ńņš. 22 |

time. In either case the computational diļ¬culty grows rapidly with the dimensionality

of the sample Hilbert space.

Quantum jumps

In general, a numerical simulation will take place in a sample Hilbert space with

some dimension M . The master equation is then an equation for an M Ć— M matrix,

and the computational cost for solving the problem scales as M 2 . This is an impor-

tant consideration, since increasing the accuracy of the simulation typically requires

enlarging the Hilbert space. On the other hand, if one could work with a state vector

instead of the density operator, the cost of a solution would only scale as M . This gain

alone justiļ¬es the development of the Monte Carlo wave function technique described

below.

The Monte Carlo wave function methodā—

18.7.3

According to eqn (18.115), the change in the density operator over a time step āt is

āt

[HS , ĻS ] + ātLdis ĻS + O āt2 .

ĻS (t + āt) = ĻS (t) + (18.144)

i

By combining the ļ¬rst two terms in eqn (18.117) for Ldis with the Hamiltonian term,

this can be rewritten as

K

iāt iāt ā ā

ĻS (t + āt) = ĻS (t) ā’ Hdis ĻS (t) + ĻS (t) Hdis + āt Ck ĻS (t) Ck , (18.145)

k=1

where the dissipative Hamiltonian is

K

i ā

= HS ā’

Hdis Ck Ck . (18.146)

2

k=1

This suggests deļ¬ning a dissipative, nonunitary time translation operator,

iāt

Udis (āt) = eā’iātHdis / = 1 ā’ Hdis + O āt2 , (18.147)

and then using it to rewrite eqn (18.145) as

K

ā ā

ĻS (t + āt) = Udis (āt) ĻS (t) Udis (āt) + āt Ck ĻS (t) Ck , (18.148)

k=1

correct to O (āt).

The ensemble deļ¬nition (2.116) of the density operator shows that this is equivalent

to

ā

|ĪØe (t + āt) Pe ĪØe (t + āt)| = Pe Udis (āt) |ĪØe (t) ĪØe (t)| Udis (āt)

e e

K

ā

Pe āt Ck |ĪØe (t) ĪØe (t)| Ck ,

+

e k=1

(18.149)

where the Pe s are the probabilities deļ¬ning the initial state, and |ĪØe (0) = |Ī˜e .

The ļ¬rst term on the right side of this equation evidently represents the dissipative

The master equation

evolution of each state in the ensemble. This is closely related to the Weisskopfā“Wigner

approach to perturbation theory, which we used in Section 11.2.2 to derive the decay

of an excited atomic state by spontaneous emission.

This is all very well, but what is the meaning of the second term on the right side

of eqn (18.149)? One way to answer this question is to ļ¬x attention on a single state

in the ensemble, say |ĪØe (t) , and to deļ¬ne the normalized states

Ck |ĪØe (t)

|Ļek (t) = , k = 1, . . . , K . (18.150)

ā

ĪØe (t) Ck Ck ĪØe (t)

With this notation, the contribution of |ĪØe (t) to the second term in eqn (18.149) is

(Ī“e (t) āt) Ļe

meas (t), where

K

Pk |Ļek (t) Ļek (t)| ,

Ļe e

(t) = (18.151)

meas

k=1

ā

ĪØe (t) Ck Ck ĪØe (t)

Pk =

e

, (18.152)

Ī“e (t)

and

K

ā

Ī“e (t) = ĪØe (t) Ck Ck ĪØe (t) (18.153)

k=1

is the total transition (quantum-jump) rate of |ĪØe (t) into the collection of normalized

states deļ¬ned by eqn (18.150). Since the coeļ¬cients Pk satisfy 0 Pk 1 and

e e

K

Pk = 1 ,

e

(18.154)

k=1

they can be treated as probabilities.

With this interpretation, Ļe

meas has the form (2.127) of the mixed state describing

the sample after a measurement has been performed, but before the particular outcome

is known. This suggests that we interpret the second term on the right side of eqn

(18.149) as a wave packet reduction resulting from a measurement-like interaction

with the reservoir.

After summing over the ensemble, eqn (18.148) becomes

ā

ĻS (t + āt) = Udis (āt) ĻS (t) Udis (āt) + Ī“ (t) āt Ļmeas (t) , (18.155)

where

P e Ļe

Ļmeas (t) = meas (t) , (18.156)

e

Pe Ī“e (t)

Pe = , (18.157)

Ī“ (t)

and

Quantum jumps

Pe Ī“e (t)

Ī“ (t) = (18.158)

e

is the ensemble-averaged transition rate.

A The Monte Carlo wave function algorithm

In quantum theory, a system evolves smoothly by the SchrĀØdinger equation until a

o

measurement event forces a discontinuous change. This feature is the basis for the

procedure described here.

It is plausible to expect that only one of the two terms in eqn (18.155)ā”dissipative

evolution or wave packet reductionā”will operate during a suļ¬ciently small time step.

We will ļ¬rst describe the Monte Carlo wave function algorithm (MCWFA) that follows

from this assumption, and then show that the density operator calculated in this way

is an approximate solution of the master equation (18.115).

In order to simplify the presentation we assume that the initial ensemble is deļ¬ned

by

states {|Ī˜1 , . . . , |Ī˜M } ,

(18.159)

probabilities {P1 , . . . , PM } ,

so that the index e = 1, 2, . . . , M .

In each time step, a choice between dissipative evolution and wave packet reduc-

tionā”i.e. a quantum jumpā”has to be made. For this purpose, we note that the prob-

ability of a quantum jump during the interval (t, t + āt) is āPe (t) = Ī“e (t) āt, where

Ī“e (t) is the total transition rate deļ¬ned by eqn (18.153). The discrete scheme will

only be accurate if the jump probability during a time step is small, i.e. āPe (t) 1.

Consequently, the time step āt must satisfy Ī“e (t) āt 1.

With this preparation, we are now ready to state the algorithm for integrating the

master equation in the interval (0, T ).

(1) Set e = 1 and deļ¬ne the discrete times tn = (n ā’ 1) āt, where 1 n N and

(N ā’ 1) āt = T .

(2) At the initial time t = 0, set |ĪØ (0) = |ĪØe (0) = |Ī˜e .

(3) For n = 2, . . . , N choose a random number r in the interval (0, 1). If āPe (tnā’1 ) < r

go to (a), and if āPe (tnā’1 ) > r go to (b). Since we have imposed āPe (t) 1,

this procedure guarantees that quantum jumps are relatively rare interruptions of

continuous evolution.

(a) In this case there is no quantum jump, and the state vector is advanced from

tnā’1 to tn by dissipative evolution followed by normalization:

Udis (āt) |ĪØe (tnā’1 )

|ĪØe (tn ) =

ā

ĪØe (tnā’1 ) Udis (āt) Udis (āt) ĪØe (tnā’1 )

1ā’ Hdis |ĪØe (tnā’1 )

iāt

= , (18.160)

1 ā’ āPe (tnā’1 )

where the last line follows from the deļ¬nition (18.147) of Udis (āt).

Ā¼ The master equation

(b) In this case there is a quantum jump, and the new state vector is deļ¬ned

by choosing k randomly from {1, 2, . . . , K}ā”conditioned by the probability

distribution Pk deļ¬ned in eqn (18.152)ā”and setting

e

|ĪØe (tn ) = |Ļek (tnā’1 ) , (18.161)

i.e. |ĪØe (tn ) jumps to one of the states permitted by the second term in eqn

(18.148).

(4) Repeat step (3) Ntraj times to get Ntraj discrete representations

{|ĪØej (tn ) , 1 N } , j = 1, . . . , Ntraj

n (18.162)

of the state vector. These representations are distinct, due to the random choices

made in each time step. The density operator that evolves from the original pure

state |Ī˜e is then given by

Ntraj

1

|ĪØej (tn ) ĪØej (tn )| .

Ļe (tn ) = (18.163)

Ntraj j=1

(5) Replace e by e + 1. If e + 1 M go to step (2). If e + 1 > M go to step (6).

(6) The density operator Ļ (t) that evolves from the initial density operator Ļ (0)ā”

deļ¬ned by the ensemble (18.159)ā”is given by

M

Pe Ļe (tn ) .

Ļ (tn ) = (18.164)

e=1

The computational cost of this method scales as Ntraj N , where N is the dimen-

sionality of the sample Hilbert space HS . Consequently, the MCWFA would not be

very useful as a technique for solving the master equation, if the required number of

trials is itself of order N . Fortunately, there are applications with large N for which

one can get good statistics with Ntraj N.

B Proof that the MCWFA generates a solution

If each of the density operators Ļe (t) satisļ¬es the master equation, then so will the

overall density operator deļ¬ned by eqn (18.164); therefore, it is suļ¬cient to give the

proof for a single Ļe (t). For a suļ¬ciently large number of trials, the evolution of the

pure state operators,

Ļej (tn ) = |ĪØej (tn ) ĪØej (tn )| , (18.165)

is eļ¬ectively given by step (2a) with probability 1 ā’ āPe (tnā’1 ) and by step (2b) with

probability āPe (tnā’1 ). In other words,

Ļej (tn ) = (1 ā’ āPe (tnā’1 )) ĪØdis (tn ) ĪØdis (tn )

ej ej

Pk (tnā’1 ) |Ļek (tnā’1 ) Ļek (tnā’1 )| ,

e

+ āPe (tnā’1 ) (18.166)

k

Ā½

Quantum jumps

where

1ā’ Hdis |ĪØej (tnā’1 )

iāt

ĪØdis (tn ) = . (18.167)

ej

1 ā’ āPe (tnā’1 )

The |Ļek (tnā’1 ) s are deļ¬ned by substituting |ĪØej (tnā’1 ) for |ĪØe (tnā’1 ) in eqn (18.150).

Using the deļ¬nitions of āPe , Pk , and Hdis in this equation and neglecting O āt2 -

e

terms leads to

Ļej (tn ) ā’ Ļej (tnā’1 ) i

= ā’ [HS , Ļej (tnā’1 )] + Ldis Ļej (tnā’1 ) . (18.168)

āt

Averaging this result over the trials, according to eqn (18.163), and taking the limit

āt ā’ 0 shows that Ļe (t) satisļ¬es the master equation (18.115).

Laser-induced ļ¬‚uorescenceā—

18.7.4

For a concrete application of the MCWFA, we return to the trapped three-level ion

considered in Section 18.7.1. For this example, however, we replace the incoherent

source driving 3 ā” 1 by a coherent laser ļ¬eld E L eā’iĻL t that is close to resonance,

i.e. |ĻL ā’ Ļ31 | ĻL . In the interests of simplicity, we also drop the ļ¬eld driving

3 ā” 2. The semiclassical approximation for the laser is applied by substituting E(+) ā’

E L eā’iĻL t in the general results (11.36) and (11.40) of Section 11.1.4.

In the resonant wave approximation, the SchrĀØdinger-picture Hamiltonian is HS =

o

HS0 + HS1 , where

HS0 = q Sqq , (18.169)

q

HS1 = ā„¦L S31 eā’iĻL t + HC , (18.170)

and ā„¦L = ā’d31 Ā· E L / is the Rabi frequency for the laser driving the 1 ā” 3 transition.

The Sqp s are the atomic transition operators deļ¬ned in Section 11.1.4, and the labels

q and p range over the values 1, 2, 3.

The form of the dissipative operator Ldis for the three-level ion can be inferred from

the result (18.44) for the two-level atom, by identifying each pair of levels connected

by a decay channel with a two-level atom. For example, the lowering operator Ļā’ in

eqn (18.44) will be replaced by S13 for the 3 ā’ 1 decay channel, and the remaining

transitions are treated in the same way.

There are two important simpliļ¬cations in the present case. The ļ¬rst is that the

phase-changing collision term in eqn (18.44) is absent for an isolated ion. The second

simpliļ¬cation is the assumption that the reservoirs coupled to the three transitionsā”

i.e. the modes of the radiation ļ¬eld near resonanceā”are at zero temperature. This

approximation is generally accurate at optical frequencies, since kB T Ļopt for any

reasonable temperature.

One can use these features to show that Ldis is deļ¬ned by

Ī“31

Ldis ĻS = ā’ (S31 S13 ĻS + ĻS S31 S13 ā’ 2S13 ĻS S31 )

2

Ī“32

ā’ (S32 S23 ĻS + ĻS S32 S23 ā’ 2S23 ĻS S32 )

2

Ī“2

ā’ (S21 S12 ĻS + ĻS S21 S12 ā’ 2S12 ĻS S21 ) . (18.171)

2

Ā¾ The master equation

This expression for Ldis can be cast into the general Lindblad form (18.117) by setting

K = 3 and deļ¬ning the operators

C1 = Ī“31 S13 , C2 = Ī“32 S23 , C3 = Ī“2 S12 , (18.172)

corresponding respectively to the decay channels 3 ā’ 1, 3 ā’ 2, and 2 ā’ 1.

The Rabi frequency ā„¦L is small compared to the laser frequency ĻL , so the

SchrĀØdinger-picture master equation,

o

ā‚

ĻS (t) = [HS , ĻS (t)] + Ldis ĻS (t) ,

i (18.173)

ā‚t

involves two very diļ¬erent time scales, 1/ĻL 1/ā„¦L . Diļ¬erential equations with this

feature are said to be stiļ¬, and it is usually very diļ¬cult to obtain accurate numerical

solutions for them (Press et al., 1992, Sec. 16.6). In the case at hand, this diļ¬culty

can be avoided by transforming to the interaction picture.

The general results in Section 4.8 yield the transformed master equation

ā‚I ā

ĻS (t) = HS1 , ĻI (t) + U0 (t) Ldis ĻS (t) U0 (t) ,

I

i (18.174)

S

ā‚t

where U0 (t) = exp (ā’iHS0 t/ ) and the transform of any operator X is X I (t) =

ā

U0 (t) XU0 (t). Applying this rule to the transition operators gives

ā

Sqp (t) = U0 (t) Sqp U0 (t) = eiĻqp t Sqp ,

I

(18.175)

and this in turn leads to

ā

U0 (t) Ldis ĻS (t) U0 (t) = Ldis ĻI (t) . (18.176)

S

Thus we arrive at the useful conclusion that Ldis has the same form in both pictures.

The transformed interaction Hamiltonian is

HS1 = ā„¦L S31 eā’iĪ“t + HC ,

I

(18.177)

where Ī“ = ĻL ā’ Ļ31 . The interaction-picture master equation (18.174) is not stiļ¬, but

it still has time-dependent coeļ¬cients. This annoyance can be eliminated by a further

transformation

ĻS (t) = eitF ĻI (t) eā’itF , (18.178)

S

where

F= fq Sqq . (18.179)

q

The algebra involved here is essentially identical to the original transformation to

the interaction picture, and it is not diļ¬cult to show that the equation of motion

Āæ

Quantum jumps

for Ļ (t) will have constant coeļ¬cients provided that the parameters fq are chosen to

satisfy

f3 ā’ f1 = Ī“ . (18.180)

The simple solution f1 = f2 = 0, and f3 = Ī“, leads to

ā‚

Ļ (t) = H S1 , ĻS (t) + Ldis ĻS (t) ,

i (18.181)

ā‚t S

where the transformed interaction Hamiltonian is

ā” ā¤

0 0 ā„¦ā— L

H S1 = ā£ 0 0 0 ā¦ . (18.182)

ā„¦L 0 ā’Ī“

We are now in a position to calculate all the bits and pieces that are needed for the

direct solution of the master equation (18.181), or the application of the MCWFA. We

leave the algebra as an exercise for the reader and proceed directly to the numerical

solution of the master equation. The density operator for this problem is represented

by a 3 Ć— 3 hermitian matrix which is determined by nine real numbers. Thus the

master equation in this case consists of nine linear, ordinary diļ¬erential equations

with constant coeļ¬cients. There are many packaged programs that can be used to

solve this problem.

Of course, this means that we do not really need the MCWFA, but it is still useful

to have a solvable problem as a check on the method. In Fig. 18.7 we compare the

direct solution to the average over 48 trials of the MCWFA. The match between the

averaged results and the direct solution can be further improved by using more trials

in the average, but it should already be clear that the MCWFA is converging on a

solution of the master equation.

Following the general practice in physics, we assumeā”on the basis of this special

caseā”that the MCWFA can be conļ¬dently applied in all cases. In particular, this

includes those applications for which the dimension of the relevant Hilbert space is

large compared to the number of trials needed.

Quantum trajectoriesā—

18.7.5

The results displayed in Fig. 18.7 show that the full-blown master equationā”whether

solved directly or by averaging over repeated trials of the MCWFAā”does no better

than the rate equations of Section 18.7.1 in describing the phenomenon of interrupted

ļ¬‚uorescence. This should not be a surprise, since the master equation describes the

evolution of the entire ensemble of state vectors for the ion.

What is needed for the description of quantum jumps (interrupted ļ¬‚uorescence)

is an improved version of the simple on-and-oļ¬ model used to derive the random

telegraph signal in Fig. 18.3. This is where single trials of the MCWFA come into

play. Each trial yields a sequence of state vectors

|ĪØ (t1 ) , |ĪØ (t2 ) , . . . , |ĪØ (tN ) , (18.183)

which is a discrete sampling of a continuous function |ĪØ (t) . This has led to the use

of the name discrete quantum trajectory for each individual trial of the MCWFA.

The master equation

2!

J

Fig. 18.7 The population of |Īµ3 as a function of time. The smooth curve represents the

direct solution of eqn (18.181) and the jagged curve is the result of averaging over 48 trials of

the Monte Carlo wave function algorithm. Time is measured in units of the decay time 1/Ī“31

for the 3 ā’ 1 transition. In these units ā„¦L = 0.5, Ī“ = 0, Ī“32 = 0.01, and Ī“21 = 0.001.

An example of the upper-level population P3 obtained from a single quantum trajec-

tory is shown in Fig. 18.8. Once again, a judicious choice from the results for several

trajectories nicely exhibits the random telegraph signal characterizing interrupted ļ¬‚u-

orescence.

According to the standard rules of quantum theory, the information from a com-

pleted measurementā”in particular, the collapse of the state vectorā”should be taken

into account immediately. In the algorithm presented in Section 18.7.3 the new infor-

2!

J

Fig. 18.8 The population of |Īµ3 as a function of time for a single quantum trajectory. The

parameter values are the same as in Fig. 18.7.

Quantum jumps

mation is not used until the next time step at tn + āt, so single trials of the Monte

Carlo wave function method are approximations to the true quantum trajectory.

A more reļ¬ned treatment involves allowing for the projection or collapse event to

occur one or more times during the interval āt, and using the dissipative Hamiltonian

to propagate the state vector in the subintervals between collapses. With this kind

of analysis, it can be shown that the Monte Carlo method is accurate to order āt.

Increasing the accuracy to order āt2 requires the inclusion of jumps at both ends of

the interval and also the possibility that two jumps can occur in succession (Plenio

and Knight, 1998).

Results like that shown in Fig. 18.8 might tempt one to believe that the Monte Carlo

techniqueā”or the more reļ¬ned quantum trajectory methodā”provides a description of

single quantum events in isolated microscopic samples. Any such conclusion would be

completely false. A large sample of trials for the Monte Carlo technique will resemble

a corresponding set of experimental runs, but the relation between the two sets is

purely statistical. Both will yield the same expectation values, correlation functions,

etc. In other words, the Monte Carlo or quantum trajectory methods are still based

on ensembles. The diļ¬erence between these methods and the full master equation is

that the ensembles are conditioned, i.e. reduced, by taking experimental results into

account.

Quantum state diļ¬usionā—

18.7.6

As explained above, the standard formulations of quantum theory do not apply to

individual microscopic samples, but rather to ensembles of identically prepared sam-

ples. Several of the founders of the quantum theory, including Einstein (Einstein et al.,

1935) and SchrĀØdinger (SchrĀØdinger, 1935b), were not at all satisļ¬ed with this feature,

o o

and there have been many subsequent eļ¬orts to reformulate the theory so that it ap-

plies to individual microscopic objects. One approach, which has attracted a great deal

of attention, is to replace the SchrĀØdinger equation for an ensemble by a stochastic

o

equationā”e.g. a diļ¬usion equation in the Hilbert space of quantum statesā”for an

individual system.

The universal empirical success of conventional quantum theory evidently requires

that the new stochastic equation should agree with the SchrĀØdinger equation when ap-

o

plied to ensembles. Many such equations are possible, but symmetry considerationsā”

see Gisin and Percival (1992) and references contained thereinā”have led to an essen-

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

ā

|ĪØ (t) = Hdis |ĪØ (t) + Ck (t) ā’ |ĪØ (t)

Ck (t) Ck (t) ĪØ

dt i 2

ĪØ

k

[Ck (t) ā’ Ck (t) ĪØ ] |ĪØ (t)

+ Ī¶k (t) , (18.184)

k

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

= ĪØ |X| ĪØ

X (18.185)

ĪØ

The master equation

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

variables, i.e.

ā—

Ī¶k (t) Ī¶k (t ) P = Ī“kk Ī“ (t ā’ t ) , (18.186)

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

variables Ī¶k .

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) ĪØ 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 diļ¬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 diļ¬usion, it is not very surprising to learn that quantum state diļ¬usion

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

that of inļ¬nitely many jumps, where each jump causes an inļ¬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

diļ¬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 ĻW (0) = ĻS (0) ĻE (0) and the assumption brĪ½ = 0 with eqn (18.14)

E

to derive eqn (18.15).

(2) Drop the assumption brĪ½ E = 0, and introduce the ļ¬‚uctuation operators Ī“brĪ½ =

brĪ½ ā’ brĪ½ E . Show how to redeļ¬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 ) = Fā (t) , G (t )

+ HC .

(2) Use the expression (18.25) for F to show that TrE Fā (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 SchrĀØdinger picture to derive eqns (18.31)ā“(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, ļ¬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 SchrĀØdinger picture to derive eqns (18.42)ā“(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 ļ¬nd eqn (18.39).

18.5 Fokkerā“Planck equation

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

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

(18.85).

(3) Derive eqn (18.93).

Lindblad form for the two-level atomā—

18.6

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

Evolution of the purity of a general stateā—

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

Bellā™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 predictionsā”even the most counterintuitive

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

iļ¬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 satisļ¬ed the principles of locality and realism.

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

of space in a given time interval could not possibly inļ¬‚uenceā”or be inļ¬‚uenced byā”

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 ļ¬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: ā˜Is the Moon there when nobody

looks?ā™ 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 theoryā”i.e. a theory satisfying a plausible inter-

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

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 Bellā™s orig-

inal approach, but the common theme continues to be an inequality satisļ¬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 otherā”or possibly even bothā”of the principles of locality

and realism (Chiao and Garrison, 1999).

The Einsteinā“Podolskyā“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 ļ¬rst experiment to test Bellā™s

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

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

principles.

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

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

theoryā”have turned out to have quite practical applications to ļ¬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 ļ¬‚ying oļ¬ in diļ¬erent directions, for example, the two photons emitted in

spontaneous down-conversion.

19.1 The Einsteinā“Podolskyā“Rosen paradox

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

the principles of locality and realism presented above. The ļ¬rst is the deļ¬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

ā

dk ik(xA ā’xB ā’L)

Ļ (xA , xB ) = e , (19.1)

ā’ā 2Ļ

which is a special case of the EPR states deļ¬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.

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

guishable spin-1/2 particles, and itā”like the original EPR argumentā”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.

Ā¼ Bellā™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 ā’ k0 , is described by the two-body state

= eik0 (zA ā’zB ) |Ī¦

|ĪØ , (19.2)

AB AB

where the spins ĻA and ĻB are described by the Bohm singlet state

1

= ā {|ā‘

|Ī¦ |ā“ ā’ |ā“ |ā‘ 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, sinceā”as seen in Exercise 6.3ā”

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 |Ī¦ AB of the two-body

state vector.

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

Sternā“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 ļ¬rst suppose that the magnetic ļ¬eldsā”and consequently the spatial quan-

tization axesā”of the two Sternā“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 Sternā“Gerlach apparatus A. Applying

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

|Ī¦ = |ā‘x |ā“x

x

, (19.4)

AB A B

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

A

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

B

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

B

B

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

)

*

ā’ ā’

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

in a singlet state, and Ī± and Ī² are the angles of orientation of the two Sternā“Gerlach magnets.

Ā½

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

|Ī¦ y = |ā‘y A |ā“y B , (19.5)

AB

and this in turn implies that the value of Sy is certainly ā’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 theoryā”which

cannot predict values for both of themā”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 deļ¬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 |Ī¦ AB by |Ī¦ x or |Ī¦ 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, ā˜No reasonable deļ¬nition of real-

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

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

to SchrĀØdingerā™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 deļ¬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 diļ¬erent parts of space; they would be justiļ¬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 ļ¬nd some way to complete it. One advantage of such a construction would

Ā¾ Bellā™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 ā’ āt/2, t + āt/2) at two distant points r1 and r2 . If the two sets of events are

space-like separated, i.e. |r1 ā’ r2 | > cāt, 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.

Einsteinā™s ā˜hidden variablesā™ would be the un-

known initial conditions of these balls.

Āæ

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

deļ¬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. Bellā™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 diļ¬erences requires a quantitative formulation of local

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

(Shimony, 1990) of Bellā™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 ļ¬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 Sternā“Gerlach

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

unpolarized right-going photons Ī³A is split by the calcite prism A into an extraordinary

ray e and an ordinary ray o. Similarly, the beam of left-going photons Ī³B 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 Ī± and Ī² respectively.

The values of Ī± and Ī²ā”which determine the division of the incident wave into e-

and o-wavesā”correspond to the direction of the magnetic ļ¬eld in a Sternā“Gerlach

apparatus.

Bellā™s theorem and its optical tests

* )

Fig. 19.3 An optical implementation of the EPR experiment. Calcite prisms replace the

Sternā“Gerlach magnets shown in Fig. 19.1. TheĀ¬source emits an entangled state of two oppo-

Ā«

sitely-directed photons, such as the Bell state Ā¬ĪØā’ . The birefringent prisms split the light

into ordinary ā˜oā™ and extraordinary ā˜eā™ rays. The vertical dotted lines inside the prisms in-

dicate the optic axes of the calcite crystals. Coincidence-counting circuitry connecting the

Geiger counters is not shown.

The counters on each side of the apparatus are mounted rigidly with respect to

the calcite prisms, so that they corotate with the prisms. Thus the four counters will

constantly monitor the o and e outputs of the calcite prisms for all values of Ī± and Ī².

The azimuthal angles Ī± and Ī² are examples of what are called parameter set-

tings, or simply parameters, of the EPR experiment. The experimentalist on the right

side of the apparatus, Alice, is free to choose the parameter setting Ī± (the azimuthal

angle of rotation of calcite prism A) as she pleases. Likewise, the experimentalist on

the left side, Bob, is free to choose the parameter setting Ī² (the azimuthal angle of

rotation of calcite prism B) as he pleases, independently of Aliceā™s choice.

19.3.2 Conditions deļ¬ning locality and realism

Bellā™s seminal paper has inspired many proposals for realizations of the metaphysical

notions of realism and locality, including both deterministic and stochastic forms of

hidden variables theories. In this section we present a general class of realizations by

specifying the conditions that a theory must satisfy in order to be called local and

realistic.

We will say that a theory is realistic if it describes all required elements of physical

reality for a system by means of a space, Ī, of completely speciļ¬ed states Ī»ā”i.e. the

states of maximum informationā”satisfying the following two conditions.

Objective reality

Ī is deļ¬ned without reference to any measurements. (19.7)

Spatial separability

The state spaces ĪA and ĪB for the spatially-separated systems

A and B are independently deļ¬ned. (19.8)

The only other condition imposed on Ī is that it must support probability distributions

Ļ (Ī») in order to describe situations in which maximum information is not available.

Local realism

The only conditions imposed on an admissible distribution Ļ (Ī») are that it be

positive deļ¬nite, i.e. Ļ (Ī») 0, normalized to unity,

dĪ»Ļ (Ī») = 1 , (19.9)

and independent of the parameter values Ī± and Ī². The last condition incorporates the

intuitive idea that the states Ī» are determined at the source S, before any encounters

with the measuring devices at A and B.

One possible example for Ī would be the classical phase space involved in the simple

model of radioactive decay presented above. In this case, the completely speciļ¬ed states

Ī» are simply points in the phase space, and a probability distribution Ļ (Ī») would be

the usual phase space distribution.

A much more surprising example comes from a disentangled version of quantum

theory, which is deļ¬ned by excluding all entangled states of spatially-separated sys-

tems. This mutilated theory violates the superposition principle, but by doing so it

allows us to identify Ī with the Hilbert space H for the local system. An individual

state Ī» is thereby identiļ¬ed with a pure state |Ļ .

According to the standard interpretation of quantum theory, this choice of Ī» gives

a complete description of the state of an isolated system. In this case Ļ (Ī») is just the

distribution deļ¬ning a mixed state. The fact that the disentangled version of quantum

theory is realistic illustrates the central role played by entanglement in diļ¬erentiating

the quantum view from the local realistic view.

We next turn to the task of developing a quantitative realization of locality. For this

purpose, we need a language for describing measurements at the spatially-separated

stations A and B, shown in Fig. 19.3. For the sake of simplicity, it is best to consider

experiments that have a discrete set of possible outcomes {Am , m = 1, . . . , M } and

{Bn , n = 1, . . . , N } at the stations A and B respectively, e.g. A1 could describe a

detector ļ¬ring at station A during a certain time interval. With each outcome Am , we

associate a numerical value, Am , called an outcome parameter. The deļ¬nition of

the output parameters is at our disposal, so they can be chosen to satisfy the following

convenient conditions:

ā’1 Am +1 and ā’ 1 Bn +1 . (19.10)

For the two-calcite-prism experiment, sketched in Fig. 19.3, the indices m and n

can only assume the values o and e, corresponding respectively to the ordinary and the

extraordinary rays emerging from a given prism. The source S emits a pair of photons

prepared at birth in some state Ī». The experimental signals in this case are clicks in

one of the counters, so one useful deļ¬nition of the outcome parameters is

Ae = 1 for outcome Ae (Aliceā™s e-counter clicks) ,

Ao = ā’1 for outcome Ao (Aliceā™s o-counter clicks) ,

(19.11)

Be = 1 for outcome Be (Bobā™s e-counter clicks) ,

Bo = ā’1 for outcome Bo (Bobā™s o-counter clicks) .

The outcome Ae occurs when a rightwards-propagating photon from the source S

is deļ¬‚ected through the e port of the calcite prism A, and subsequently registered by

Bellā™s theorem and its optical tests

Aliceā™s e-counter, etc. In this thought experiment we imagine that all counters have

100% sensitivity; consequently, if an e-counter does not click, we can be sure that the

corresponding o-counter will click.

The following conditional probabilities will be useful.

p(Am |Ī», Ī±, Ī²) ā” probability of outcome Am , given

the system state Ī» and parameter settings Ī±, Ī² . (19.12)

p(Bn |Ī», Ī±, Ī²) ā” probability of outcome Bn , given

the system state Ī» and parameter settings Ī±, Ī² . (19.13)

p(Am |Ī», Ī±, Ī², Bn ) ā” probability of outcome Am , given

the system state Ī», parameter settings Ī±, Ī² ,

and outcome Bn . (19.14)

p(Bn |Ī», Ī±, Ī², Am ) ā” probability of outcome Bn , given

the system state Ī», parameter settings Ī±, Ī² ,

and outcome Am . (19.15)

p(Am , Bn |Ī», Ī±, Ī²) ā” joint probability of outcomes Am and Bn ,

given the system state Ī» and

the parameter settings Ī±, Ī² . (19.16)

Following the work of Jarrett (1984), as presented by Shimony (1990), we will say

that a theory is local if it satisļ¬es the following conditions.

Parameter independence

p(Am |Ī», Ī±, Ī²) = p(Am |Ī», Ī±) , (19.17)

p(Bn |Ī», Ī±, Ī²) = p(Bn |Ī», Ī²) . (19.18)

Outcome independence

p(Am |Ī», Ī±, Ī², Bn ) = p(Am |Ī», Ī±, Ī²) , (19.19)

p(Bn |Ī», Ī±, Ī², Am ) = p(Bn |Ī», Ī±, Ī²) . (19.20)

Parameter independence states that the parameter settings chosen by one observer

have no eļ¬ect on the outcomes seen by the other. For example, eqn (19.17) tells us that

the probability distribution of the outcomes observed by Alice at A does not depend

on the parameter settings chosen by Bob at B.

This apparently innocuous statement is, in fact, extremely important. If parameter

independence were violated, then Bobā”who might well be space-like separated from

Aliceā”could send her an instantaneous message by merely changing Ī², e.g. twisting

his calcite crystal. Such a possibility would violate the relativistic prohibition against

sending signals faster than light. Likewise, eqn (19.18) prohibits Alice from sending

instantaneous messages to Bob.

The principle of outcome independence states that the probability of outcomes seen

by one observer does not depend on which outcomes are actually seen by the other.

This is what one would expect for two independent coin tossesā”since the outcome of

one coin toss is clearly independent of the outcome of the otherā”but eqns (19.19) and

(19.20) also seem to prohibit correlations due to a common cause, e.g. in the source S.

Local realism

This incorrect interpretation stems from overlooking the assumption that Ī» is a

complete description of the state, including any secret mechanism that builds in corre-

lations at the source (Bub, 1997, Chap. 2). With this in mind, the conditions (19.19)

and (19.20) simply reļ¬‚ect the fact that the actual outcomes Bn or Am are superļ¬‚uous,

if Ī» is given as part of the conditions. We will return to the issue of correlations after

deriving Bellā™s strong-separability condition.

It is also important to realize that the individual events at A and B can be truly

random, even if they are correlated. This situation is exhibited in the experiment

sketched in Fig. 19.3. When the polarizations of photons Ī³A and Ī³B , in the Bell state

|ĪØā’ , are measured separatelyā”i.e. without coincidence countingā”they are randomly

polarized; that is, the individual sequences of e- or o-counts at A and B are each as

random as two independent sequences of coin tosses.

Finally, we note that a violation of outcome independence does not imply any viola-

tions of relativity. The conditional probability p(Am |Ī», Ī±, Ī², Bn ) describes a situation

in which Bob has already performed a measurement and transmitted the result to Al-

ice by a respectably subluminal channel. Thus protecting the world from superluminal

messages and the accompanying causal anomalies is the responsibility of parameter

independence alone.

19.3.3 Strong separability

Bellā™s theorem is concerned with the strength of correlations between the random out-

comes at A and B, so the ļ¬rst step is to ļ¬nd the constraints imposed by the combined

eļ¬ects of realism and localityā”in the form of parameter and outcome independenceā”

on the joint probability p(Am , Bn |Ī», Ī±, Ī²) deļ¬ned by eqn (19.16).

We begin by applying the compound probability rule (A.114) to ļ¬nd

p(Am , Bn |Ī», Ī±, Ī²) = p(Am |Ī», Ī±, Ī², Bn )p(Bn |Ī», Ī±, Ī²) . (19.21)

In other words, the joint probability for outcome Am and outcome Bn is the product

of the probability for outcome Am (conditioned on the occurrence of the outcome Bn )

with the probability that outcome Bn actually occurred. All three probabilities are

conditioned by the assumption that the state of the system was Ī» and the parameter

settings were Ī± and Ī². The situation is symmetrical in A and B, so we also ļ¬nd

p(Am , Bn |Ī», Ī±, Ī²) = p(Bn |Ī», Ī±, Ī², Am )p(Am |Ī», Ī±, Ī²) . (19.22)

Applying outcome independence, eqn (19.19), to the right side of eqn (19.21) yields

p(Am , Bn |Ī», Ī±, Ī²) = p(Am |Ī», Ī±, Ī²)p(Bn |Ī», Ī±, Ī²) , (19.23)

and applying parameter independence to both terms on the right side of this equation

results in the strong-separability condition:

p(Am , Bn |Ī», Ī±, Ī²) = p(Am |Ī», Ī±)p(Bn |Ī», Ī²) . (19.24)

This is the mathematical expression of the following, seemingly common-sense,

statement: for a given speciļ¬cation, Ī», of the state, whatever Alice does or observes

Bellā™s theorem and its optical tests

must be independent of whatever Bob does or observes, since they could reside in

space-like separated regions.

Before using the strong-separability condition to prove Bellā™s theorem, we return

to the question of correlations that might be imposed by a common cause. In typical

experiments, the complete speciļ¬cation of the state represented by Ī» is not availableā”

for example, the values of the hidden variables cannot be determinedā”so the strong-

separability condition must be averaged over a distribution Ļ (Ī») that represents the

experimental information that is available.

The result is

p(Am , Bn |Ī±, Ī²) = dĪ»Ļ (Ī») p(Am |Ī», Ī±)p(Bn |Ī», Ī²) , (19.25)

where

p(Am , Bn |Ī±, Ī²) = dĪ»Ļ (Ī») p(Am , Bn |Ī», Ī±, Ī²) . (19.26)

The corresponding averaged probabilities for single outcomes are

p(Am |Ī±) = dĪ»Ļ (Ī») p(Am |Ī», Ī±) ,

(19.27)

p(Bn |Ī²) = dĪ»Ļ (Ī») p(Bn |Ī», Ī²) ;

consequently, the condition for statistical independence,

p(Am , Bn |Ī±, Ī²) = p(Am |Ī±)p(Bn |Ī²) , (19.28)

can only be satisļ¬edā”for general choices of Am and Bn ā”when Ļ (Ī») = Ī“ (Ī» ā’ Ī»0 ).

A closer connection with experiment is aļ¬orded by deļ¬ning Bellā™s expectation

values.

(1) The expectation value of outcomes seen by Alice is

p(Am |Ī», Ī±)Am .

E(Ī», Ī±) = (19.29)

m

(2) The expectation value of outcomes seen by Bob is

p(Bn |Ī», Ī²)Bn .

E(Ī», Ī²) = (19.30)

n

(3) The expectation value of joint outcomes seen by both Alice and Bob is

p(Am , Bn |Ī», Ī±, Ī²)Am Bn .

E(Ī», Ī±, Ī²) = (19.31)

m,n

The quantity E(Ī», Ī±, Ī²) is the average value of joint outcomes as measured, for

example, in a coincidence-counting experiment. The bounds |Am | 1 and |Bn | 1,

together with the normalization of the probabilities, imply that the absolute values of

all these expectation values are bounded by unity.

Bellā™s theorem

From Bellā™s strong-separability condition, it follows that the joint expectation

valueā”for a given complete state Ī»ā”also factorizes:

E(Ī», Ī±, Ī²) = E(Ī», Ī±)E(Ī», Ī²) , (19.32)

but in the absence of complete state information, the relevant expectation values are

E (Ī±) ā” p(Am |Ī±)Am ,

dĪ»Ļ (Ī») E(Ī», Ī±) = (19.33)

m

etc. Thus the correlation function

C (Ī±, Ī²) = E(Ī±, Ī²) ā’ E (Ī±) E (Ī²) (19.34)

can only vanish in the extreme case, Ļ (Ī») = Ī“ (Ī» ā’ Ī»0 ), of perfect information.

19.4 Bellā™s theorem

An evaluation of any one of Bellā™s expectation values, e.g. E(Ī», Ī±), would depend

on the details of the particular local realistic theory under consideration. One of the

consequences of Bellā™s original work (Bell, 1964) has been the discovery of various

linear combinations of expectation values, which have the useful property that upper

and lower bounds can be derived for the entire class of local realistic theories deļ¬ned

above. We follow Shimony (1990), by considering the particular sum

S (Ī») ā” E(Ī», Ī±1 , Ī²1 ) + E(Ī», Ī±1 , Ī²2 ) + E(Ī», Ī±2 , Ī²1 ) ā’ E(Ī», Ī±2 , Ī²2 ) , (19.35)

which was ļ¬rst suggested by Clauser et al. (1969). With a ļ¬xed value, Ī», of the hid-

den variables, the four combinations (Ī±1 , Ī²1 ), (Ī±1 , Ī²2 ), (Ī±2 , Ī²1 ), and (Ī±2 , Ī²2 ) represent

independent choices Ī±1 or Ī±2 by Alice and Ī²1 or Ī²2 by Bob, as shown in Fig. 19.4.

For the typical situation in which the complete state Ī» is not known, S (Ī») should

be replaced by the experimentally relevant quantity:

S ā” E(Ī±1 , Ī²1 ) + E(Ī±1 , Ī²2 ) + E(Ī±2 , Ī²1 ) ā’ E(Ī±2 , Ī²2 ) . (19.36)

Bellā™s theorem is then stated as follows.

Bob Alice

Correlation -(Ī±1,Ī²1)

Ī±1

Ī²1

Correlation -(Ī±2,Ī²1) Correlation -(Ī±1,Ī²2)

Ī²2 Ī±2

Anticorrelation ā’-(Ī±2,Ī²2)

Two choices of Two choices of

Bob's settings Alice's settings

Fig. 19.4 The four terms in the sum S deļ¬ned in eqn (19.35). The dependence of the

expectation values E(Ī», Ī±, Ī²) on the system state Ī» has been suppressed in this ļ¬gure.

Ā¼ Bellā™s theorem and its optical tests

Theorem 19.1 For all local realistic theories,

ā’2 E(Ī», Ī±1 , Ī²1 ) + E(Ī», Ī±1 , Ī²2 ) + E(Ī», Ī±2 , Ī²1 ) ā’ E(Ī», Ī±2 , Ī²2 ) +2 . (19.37)

Averaging over the distribution of states produces the Bell inequality:

ā’2 E(Ī±1 , Ī²1 ) + E(Ī±1 , Ī²2 ) + E(Ī±2 , Ī²1 ) ā’ E(Ī±2 , Ī²2 ) +2 . (19.38)

This result limits the total amount of correlation, as measured by S, that is allowed

for a local realistic theory. Experiments using coincidence-detection measurements

performed on two-photon decays have shown that this bound can be violated.

19.4.1 Merminā™s lemma

In order to prove Bellā™s theorem, we ļ¬rst prove the following lemma due to Mermin.

Lemma 19.2 If x1 , x2 , y1 , y2 are real numbers in the interval [ā’1, +1], then the sum

S ā” x1 y1 + x1 y2 + x2 y1 ā’ x2 y2 lies in the interval [ā’2, +2], i.e. |S| 2.

Proof Since S is a linear function of each of the four variables x1 , x2 , y1 , y2 , it must

take on its extreme values when the arguments of the function themselves are extrema,

i.e. when (x1 , x2 , y1 , y2 ) = (Ā±1, Ā±1, Ā±1, Ā±1), where the four Ā±s are independent. There

are four terms in S, and each term is bounded between ā’1 and +1; consequently,

|S| 4. However, we can also rewrite S as

S = (x1 + x2 ) (y1 + y2 ) ā’ 2x2 y2 . (19.39)

The extrema of x1 + x2 are 0 or Ā±2, and similarly for y1 + y2 . Therefore the extrema

of the product (x1 + x2 ) (y1 + y2 ) are 0 or Ā±4. The extrema for 2x2 y2 are Ā±2. Hence

the extrema for S are Ā±2 or Ā±6. The latter possibility is ruled out by the previously

determined limit |S| 4; therefore, the extrema of S are Ā±2, i.e. |S| 2.

19.4.2 Proof of Bellā™s theorem

Proof Bellā™s theorem now follows as a corollary of Merminā™s lemma. With the iden-

tiļ¬cations

x1 = E(Ī», Ī±1 ) , where |E(Ī», Ī±1 )| 1 ,

x2 = E(Ī», Ī±2 ) , where |E(Ī», Ī±2 )| 1 ,

(19.40)

y1 = E(Ī», Ī²1 ) , where |E(Ī», Ī²1 )| 1 ,

where |E(Ī», Ī²2 )|

y2 = E(Ī», Ī²2 ) , 1,

Lemma 19.2 implies

|E(Ī», Ī±1 )E(Ī», Ī²1 ) + E(Ī», Ī±1 )E(Ī», Ī²2 ) + E(Ī», Ī±2 )E(Ī», Ī²1 ) ā’ E(Ī», Ī±2 )E(Ī», Ī²2 )| 2 .

(19.41)

Using the strong-separability condition (19.32) for each term, i.e. E(Ī», Ī±, Ī²) =

E(Ī», Ī±)E(Ī», Ī²), we now arrive at

ā’2 E(Ī», Ī±1 , Ī²1 ) + E(Ī», Ī±1 , Ī²2 ) + E(Ī», Ī±2 , Ī²1 ) ā’ E(Ī», Ī±2 , Ī²2 ) +2 , (19.42)

and averaging over Ī» yields eqn (19.38).

Ā½

Quantum theory versus local realism

19.5 Quantum theory versus local realism

As a prelude to the experimental tests of local realism, we ļ¬rst support our previous

claim that quantum theory violates outcome independence and satisļ¬es parameter in-

dependence. In addition, we give an explicit example for which the quantum prediction

of the correlations violates Bellā™s theorem.

19.5.1 Quantum theory is not local

The issues of parameter independence and outcome independence will be studied by

considering an experiment simpler than the one presented in Section 19.3.1. In this

arrangement, shown in Fig. 19.5, pairs of polarization-entangled photons are produced

by down-conversion, and Alice and Bob are supplied with linear polarization ļ¬lters and

a single counter apiece. This reduces the outcomes for Alice to: Ayes (Aliceā™s detector

clicks) and Ano (there is no click). The corresponding outcome parameters are Ayes = 1

and Ano = 0. Bobā™s outcomes and outcome parameters are deļ¬ned in the same way.

We begin by assuming that the source produces the entangled state

|Ļ = F |hA , vB + G |vA , hB , (19.43)

where

|hA , vB ā” aā A h aā B v |0 , |vA , hB ā” aā A v aā B h |0 , (19.44)

k k k k

kA and kB are directed toward Alice and Bob respectively, and h and v label orthog-

onal polarizations: eh (horizontal ) and ev (vertical ). The parameters are the angles Ī±

and Ī² deļ¬ning the linear polarizations eĪ± and eĪ² transmitted by the polarizers.

Since akA h ā ekA h Ā· E (+) , etc., the annihilation operators in the (h, v)-basis are

related to the annihilation operators in the (Ī±, Ī± = Ļ/2 ā’ Ī±)-basis by

ak A h

ak A Ī± cos Ī± sin Ī±

= . (19.45)

ā’ sin Ī±

ak A Ī± cos Ī± ak A v

The corresponding relation for Bob follows by letting Ī± ā’ Ī² and kA ā’ kB .

A Parameter independence

For this experiment, the role of p(Am |Ī», Ī±, Ī²) in eqn (19.17) is played by p(Ayes |Ļ, Ī±, Ī²),

the probability that Aliceā™s detector clicks for the given state and parameter settings.

This is proportional to the detection rate for eĪ± -polarized photons, i.e.

* )

Fig. 19.5 Schematic of an apparatus to measure the polarization correlations of the entan-

gled photon pair Ī³A and Ī³B emitted back-to-back from the source S. The coincidence-counting

circuitry connecting the two Geiger counters is not shown.

Ā¾ Bellā™s theorem and its optical tests

p(Ayes |Ļ, Ī±, Ī²) ā G(1) (rA , tA ; rA , tA ) ā Ļ aā A Ī± akA Ī± Ļ . (19.46)

k

Ī±

A calculationā”see Exercise 19.2 ā”using eqns (19.43)ā“(19.45) yields

p(Ayes |Ļ, Ī±, Ī²) ā |F |2 cos2 Ī± + |G|2 sin2 Ī± . (19.47)

Thus the quantum result for the probability of a click of Aliceā™s detector is independent

of the setting Ī² of Bobā™s polarizer, although it can depend on her own polarizer setting

Ī±. In other words, quantum theoryā”at least in this exampleā”satisļ¬es parameter

independence. The symmetry of the experimental arrangement guarantees that the

probability, p(Byes |Ļ, Ī±, Ī²), seen by Bob is independent of Ī±.

This single example does not constitute a general proof that quantum theory sat-

isļ¬es parameter independence, but the features of the calculation provide guidance

for crafting such a proof. In general, the calculation of outcome probabilities for Alice

take the same form as in the example, i.e. the expectation value of an operatorā”which

may well depend on Aliceā™s parameter settingsā”is evaluated by using the state vector

determined by the source. Neither Aliceā™s operator nor the state vector depend on

Bobā™s parameter settings; therefore, parameter independence is guaranteed for quan-

tum theory. ā

For the special values F = ā’G = 1/ 2, the entangled state |Ļ becomes the

singlet-like Bell state

1

ĪØā’ = ā {|hA , vB ā’ |vA , hB } , (19.48)

2

ļ¬rst deļ¬ned in Section 13.3.5. In this case, p(Ayes |Ļ, Ī±, Ī²) is independent of Ī± as well

as Ī², so that Aliceā™s singles-counting measurements are the same as expected from an

unpolarized beam. This supports our previous claim that the individual measurements

can be as random as coin tosses.

B Outcome independence

Checking outcome independence requires the evaluation of the conditional probability

p(Ayes |Ī», Ī±, Ī², Brslt ) that Alice hears a click, given that Bob has observed the outcome

Brslt , where rslt = yes, no. In this case, we will simplify the calculation by setting

|Ļ = |ĪØā’ at the beginning.

With the usual assumption of 100% detector sensitivity, both possible outcomes

for Bobā”Byes (click) or Bno (no click)ā”constitute a measurement. According to von

Neumannā™s projection postulate, we must then replace the original state |ĪØā’ by the

reduced state |ĪØā’ rslt , to ļ¬nd

p(Ayes |Ī», Ī±, Ī², Brslt ) ā rslt ĪØā’ aā A Ī± akA Ī± ĪØā’ . (19.49)

k rslt

The reduced state for either of Bobā™s outcomes can be constructed by inverting

Bobā™s version of eqn (19.45) to express the creation operators in the (h, v)-basis in

terms of the creation operators in the Ī², Ī² -basis:

Āæ

Quantum theory versus local realism

aā B Ī²

aā B h ā’ sin Ī²

cos Ī² k

k = . (19.50)

aā Ī²

aā B v ā’ sin Ī² cos Ī²

k kB

Using this in the deļ¬nition (19.44) exhibits the original states as superpositions of

states containing Ī²-polarized photons and states containing Ī²-polarized photons.

For the outcome Byes ā”Bob heard a clickā”the projection postulate instructs us

to drop the states containing the Ī²-polarized photons, since they are blocked by the

polarizer. This produces the reduced state

1

ĪØā’ = ā {sin Ī² |hA , Ī²B ā’ cos Ī² |vA , Ī²B } , (19.51)

yes

2

where |Ī²B = aā B Ī² |0 . Substituting this into eqn (19.49) leadsā”by way of the calcu-

k

lation in Exercise 19.3ā”to the simple result

p(Ayes |Ī», Ī±, Ī², Byes ) ā sin2 (Ī± ā’ Ī²) . (19.52)

For the opposite outcome, Bno , the projection postulate tells us to drop the states

containing Ī²-polarized photon states instead, and the result is

p(Ayes |Ī», Ī±, Ī², Bno ) ā cos2 (Ī± ā’ Ī²) . (19.53)

The conclusion is that quantum theory violates outcome independence, since the

probability that Alice hears a click depends on the outcome of Bobā™s previous mea-

surement. The fact that Aliceā™s probabilities only depend on the diļ¬erence in polarizer

settings follows from the assumption that the source produces the special state |ĪØā’ ,

which is invariant under rotations around the common propagation axis.

The violation of outcome independence implies that the two sets of experimental

outcomes must be correlated. The probability that both detectors click is proportional

to the coincidence-count rate, whichā”as we learnt in Section 9.2.4ā”is determined by

the second-order Glauber correlation function; consequently,

p Ayes , Byes ĪØā’ , Ī±, Ī² ā GĪ±Ī² (r1 t1 ; r2 t2 )

(2)

ā ĪØ ā’ aā A Ī± aā B Ī² ak B Ī² ak A Ī± ĪØ ā’ . (19.54)

k k

The techniques used above give

p Ayes , Byes ĪØā’ , Ī±, Ī² = sin2 (Ī± ā’ Ī²)

11

= ā’ cos(2Ī± ā’ 2Ī²) , (19.55)

22

which describes an interference pattern, e.g. if Ī² is held ļ¬xed while Ī± is varied. Fur-

thermore, this pattern has 100% visibility, since perfect nulls occur for the values

Ī± = Ī², Ī² + Ļ, Ī² + 2Ļ, . . ., at which the planes of polarization of the two photons are

parallel. The surprise is that an interference pattern with 100% visibility occurs in the

(2) (1) (1)

second-order correlation function GĪ±Ī² while the ļ¬rst-order functions GĪ± and GĪ²

display zero visibility, i.e. no interference at all.

Bellā™s theorem and its optical tests

19.5.2 Quantum theory violates Bellā™s theorem

The results (19.52), (19.53), and (19.55) show that quantum theory violates outcome

independence and the strong-separability principle; consequently, quantum theory does

not satisfy the hypothesis of Bellā™s theorem. Nevertheless, it is still logically possible

that quantum theory could satisfy the conclusion of Bellā™s theorem, i.e. the inequality

(19.37). We will now dash this last, faint hope by exhibiting a speciļ¬c example in

which the quantum prediction violates the Bell inequality (19.38).

For the experiment depicted in Fig. 19.3, let us now calculate what quantum theory

predicts for S (Ī») when Ī» is represented by the Bell state |ĪØā’ . For general parameter

settings Ī± and Ī², the deļ¬nition (19.31) for Bellā™s joint expectation value can be written

as

E(Ī±, Ī²) = pee (Ī±, Ī²)Ae Be + peo (Ī±, Ī²)Ae Bo + poe (Ī±, Ī²)Ao Be + poo (Ī±, Ī²)Ao Bo , (19.56)

where we have omitted the Ī»-dependence of the expectation value, and adopted the

simpliļ¬ed notation

pmn (Ī±, Ī²) ā” p(Am , Bn |Ī», Ī±, Ī²) (19.57)

for the joint probabilities.

In Exercise 19.4, the calculation of the probabilities is done by using the techniques

leading to eqn (19.55), with the result

1

sin2 (Ī± ā’ Ī²) ,

pee (Ī±, Ī²) = poo (Ī±, Ī²) = (19.58)

2

1

cos2 (Ī± ā’ Ī²) .

peo (Ī±, Ī²) = poe (Ī±, Ī²) = (19.59)

2

After combining these expressions for the probabilities with the deļ¬nition (19.11) for

the outcome parameters, Bellā™s joint expectation value (19.56) becomes

E(Ī±, Ī²) = sin2 (Ī± ā’ Ī²) ā’ cos2 (Ī± ā’ Ī²) = ā’ cos (2Ī± ā’ 2Ī²) . (19.60)

Our objective is to choose values (Ī±1 , Ī²1 , Ī±2 , Ī²2 ) such that S violates the inequality

|S| 2. A set of values that accomplishes this,

Ī±1 = 0ā—¦ , Ī±2 = 45ā—¦ , Ī²1 = 22.5ā—¦ , Ī²2 = ā’22.5ā—¦ , (19.61)

is illustrated in Fig. 19.6.

Ī±

Ī²

Fig. 19.6 A choice of angular settings

Ī± =

Ī±1 , Ī±2 , Ī²1 , Ī²2 in the calcite-prism-pair experi-

ment (see Fig. 19.3) that maximizes the viola-

tion of Bellā™s bounds (19.42) by the quantum

Ī²

theory.

Quantum theory versus local realism

For these settings, the expectation values are given by

1

E(Ī±1 = 0, Ī²1 = 22.5ā—¦ ) = ā’ cos (45ā—¦ ) = ā’ ā ,

2

1

E(Ī±1 = 0, Ī²2 = ā’22.5ā—¦) = ā’ cos (ā’45ā—¦ ) = ā’ ā ,

2

(19.62)

1

ā—¦ ā—¦ ā—¦

E(Ī±2 = 45 , Ī²1 = 22.5 ) = ā’ cos (ā’45 ) = ā’ ā ,

2

1

E(Ī±2 = 45ā—¦ , Ī²2 = ā’22.5ā—¦) = ā’ cos (ā’135ā—¦) = + ā ,

2

ā ā

so that S = ā’2 2. This violation of the bound |S| 2 by a factor of 2 shows that

quantum theory violates the Bell inequality (19.38) by a comfortable margin.

Motivation for the deļ¬nition of the sum S

19.5.3

What motivates the choice of four terms and the signs (+, +, +, ā’) in eqn (19.35)?

The answers to this question now becomes clear in light of the above calculation. The

independent observers, Alice and Bob, need to make two independent choices in their

respective parameter settings Ī± and Ī², in order to observe changes in the correlations

between the polarizations of the photons Ī³A and Ī³B . This explains the four pairs of

parameter settings appearing in the deļ¬nition of S, and pictured in Fig. 19.4.

The motivation for the choice of signs (+, +, +, ā’) in S can be explained by refer-

ence to Fig. 19.6. Alice and Bob are free to choose the ļ¬rst three pairs of parameters

settings, (Ī±1 , Ī²1 ), (Ī±1 , Ī²2 ), and (Ī±2 , Ī²1 ), so that all three pairs have the same setting

diļ¬erence, 22.5ā—¦ , and negative correlations. In the quantum theory calculation of S for

ā

the Bell state |ĪØā’ , these choices yield the same negative correlation, ā’1/ 2, since

the expectation values only depend on the diļ¬erence in the polarizer settings.

By contrast, the fourth pair of settings, (Ī±2 , Ī²2 ), describes the two angles that are

the farthest away fromā each other in Fig. 19.6, and it yields a positive expectation

value E(Ī±2 , Ī²2 ) = +1/ 2. This arises from the fact that, for this particular pair of

angles (Ī±2 = 45ā—¦ , Ī²2 = ā’22.5ā—¦), the relative orientations of the planes of polarization

of the back-to-back photons Ī³A and Ī³B are almost orthogonal. The opposite sign of

this expectation value compared to the ļ¬rst three can be exploited by deliberately

choosing the opposite sign for this term in eqn (19.35). This stratagem ensures that

all four terms contribute with the same sign, and this gives the best chance of violating

the inequality.

It should be emphasized that the violation of this Bell inequality by quantum

theory is not restricted to this particular example. However, it turns out that this

special choice of angular settings deļ¬nes an extremum for S in the important case

of maximally entangled states. Consequently, these parameter settings maximize the

quantum theory violation of the Bell inequality (Su and WĀ“dkiewicz, 1991).

o

Bellā™s theorem and its optical tests

19.6 Comparisons with experiments

19.6.1 Visibility of second-order interference fringes

For comparison with experiments with two counters, such as the one sketched in Fig.

19.5, the visibility of the second-order interference fringes observed in coincidence

detection can be deļ¬nedā”by analogy to eqn (10.26)ā”as

(2) (2)

ā’ GĪ±Ī²

GĪ±Ī²

Vā” max min

, (19.63)

(2) (2)

GĪ±Ī² + GĪ±Ī²

max min

(2) (2)

where GĪ±Ī² max and GĪ±Ī² min are respectively the maximum and minimum, with respect

to the angles Ī± and Ī², of the second-order Glauber correlation function. Let us assume

that data analysis shows that an empirical ļ¬t to the second-order interference fringes

has the form

(2)

ńņš. 22 |