Foundations of Physics Letters, Vol. 3, No. 6, 1990

Q U A N T U M - O P T I C A L P R E D I C T I O N S F O R A N E X P E R I M E N T O N T H E D E B R O G L I E W A V E S D E T E C T I O N 1

J. R. Croca, 2 A . Garuccio , 3 V. L. Lepore, 3 and R. N. Moreira 2

2Departamento de Fisica, Universidade de Lisboa Campo Grande, Ed. C1, 1700 Lisboa, Portugal

3Dipartimento di Fisica, Universitg7 di Bari L N. F. N., Sezione de Bari Via Amendola 173, 70126 BarL Italy

Received April 2, 1990

An experimental apparatus to detect de Broglie waves is discussed. The wave packets of two photons generated in the parametric-down conversion are overlapped in a modified Mach-Zehnder interferometer. The coincidence photodetection rate of photon pairs is evaluated, as a function of path-length of two interferometer arms, both by using the de Broglie concept of a real quantum wave and by the quantum optical approach. The different results of these two theories are compared, and it is shown that the proposed experiment can disprove either the theories.

Key words: quantum waves, wave-particle dualism, de Broglie's wave detection, fundamental law in quantum optics, two-photon interfer- ence.

1. I N T R O D U C T I O N

Until recently, as is well known, the problem concerning the re- ality of de Broglie waves [1, 2] could not be settled in a conclusive way. The discussions on this subject were mainly done on theoretical grounds. Essentially, the question is related to the "correct" inter- pretation of the wave function ~, whether it must be interpreted as a simple tool for predicting mathematical probabilities or, on the

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558 Croca et aL

contrary, whether it represents, as suggested by de Brogtie, a real physical wave propagating in spacetime. More recently, several ex- perimental methods to detect de Broglie waves have beer= discussed [3, 4, 5]. Following the ideas of Ref. 5 and 6, Croca, Oaruccio, and Selleri [7] proposed another experimental setup of the same experi- ment [6] to discriminate the two interpretations of wave function @, based on the neutron interferometry. Unfortunately, sources of neu- tron pairs are not available and, moreover, there is no experimental evidence of any interference effect generated by the overlapping of waves belonging to two different neutrons. Now, recent interference experiments of two photons [8] have opened the way to clarify the problem.

In the following we shall discuss a new version of the experiment using photon pairs. Starting with the de Broglie model of a real wave, we shall carry out the calculations of expected joint detection prob- abilities in the experiment and compare these results with quantum optical predictions.

In Fig. 1 is sketched the experimental setup. A parametric- down converter, pumped by u v laser light, produces pairs of linearly polarized photons; the two photons are generated simultaneously [14, 15] and, following different paths, form two beams, the signal and idler beams.

a aLric ( down-converter%

BS "I ¢i

¢2

BS I

A / \ pS

BS

-he2 [___] " / ~ J coilcidence

BS counter

Fig. 1. Outline of the experiment, with de Broglie's fields ¢51, ~2, ~1, and ~2.

The beams go through a modified Mach-Zehnder interferometer, in which all the mirrors are semi-transparent and the optical lengths can be varied by the phase-shifter PS. We shall consider the events in which the idler photon, after traversing BS1 and BS4, is detected by the photomultiplier D2, and the signal photon, after traversing BS2 and BS3, is detected by D1. The measured quantity is the joint-

Quantum-Optical Predictions 559

detection probability in D1 and D2 as a function of the optical length difference between the two paths, BS1-BS4-BS3 and BS1-BS2-BSa.

2. T H E D E B R O G L I E A P P R O A C H T O T H E E X P E R I M E N T

Let us state explicitly the basic assumptions within the de Broglie model before carrying out the calculation of the joint- detection probability.

1. A photon is composed of a localized particle and a real wave ~(x, y, z, t) propagating in spacetime in accordance with d'Alembert 's equation

1 02~5 A2q~ d oT=O" (1)

A consequence of assumption 1 is that, when a photon impinges on a semitransparent mirror, the associated wave is partly transmitted and partly reflected, while the particle is either t ransmitted or reflected. Naturally, the detection of the particle in one of the two channels does not induce the collapse of the wave in the other channel.

2. If, in a region of space, n waves (I)1, (I)2,..., (I)n are simulta- neously present the total wave ~ is given by the sum

(1} = ~1 + ~2 + "'" + On. (2)

We stress that the previous assumption holds even when the waves ~ 1 , . . . , ~n belong to distinct photons, as shown by the experiments on interference of light beams coming from different sources [9, 10]. L. de Broglie and J. Andrade e Silva [11], commenting on the results of Pfleegor and Mandel's experiment [9], wrote: " . . .A photon coming from one laser or the other and arriving in the interference zone is guided--and this seems to us physically cer tain--by the superposition of the waves emitted by the two lasers . . . . "

3. If a photon at a time can be presented in the same region of space, the amplitude of the real wave q) expresses the photon detection probability through the formula

PD = 1 12, (3)

where c~ is a constant depending on the detection process. We are dealing thus with a double interpretation of the wave

function q): It represents a real wave propagating in spacetime, and

560 Croca e~ aL

its ampli tude squared gives the detection probability measured as frequencies or rates in repeated events.

Using the previous model, we now calculate the joint-detection probability in D1 and D2. Supposing that the beamsplit ters are loss- less, the transmission and reflection coefficients t and r satisfy the relation [12]

Irl 2 + It[ 2 = 1, (4)

rt* + r*t = O.

The outgoing waves @1 and ~2 can be written in terms of the waves q51 and ~2 associated with the signal photon and the idler photon, respectively, as

q21 = t2621 -~- r2t~2 + tr202e i8,

~2 = t2~2 eiS, (5)

where 5 is the phase difference between the two optical lengths. When the idler photon is detected in D2, we are sure that the photomulti- plier D1 can reveal only the twin photon going along the signal beam. Then, according to the assumption 3 of the previous model, the con- ditional detection probability in D2 is proportional to !q/112. One can therefore write

P(D1 I D2) = C~l I/Ill ]2. (6)

The coincidence probability is given by

r ( D l , D 2 ) = P ( D 1 [ U 2 ) P ( D 2 ) = ttlff2]kl/ll2[/I/2l 2. (7)

Inserting (5) in (7), one obtains

P (Dl ,D2) = c~l~elt14[~212[lt[41¢112 + [O2122[TI41t]2(1 + cos 5)1, (8)

since 11) 1 and q52 have no phase relation. The joint-detection proba- bility do depend upon the phase difference between the two optical lengths of the interferometer. This effect is essentially due to the fact that in the path BSa-D1 there is an overlapping of the signal wave with the empty wave generated by the idler photon going through BS1 and BS4.

Quantum-Optical Predictions 5 6 1

3. Q U A N T U M OPTICAL P R E D I C T I O N S

In quantum optics [12, 13] the reflection coefficient r and the transmission coefficient t of a lossless beamsplitter connect the input annihilation operators ~1 and ~2 to the output annihilation operators 31 and 82 through the relations

81 = r~l + t~2, o2 = H1 + r~2. (9)

where the relations (4) obviously hold. In Fig. 2 we report the annihilation operators for each branch

of the experimental apparatus; ~ and b refer to the signal and idler photons; ~ and 5" are introduced according to the optical quantum theory of beamsplitters and describe the incoming mode of BS1 and BS4.

r BS 2

parametric A down-converter% b /

BS~

a"

V

Z~

z 2

/ f za

ps

,,.%

BS3/z//I rAf ~..'~ I D1 ~ - - coincidence

B S 4 counter

2% u

Fig. 2. Outline of the experiment, with the annihilation operators and optical lengths for each branch.

By using relation (9) for each beamsplitter and indicating by Zl, . . . , z4 the optical path lengths as in Fig. 2, one can write the outgoing annihilation operators as linear combinations of the ingoing annihilation operators:

F = t~e ikza + r~ = t2eikZ3"b+ treikz3~ + r~ (10)

562 C r o e a et at .

= t2eikz2"d + tr2[eik(z~+z~) + eik(z3+z4)]b

+ [rt2e ik(z~+z~) + r3eik(za+z4)]~ A- rteikZ4u.

(11)

The joint probability of detection of two photons one in the output mode i and the other in the output mode r, is given [16,17] by

Pir(Ri, ti; RT, tr) (12)

in which the single-mode field operators are

( 2 7 r h c o ~ 1 / 2 . ^ ~}+)(Ri , t~)=i \ y / e['(~"R'-"~')]h'

2~rhw 1 / 2 . A = i(__c_ ) (13)

C is a constant characteristic of the detectors and of coincidence time, and I ~) is the input state.

As is known, the Fock state ] la, lb) describes to a good ap- proximation the photon pairs emitted in parametric-down conversion [18]. Then the input state for our experiment is

I ~) =] la, lb,O.,Ov), (14)

where we assume the vacuum state for the empty modes ~ and ~. On inserting (13) in (12) and using the formulae (10) and (11)

for F and ~, respectively, and (14) for ] kg), one obtains

(2zrhw ) Fir = c ,-V-j2(lti2) 4. (15)

It can be seen that in the formula (15) the joint-detection prob- ability does not depend upon the difference of the optical path lengths of the Mach-Zehnder interferometer, i.e., no interference arises.

Quantum-Optical Predictions 563

4. CONCLUSION

Equation (8), derived from the de Broglie model, contains a different prediction if compared to quantum optical results for the coincidence rate in the proposed experiment. For the particular case of 50/50 lossless beamsplitters and the ingoing fields q51 and ~2 of equal intensity, we report in Fig. 3 the joint-detection probability as a function of the phase difference between the two arms of the interferometer.

1

0 0 ' 4 ' ' ' ' ' ' ' ' d 8 12

Fig. 3. Predicted results of de Broglie's model (continuous line) and quantum optics (dotted line).

We want to stress that the different results given by the two approaches to the experiment are essentially due to the assumption that de Broglie waves do really exist. Thus, the proposed experiment can be considered as a tool for disproving either de Broglie's model or the quantum optical predictions. Moreover, we wish to recall that ex- periments using a similar technology have already been performed [8, 19]; therefore, the difficulties of the previous experimental proposal, based on neutron interference, have been overcome.

The authors wish to thank Prof. F. Selleri for useful discussions concerning the content of this article.

564 Croca ei al.

R E F E R E N C E S

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101 (1988). 8. C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044

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199 (1987). 14. D. C. Buruham and D. L. Weinberg, Phys. Rev. Lett. 25, 84

(1970). 15. S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011

(1985). 16. R. J. Glauber, Phys. Rev. 130, 2529 (1963). 17. R. J. Glauber, Phys. Rev. 131, 2766 (1963). 18. C. K. Hong and L. Mandel, Phys. Rev. A 31, 2409 (1985); R. Gosh,

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19. J. Brendel, S. Schumtrumpf, R. Lange, W. Martiessen, and M. O. Scully, Europhys. Lett. 5,223 (1988).

N O T E S

1. This work was supported by a collaboration between I. N. 1. C. (Portugal) and C. N. R. (Italia).