tially unique form.

For a sample described by the Lindblad master equation (18.115) the stochastic

equation for the state vector can be written as

d 1 1

†

|Ψ (t) = Hdis |Ψ (t) + Ck (t) ’ |Ψ (t)

Ck (t) Ck (t) Ψ

dt i 2

Ψ

k

[Ck (t) ’ Ck (t) Ψ ] |Ψ (t)

+ ζk (t) , (18.184)

k

where Hdis is the dissipative Hamiltonian de¬ned by eqn (18.146), and

= Ψ |X| Ψ

X (18.185)

Ψ

The master equation

is the expectation value in the state. The c-numbers ζk (t) are delta-correlated random

variables, i.e.

—

ζk (t) ζk (t ) P = δkk δ (t ’ t ) , (18.186)

where the average · · · P is de¬ned by the probability distribution P for the random

variables ζk .

We have chosen to write the stochastic equation for the state vector so that it

resembles the operator Langevin equations discussed in Chapter 14, but most authors

prefer to use the more mathematically respectable Ito form (Gardiner, 1991). The

presence of the averages Ck (t) Ψ makes this equation nonlinear, so that analytical

solutions are hard to come by.

In this approach, quantum jumps appear as smooth transitions between discrete

quantum states. The transitions occur on a short time scale, that is determined by

the equation itself. Physical interactions describing measurements of an observable

lead to irreversible di¬usion toward one of the eigenstates of the observable, so that

no separate collapse postulate is required. In applications, the numerical solution of

eqn (18.184) has the same kind of advantage over the direct solution of the master

equation as the Monte Carlo wave function method.

Given the close relation between the master equation, quantum jumps, and quan-

tum state di¬usion, it is not very surprising to learn that quantum state di¬usion

can be derived as a limiting case of the quantum-jump method. The limiting case is

that of in¬nitely many jumps, where each jump causes an in¬nitesimal change in the

state vector. This mathematical procedure is related to the experimental technique

of balanced heterodyne detection discussed in Section 9.3. Thus the quantum state

di¬usion method can be regarded as a new conceptual approach to quantum theory,

or as a particular method for solving the master equation.

18.8 Exercises

18.1 Averaging over the environment

(1) Combine ρW (0) = ρS (0) ρE (0) and the assumption brν = 0 with eqn (18.14)

E

to derive eqn (18.15).

(2) Drop the assumption brν E = 0, and introduce the ¬‚uctuation operators δbrν =

brν ’ brν E . Show how to rede¬ne HS and HE , so that eqn (18.15) will still be

valid.

18.2 Master equation for a cavity mode

(1) Use the discussion in Section 18.4.1 to argue that the general expression (18.20)

for the double commutator C2 (t, t ) can be replaced by C2 (t, t ) = F† (t) , G (t )

+ HC .

(2) Use the expression (18.25) for F to show that TrE F† (t) , G (t ) can be expressed

in terms of the correlation functions in eqns (18.28) and (18.29).

(3) Put everything together to derive eqn (18.30). Do not forget the end-point rule.

(4) Transform back to the Schr¨dinger picture to derive eqns (18.31)“(18.33).

o

Exercises

18.3 Master equation for a two-level atom

(1) Use the Markov assumptions (14.142) and (14.143) to verify eqns (18.40) and

(18.41).

(2) Use these expressions to evaluate the double commutator G2 .

(3) Given the assumptions made in Section 18.4.2, ¬nd out which terms in G2 have

vanishing traces over the environment.

(4) Evaluate the traces of the surviving terms and thus derive the master equation in

the environment picture.

(5) Transform back to the Schr¨dinger picture to derive eqns (18.42)“(18.44).

o

18.4 Thermal equilibrium for a cavity mode

(1) Derive eqn (18.34) from eqn (18.31).

(2) Solve the recursion relation (18.37), subject to eqn (18.38), to ¬nd eqn (18.39).

18.5 Fokker“Planck equation

(1) Carry out the chain rule calculation needed to derive eqn (18.81).

(2) Derive and solve the di¬erential equations for the functions introduced in eqn

(18.85).

(3) Derive eqn (18.93).

Lindblad form for the two-level atom—

18.6

Determine the three operators C1 , C2 , and C3 for the two-level atom.

Evolution of the purity of a general state—

18.7

(1) Use the cyclic invariance of the trace operation to deduce eqn (18.119) from eqn

(18.115).

(2) Suppose that a single cavity mode is in thermal equilibrium with the cavity walls at

temperature T . At t = 0 the cavity walls are suddenly cooled to zero temperature.

Calculate the initial rate of change of the purity.

19

Bell™s theorem and its optical tests

Since this is a book on quantum optics, we have assumed throughout that quantum

theory is correct in its entirety, including all its strange and counterintuitive predic-

tions. As far as we know, all of these predictions”even the most counterintuitive

ones”have been borne out by experiment. Einstein accepted the experimentally ver-

i¬ed predictions of quantum theory, but he did not believe that quantum mechanics

could be the entire story. His position was that there must be some underlying, more

fundamental theory, which satis¬ed the principles of locality and realism.

According to the principle of locality, a measurement occurring in a ¬nite volume

of space in a given time interval could not possibly in¬‚uence”or be in¬‚uenced by”

measurements in a distant volume of space at a time before any light signal could

connect the two localities. In the language of special relativity, two such localities are

said to be space-like separated.

The principle of realism contains two ideas. The ¬rst is that the physical properties

of objects exist independently of any measurements or observations. This point of view

was summed up in his rhetorical question to Abraham Pais, while they were walking

one moonless night together on a path in Princeton: ˜Is the Moon there when nobody

looks?™ The second is the condition of spatial separability: the physical properties

of spatially-separated systems are mutually independent.

The combination of the principles of locality and realism with the EPR thought

experiment convinced Einstein that quantum theory must be an incomplete description

of physical reality.

For many years after the EPR paper, this discussion appeared to be more concerned

with philosophy than physics. The situation changed dramatically when Bell (1964)

showed that every local realistic theory”i.e. a theory satisfying a plausible inter-

pretation of the metaphysical principles of locality and realism favored by Einstein”

predicts that a certain linear combination of correlations is uniformly bounded. Bell

further showed that this inequality is violated by the predictions of quantum mechan-

ics.

Subsequent work has led to various generalizations and reformulations of Bell™s orig-

inal approach, but the common theme continues to be an inequality satis¬ed by some

linear combination of correlations. We will refer to these inequalities generically as

Bell inequalities. Most importantly, two-photon, coincidence-counting experiments

have shown that a particular Bell inequality is, in fact, violated by nature. One must

therefore give up one or the other”or possibly even both”of the principles of locality

and realism (Chiao and Garrison, 1999).

The Einstein“Podolsky“Rosen paradox

Bell thereby successfully transformed what seemed to be an essentially philosoph-

ical problem into experimentally testable physical propositions. This resulted in what

Shimony has aptly called experimental metaphysics. The ¬rst experiment to test Bell™s

theorem was performed by Freedman and Clauser (1972). This early experiment al-

ready indicated that there must be something wrong with Einstein™s fundamental

principles.

One of the most intriguing developments in recent years is that the Bell inequalities

”which began as part of an investigation into the conceptual foundations of quantum

theory”have turned out to have quite practical applications to ¬elds like quantum

cryptography and quantum computing.

Quantum optics is an important tool for investigating the phenomenon of quantum

nonlocality connected with EPR states and the EPR paradox. Although Einstein,

Podolsky, and Rosen formulated their argument in the language of nonrelativistic

quantum mechanics, the problem they posed also arises in the case of two relativistic

particles ¬‚ying o¬ in di¬erent directions, for example, the two photons emitted in

spontaneous down-conversion.

19.1 The Einstein“Podolsky“Rosen paradox

The Einstein“Podolsky“Rosen paper (Einstein et al., 1935) adds two further ideas to

the principles of locality and realism presented above. The ¬rst is the de¬nition of an

element of physical reality:

If, without in any way disturbing a system, we can predict with certainty (i.e. with

probability equal to unity) the value of a physical quantity, then there exists an

element of physical reality corresponding to this physical quantity.

The second is a criterion of completeness for a physical theory:

. . .every element of the physical reality must have a counterpart in the physical

theory.

The argument in the EPR paper was formulated in terms of the entangled two-body

wave function

∞

dk ik(xA ’xB ’L)

ψ (xA , xB ) = e , (19.1)

’∞ 2π

which is a special case of the EPR states de¬ned by eqn (6.1), but we will use a

simpler example due to Bohm (1951, Chap. 22), which more closely resembles the

actual experimental situations that we will study. Hints for carrying out the original

argument can be found in Exercise 19.1.

Bohm™s example is modeled on the decay of a spin-zero particle into two distin-

guishable spin-1/2 particles, and it”like the original EPR argument”is expressed in

the language of nonrelativistic quantum mechanics. In the rest frame of the parent

particle, conservation of the total linear momentum implies that the daughter parti-

cles are emitted in opposite directions, and conservation of spin angular momentum

implies that the total spin must vanish.

¼ Bell™s theorem and its optical tests

In this situation, the decay channel in which the particles travel along the z-axis,

with momenta k0 and ’ k0 , is described by the two-body state

= eik0 (zA ’zB ) |¦

|Ψ , (19.2)

AB AB

where the spins σA and σB are described by the Bohm singlet state

1

= √ {|‘

|¦ |“ ’ |“ |‘ B} , (19.3)

AB A B A

2

which is expressed in the notation introduced in eqns (6.37) and (6.38).

The choice of the quantization axis n is left open, since”as seen in Exercise 6.3”

the spherical symmetry of the Bohm singlet state guarantees that it has the same form

for any choice of n. Since only spin measurements will be considered, the following

discussion will be carried out entirely in terms of the spin part |¦ AB of the two-body

state vector.

The spins of the daughter particles can be measured separately by means of two

Stern“Gerlach magnets placed to intercept them, as shown in Fig. 19.1. Correlations

between the spatially well-separated spin measurements can then be determined by

means of coincidence-counting circuitry connecting the four counters.

Let us ¬rst suppose that the magnetic ¬elds”and consequently the spatial quan-

tization axes”of the two Stern“Gerlach magnets are directed along the x-axis, i.e.

A

n = ux . A measurement of the spin component Sx with the result +1/2 is signalled

by a click in the upper Geiger counter of the Stern“Gerlach apparatus A. Applying

von Neumann™s projection postulate to the Bohm singlet state yields the reduced state

|¦ = |‘x |“x

x

, (19.4)

AB A B

where |‘x A is an eigenstate of Sx with eigenvalue +1/2, etc.

A

The reduced state is also an eigenstate of Sx with eigenvalue ’1/2; therefore, any

B

measurement of Sx would certainly yield the value ’1/2, corresponding to a click

B

B

in the lower counter of apparatus B. Since this prediction of a de¬nite value for Sx

)

*

’ ’

Fig. 19.1 The Bohm singlet version of the EPR experiment. σA and σB are spin-1/2 particles

in a singlet state, and ± and β are the angles of orientation of the two Stern“Gerlach magnets.

½

The nature of randomness in the quantum world

does not require any measurement at all, the system is not disturbed in any way.

B

Consequently, Sx is an element of physical reality at B.

Now consider the alternative scenario in which the quantization axes are directed

A

along y. In this case, a measurement of Sy with the outcome +1/2 leaves the system

in the reduced state

|¦ y = |‘y A |“y B , (19.5)

AB

and this in turn implies that the value of Sy is certainly ’1/2. This prediction is also

B

B

possible without disturbing the system; therefore, Sy is also an element of physical

reality at B.

From the local-realistic point of view, a believer in quantum theory now faces a

B B

dilemma. The spin components Sx and Sy are represented by noncommuting opera-

tors:

B B B

Sx , Sy = i S z = 0 , (19.6)

so they cannot be simultaneously predicted or measured. This leaves two alternatives.

B B

(1) If Sx and Sy are both elements of physical reality, then quantum theory”which

cannot predict values for both of them”is incomplete.

B B

(2) Two physical quantities, like Sx and Sy , that are associated with noncommuting

operators cannot be simultaneously real.

The latter alternative implies a more restrictive de¬nition of physical reality in

which, for example, two quantities cannot be simultaneously real unless they can be

simultaneously measured or predicted. This would, however, mean that the physical

B B

reality of Sx or Sy at B depends on which measurement was carried out at the distant

apparatus A.

The state reductions in eqns (19.4) or (19.5), i.e. the replacement of the original

state |¦ AB by |¦ x or |¦ y respectively, occur as soon as the measurement at

AB AB

A is completed. This is true no matter where apparatus B is located; in particular,

when the light transit time from A to B is larger than the time required to complete

the measurement at A. Thus the global change in the state vector occurs before any

signal could travel from A to B. This evidently violates local realism.

In the words of Einstein, Podolsky, and Rosen, ˜No reasonable de¬nition of real-

ity could be expected to permit this.™ On this basis, they concluded that quantum

theory is incomplete. In this connection, it is interesting to quote Einstein™s reaction

to Schr¨dinger™s introduction of the notion of entangled states. In a letter to Born,

o

written in 1948, Einstein wrote the following (Einstein, 1971):

There seems to me no doubt that those physicists who regard the descriptive meth-

ods of quantum mechanics as de¬nitive in principle would react to this line of thought

in the following way: they would drop the requirement for the independent existence

of the physical reality present in di¬erent parts of space; they would be justi¬ed in

pointing out that the quantum theory nowhere makes explicit use of this require-

ment. [Emphasis added]

19.2 The nature of randomness in the quantum world

If the EPR claim that quantum theory is incomplete is accepted, then the next step

would be to ¬nd some way to complete it. One advantage of such a construction would

¾ Bell™s theorem and its optical tests

be that the randomness of quantum phenomena, e.g. in radioactive decay, might be

explained by a mechanism similar to ordinary statistical mechanics.

In other words, there may exist some set of hidden variables within the radioac-

tive nucleus that evolve in a deterministic way. The apparent randomness of radioactive

decay would then be merely the result of our ignorance of the initial values of the hid-

den variables. From this point of view, there is no such thing as an uncaused random

event, and the characteristic randomness of the quantum world originates at the very

beginning of each microscopic event.

This should be contrasted with the quantum description, in which the state vector

evolves in a perfectly deterministic way from its initial value, and randomness enters

only at the time of measurement.

A simple example of a hidden variable theory is shown in Fig. 19.2. Imagine a

box containing many small, hard spheres that bounce elastically from the walls of the

box, and also scatter elastically from each other. The properties of such a system of

particles can be described by classical statistical mechanics.

Cutting a small hole into one of the walls of the box will result in an exponential

decay law for the number of particles remaining in the box as a function of time. In this

model for a nucleus undergoing radioactive decay, the apparent randomness is ascribed

to the observers ignorance of the initial conditions of the balls, which obey completely

deterministic laws of motion. The unknown initial conditions are the hidden variables

responsible for the observed phenomenon of randomness.

For an alternative model, we jump from the nineteenth to the twentieth century,

and imagine that the box is equipped with a computer running a program generating

random numbers, which are used to decide whether or not a particle is emitted in a

given time interval. In this case the apparently random behavior is generated by a

deterministic algorithm, and the hidden variables are concealed in the program code

and the seed value used to begin it.

Let us next consider a series of random events occurring in a time interval

(t ’ ∆t/2, t + ∆t/2) at two distant points r1 and r2 . If the two sets of events are

space-like separated, i.e. |r1 ’ r2 | > c∆t, then the principle of local realism requires

that correlations between the random series can only occur as a result of an earlier,

common cause. We will call this the principle of statistical separability.

In the absence of a common cause, the separated random events are like inde-

pendent coin tosses, located at r1 and r2 , so it would seem that they must obey a

common-sense factorization condition. For example, the joint probability of the out-

comes heads-at-r1 and heads-at-r2 should be the product of the independent proba-

bilities for heads at each location.

Fig. 19.2 A simple model for radioactive de-

cay, consisting of small balls inside a large box

with a small hole cut into one of the walls.

Einstein™s ˜hidden variables™ would be the un-

known initial conditions of these balls.

¿

Local realism

In quantum mechanics, the factorizability of joint probabilities implies the factor-

izability of joint probability amplitudes (up to a phase factor); for example, a situation

in which measurements at r1 and r2 are statistically independent is described by a

separable two-body wave function, i.e. the product of a wave function of r1 and a

wave function of r2 . Conversely, the absolute square of a product wave function is the

product of two separate probabilities, just as for two independent coin tosses at r1 and

r2 .

By contrast, an entangled state of two particles, e.g. a superposition of two prod-

uct wave functions, is not factorizable. The result is that the probability distribution

de¬ned by an entangled state does not satisfy the principle of statistical separability,

even when the parts are far apart in space.

The EPR argument emphasizes the importance of these disparities between the

classical and quantum descriptions of the world, but it does not point the way to an

experimental method for deciding between the two views. Bell realized that the key is

the fact that the nonfactorizability of entangled states in quantum mechanics violates

the common-sense, independent-coin-toss rule for joint probabilities.

He then formulated the statistical separability condition in terms of a factoriz-

ability condition on the joint probability for correlations between measurements on

two distant particles. Bell™s analysis applies completely generally to all local realistic

theories, in a sense to be explained in the next section.

19.3 Local realism

Converting the qualitative disparities between the classical and quantum approaches

into experimentally testable di¬erences requires a quantitative formulation of local

realism that does not depend on quantum theory. We will follow Shimony™s version

(Shimony, 1990) of Bell™s solution for this problem. This analysis can be presented in

a very general way, but it is easier to understand when it is described in terms of a

concrete experiment. For this purpose, we ¬rst sketch an optical version of the Bohm

singlet experiment.

19.3.1 Optical Bohm singlet experiment

As shown in Fig. 19.3, the entangled pair of spin-1/2 particles in Fig. 19.1 is replaced

by a pair of photons emitted back-to-back in an entangled state, and the Stern“Gerlach

magnets are replaced by calcite prisms that act as polarization analyzers. The beam of

unpolarized right-going photons γA is split by the calcite prism A into an extraordinary

ray e and an ordinary ray o. Similarly, the beam of left-going photons γB is split by

calcite prism B into e and o rays.

The ordinary-ray and extraordinary-ray output ports of the calcite prisms are

monitored by four counters. The two calcite prisms A and B can be independently

rotated around the common decay axis by the azimuthal angles ± and β respectively.

The values of ± and β”which determine the division of the incident wave into e-

and o-waves”correspond to the direction of the magnetic ¬eld in a Stern“Gerlach

apparatus.

Bell™s theorem and its optical tests

* )