Foundations o f Physics Letters, Vol. 5, No. 5, 1992

HOMODYNE DETECTION OF DE BROGLIE WAVES

Vito Luigi Lepore

Dipartimento di Fisica, Universitä di Bari INFN, Sezione di Bari Via G. Amendola 173, 1-70126 Bari, ItaIy

Received February 4, 1992

According to de Broglie, a quantum object is composed of a wave which guides a corpuscle, both existing objectively in the space and time. The Copenhagen interpretation of quantum mechanics is completely different. The wavefunction is considered as a mathematical tool to calculate probabilities which collapses whenever a measurement is performed. In this letter we propose an experiment which can distinguish between the two approaches.

Key words: quantum waves, wave-corpuscle dualism, fundamental law in quantum optics.

Recenfly there has been renewed interest in the physical meaning of quantum waves and the wave-particle dualism. According to de Broglie, a quantum object is made .up of a corpuscle surrounded by a wave, both objectively emstmg in space and time, and the role of the wave is to guide the corpuscle [1]. By contrast, according to the Copenhagen interpretation, the ware aspect and the corpuscular aspect are not contemporary but complementary; in the sense that the two descriptions cannot be employed simuhaneously but are mutually exclusive: When considering the propagation in space and time, the suitable conceptual tool is the wave function, whose modulus squared gives the probability density for observing the position of the corpuscle at a certain position; however, whenever a measurement of the position is performed, the ware function undergoes an irreversible change, the "collapse" of the wave function. The difference between the two approaches has been extensively

469

0894-9875/92/1000-0469506.50/0 © 1992 Plenum Publishing Corporation

470 Lepore

• BS1 %

,~ BS2 %

1394

a s 3 \

COINCIDENCE COUNTER

Fig. 1. Outline of the proposed experiment.

studied in recent years [2], and a number of experimental setups have been proposed to discriminate between them [3,4,5]. Following the proposal of Ref. 5, Wang, Zou, and Mandel [6] carried out an optical experiment to detect de Broglie waves; however, no evidence for such waves were found.

In the present paper we discuss a different experimental proposal based on the joint detection probability of two homodyne detectors, similar to another homodyne scheme proposed for studying violations of Bell's inequality [7,8]. A homodyne detector is an apparatus in which the signal under measurement is summed with a reference signal. In our case ^the re~rer/ce signals are the plane wave incoming along the modes m and n (see Fig. 2).

Homodyne Deteetion of de Broglie Waves 471

!%

I1>, A i

A

l

\

A a

I%

A b

I~~ >

A

0

A m

\

Fig. 2. Quantum-optical . not ations. The incomi_ng annihilation operators are ~i ~, ~\, I~, ^ ^ " m, and n. The outgomg annihilation operators are ~, ~, and ~.

We will analyze the experiment by using both quantum optics and de Broglie model and show that the two approaches give rise to different predictions.

Consider the apparatus shown in Fig. 1. An incoming single-photon beam impinging on the beamsplitter BS 1 is split in

a reflected ray and a transmitted ray. The reflected ray is directed on the detector D , while the transmitted one is divided

along the outgoing ports of the beamsplitter BS e. These two rays,

after being separately superimposed on the beamsplitters BS 3 and

BS 4 with two local oscillators, i.e., t w o plane waves correlated

in phase, fall on the detectors D b and D~, respectivety.

The description of the experiment within the de Broglie model nms as follows: When a photon impinges on the beamsplitter BSI, the wave is divided into two different patts, but the

corpuscle follows either one of the outgoing paths. Therefore, if a corpuscle is revealed by the detector D , a

472 L e p o r e

wave-without-corpuscle is directed on the beamsplitter BS 2. After

splitting along the outgoing routes of BS2, the wave is summed

with the plane waves ot 1 and ot 2 at BS 3 and BS4, respectively. The

coincidence detection probability shows a sinusoidal dependence on the phase difference of the local oscillators due to the correlation induced by the empty waves. The Copenhagen interpretation leads to completely different predictions. On revealing the corpuscle in Da, the wave function "collapses," and

no correlation is predicted between D b and D . c

In the following we shall carry out the calculation by using the quantum optics and compare the results with the prediction of de Broglie model.

Referring to Fig. 2 for the notations and assuming that all the beamsplitters are described by the same reflection and

i 1 transmission coefficients, r - and t - , one can write erg / 7

the quanmm-optical expressions for the outgoing annihflation operators in terms of the incoming operators as foUows:

ä _ i ,~,+ 1 ~,

¢ 2 ¢ 7 ~ _ i ~ _ 1 ~ ,+ 1_ ~ + i ~, (t)

2,/7 2J7 2 e~7 - - " m ,

2",/"7 ̀ 2v / 2" 2 ' 7 2

The incoming state is given by the product of the single-mode states

Ix[r> = I1> 10> 10> Io~ > to~ > (2) i j k 1 m 2 n

where it is assumed that a single-photon state is entering the apparatus along the mode i, a vacuum state is assumed for the modes j and k and the coherent state la > and Ic~ > , with

. 1 m 2 n

= o~eZ~l = ouei~z, respectively, for the eigenvalues otl and ot2

local osciUators. The photon-count probabilities at the detectors D b and D c

are proportional to [9]

Homodyne Detection of de Broglie Waves 473

gQo

goo <Vt~+~lv> (3)

Inserting in (3) the expressions (1) for the operators and (2) for the state, one obtains

b 1 tZ 2 gQO= --8--- + -2--,

c 1 2

g o o = --8-- + -2 --.

(4)

The coincidence counting probability between D b proportional to

b c gQo = <vl~+cA+c%lv>

and D is

(5)

On using (1) and (2), Eq. (5) becomes

be ~2(1+2a2) ['1- 1 gQo = 8 L 1+2¢~2 s~ 102 0,)] ~6)

A similar experimental semp has been considered in Rel. [8], and it has been shown that the joint detection probability

as a function of (~2 t~l ) allows a violation of Bell's

inequality for suitable values of tz. However, the most interesting quantity for the comparison with the de Broglie model is given by the joint probability of counting photons at the detectors D,a Db and De, which is proportional to

a b c gQo = <VIä+ü+~+~älw>' (7/

i.e.,

a b c 0~ 4

gQo = -8~" (8)

As follows from (8), the joint photon counting probability at D a,

Db, and De does not depend on the phase-difference [~2-~1)"

474 Lepore

13e

rl

\

1

O~ 2

rl b

rl

\

c

Fig. 3. Notation according to the de Broglie mode1. The

incoming photon is represented by the wave ~3e iX, with 3( changing from one photon to another. The local oscillators are represented by two wave amplitudes

Beiß1 ocei~2, Œ = and c~ 2 = and the outgoing fields are

'[la , TIb , and Tl.

Let us carry out the photon counting probabilities at D , D b

and D , within the de Broglie model. With reference to Fig. 3 for

the notations, we indicate by 11, T1 b, and rl¢ the fields at the

detectors, with ~e iX the wave associated with the incoming photon

and with c~ = c~eißl and ~ = o~ei~2 the fields for the local 1 2

oscillators. According to de Broglie's ideas, the detection probability is proportional to the amplitude of the wave [5]. In the experiment under discussion one, therefore, has

b (9) gdB = <lTlbl2> X

and

Homodyne Detection of de Broglie Waves 475

c <11.1c!2>% gdB = (10)

where <...>% is the average over all the possible values of the

phase Z of the incoming photon. Similarly, for the joint detection probability, orte has

bc <ll]bl21Tic[2>Z ' gaß = (11)

and for the case in which a photon is detected in D , the joint a

detection probability at the three detectors is

c <l.qcl2>) ~ gdB = (10)

The fields at the detectors, rla, rlb, and Tl, can be expressed in

terms of the incoming fields:

rl ~ _ i ~ ei%,

72 Bb -- i ~ eiZ + i et ei~2, (13)

2v,/2 ' v/E "

Tlc _ i ~ ei% + 1 et eißl.

2v/2 " v/2

On inserting (13) in (9) and (10), one can calculate the single detection probabilities:

b d Z irlbl2 + et gdB = ~ = --'-g'-- -"2---

( 1 4 )

0

and

9re dx Il"Icl2 ~2 et2 g«B = ~ = - -8 - - - + - 2 - - "

0

(15)

Similarly, for the joint detection probabilities, it follows from (11) that

476 Lepore

o

e, tu

1.6 "_--

O.a

0.6 _ 0.4 _:-

O.2 ~- 0 ~ I , r i ~ ~ I

0 2 '~ ~ 8 10 72 14

(*Z * ) I

Fig. 4. The continuous line represents the de Broglie prediction when the "visibility" of the dependence upon the phase difference is maximum. According to quantum optics, no oscillation in the joint detection probability should be observed.

bc d X [Tib[2 h~c[2 gdB =

0

1 ~ 4 "4- _ O~ 4 ~2(Z2

(16)

and, from (12),

abc ?Tl; dx gdB = J ~ Irlal2 ITIbl2 [T]c]2

0

(17)

Assuming ~ = 1, the predictions of the de Broglie mode1 for

single detection probabilities [Eqs. ( 1 4 ) a n d (15) I give the same t

results as those deduced fromt quantum optics (-)[Eqs. (4) t . But the ~t

J

joint detection probability at D a, D b, and De becomes

Homodyne Detection of de Broglie Waves 477

gaß = -8-- + ---7-- + ~ - - 7 - - s i n ~ 2 - ~ 1 " (18)

The dependence upon the phase difference / ~ b 2 - , ] in (18) k )

is due to the fact that the wave associated with the corpuscle is considered as objectively existing in space and time, with appropriate physical properties. By contrast, according to

quantum-optical predictionr ~ (8), g~¢ is independem of the

phase-difference ]~b 2 - ~11" The "visibility" of the sinusoidal k ) 1 oscillation in (18) is maximum for c~ = ~2-' and Eq. (18) becomes

abe [ 1 ( / ] gaß = g0 1 - --2- sin ~a - ~1 ' (19)

with go denoting the mean value of abe gdB " ThUS, the experiment can

be used as a tool to detect de Broglie waves. Moreover, since the quantum-optical predictions lead to different results, the experiment allows one to discriminate between the two theories. Finally, it worths stressing that similar experimental setups, either using a two-photon atornic cascade [10] or a parametric down-conversion [6,11], have already been employed to study other fundamental properties of light. Thus, the technological difficulties in carrying out the experiment may be overcome in the near future.

The author wish to thank Saverio Pascazio for a critical reading of the manuscript.

REFERENCES 1. L. de Broglie, Non-Linear Wave Mechanics, A Causal

Interpretation (Elsevier, Amsterdam, 1960). 2. F. SeUeri, Quantum Paradoxes and Physical Reality

(Kluwer, Dordrecht, 1990). 3. A. Garuccio, K. Popper, and J.-P. Vigier, Phys. Leit. A86,

397 (1981). 4. J.R. Croca, A. Garuccio, and F. Selleri, Found. Phys. Lett. 1,

101 (1988). 5. J.R. Croca, A. Garuccio, V.L. Lepore, and R.N. Moreira,

Found. Phys. Lett. 3, 557 (1990).

478 Lepore

6. L.J. Wang, X.Y. Zou, and L. Mandel, Phys. Rer. Lett. 66, 1111 (1991).

7. P. Grangier, M.J. Potasek, and B. Yurke, Phys. Rev. A 38, 3132 (1988).

8. S.M. Tan, D.F. Walls, and M.J. Collett, Phys. Rer. Lett. 66, 252 (1991).

9. R.J. Glauber, Phys. Rev. 130, 2529 (1963); R. J. Glauber, Phys. Rer. 131, 2766 (1963).

10. A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 47, 460 (1981). A. Aspect, P. Grangier, and G. Roger, Phys. Rer. Lett. 49, 91 (1982). P. Grangier, G. Roger, and A. Aspect, Europhys. Lett. 1, 173 (1986).

11. J.G. Rarity and P.R. Tapster, Phys. Rev. A 41, 5139 (1990).