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Locality and nonlocality in correlated two-particle interferometry
Volume 150, number 3.4
PHYSICS LETTERS A
5 November ! 990
Locality and nonlocality in correlated two-particle interferometry IM.M. L a i n a n d C. D e w d n e y Department of Applied Physics, Portsmouth Polytechnic, King Henry I Street, Portsmouth, P01 2DZ, UK
iReceived 7 August 1990; accepted for publication 5 September 1990 Communicated by J.P. Vigier
By using the de Broglie-Bohm interpretation of quantum mechanics, we can explore the nature of nonlocality by modelling one iof the recent nonlocally-correlated two-photon intefferometry experiments. The system and its entangled wavefunction evolve ideterministically with unique initial positions of the particles producing unique and locally determined system trajectories in configuration space. The individual particle motions in real space are obtainable from the system trajectories. i
1. Introduction
with the single-particle wavefunctions A ( 1 ), B (2), C ( 2 ) and D( 1 ) respectively where 1 and 2 label the particles. The particles are emitted so that either (i) particle one follows path A and particle two follows path C or (ii) particle one follows path D and particle two follows path B. Particles on path A experience a variable phase shift 01, whilst those on path B receive a shift 02. Each particle encounters a barrier (H1 or H2) through which it m a y be transmitted or reflected with equal chance, regardless o f the settings o f the phase shif-
Nonlocal correlations in the observed positions o f two particles have been displayed recently in twophoton interferometry experiments. We shall be examining the specific example o f correlated two-particle !nterferometry described by H o m e , Shimony and Zeilinger [ 1 ] and illustrated in fig. 1, where we shall be using particles o f nonzero mass rather than photqns. The source S emits two distinguishable noninteracting particles, one and two, into the spatially distinct paths A, B, C and D. These are associated
MD Fig. 1~The arrangement of the two-particle interferometry experiment. The two particles are emitted by the source, S, into two of the four p~ths, A, B, C and D, with the entangled wavefunction given by ( 1 ). Any particles on paths A andB are given variable phase shifts, 01 and 02, respectively, before the four paths are recombined at the beam splitters, HI and H2. The two-particle coincident couni rate is recorded by use of the detectors Ut, U2, LI and L2. i
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ters. The particles are then detected by L~, U~, L 2 o r U2. The wavefunction on emission from the source is the entangled wavefunction ~(1, 2 ) = ~ 2 2 [A(1)C(2)+D(1)B(2)].
(1)
If attention is focused on only one of the particles at a time, then the behavior of that particle does not appear to be affected by the settings of the phase shifters. However, both theory and experiment show a nonlocal correlation between the choice of detectors made jointly by the two particles. The probabilities for joint detection by the detector pairs (U~, U2), (L~, LE), (U,, L2) and (Ll, U2) respectively, are for ideal detectors P(Wl, U2101, 02) =p(L1, L210x, 02)
(2a)
=~+~ c o s ( 0 2 - 0 ~ ) , P(Ul, L2101, 0 2 ) = p ( L l , U2101, 02)
(2b)
= ~ - ~ c o s ( 0 ~ - 0 , ). The particle detectors measure the correlation between the final position of the particles. They do not measure the precise position of the particles, but simply their position in relation to the beam splitters H~ and Hz. The usual interpretation of quantum mechanics cannot provide much insight into the nature of the nonlocal interactions found in quantum systems, since it regards physical processes at this level as fundementally indeterministic. It does not seek any further explanation of quantum behaviour which would extend beyond a probabilistic interpretation of the wavefunction and there is no provision for a physical description of the system which is deterministic and independent of measurement. The causal interpretation [2 ] of de Brogfie and Bohm, however, does provide such a description and consequently it is able to provide more information about how the nonlocal interaction works.
5 November 1990
has been transmitted or reflected at the barrier. So all features of the evolution of the wavefunction, relevant to this discussion, take place in the configuration space (Xl, x2) formed by the x axes of the two particles. The wavefunctions associated with each path in the apparatus are labelled a(x~ ), b(x2), c(x2) and d(Xl ), and are taken to be finite wavepackets as this permits some discussion of the time of occurrence of specific events. Each wavepacket undergoes reflection at a mirror Ma, Mb, Me or Md and packets a(x~) and b(x2) undergo phase shifts 01 and 0z respectively. Only the final part of the experiment need be modelled, when the two packets in configuration space, represented by the two terms in (1), have been shifted in phase and are converging on the beam splitters. Hence the model is that of two massive particles moving in one dimension and scattering from a narrow potential of transmission coefficient 0,5. It is interesting and useful to note when trying to gain familiarity with the behaviour of this model that the wave equation describing the evolution of two particles in one dimension, x, is identical in form to the equation describing a single particle in two dimensions, y and z, if y is replaced by xt and z is replaced by x2. We started to model behaviour after interaction with the phase shifters had occurred, at time to when the wave function is of the form
i/l(x,,x2)=a(xl)c(x2)eiOl-Fd(xl)b(x2)e i¢~
and the relative positions of the wavepackets are as shown in fig. 2. The initial individual particle wavepackets are chosen to be Gaussian, with equal magnitude average momenta. The initial two-particle wavefunction is chosen so that the wavepackets b (x 2 ) and c(x2) reach HE simultaneously and before a (x~) and d(x~ ) reach H1, also simultaneously. Its evolution is found by solving numerically the two-particle time-dependent Schrrdinger equation [ 3 ] RE
02
2ml OxZl
ihO ~u(xl, x2, t) = 2. The model
128
RE
02
2m20x2z
\
+ V(xt ) + V(x2)) ~u(x,, x2, t ) , The experiment is modelled in one dimension, x, normal to the plane of the beam splitters, as it is motion in this direction which shows whether a particle
(3)
(4)
where V(x~ ) and V(x2) are potentials chosen so as to represent the beam splitters H~ and H2 respec-
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5 November 1990
a(x~)eii~ ,,,~a
- ~
b(X2)eill
XI, ~-0.8
$
0.§ 4
"~
d(X~)
O.2
MB
Fig. 2. This shows the initial positions of the wavepacketsa(x~ ), b(x2), c(x2) and d(x~ ) which are componentsof the initial system wavefunction.They are Gaussian in form and are movingwith velocitiesof equal magnitude, in the directionsindicated in such a way that particle two willbe scatteredbeforeparticle one. tively. T h e wavefunction is calculated on a finite mesh in the two-dimensional configuration space, at appropriate time steps and for discrete time intervals. From this the joint probability density, P(xl, X 2 ) ~---I~¢(XI, X 2 ) 12is plotted in the configuration space (x~, x2) at various stages of the experiment in figs. 3a-3e !where in each figure the probability density has been scaled to fill the picture. Also displayed here are th~ marginal distributions of the joint probability density on the axes Xl and x2. Stage one of the model covers the interval from time to, just after the packets have received their phase ~hifts, until the scattering of particle two at H2, at time t~. Figs. 3a and 3b show the probability density, atl the beginning and at the end of stage one, respectively. During this stage the probability density does not depend on 01 o r ~2Now consider the nature of the configuration space wavefunction just after particle two has scattered as shown fin fig. 3b. Each of the packets c(x2) and b(x2) has split into a reflected part, c,(x2) and br(x2) respectively, and a transmitted part, ct (x2) and bt (x2) respeciively, with the reflected packets undergoing a phase Shift of n/2. The packets Cr(X2) and bt(x2) overlap exactly as do ct(x2) and b~(x2). Writing
c,(x2) =b,(x2)=rt(x2) and c,(x2) --br(x2) =tr(x2)
the equation for the wavefunction may be rearranged into the form ~(xl, x2 ) = rt (x2) [a (Xl)e i.1 + id(xl )e i~ ] + tr (x2) [ia (x~)e i~' +d(x~ )e i~2].
(5)
The entanglement of the two-particle wavefunction can be clearly seen in this expression since the phase ~2 is transferred to what we may call, with due caution, the wavefunction of particle one, i.e.: to that part of the wavefunction which is dependent on x~ and not on x2. Actually a wavefunction for one particle cannot strictly be defined independently of the other, but given the position of one particle there does exist a conditional wavefunction for the other particle. Stage two covers the interval from time tl, just after particle two has scattered from H2, until time t2 when particle one is scattered from HI. During the first part of stage two, the joint probability density is still independent of the settings of the phase shifters and it is only at time t2, when interference occurs between packets of different phase shift, that it becomes dependent on A¢. If we think of the single-particle twodimensional analogue, t2 corresponds to the point in time when two initially separated coherent beams in real space overlap to form an interference pattern, which may be shifted in position by varying the phase difference between the two coherent beams. Figs. 3c, 3d and 3e show the probability density at the end of stage two, for phase differences A~ = ~2 - ~1 129
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PHYSICS LETTERS A
5 November 1990
a(Xt)ct.~XtIeillt x,\~\ ~
0.2'
~
(x') e'li'
Y
7-0.5
to.4
x,\
c
ee~ Y
Jl
>"
d
U.U
i.o
Fig. 3. The time development o f the probability density in configuration space at (a) time to, (b) after particle two has scattered from the beam splitter H2, at time t nand (c) - (e) after both particles have scattered from the beam splitters, at time t2 for phase differences A¢ = ¢1 - ~ of 0, ~/2 and g, respectively.
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of 0, n/2 and x respectively. The probability density associated with each particle in real space, irrespective of the position of the other particle, is represented by the marginal distributions of the probability density in configuration space, on the axes x~ and x2. The figures show that these are always independent of the setting of the phase shifters, even though: the joint probability densities from which they are derived are not.
3. Configuration space system trajectories According to the de Broglie-Bohm interpretation of quantum mechanics [2 ], the motion of both particles is represented simultaneously in configuration space by the motion of the system particle of coordinate (x~, x2), along a system trajectory. If the twoparticle wavefunction in configuration space is expressed in the polar form
~/(xt,x2, t)=R(Xl,X2,t)exp[iS(xl,x2, t)],
(6)
where R and S are real and fi= 1, then the system trajectory followed up until the time t is defined by integrating the following equations for the velocity of each particle, from the initial system particle position up until time t,
1 vl S(x~, x2, t) vl -~ m--~
(7a)
v2 = ~ V2S(x~ , x2, t).
(7b)
Eqs. (Ta) and (7b) define the components of velocity for the system particle in configuration space. The two-particle wavefunction for the interferometry experiment is an entangled wavefunction ~(x~, x2) which means that it cannot be factorised into a product of two single-particle wavefunctions, each describing independent particle behaviour. In par, ticular S(x~, x2, t) is not factorisable so that the velocity field for each particle will be dependent on the variables of the other particle as well as its own. Figs. 4a, 4b and 4c show some possible system trajectorie s for an identical set of initial particle positions but for the three phase differences, A#=0, lr/2 and n respectively. Any differences between the trajectories followed from a given initial position, for
5 November 1990
the three values of A~, only occur after the path has reached H~, where wavepackets of different phase start to overlap. After this point the velocity of each particle is dependent on the local value of the wavefunction, now a function of the relative values of the phase shifts, ~1 and ~2. Initially, the two packets in configuration space are centred at (0.2, 0.6) and (0.8, 0.4). For the first packet, an initial value of 0.2 is chosen for x~ along with a range of initial values for x2 centred on 0.6. Similarly, for the second packet, the initial value of x~ is 0.8 and the initial values of x2 are centred on 0.4. Since the beam splitters are placed at x~=0.5 and x2=0.5, in real space, their position in configuration space divides it into four equally sized quadrants. Each quadrant represents, with a system particle within its boundaries at the end of the experiment, the firing of a different combination of two detectors. The top left quadrant represents UIL2, the top right UIU2, the bottom left L~L2 and the bottom right L~U2. The system trajectory and so the outcome of the joint detection is determined from the outset by the initial position of the system particle, the initial wavefunction and the experimental arrangement including the setting of the phase shifters. Since the evolution of the wavefunction in configuration space is locally determined, the system trajectory also evolves in a local manner. Distant perturbations of the wavefunction only affect the system trajectory when the resulting disturbance has propagated locally through the wavefield, according to the time-dependent Schr~Sdinger equation, to the system particle's position. So we see that nonlocal interactions need not take effect instantaneously, although they can do as in Aspect's experiments. In this example, a change in the value of the phase shift ¢~2 will only be introduced into the system when a packet passes through the phase shifter ~2. This change will then only alter the joint detection probability and so have a nonlocal effect on particle one at time t2, when particle one is scattered from H1.
4. Individual particle spacetime trajectories The individual particle trajectories i n real space are given by the projection of the system trajectory onto the individual particle coordinate axes and are 131
H2 I . 00
Ut U2
UI L 2 0.80
0.60
HI 0.40
0.20
LI Uz 0.00 0.00
0.20
0,40
0.60
0.60
I .00
X2
HSZ TRAJECTORIES OPl
H2 1.00
Ua L2
Ut U t
0.80
0.60
H1
x 0.40
0,20
L1 L2 o.oo b
,00
Lt U 2 o' .20
'
0.40
'
0,60
:
,80
.00
X2 HSZ TRAJECTORIES OPI
Fig. 4. Possible system trajectories with three identical sets of initial positions but for the three different phase differences (a) 0, (b) ~/
2 and (c) ~. Each quadrant of (a)-(c ) represents the joint f ~ n g of two of the detectors. The top left quadrant represents U~L2, the top right U~U2, the bottom left L~I~ and the bottom right L~U2.
Volume 150, number 3.4
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5 November 1990
H2 I .00
I
I
I
I
f
Ui Uz
U1L2
0.80
0.60
Hi 0.40
0.20
LIU2
LI L2 I
0.00 0 , O0
0.20
I 0.40
I
I 0.60
I o.0o
1.00
X2
C
HSZ TRAJECTORIES OPI Fig. 4. (Continued).
defined by (7a) and (7b). These equations show that, for an entangled wavefunction, the velocity of each Particle is dependent in some way on the other particle, for instance the velocity of particle one will in general depend on x2 and 02, as well as on x~ and 0~. So the use of configuration space to provide a complete physical description of the system is a necessity, not a matter of convenience as it is for classical mechanics. In the latter ease the individual particle trajectories can still be calculated from the individual Newtonian equations. The nonlocal nature of the system reflects the need to use configuration space and the impossibility of assigning individual equations of motion to the particles. Let us examine the nature of the two-particle interdependence for the interferometer. Consider the ease in which the wavefunction evolves in such a way that it can be written as
7t(x~,x2, t ) = ~ ( x ,
x2, t)+~/p(x~,x2, t) ,
(8)
where ~ and ¢tp are separated packets in configuration space. Writing ¥,~,p= R,~,pexp (iS,~,p)
(9 )
the velocities may be written as
1 vs=! [R~ vs~ +R~ VSp
ml,2
p
+ (Rp VRa -R,~VRp) sin (S,~ - S p )
+R~RB(VS~ + VSp)cos(S~-Sp) ],
(10)
where
p = R 2 +R$+ZR,~Rp c o s ( S # - S , ) ,
/~1 ----~1
v l St~
and
( 11 )
v2= 1 V 2 S ~
(12a)
v2= --~-1V2Sp.
(12b-)
or
v, = m~V, Sp and
m2
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Further, if ~/~ and ~p are simple products of functions of the individual particle coordinates, for example ¢/~ =R~ exp (iS,)
=R~(x~)R,(x2) exp{i[Sr(x,)+S,(x2)l},
(13)
then •1 = 1VIS)'(XI)m~
and
I12= 1 V 2 S r / ( X 2 ) ,
(14)
and the individual panicle trajectories, associated with each of the separated packets in configuration space, are approximately independent o f the coordinate of the other particle. As far as the calculation of the individual particle trajectories is concerned, while the system is under these conditions, the pure state (8) gives the same result as a statistical mixture of systems, some with the wavefunction ¥,~ and the others with the wavefunction yp. If the wavefunction retains this form (a sum of nonovcrlapping wavepackets), then the individual particle motions associated with each packet remain independent. During stage one of the model, the motion of the two particles will clearly be independent, since the wavefunction is a sum of the nonoverlapping packets in configuration space, a(x~)c(x2) and d(x, )b(x2). Each of figs. 4a-4c shows that for a given initial position of particle one, the motion of particle one during stage one will be identical for a range of initial values for particle two. When particle two is scattered from H2 (fig. 3b), the two original configuration space wavepackets still do not overlap and the wavefunction is still of the form ofeq. (8), even though each term in the equation now consists of two packets in configuration space. Figs. 4a-4c show that whether particle two is reflected or transmitted is determined purely by its initial position and not by any properties of particle one or by ¢2. The wavefunction which exists after particle two has been scattered may be rearranged into the form of eq. (5), by pairing together packets which will overlap at H~. This could not be done if the initial state of the system was in fact a statistical mixture of the two packets. Eq. (5) is also of the form of eq. (8), with the crucial difference that the phase 02 is now present in terms that we would associate with particle one. So as far as the calculation of the tra134
5 November 1990
jectories is concerned, the function in eq. (5) gives the same results as a statistical mixture of the two separated parts, but the statistical mixture which may be used at this stage is different to the one which may be used when particle two scatters from H2. In particular the wavefunction that would be assigned to particle one in each term of the mixture is dependent on ~2 as well as ~ . At the beginning of stage one the particles are associated with one of the packets in configuration space. The behaviour during stage one is then determined by the initial position of particle two as this determines whether particle two will be transmitted or reflected and also which member of the mixture ofeq. (5) the particles will be associated with. During stage two it is the phase difference A¢ which determines the motion as this determines whether particle one will subsequently be reflected or transmitted at H,. Let us now examine the effect of detection on the wavefunction of particle two. If the detector wavefunctions are U2o and L20 in their ground states and U2fand L2f in their excited states then we may write the wavefunction after detection as ~V(xl, x2) =L2fU20 rt (x2) [a(xl )e i*t + id(xl )e i°2]
+L20U2ftr(xE)[ia(xl)ei*~+d(xl)e i*~] .
(15)
Evidently particle two may already have been detected an arbitrary time before particle one reaches Ht, without affecting the results. Before particle two is detected, it is still possible to recombine rt (x2) and tr(x2) and reintroduce the interdependence. After the functioning of the detectors the chances of bringing the packets back so that they overlap in the configuration space, now spanned in addition by all the measuring devices coordinates, is effectively zero. In the causal interpretation there is no need to use the ill-defined concept of wavepacket coUapsc or of decoherence of the wavefunction. Even after detection the wavefunction is still a pure state, and as before the behaviour of this pure state may be approximated by a statisticalmixture. The system particle however isassociated with only one statein thismixture for any individual two-particleevent, and so the pure superposition statefor the two particlesappears to have "collapsed" into one state in the supcrposition. The trajectory of particle one at H~ depends on
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PHYSICS LETTERSA
whether particle two has entered U2 or L2 and on the phase shift applied to b(x2) at some arbitrary time in the past. In the case where the wavefunction takes the form of (13) and the pure state gives the same trajectories as a statistical mixture, it is possible to think of the individual particles as if they had one of two possible individual wavefunctions. What the wavefunction actually is, in a particular case, is given by the position of the other particle. In this case which wavefunction particle one has as it approaches H~ then depends on which detector particle two has entered. The wavefunction that we assign to particle one is derived from the wavefunction of the whole system and its evolution cannot be calculated separately. The entanglement is manifested through the presence of ¢2 in what we have called the wavefunction of particle one, when it is localised at H1. Altering 02 alters the wavefunction of the whole system, but this change is not apparent until particle one scatters from H1. In the usual approach, since particles do not have positions, it is not possible to introduce individual wavefunctions in this manner.
5 November 1990
encompasses all the relevant information about the two particles and their environment. An interaction with one of the particles may or may not come to affect the motion of the other particle, depending on how the configuration space wavefunction evolves. Some interactions may affect both particles simultaneously, as in the case of the measurement of one particle in the E P R - B o h m experiment, but this need not be the general character of the nature of the nonlocality. The analysis given here of correlated two-particle interferometry experiments illustrates Bohm and Riley's [ 5 ] discussion of nonlocal interaction in terms of active and inactive information. Information concerning the whole system is contained in the configuration space wavefunction, from which the individual particle trajectories a r e derived. This information may be active or inactive for an individual particle at a particular time, so that it may or may not play a role in determining the particle trajectory.
Acknowledgement 5. Conclusion
The individual panicle real spacetime trajectories are clearly determined in a nonlocal manner in the sense Ithat what happens to particle one will be correlated with what has already happened to particle two, regardless of their separation. The nonlocal correlations do not arise as a result of the operation of a crude mechanistic nonlocal interaction. That is, we should not think of the situation described here in terms of an action at a distance in the sense that Newtonian gravitation or the Coulomb interaction couldl be so described. The two-particle system has a wavefunction in configuration space which develops in a way which
The authors wish to thank G. Horton for helpful discussions and R. Mortlock for figures 1 and 2.
References
[ 1] M.A. Home, A. Shimony and A. Zeilinger, Phys. Rev. Lett. 62 (1989) 2209. [2] D. Bohm, Phys. Rev. 85 (1952) 166, 180; L. de Broglie, Nonlinear wave mechanics (Elsevier, Amsterdam, 1960). [3] I. Galbraith, Y.S. Ching and E. Abraham, Am. J. Phys. 52 (1984) 1. [4] C. Dewdney, P.R. Holland, A. Kyprianidis and J.P. Vigier, Nature 336 (1988) 536. [ 5 ] D. Bohm and B.J. Hiley, Phys. Rep. 144 (1987) 323.
135
PHYSICS LETTERS A
5 November ! 990
Locality and nonlocality in correlated two-particle interferometry IM.M. L a i n a n d C. D e w d n e y Department of Applied Physics, Portsmouth Polytechnic, King Henry I Street, Portsmouth, P01 2DZ, UK
iReceived 7 August 1990; accepted for publication 5 September 1990 Communicated by J.P. Vigier
By using the de Broglie-Bohm interpretation of quantum mechanics, we can explore the nature of nonlocality by modelling one iof the recent nonlocally-correlated two-photon intefferometry experiments. The system and its entangled wavefunction evolve ideterministically with unique initial positions of the particles producing unique and locally determined system trajectories in configuration space. The individual particle motions in real space are obtainable from the system trajectories. i
1. Introduction
with the single-particle wavefunctions A ( 1 ), B (2), C ( 2 ) and D( 1 ) respectively where 1 and 2 label the particles. The particles are emitted so that either (i) particle one follows path A and particle two follows path C or (ii) particle one follows path D and particle two follows path B. Particles on path A experience a variable phase shift 01, whilst those on path B receive a shift 02. Each particle encounters a barrier (H1 or H2) through which it m a y be transmitted or reflected with equal chance, regardless o f the settings o f the phase shif-
Nonlocal correlations in the observed positions o f two particles have been displayed recently in twophoton interferometry experiments. We shall be examining the specific example o f correlated two-particle !nterferometry described by H o m e , Shimony and Zeilinger [ 1 ] and illustrated in fig. 1, where we shall be using particles o f nonzero mass rather than photqns. The source S emits two distinguishable noninteracting particles, one and two, into the spatially distinct paths A, B, C and D. These are associated
MD Fig. 1~The arrangement of the two-particle interferometry experiment. The two particles are emitted by the source, S, into two of the four p~ths, A, B, C and D, with the entangled wavefunction given by ( 1 ). Any particles on paths A andB are given variable phase shifts, 01 and 02, respectively, before the four paths are recombined at the beam splitters, HI and H2. The two-particle coincident couni rate is recorded by use of the detectors Ut, U2, LI and L2. i
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ters. The particles are then detected by L~, U~, L 2 o r U2. The wavefunction on emission from the source is the entangled wavefunction ~(1, 2 ) = ~ 2 2 [A(1)C(2)+D(1)B(2)].
(1)
If attention is focused on only one of the particles at a time, then the behavior of that particle does not appear to be affected by the settings of the phase shifters. However, both theory and experiment show a nonlocal correlation between the choice of detectors made jointly by the two particles. The probabilities for joint detection by the detector pairs (U~, U2), (L~, LE), (U,, L2) and (Ll, U2) respectively, are for ideal detectors P(Wl, U2101, 02) =p(L1, L210x, 02)
(2a)
=~+~ c o s ( 0 2 - 0 ~ ) , P(Ul, L2101, 0 2 ) = p ( L l , U2101, 02)
(2b)
= ~ - ~ c o s ( 0 ~ - 0 , ). The particle detectors measure the correlation between the final position of the particles. They do not measure the precise position of the particles, but simply their position in relation to the beam splitters H~ and Hz. The usual interpretation of quantum mechanics cannot provide much insight into the nature of the nonlocal interactions found in quantum systems, since it regards physical processes at this level as fundementally indeterministic. It does not seek any further explanation of quantum behaviour which would extend beyond a probabilistic interpretation of the wavefunction and there is no provision for a physical description of the system which is deterministic and independent of measurement. The causal interpretation [2 ] of de Brogfie and Bohm, however, does provide such a description and consequently it is able to provide more information about how the nonlocal interaction works.
5 November 1990
has been transmitted or reflected at the barrier. So all features of the evolution of the wavefunction, relevant to this discussion, take place in the configuration space (Xl, x2) formed by the x axes of the two particles. The wavefunctions associated with each path in the apparatus are labelled a(x~ ), b(x2), c(x2) and d(Xl ), and are taken to be finite wavepackets as this permits some discussion of the time of occurrence of specific events. Each wavepacket undergoes reflection at a mirror Ma, Mb, Me or Md and packets a(x~) and b(x2) undergo phase shifts 01 and 0z respectively. Only the final part of the experiment need be modelled, when the two packets in configuration space, represented by the two terms in (1), have been shifted in phase and are converging on the beam splitters. Hence the model is that of two massive particles moving in one dimension and scattering from a narrow potential of transmission coefficient 0,5. It is interesting and useful to note when trying to gain familiarity with the behaviour of this model that the wave equation describing the evolution of two particles in one dimension, x, is identical in form to the equation describing a single particle in two dimensions, y and z, if y is replaced by xt and z is replaced by x2. We started to model behaviour after interaction with the phase shifters had occurred, at time to when the wave function is of the form
i/l(x,,x2)=a(xl)c(x2)eiOl-Fd(xl)b(x2)e i¢~
and the relative positions of the wavepackets are as shown in fig. 2. The initial individual particle wavepackets are chosen to be Gaussian, with equal magnitude average momenta. The initial two-particle wavefunction is chosen so that the wavepackets b (x 2 ) and c(x2) reach HE simultaneously and before a (x~) and d(x~ ) reach H1, also simultaneously. Its evolution is found by solving numerically the two-particle time-dependent Schrrdinger equation [ 3 ] RE
02
2ml OxZl
ihO ~u(xl, x2, t) = 2. The model
128
RE
02
2m20x2z
\
+ V(xt ) + V(x2)) ~u(x,, x2, t ) , The experiment is modelled in one dimension, x, normal to the plane of the beam splitters, as it is motion in this direction which shows whether a particle
(3)
(4)
where V(x~ ) and V(x2) are potentials chosen so as to represent the beam splitters H~ and H2 respec-
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5 November 1990
a(x~)eii~ ,,,~a
- ~
b(X2)eill
XI, ~-0.8
$
0.§ 4
"~
d(X~)
O.2
MB
Fig. 2. This shows the initial positions of the wavepacketsa(x~ ), b(x2), c(x2) and d(x~ ) which are componentsof the initial system wavefunction.They are Gaussian in form and are movingwith velocitiesof equal magnitude, in the directionsindicated in such a way that particle two willbe scatteredbeforeparticle one. tively. T h e wavefunction is calculated on a finite mesh in the two-dimensional configuration space, at appropriate time steps and for discrete time intervals. From this the joint probability density, P(xl, X 2 ) ~---I~¢(XI, X 2 ) 12is plotted in the configuration space (x~, x2) at various stages of the experiment in figs. 3a-3e !where in each figure the probability density has been scaled to fill the picture. Also displayed here are th~ marginal distributions of the joint probability density on the axes Xl and x2. Stage one of the model covers the interval from time to, just after the packets have received their phase ~hifts, until the scattering of particle two at H2, at time t~. Figs. 3a and 3b show the probability density, atl the beginning and at the end of stage one, respectively. During this stage the probability density does not depend on 01 o r ~2Now consider the nature of the configuration space wavefunction just after particle two has scattered as shown fin fig. 3b. Each of the packets c(x2) and b(x2) has split into a reflected part, c,(x2) and br(x2) respectively, and a transmitted part, ct (x2) and bt (x2) respeciively, with the reflected packets undergoing a phase Shift of n/2. The packets Cr(X2) and bt(x2) overlap exactly as do ct(x2) and b~(x2). Writing
c,(x2) =b,(x2)=rt(x2) and c,(x2) --br(x2) =tr(x2)
the equation for the wavefunction may be rearranged into the form ~(xl, x2 ) = rt (x2) [a (Xl)e i.1 + id(xl )e i~ ] + tr (x2) [ia (x~)e i~' +d(x~ )e i~2].
(5)
The entanglement of the two-particle wavefunction can be clearly seen in this expression since the phase ~2 is transferred to what we may call, with due caution, the wavefunction of particle one, i.e.: to that part of the wavefunction which is dependent on x~ and not on x2. Actually a wavefunction for one particle cannot strictly be defined independently of the other, but given the position of one particle there does exist a conditional wavefunction for the other particle. Stage two covers the interval from time tl, just after particle two has scattered from H2, until time t2 when particle one is scattered from HI. During the first part of stage two, the joint probability density is still independent of the settings of the phase shifters and it is only at time t2, when interference occurs between packets of different phase shift, that it becomes dependent on A¢. If we think of the single-particle twodimensional analogue, t2 corresponds to the point in time when two initially separated coherent beams in real space overlap to form an interference pattern, which may be shifted in position by varying the phase difference between the two coherent beams. Figs. 3c, 3d and 3e show the probability density at the end of stage two, for phase differences A~ = ~2 - ~1 129
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a(Xt)ct.~XtIeillt x,\~\ ~
0.2'
~
(x') e'li'
Y
7-0.5
to.4
x,\
c
ee~ Y
Jl
>"
d
U.U
i.o
Fig. 3. The time development o f the probability density in configuration space at (a) time to, (b) after particle two has scattered from the beam splitter H2, at time t nand (c) - (e) after both particles have scattered from the beam splitters, at time t2 for phase differences A¢ = ¢1 - ~ of 0, ~/2 and g, respectively.
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of 0, n/2 and x respectively. The probability density associated with each particle in real space, irrespective of the position of the other particle, is represented by the marginal distributions of the probability density in configuration space, on the axes x~ and x2. The figures show that these are always independent of the setting of the phase shifters, even though: the joint probability densities from which they are derived are not.
3. Configuration space system trajectories According to the de Broglie-Bohm interpretation of quantum mechanics [2 ], the motion of both particles is represented simultaneously in configuration space by the motion of the system particle of coordinate (x~, x2), along a system trajectory. If the twoparticle wavefunction in configuration space is expressed in the polar form
~/(xt,x2, t)=R(Xl,X2,t)exp[iS(xl,x2, t)],
(6)
where R and S are real and fi= 1, then the system trajectory followed up until the time t is defined by integrating the following equations for the velocity of each particle, from the initial system particle position up until time t,
1 vl S(x~, x2, t) vl -~ m--~
(7a)
v2 = ~ V2S(x~ , x2, t).
(7b)
Eqs. (Ta) and (7b) define the components of velocity for the system particle in configuration space. The two-particle wavefunction for the interferometry experiment is an entangled wavefunction ~(x~, x2) which means that it cannot be factorised into a product of two single-particle wavefunctions, each describing independent particle behaviour. In par, ticular S(x~, x2, t) is not factorisable so that the velocity field for each particle will be dependent on the variables of the other particle as well as its own. Figs. 4a, 4b and 4c show some possible system trajectorie s for an identical set of initial particle positions but for the three phase differences, A#=0, lr/2 and n respectively. Any differences between the trajectories followed from a given initial position, for
5 November 1990
the three values of A~, only occur after the path has reached H~, where wavepackets of different phase start to overlap. After this point the velocity of each particle is dependent on the local value of the wavefunction, now a function of the relative values of the phase shifts, ~1 and ~2. Initially, the two packets in configuration space are centred at (0.2, 0.6) and (0.8, 0.4). For the first packet, an initial value of 0.2 is chosen for x~ along with a range of initial values for x2 centred on 0.6. Similarly, for the second packet, the initial value of x~ is 0.8 and the initial values of x2 are centred on 0.4. Since the beam splitters are placed at x~=0.5 and x2=0.5, in real space, their position in configuration space divides it into four equally sized quadrants. Each quadrant represents, with a system particle within its boundaries at the end of the experiment, the firing of a different combination of two detectors. The top left quadrant represents UIL2, the top right UIU2, the bottom left L~L2 and the bottom right L~U2. The system trajectory and so the outcome of the joint detection is determined from the outset by the initial position of the system particle, the initial wavefunction and the experimental arrangement including the setting of the phase shifters. Since the evolution of the wavefunction in configuration space is locally determined, the system trajectory also evolves in a local manner. Distant perturbations of the wavefunction only affect the system trajectory when the resulting disturbance has propagated locally through the wavefield, according to the time-dependent Schr~Sdinger equation, to the system particle's position. So we see that nonlocal interactions need not take effect instantaneously, although they can do as in Aspect's experiments. In this example, a change in the value of the phase shift ¢~2 will only be introduced into the system when a packet passes through the phase shifter ~2. This change will then only alter the joint detection probability and so have a nonlocal effect on particle one at time t2, when particle one is scattered from H1.
4. Individual particle spacetime trajectories The individual particle trajectories i n real space are given by the projection of the system trajectory onto the individual particle coordinate axes and are 131
H2 I . 00
Ut U2
UI L 2 0.80
0.60
HI 0.40
0.20
LI Uz 0.00 0.00
0.20
0,40
0.60
0.60
I .00
X2
HSZ TRAJECTORIES OPl
H2 1.00
Ua L2
Ut U t
0.80
0.60
H1
x 0.40
0,20
L1 L2 o.oo b
,00
Lt U 2 o' .20
'
0.40
'
0,60
:
,80
.00
X2 HSZ TRAJECTORIES OPI
Fig. 4. Possible system trajectories with three identical sets of initial positions but for the three different phase differences (a) 0, (b) ~/
2 and (c) ~. Each quadrant of (a)-(c ) represents the joint f ~ n g of two of the detectors. The top left quadrant represents U~L2, the top right U~U2, the bottom left L~I~ and the bottom right L~U2.
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H2 I .00
I
I
I
I
f
Ui Uz
U1L2
0.80
0.60
Hi 0.40
0.20
LIU2
LI L2 I
0.00 0 , O0
0.20
I 0.40
I
I 0.60
I o.0o
1.00
X2
C
HSZ TRAJECTORIES OPI Fig. 4. (Continued).
defined by (7a) and (7b). These equations show that, for an entangled wavefunction, the velocity of each Particle is dependent in some way on the other particle, for instance the velocity of particle one will in general depend on x2 and 02, as well as on x~ and 0~. So the use of configuration space to provide a complete physical description of the system is a necessity, not a matter of convenience as it is for classical mechanics. In the latter ease the individual particle trajectories can still be calculated from the individual Newtonian equations. The nonlocal nature of the system reflects the need to use configuration space and the impossibility of assigning individual equations of motion to the particles. Let us examine the nature of the two-particle interdependence for the interferometer. Consider the ease in which the wavefunction evolves in such a way that it can be written as
7t(x~,x2, t ) = ~ ( x ,
x2, t)+~/p(x~,x2, t) ,
(8)
where ~ and ¢tp are separated packets in configuration space. Writing ¥,~,p= R,~,pexp (iS,~,p)
(9 )
the velocities may be written as
1 vs=! [R~ vs~ +R~ VSp
ml,2
p
+ (Rp VRa -R,~VRp) sin (S,~ - S p )
+R~RB(VS~ + VSp)cos(S~-Sp) ],
(10)
where
p = R 2 +R$+ZR,~Rp c o s ( S # - S , ) ,
/~1 ----~1
v l St~
and
( 11 )
v2= 1 V 2 S ~
(12a)
v2= --~-1V2Sp.
(12b-)
or
v, = m~V, Sp and
m2
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Further, if ~/~ and ~p are simple products of functions of the individual particle coordinates, for example ¢/~ =R~ exp (iS,)
=R~(x~)R,(x2) exp{i[Sr(x,)+S,(x2)l},
(13)
then •1 = 1VIS)'(XI)m~
and
I12= 1 V 2 S r / ( X 2 ) ,
(14)
and the individual panicle trajectories, associated with each of the separated packets in configuration space, are approximately independent o f the coordinate of the other particle. As far as the calculation of the individual particle trajectories is concerned, while the system is under these conditions, the pure state (8) gives the same result as a statistical mixture of systems, some with the wavefunction ¥,~ and the others with the wavefunction yp. If the wavefunction retains this form (a sum of nonovcrlapping wavepackets), then the individual particle motions associated with each packet remain independent. During stage one of the model, the motion of the two particles will clearly be independent, since the wavefunction is a sum of the nonoverlapping packets in configuration space, a(x~)c(x2) and d(x, )b(x2). Each of figs. 4a-4c shows that for a given initial position of particle one, the motion of particle one during stage one will be identical for a range of initial values for particle two. When particle two is scattered from H2 (fig. 3b), the two original configuration space wavepackets still do not overlap and the wavefunction is still of the form ofeq. (8), even though each term in the equation now consists of two packets in configuration space. Figs. 4a-4c show that whether particle two is reflected or transmitted is determined purely by its initial position and not by any properties of particle one or by ¢2. The wavefunction which exists after particle two has been scattered may be rearranged into the form of eq. (5), by pairing together packets which will overlap at H~. This could not be done if the initial state of the system was in fact a statistical mixture of the two packets. Eq. (5) is also of the form of eq. (8), with the crucial difference that the phase 02 is now present in terms that we would associate with particle one. So as far as the calculation of the tra134
5 November 1990
jectories is concerned, the function in eq. (5) gives the same results as a statistical mixture of the two separated parts, but the statistical mixture which may be used at this stage is different to the one which may be used when particle two scatters from H2. In particular the wavefunction that would be assigned to particle one in each term of the mixture is dependent on ~2 as well as ~ . At the beginning of stage one the particles are associated with one of the packets in configuration space. The behaviour during stage one is then determined by the initial position of particle two as this determines whether particle two will be transmitted or reflected and also which member of the mixture ofeq. (5) the particles will be associated with. During stage two it is the phase difference A¢ which determines the motion as this determines whether particle one will subsequently be reflected or transmitted at H,. Let us now examine the effect of detection on the wavefunction of particle two. If the detector wavefunctions are U2o and L20 in their ground states and U2fand L2f in their excited states then we may write the wavefunction after detection as ~V(xl, x2) =L2fU20 rt (x2) [a(xl )e i*t + id(xl )e i°2]
+L20U2ftr(xE)[ia(xl)ei*~+d(xl)e i*~] .
(15)
Evidently particle two may already have been detected an arbitrary time before particle one reaches Ht, without affecting the results. Before particle two is detected, it is still possible to recombine rt (x2) and tr(x2) and reintroduce the interdependence. After the functioning of the detectors the chances of bringing the packets back so that they overlap in the configuration space, now spanned in addition by all the measuring devices coordinates, is effectively zero. In the causal interpretation there is no need to use the ill-defined concept of wavepacket coUapsc or of decoherence of the wavefunction. Even after detection the wavefunction is still a pure state, and as before the behaviour of this pure state may be approximated by a statisticalmixture. The system particle however isassociated with only one statein thismixture for any individual two-particleevent, and so the pure superposition statefor the two particlesappears to have "collapsed" into one state in the supcrposition. The trajectory of particle one at H~ depends on
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whether particle two has entered U2 or L2 and on the phase shift applied to b(x2) at some arbitrary time in the past. In the case where the wavefunction takes the form of (13) and the pure state gives the same trajectories as a statistical mixture, it is possible to think of the individual particles as if they had one of two possible individual wavefunctions. What the wavefunction actually is, in a particular case, is given by the position of the other particle. In this case which wavefunction particle one has as it approaches H~ then depends on which detector particle two has entered. The wavefunction that we assign to particle one is derived from the wavefunction of the whole system and its evolution cannot be calculated separately. The entanglement is manifested through the presence of ¢2 in what we have called the wavefunction of particle one, when it is localised at H1. Altering 02 alters the wavefunction of the whole system, but this change is not apparent until particle one scatters from H1. In the usual approach, since particles do not have positions, it is not possible to introduce individual wavefunctions in this manner.
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encompasses all the relevant information about the two particles and their environment. An interaction with one of the particles may or may not come to affect the motion of the other particle, depending on how the configuration space wavefunction evolves. Some interactions may affect both particles simultaneously, as in the case of the measurement of one particle in the E P R - B o h m experiment, but this need not be the general character of the nature of the nonlocality. The analysis given here of correlated two-particle interferometry experiments illustrates Bohm and Riley's [ 5 ] discussion of nonlocal interaction in terms of active and inactive information. Information concerning the whole system is contained in the configuration space wavefunction, from which the individual particle trajectories a r e derived. This information may be active or inactive for an individual particle at a particular time, so that it may or may not play a role in determining the particle trajectory.
Acknowledgement 5. Conclusion
The individual panicle real spacetime trajectories are clearly determined in a nonlocal manner in the sense Ithat what happens to particle one will be correlated with what has already happened to particle two, regardless of their separation. The nonlocal correlations do not arise as a result of the operation of a crude mechanistic nonlocal interaction. That is, we should not think of the situation described here in terms of an action at a distance in the sense that Newtonian gravitation or the Coulomb interaction couldl be so described. The two-particle system has a wavefunction in configuration space which develops in a way which
The authors wish to thank G. Horton for helpful discussions and R. Mortlock for figures 1 and 2.
References
[ 1] M.A. Home, A. Shimony and A. Zeilinger, Phys. Rev. Lett. 62 (1989) 2209. [2] D. Bohm, Phys. Rev. 85 (1952) 166, 180; L. de Broglie, Nonlinear wave mechanics (Elsevier, Amsterdam, 1960). [3] I. Galbraith, Y.S. Ching and E. Abraham, Am. J. Phys. 52 (1984) 1. [4] C. Dewdney, P.R. Holland, A. Kyprianidis and J.P. Vigier, Nature 336 (1988) 536. [ 5 ] D. Bohm and B.J. Hiley, Phys. Rep. 144 (1987) 323.
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