H1 — H2 — · · · — HNres , (14.4)

of the Hilbert spaces for the Nres independent reservoirs that de¬ne the environment.

Pure states, |χ , in HW are linear combinations of product states:

|χ = C1 |Ψ1 |Λ1 + C2 |Ψ2 |Λ2 + · · · , (14.5)

where |Ψj and |Λj belong respectively to HS and HE . In most situations, however,

both the sample and the reservoirs must be described by mixed states.

In general, the sample may be acted on by time-dependent external classical ¬elds

or currents, and its constituent parts may interact with each other. Thus the total

Schr¨dinger-picture Hamiltonian for the sample is

o

HS (t) = HS0 + HS1 (t) , (14.6)

where HS0 is the noninteracting part of the sample Hamiltonian. The interaction term

HS1 (t) is

HS1 (t) = HSS + VS (t) , (14.7)

where HSS describes the internal sample interactions and VS (t) represents any inter-

actions with external classical ¬elds or currents. The time dependence of the external

¬elds is the source of the explicit time dependence of VS (t) in the Schr¨dinger picture.

o

In typical cases, VS (t) is a linear function of the sample operators. The Hamiltonian

for the isolated sample is

HS = HS0 + HSS . (14.8)

The total Schr¨dinger-picture Hamiltonian for the world is then

o

HW = HS (t) + HE + HSE , (14.9)

where

¾ Quantum noise and dissipation

Nres

HE = HJ (14.10)

J=1

is the free Hamiltonian for the environment, HJ is the Hamiltonian for the Jth reser-

voir,

Nres

(J)

HSE = HSE (14.11)

J=1

is the total interaction Hamiltonian between the sample and the environment, and

(J)

HSE is the interaction Hamiltonian of the sample with the Jth reservoir. The world

is, by de¬nition, a closed system.

We will initially use a box-quantization description of the reservoir oscillators that

parallels the treatment of the radiation ¬eld in Section 3.1.4, i.e. each family of oscil-

lators will be labeled by a discrete index ν. The free Hamiltonian for reservoir J is

therefore given by

„¦ν b† bJν ,

HJ = (14.12)

Jν

ν

where bJν is the annihilation operator for the νth mode of the Jth reservoir. We have

simpli¬ed the model by assuming that each reservoir has the same set of fundamen-

tal frequencies {„¦ν }, rather than a di¬erent set {„¦Jν } for each reservoir. This is not

a serious restriction, since in the continuum limit each „¦Jν is replaced by a contin-

uous variable „¦. The kinematical independence of the reservoirs is imposed by the

commutation relations

bJν , b†

[bJν , bKµ ] = 0 , Kµ = δJK δνµ .

In typical applications, the sample is coupled to the environment through sample

operators, OJ , that can be chosen to satisfy

[OJ , HS0 ] ≈ ωJ OJ , (14.13)

where ωJ 0. For ωJ > 0, this means that OJ is an approximate energy-lowering

operator for the unperturbed sample Hamiltonian HS0 . We will also need the limiting

case ωJ = 0, which means that OJ is an approximate constant of the motion.

In the resonant wave approximation, the sample“environment interaction can be

written as

vJ („¦ν ) OJ bJν ’ b† OJ ,

†

(J)

HSE = i (14.14)

Jν

ν

where vJ („¦ν ) is a real, positive coupling frequency. This ansatz incorporates the

assumption that each sample“reservoir interaction Hamiltonian is a linear function of

the reservoir operators. The restriction to real coupling frequencies is not signi¬cant, as

shown in Exercise 14.1. Each coupling frequency is a candidate for the characteristic,

sample-response frequency „¦S , so it must satisfy the condition

vJ („¦ν ) ∆E ωS . (14.15)

The choice of the sample operator OJ is determined by the physical damping mecha-

nism associated with the Jth reservoir.

¾

The world as sample and environment

B The world density operator

The probability distributions relevant to experiments are determined by the Schr¨- o

S

dinger-picture density operator, ρW (t), that describes the state of the world. We

must, therefore, begin by choosing an initial form, ρS (t0 ), for the density operator.

W

The natural assumption is that the sample and the reservoirs are uncorrelated for

a su¬ciently early time t0 . Since the time-independent, Heisenberg-picture density

operator, ρH , satis¬es ρH = ρS (t0 ), this is equivalent to assuming that

W W W

ρW = ρS ρE , (14.16)

where ρS acts on HS , and ρE acts on HE . We have dropped the superscript H, since

the remaining argument is conducted entirely in the Heisenberg picture. Furthermore,

it is equally natural to assume that the various reservoirs are mutually uncorrelated

at the initial time, so that

ρE = ρ1 ρ2 . . . ρNres , (14.17)

where ρJ acts on HJ for J = 1, 2, . . . , Nres . One or more of the density operators ρJ is

often assumed to describe a thermal equilibrium state, in which case the corresponding

reservoir is called a heat bath.

The average value of any observable O is given by

O = TrW (ρW O) , (14.18)

where TrW is de¬ned by the sum over a basis set for HW = HS — HE . By using the

de¬nition of partial traces in Section 6.3.1, it is straightforward to show that

TrW (SR) = (TrS S) (TrE R) , (14.19)

if S acts only on HS and R acts only on HE . The average of an operator product, SR,

with respect to the world density operator ρW = ρS ρE is then

SR = TrW [(ρS ρE ) (SR)]

= TrW [ρS S ρE R]

= [TrS (ρS S)] [TrE (ρE R)]

=S R, (14.20)

where the identities (6.17) and (14.19) were used to get the second and third lines.

Applying this relation to S = 1 (more precisely, S = IS , where IS is the identity

operator for HS ), and R = RJ RK , where RJ acts on HJ , RK acts on HK , and J = K,

yields

RJ RK = RJ RK . (14.21)

In other words, distinct reservoirs are statistically independent.

C Noise statistics

The statistical independence of the various reservoirs allows them to be treated indi-

vidually, so we drop the reservoir index in the present section. For most experimental

¾ Quantum noise and dissipation

arrangements, the reservoir is not subjected to any special preparation; therefore, we

will assume that distinct reservoir modes are uncorrelated, i.e. the reservoir density

operator is factorizable:

ρ= ρν , (14.22)

ν

where ρν is the density operator for the νth mode. For operators Fν and Gµ that are

respectively functions of bν , b† and bµ , b† , this assumption implies Fν Gµ = Fν Gµ

ν µ

for µ = ν.

For the discussion of quantum noise, only ¬‚uctuations around mean values are of

interest. We will say that a factorizable density operator ρ is a noise distribution if

the natural oscillator variables bν and b† satisfy

ν

b† = bν = 0 for all ν . (14.23)

ν

These conditions can always be achieved by using the ¬‚uctuation operator δbν =

bν ’ bν in place of bν . By means of suitable choices of the operators Fν and Gµ ,

the combination of eqns (14.23) and (14.22) can be used to derive restrictions on the

moments of a noise distribution ρ. For example, the results

b† bµ = bµ b† = b† bµ = 0 and b ν bµ = bν bµ = 0 for µ = ν (14.24)

ν ν ν

lead to the useful rules

b† bµ = δνµ b† bν , bµ b† = δνµ bν b† , (14.25)

ν ν ν ν

bν bµ = δνµ b2 (14.26)

ν

for the fundamental second-order moments of a noise distribution ρ. For some appli-

cations it is more convenient to employ symmetrically-ordered moments, e.g.

1†

b bµ + bµ b† = Nµ δµν , (14.27)

2ν ν

where

1

Nµ = b† bµ + (14.28)

µ

2

is the noise strength. One virtue of this choice is that the lower bound in the in-

equality Nµ 1/2 represents the presence of vacuum ¬‚uctuations.

If we neglect the weak reservoir“sample interaction, the time-domain analogue of

these relations can be expressed in terms of the Heisenberg-picture noise operator,

ξ (t), de¬ned as a solution of the Heisenberg equation,

d

i ξ (t) = [ξ (t) , Hres ] , (14.29)

dt

where Hres is given by eqn (14.12). The value of ξ (t) at the initial time t = t0 ”when

the Schr¨dinger and Heisenberg pictures coincide”is taken to be

o

ξ (t0 ) = Cν bν , (14.30)

ν

¾

The world as sample and environment

where Cν is a c-number coe¬cient. The explicit solution,

Cν bν e’i„¦ν (t’t0 ) ,

ξ (t) = (14.31)

ν

leads to the results

G (t, t ) = ξ † (t) ξ (t ) = |Cν |2 b† bν ei„¦ν (t’t ) (14.32)

ν

ν

and

Cν b2 e’i„¦ν (t+t ’2t0 )

2

F (t, t ) = ξ (t) ξ (t ) = (14.33)

ν

ν

for the second-order correlation functions G (t, t ) and F (t, t ).

The factorizability assumption (14.22) alone is su¬cient to show that G (t, t ) is

invariant under the uniform time translation (t ’ t + „ , t ’ t + „ ) for any set

of coe¬cients Cν , but the same cannot be said for F (t, t ). The only way to ensure

time-translation invariance of F (t, t ) is to impose

b2 = 0 , (14.34)

ν

which in turn implies F (t, t ) ≡ 0. A distribution satisfying eqns (14.27) and (14.34) is

said to represent phase-insensitive noise. It is possible to discuss many noise prop-

erties using only the second-order correlation functions F and G (Caves, 1982), but for

our purposes it is simpler to impose the stronger assumption that the distribution ρ is

stationary. From the general discussion in Section 4.5, we know that a stationary den-

sity operator commutes with the Hamiltonian. The simple form (14.12) of H in turn

implies that each ρν commutes with the mode number operator Nν ; consequently, ρν

is diagonal in the number-state basis. This very strong feature subsumes eqn (14.34)

in the general result

n n

b† b†

m n

(bν ) = δnm (bν ) , (14.35)

ν ν

which guarantees time-translation invariance for correlation functions of all orders.