V O L U M E 66, N U M B E R 9 PHYSICAL REVIEW LETTERS 4 M A R C H 1991

Experimental Test of the de Broglie Guided-Wave Theory for Photons

L. J. Wang, X. Y. Zou, and L. Mandel Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627

(Received 13 August 1990)

A two-photon interference experiment has been carried out, based on an idea recently proposed by Croca, Garuccio, Lepore, and Moreira. The experiment is designed to test a prediction of the de Broglie guided-wave theory, according to which the waves have a physical reality in addition to yielding photon detection probabilities. The results clearly contradict the de Broglie theory, but are consistent with con­ventional quantum mechanics.

PACS numbers: 03.65.Bz, 42.50.Wm

A number of optical interference experiments with photon pairs have recently been reported that are under­standable only in quantum-mechanical terms; classical optics predicts no interference under similar condi­tions. 1 -3 By contrast Croca et al.4 have recently pro­posed a two-photon interference experiment with the op­posite characteristics, viz., interference effects are ex­pected classically but not on the basis of quantum mechanics. They have analyzed the proposed experi­ment within the framework of the de Broglie guided-wave theory,5,6 which is a hybrid of classical and quan­tum concepts. According to this theory there exist elec­tromagnetic waves as well as photons, with the former serving as a guide for the latter. The waves not only yield the probabilities for detecting photons, but they are supposed to have a certain physical reality, to the extent that an "empty" wave can cause photodetection. A number of experiments for testing the predictions of the guided-wave theory have been suggested,7,8 but the one recently proposed by Croca et al.4 appears simpler and more conclusive than others. From their analysis, Croca et al. conclude that the proposed experiment should re­sult in observable interference effects in two-photon coin­cidence counting according to the de Broglie theory, but not according to quantum theory. The experiment should therefore serve as a decisive experimental test of the de Broglie guided-wave theory.

We wish to report the results of an experiment of this kind that clearly contradict the predictions of the de Bro­glie theory, but are in agreement with conventional quantum mechanics.

The basic idea of the experiment is illustrated in Fig. 1(a), while Fig. 1(b) shows the actual realization. The arrangement of Fig. 1 (a) is somewhat similar, in princi­ple, to that proposed by Croca et al.,4 but differs in several details. Only three beam splitters (BS) forming a rudimentary Michelson interferometer are used, in­stead of four. The main difference between the actual geometry shown in Fig. 1 (b) and the simplified geometry of Fig. 1(a) is that the light does not fall normally on BS2 and BS3 in the former case. This makes it possible to insert an auxiliary detector D3 as shown, to collect the

light that penetrates BS3 in the direction PR, without in­terfering with the rest of the setup. However, the princi­ple can be understood by reference to Fig. 1 (a).

Pairs of signal and idler photons produced simultane­ously in the process of parametric down-conversion within a crystal of LiIC>3 (NLC) , 9 serve as the two in­puts to the interferometer formed by 50%:50% beam splitters BSi,BS2,BS3. The crystal is optically pumped by the light of an argon-ion laser oscillating in the UV at 351.1 nm. Signal (s) and idler (1) photons at the subharmonic wavelength of 702 nm enter the interferom­eter at BSi (point P) and at BS3 (point R), as shown. Two similar interference filters IF with a passband of 1012 Hz restrict the bandwidth of the detected light

(a)

(b)

uv NLC

FIG. 1. (a) Principle of the experiment, (b) Outline of the experimental setup.

© 1 9 9 1 The American Physical Society 1111

V O L U M E 66, N U M B E R 9 P H Y S I C A L R E V I E W L E T T E R S 4 M A R C H 1991

beams, so as to make the coherence length of the order of 0.3 mm.

The light emerging from BSi (at U) and from BS2 (at Q) falls on the photon counting detectors Di and D2, re­spectively, whose output pulses, after amplification and shaping, are counted for a predetermined counting inter­val of order j min. At the same time the detector pulses are fed to a coincidence counter that registers photon detections which are simultaneous within the 7-ns coincidence resolving time TR. TR is measured in an auxiliary experiment with two uncorrelated thermal light beams. The experiment is designed to test the prediction of the guided-wave theory as it relates to the coincidence counting rate as a function of interferometer arm length.

As is apparent from Fig. 1 (a) or 1 (b), only signal pho­tons are able to reach detector D2, whereas both signal and idler photons can reach and be registered by D\. Signal light reaches D2 after passing through BSi and BS2 (along the path SPQV), and idler light reaches Di after passing through BS3 and BSi (along the path IEFRUW). On the other hand, signal light can reach Di either along the path BSi to BS3 to BSi (SPRUW) or via BSi to BS2 to BS, (SPQUW). When the path difference is much shorter than the 0.3-mm coherence length of the light, these two light waves—or probability amplitudes — interfere, no matter whether the experiment is de­scribed in terms of classical waves or quantum mechani­cally. The rate of photodetection R\ by D\ is therefore expected to exhibit a periodic variation with the optical path difference through the interferometer. Provision is made for varying the path difference by mounting BS2

on a piezoelectric transducer, that allows microscopic displacements Ax of BS2 to be made. The transducer is calibrated by making it part of a He:Ne interferometer. BS2 with the transducer and BS3 are both attached to two motor-driven translators that can be used for making larger displacements. In the experiment the average counting rates R\,R2 of detectors Di,D2 as well as the coincidence counting rate Rn are measured as a func­tion of the piezoelectric displacement Ax of BS2, over a range of about 1 jum, so as to record the interference pattern. It is in the calculation of the expected coin­cidence rate R\2 that the de Broglie theory and quantum mechanics differ substantially.

At the first stage of the experiment the interferometer has to be balanced so that the optical paths of the inter­ferometer arms BSj to BS3 (PR + RU) and BSi to BS2

(PQ + QU) are made equal to within the y -mm coher­ence length, and similarly for the paths of the signal (SPRU) and idler (IEFRU) to BSi. This is accom­plished in two stages. First, beam splitter BS2 is blocked and an auxiliary detector D3 is inserted along the line PR, as shown in Fig. 1(b). The two-photon coincidence rate R\3 between detectors Di and D3 is then measured as BS3 is moved in 25-^m steps until a minimum in the coincidence rate is found, as illustrated in Fig. 2. As has

1 o

PS

1 •3 o u

15 h

10

i , "

L

1 . 1

\

i i .

. 1 J-,

} H }

-500 -300 -100 100 300 Displacement of BS3 i n pum

5 0 0

FIG. 2. The measured coincidence counting rate R\3 as a function of displacement of BS3, with BS2 blocked. The error bars correspond to 1 standard deviation.

been shown before,10 this corresponds to the position in which the optical paths SPR and IEFR of the signal and idler to BS3 are equal. Then BS2 is unblocked, and the counting rate R \ is measured as a function of piezoelec­tric displacements Ax for different positions of BS2 until an interference pattern of maximum visibility is regis­tered. This is the position in which the interferometer arms are equal, i.e., path length PRU equals path length PQU so that all three distances SPRU, SPQU, and IEFRU are then equal.

Let Jij,Tj be the complex amplitude reflectivity and transmissivity of the beam splitter BS7 (j = 1,2,3). Then it is not difficult to show by reference to Fig. 1 whether the problem is treated classically or quantum mechani­cally, that the visibility °V of the interference registered by detector Di is given by

<V = 2 | # i 2 # 2 # 3

| T 3 | 2 + | # 1 # 3 | 2 + I # 1 # 2 (1)

provided signal and idler beams are mutually incoherent to the second order and of equal intensities. With identi­cal 50%:50% beam splitters <V =0 .5 , but <¥ can be larger or smaller than 0.5 in general.

However, differences between different theories are manifest in the coincidence counting rate Rn, which is calculated in the de Broglie guided-wave theory very much as in classical optics. According to Croca et al. 4

counting rate of detector Dy (7 = 1,2) the average given by

/?y-A)< |*y | 2 >,

and the coincidence rate R j 2 is given by

/ 2 l 2 - * < l * l | 2 | * 2 | 2 > ,

(2)

(3)

where ^1 and ¥ 2 are the wave amplitudes at the two detectors Dj,D2 , respectively, and Kj,K are constants.

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Reference to Fig. 1(a) or 1(b) shows that ^1 ,^2 are re­lated to the wave amplitudes 0 5 , 0 / of the incoming sig­nal and idler photons by

* 2 = < ^ T , T 2 , (4)

where 0=4/rAx/L is the optical phase shift of periodicity L associated with the displacement Ax of BS2. From Eqs. (2 ) - (4 ) , with the assumption that | 0 5 | 2 = | 0 , | 2 and that the phases of <ps and 0, are uncorrected, it follows immediately that the rate R \ is periodic in 0 with visibil­ity given by Eq. (1), whereas Ri is constant. Similarly, we find from Eqs. (3) and (4) that the coincidence counting rate Rn should exhibit interference with much the same visibility. Even if one argues that, for compar­ison with experiment, accidental coincidences at the rate A:<|^i |2X|^2l2) should be subtracted from the rate Ar<|^i | 2 |^ 2 | 2>, Eqs. (3) and (4) still predict interference with visibility close to V. The reason is that, as is known from previous measurements,11 when signal and idler photons fall directly on two photodetectors, the resulting coincidence counting rate A T ( | ^ | 2 | 0 / | 2 ) greatly exceeds the accidental rate AT<|0^ | 2><|0/12>, so that we must have < I ^ J 2 I ^ / I 2 ) > < l ^ 5 l 2 ) < | 0 / |

2 ) . The interference is a consequence of the fact that the de Broglie waves are supposed to have a reality independent of the photons, that can result in photodetection.

On the other hand, from the point of view of quantum mechanics the wave amplitude has no such significance and only refers to a probability wave. Suppose that a coincidence count is registered with a signal photon entering the interferometer at BSi (point P) and an idler photon entering at BS3 (point R). Then it is apparent

from Fig. 1 (a) or 1 (b) that the idler photon cannot reach D2. Hence D2 must have detected the signal photon fol­lowing the straight-line path SPQV, and Dj must have detected the idler photon following path IEFRUW. The wave function therefore collapses along all other paths. As there are no ambiguities in the paths of the photons, there are no alternative probability amplitudes to be add­ed, and no interference then shows up in the coincidence rate Rn.

Alternatively, if a more mathematical analysis is pre­ferred, one may argue that in a quantum treatment <PS

and O/ must be interpreted as signal and idler photon an­nihilation operators 4>5,<E>/. Then the amplitudes 4r\ and ^2 become Hilbert-space operators also, and both <|*i|2> and < |^ i | 2 | * 2 | 2 > in Eqs. (2) and (3) must be treated as normally ordered expectations. It then follows immediately that the interference term in ( ^ f ^ l ^ ^ i ) vanishes when OyOj acts on the two-photon Fock state |lj,l,->. Hence <^jr^l^2

1Jri> is independent of 0, and there is no interference.

Typical counting rates R\,R2 are of order 104/sec, with dark counting rates of order 100/sec. The contribu­tions from the cooled detectors of dark counts are there­fore sufficiently small to be unimportant. Figure 3 shows the measured counting rate R \ registered by photodetec-tor Dj as a function of the displacement Ax, or the phase difference 0. An interference pattern is evidently present, and the visibility is roughly consistent with Eq. (1) when | ^ i | 2 , | ^ 2 | 2 , l ^ 3 l 2 are all near 0.5. By con­trast, the counting rate R2 is found to be independent of 6 with mean value 5700/sec.

Figure 4 shows the measured two-photon coincidence counting rate R\i, after accidental coincidences at the average rate RA=R\R2TR have been subtracted for

Displacement of BS2 i n ixm 0.2 0.4 0.6 0.8

Displacement of BS2 i n jum 0.2 0.4 0.6 0.8

Phase i n mult iples of n

FIG. 3. Measured counting rate R\ of detector Di as a function of the displacement Ax, or the phase difference 9. The standard deviations are smaller than the dot size. The full curve is the best-fitting sinusoidal function of the same period.

Phase i n mul t ip les of rr

FIG. 4. Measured two-photon coincidence rate R\i, after subtraction of accidentals, as a function of the displacement Ax, or the phase difference 6. The error bars correspond to 1 standard deviation.

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each data point, as a function of Ax or 0. This time there is no detectable interference, let alone with visibili­ty <V.

We conclude that the experimental results clearly con­tradict what is expected on the basis of the de Broglie guided-wave theory, but are in good agreement with the predictions of standard quantum theory, including the implied collapse of the wave function along the alterna­tive photon paths SPRU and SPQU whenever a coin­cidence is registered.

This research was supported by the National Science Foundation and by the U.S. Office of Naval Research.

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