LETT]BRE AL NUOVO CIMENTO VOL. 39, N. I1 17 M a r z o 1984

Possible Experimental Test of the Wave Packet Collapse.

A. GARUCCIO

Istituto di _Fisica dell'Universith - Bari, Italia Istituto -Wazionale di Eisica .Nucleate - Sezione di Bari, Italia

A. KYPRIA~mXS (*), D. S~RDEmS (*) and J. P. V1GI~R

Institut H. Poincard, f~aboratoire de Physique Theorique 11, rue P. et M. Curie, 75231 Paris Cedez 05, France

(ricevuto il 23 Novembre 1983)

PACS. 0 3 . 6 5 - Quantum theory; quantum mechanics.

Summary. - A quantum analysis is presented of combined first-order/second-order optical-interference experiments with or without the wave packet collapse concept. I t is shown that the Bohr-tIcisenberg model yields testable predictions which differ from those of the Einstein-de Broglie theory of light.

One of the s tar t ing points of the Bohr-Einstein controversy is evidently the wave packet collapse concept introduced by Bolm as a basis of quantum measurement theory. As one knows Einstein never accepted it, since he believed

1) tha t this process, assumed to be instantaneous in all frames, is evidently in contradiction with relat ivi ty theory,

2) tha t i t contradicts his realistic conception of the wave-particle dualism; for him electromagnetic waves for example are considered to present simultaneously wave and particle (photons) aspects, while Bolm holds that particles are waves or particles, never the two simultaneously; for him the photons materialize in observation when their probabili ty waves collapse instantaneously.

Two recent developments, however, have confirmed Einstein and de Broglie's opinions. The first is the discovery by CI~I (1) tha t one can construct a quantum measurement

theory without wave packet collapse ... so that one of the pillars of the Copenhagen interpretat ion is now endangered. The second is a set of recent proposals by GARUCCIO,

(*) On leave from the University of Crete, Physics Department, Herakllan, Crete, Greece. (1) M. CINX: _hTuovo Cimento B, 73, 27 (1983).

225

226 A. GARUCCIO, A. KYPRIANIDIS ETC.

I)OI, FEIr (2) SELLEI{I (3) RAPIS&I~DA (~) ANDRADE a n d VmI~R (5) to t e s t t h e r ea l inde- p e n d e n t e x i s t e n c e of de Brog l i e ' s w a v e . . , in confl ict w i t h t he wave p a c k e t col- l apse concep t .

T h e s e p roposa l s h a v e e n g i n e e r e d a n u m b e r of c o m m e n t s (6) a n d c r i t i c i s m s (ns). The a im of t h e p r e s e n t l e t t e r is to p r e s e n t a comple t e q u a n t u m - m e c h a n i c a l t r e a t m e n t of t he p roposed se t -ups a n d to show t h a t t h e B o h r - I I e i s e n b e r g i n t e r p r e t a t i o n i n d e e d yie lds t e s t a b l e p r e d i c t i o n s wh ich differ f rom those of t he E i n s t e i n - d e Brogl ic t h e o r y of l igh t in c o m b i n e d f i r s t - o r de r / s econd - o r de r o p t i c a l - i n t e r c r c n c e e x p e r i m e n t s ... w h i c h can t h u s be c o n s i d e r e d as t e s t s for or a g a i n s t t h c i n t r o d u c t i o n of t h e w a v e p a c k e t col lapse in t he i n t e r p r e t a t i o n of q u a n t u m theo ry .

"We s t a r t w i t h t he Michelson device r e p r e s e n t e d in fig. 1. The q u a n t u m t h e o r y of such a dev ice h a s been g iven by LouDo.~ (7) in some de ta i l . W e sha l l r ed i scuss i t he r e in h i s f o r m a l i s m , s ince as we shal l show, he i nco r r ec t l y a s sumes t h e ]VIaxwell-Boltzmann s t a t i s t i c s for i n c o h e r e n t beams .

Fig. 1.

1 ~ detecton 1+2 I

We r e p r e s e n t b y at , a t h e c r e a t i o n , t m n i h i l a t i o n o p e r a t o r s of t h e i n p u t l i gh t w h i c h is sp l i t b y t he s e n d - t r a n s p a r e n t m i r r o r in to e q u a l - a m p l i t u d e c o m p o n e n t s ax, a2 (see iig. 1). T h e a o u t p u t b e a m a t t h e d e t e c t o r d is g i v e n b y t h e exp re s s ion <dfd>, where

a n d

a = (as-{- a2)b(na + n , - - ~ ' ) ,

d = [a 1 exp [ iKz l ] -- a~ exp [iKz2]] b(n , + n ~ - - N )

n~ = <a~a,>, i = 1, 2 ,

(:) K. R. I~OFPER, A. GARUCCIO add J. ] ). VIGIER: P]tys. Let t . .4 , 86, 326 (1982). (') F. SF.U.ERI: Found. Phys., 12, 1087 (1982). (~) V. A. t{AP[SARDA, A. GARUCCIO and J. P. VIOIER: Phy8, Lett. A, 90, 17 (1982). (s) J. ANDRADE E SILV~k, F. SELLERI and J. P. VIOIEB: Left. Nvovo Cimento, 36, 503 (1983). (6) ~V. i~. DE l~[UYNCK: Epistem. Lett., 33, 7, 13 (1982). (7) R. LOWDON: Quantum lheory ol combined first order]second order optical interference experintents, preprint to appear In Opt. Commun. (a) O. COSTA DE BEAUICE(~ARD: NuaVO Cimer~to B, 42, 41 (1977); $1, 267 (1979).

P O S S I B L E F.XPERIM:E~NTAL TEST OF THE WA.V:E P A C K E T COLLAPSE 227

which yields the following resul t :

( d t d ) =

----- ([a~ exp [--i Kz,] + a~ exp [- - iKz2]] [a~ exp [ igz~] + a~ cxp [ igz2]l~(n~ § n~-- N) ) .

The res t r ic t ion imposed by the &function, i.e. the definite number N of photons, can be expl ic i t ly evalua ted as a factor in the composit ion of the s ta te vector I~). This resul ts in a weighing of the s tates , out of which ]~) is composed, according to Bose- Einstein s tat is t ics appropr ia te for correlated par t ic les submit ted to random stochastic nonlocal actions at a distance (g). As we have shown (~0) this leads to different predic- t ions for the interference pa t te rn . The Loudon formula for the in tens i ty

I , - ~ ( 1 § e o s ~ ) ,

which exhibi ts maximum fringe v is ib i l i ty has to be replaced by

I - - :V(1 + ~(:V) cos ~ ) , ~(5r) < ~.

Only for 37 = 1 the two formulae coincide, 2(1) being 1. For N = 2 one a l ready obtains a reduced fringe vis ib i l i ty corresponding to the ~, �89 ~ probabi l i ty for the oc- currence of the s ta tes 10, 2), I1, 1) and [2, 0). For _N-~ ~ , 2(h r) approaches M/4, and the in tens i ty pa t t e rn exhibi ts a reduced fringe visibil i ty.

We now pass to the set-up of fig. 2. We introduce on arm 1, amplifier A, which is assumed to produce (when s t imula ted by one photon only) an outcoming photon with the same phase, general ly accompanied by a spontaneous photon with a random

F i g . 2.

/ /

/ ./ /

(*) A. KYPRIANIDIS, I). SARDELIS a nd J . P. VxGI~a: Causal nonlocal character el quantum sfallsttcsj s u b m i t t e d to Phys. Rev. Left. (z,) F. DE MARTINIs A, KVPRtAr~rDIs, D. SARDE~S a n d J . P. VI(HER: Quantum-mechartical causal actions-at-a-distaace correlations in optical-beam splitting device.~ and inter/erenv~ experiments, sub- m i t t e d to Nuovo Cimento.

228 A. GA,RUCCIO, A. K~PRI&NIDIS "ETC.

phase (n). In the coherent state the photon operators are transformed according to

where

exp [ - - A ] a x exp [A] = al -4- ~1,

exp [--A]a~ exp [A] = a~ + ~ ,

A = ~1 a~ - - ~* al

and ~ is determined by the geometry of the amplifier and its effective current flow. At this stage one must introduce the physical properties of the amplifier. There

are evidently two different models tha t can be applied to interpret the results. The first model is the usual Bohr-type Copenhagen interpretat ion, which implies wave- packet collapse when the photon is located somewhere, i .e. when i t triggers the am- plifier. This, however, implies the use of the amplifier explicitly as a measuring device, by means of an observer. If the amplifier is not under observation, then according to Bo~ra one cannot know if it has been triggered or not, and consequently there is a lack of information about the state of the system. Hence, the state of the system after the amplifier cannot be a pure state, but only a mixture of states with probabilities of occurrence~ The predictabil i ty of the experimental results restricts itself to statistical predictions. On the other hand, the second model, the Einstein-de Broglie theory of light, has no wave packet collapse, which is exactly the assumption made by CINI (1) in his proposed description of quantum measurement theory. The Einstein-de Broglie model considers photons to be waves and particles and consequently the amplifier does change the state of the system in branch 1 of fig. 2, but does not affect our knowledge of the system, because i t does not create mutual ly exclusive states as the Bohr-CIQM does. The system is now in a modified state with respect to the case examined previously, but still in a pure state.

Apparently, the state of the wave field depends on the physical model used. The wave packet collapse concept of Bohr introduces the following quantum mixture as the appropriate way to account for the behaviour of the system:

1 state [I> = 2V7~{[I, 0 > -+- 10, I>},

1 state [2>

and with a density matr ix

{19,, o> + ~LI,~>},

= p,[l><i{ + p~12><2{,

with probabil i ty Pl ,

with probabil i ty P2,

pl -t- p2 = 1 ,

where ]..7.) denotes spontaneous emission. In the Einstein-de Broglie model, the amplifier has a nonzero probability" s for

any given photon to pass through without being triggered and a probabili ty 1 - s to double the photon (i.e. st imulated emission). This process is always accompanied by spontaneous emission. The state of the system can be wri t ten now as a pure state ]~>

(11) 1:~. J . GLAUBER: Phys . R ~ . , 131, 2766 (1963).

POSSIBLE ~EXPERIMENTA.L TJ~ST OF THE WAIVE PAICKET COLLAPSE 2 2 9

and takes the form

1 <v> - - -

N {v~ll , o> + V i - ~ 1 2 , o> + ,,11, o> + Io, ~>}

with a density matr ix of a pure state ~ = [~><~p[. We can proceed now to calculate the interference in tens i ty by evaluating the fol-

lowing expression :

I ~ T r [~{(a~ + a*) exp [-- iKz~] + a~ exp [-- iKz~l} ( (a~ + ~) exp [ iKz a] + a~ exp [iKza]}]

with either of the proposed models, respectively.

A) In the Bohr model a straightforward calculation yields the following result:

2p~ I ~ i . [ ~ + - - + v~{1 + cos ~}.

1 + I~1 ~

This result is a statistical prediction of the interference pat tern. I t entails the predictions

a) tha t if the amplifier is triggered (p~ = 0, p~ = 1) no interference is observed;

b) that in the absence of the amplifier (p~ ~ 0, ~ = 0), we recover the well- known quantum-mechanical result I N (1 + eos~0).

B) In the Einstein-de Broglie model we obtain

I ~ I~[ 2 -+- - - 2 + lal ~

3-~ ~1 2~/~ } V~V~(1-~) + 2 + [~[~( + 2 - ~ e ~ '

which is a definite prediction depending on the efficiency of the amplifier. We can still deduce some l imit ing cases e = 0 and s = 1 and an intermediate situation e = �89

2 8 = 1 (free pass-through) I N ]a[9. + _ _ (I + cos ~0),

2 + I~I 2

or with , = 0 I ~ 1 + c o s ~ ,

which is the quantum-mechanical result with no amplifier;

i.e.

8 = 0 (full absorption)

absence of intereferenee, and

I ~ I.I ~ + - - - - 2 + 1~1 ~ '

e ~ 1]2, I~1~1 ~ + 2 + [~I ~ 2 + 2 ( 2 + t~[ 2) 1 + cosec .

This, in our opinion, is an impor tant result, since it shows tha t the Bohr and Einstein- de Broglie models give different and testable predictions in some particular simple ex- perimental set-ups.

230 X. G A R U C C I 0 , A. K Y P R I A N I D I S ETC.

We conclude with a full quantum-mechanical t r ea tment of the experiments proposed by ]'OPPER et al. (2) and GARL'CCrO et al. (t) and eri t isized by LOUDON (7) and COSTA ])~ BEAUREGARD (s). The set-up is described in fig. 3 wi th an input of simple isolated photons, t Iere also the Bohr "u~d the Finste in-de Broglie models differ, respectively in the uti l ized wave field, which is now represented as a three-s ta te ket ]a, b, c> cor- responding to the branches 1, 2 and 3 of fig. 3.

/ /

/ /

J ] / / /

/ /

3

/

J d.et:ect.or 1 | "1. J

q I cLetector 2+3 I

Fig. 3.

The operators acting on the three s tates of the ket are, respectively,

~, + 7 ~ ! ~ + ~ , ~ ~ ~1~.

Evidently, this set-up now provides both interferences and correlations which can and will be tes ted exper imental ly (~2).

A) In the Bohr model the s ta te vector takes the form

1 {0 1 1 ]0,1,0>} state I 1 > = ~ - ~ I , O , l > - ' - ~ L l , O,O>-4 V--~

1 ~ { 1 t 1 Io, 2, o> + state 12> = V' I -k- ~ ]2, O, O> -~- ~ II, 1, O> § ~-~

"" ~ IO, 1, O> + ~-~ I1, o, o> + ~--~

and

with Pl ,

wi th p~,

0 = P t [ l > < l l -4- p212><21 with Pl-4-p~ ~ 1.

(1,) A. GozztNl: Communication at the ,gymposium on Wavc-Partlcle Dualism, Apri l 1982, Perugia.

P O S S I B L E E X P E R I M E N T A L T E S T OF T H E W A V E P A C K E T C O L L A P S E 231

The interference intensi ty calculated as

~ a S

l~Tr o[(a:A-~)(a,-4-~)-,-alaa+(al-4-~)aaexp[iq]-"

]} + a 3 ~ exp [-- i~]

yields the following result:

I ~ l a 1 2 A - p 1 3 1 - 4 - - - c o s q - 4 - - - . 2 ~ 3 1 + I~1 ~

This formula for the in tens i ty is again composed of onc interference term where the amplifier is not triggered (P2 = O, Pl = 1) and a constant term. In the absence of amplifier (p~ = O, a = 0), we obtain the quantum-mechanical result

I ~ 3 1 + . . . . cosq . 4 3

The predictions of this model are simply of statistical nature. The correlation between two counters put on the outputs of beam 1 and beams 2 and 3 can be easily evaluated by means of the correlation function

A lengthy but straightforward calculation yields

C I~[~ 31~I ~ 4 +Px~-+Pl- - 1 ~- [~1~(3 + 2 x/2)

3(1 A-I~l ~) -1- p~ ~ cos (p.

In the absence of an amplfier we obtain, as expected, no correlation at all. In the general case, this formula makes statistical predictions with the probabil i ty of getting an oscil- latory behaviour with path length difference of beams 2 and 3.

B) The Einstein-do Broglie model has the following wave field for the arrange- ment of fig. 3:

1 { 1 = - - ~ - - = [11, o, o> + Io, 1, o>] +

I~> ~ 2 + 1 ~ 1 ~ V2

+ VI -~-~ ~ [12, o, o> + I1, 1, o> + IO, 2, o>2 +

+ ~ [11, o, o> + IO, 1, o>] n u Io, o, l>/"

2 ~ 2 A. GAR~'CCIO, A. KYPRIANIDI8 ~.TC.

The calcula t ions can be per formed a long s imilar l ines as before and yield the following p a t t e r n for the i n t e n s i t y :

2 2-(-2---~ I ~ l q V - ~ - - ( 1 ,~ v~) + - - -

This gives for the special cases

3 { 2 V ~ c o s } , e = 1 / ~ [ a [ z + - - - - 1 + . . . . ' 2 2(2 +--1~1 =) 3

2 2 + I~[ 2'

l + V ~ ( ~ + ~ * ) 7 f 1 i [~e]a+ . . . . e = ~ , 2 2 ~/6 �89 :-]--].:q=) q'- 4(2 -[- I~j ~) {1

4} + ~ cosq .

The resul t for the in terference pa t t e rn of the E ins te in de Broglie model wi thou t wave packet collapse has some s t r ik ing differences from the preceding one obta ined wi th the Bohr model: Ins t ead of predic t ing probabi l i t ies of occurrence, we obta in here definite predic t ions for the i n t e n s i t y depending on the amplifier character is t ics e. Fur the rmore , the in terference te rm ~ c o s 9 differs essent ia l ly in the two models: while the Bohr model gives always a probabi l i ty of ge t t ing the pa t t e rn of the no-amplifier-case (i.e.

(1 -F (2 x/2/3) cos g) in addi t ion to background terms, the E ins te in model exhibi ts an in terference behaviour of va ry ing ampl i tude (i.e. 1 + (~/2-}/(2 - - s/2)) cos ~v) in addi- t ion to different background terms. This is a testable predic t ion t ha t could be sub- rat t led to expmin :en ta l evidence.

F ina l ly the corresponding counter correlat ions can be eva lua ted in a s traight- forward way. The resul t be ing ra ther lengthy we jus t quote the three in te res t ing cases.

For s = l, we obta in as expected no correlat ion at all. For s =- 0, the expression for G can be wa'ittcn as follows:

C ~[~[4 t 31~i" . 2x~2 1 ~ eos(~0 + 0D, 4 2 ( 2 - I~lq-: 3(2 ~ ] ~ ) I~1~ + 3(2 + I~l ~ + I~1V~(2 + I~i ~)

where ~ = [a] exp [tOil. For the ~ = ~ case we ob ta in

[a]2 cos9 !_~_~ cos~

C ~ ]'(:~) + ~/---~ 2(2 -7 1~12i 2 + I~1 ~ . . . . . i l COS( ) -t- ol).

F r o m the above formulae for C some s t r ik ing differences compared wi th the Bohr model can be easily po in ted out : the osci l la t ing terms ~ cos (9 + 01) are due to ex- pec ta t ion values of the form

<1, 1, O[a~a~aaoe[O, O, 1>.

These t e rms do not exist in the Bohr model. There , the osci l la t ing t e r m is connected wi th the p robab i l i ty PI ascribed to the s ta te wi thou t wave packet collapse in the mix-

POSSIBLE EXPERII~IENTAL TEST 0]? TIt:E WAV'E PACKET COLL~kPSE ~

ture. I-[cre, the oscillating term is a natural consequence of the formalism always present ill the result, while in the Bohr model it is t ied with the probabili ty of occurence of one state in the mixture. This ctIect manifests i tself also in the general case (~ r 0 or 1) where the oscillatory pattern of the correlation has a much more complex structure.

We thus conclude this let ter with the remark that the correlation pat tern in the Einstein-de Broglie model without wave-packet collapse, exhibits in all cases examined (apart from the e = 1 case where it vanishes) an overall oscillatory behaviour that does not vanish as we come to the full absorption case. This is of course, an important feature that yields the possibility to distinguish between the two models and sholrld and will be tested ext)erimcntally.

Finally, one last remark: The authors do not believe that the formalism of quantum theory is erroneous, but only that the wave packet collapse concept introduced by BOH.~ in the Copenhagen interpreta t ion of quantum mechanics is not correct and contlicts with experiment in certain specific situations, such as the experiments discus- sed or Rauch's et al . experiments on neutron interference (xa). This view does not conflict with facts, since CIsI has shown (~) that one could construct a realistic quantum measurement device without wave packet collapse, the la t ter being a concept which evidently conflicts with re la t iv i ty thcory.

The price to pay for the construction of such a reinterpretat ion of quantum theory is evidently the introduction of subquantal random superfiuid aether along Dirac 's initial suggestion (~a). In this sense of course, quantum theory would not be complete and EI~ST]~IN would be r ight in the Bohr-Einstein controversy.

Two of the authors (AK and DS) want to thank the French Government for a grant which made this research possible.

(~) H. RAUCl~: Proceedings o/ the Bari Conference o*t , Open Questions in Quantum Phys ics * (Dor- drecht, 1983) ~nd references quoted herein. (") P. A. M. I)IRAC: Nature, 168, 906 (1951); I69, 702 (1952).