JOSA LETTERS

"Test for the nonexistence of photons" and "Photoelectric counting measurements as a test for the existence of

photons" Sherman Karp

Naval Ocean Systems Center, San Diego, California 92152 (Received 11 October 1977; revised 13 October 1978)

In recent articles by this author, [S.Karp, Statistical properties of ensembles of classical wave packets, J. Opt. Soc. Am. 65, 421 (1975); S.Karp, Test for the nonexistence of photons, J. Opt. Soc. Am. 66, 1421 (1976)], a radiation model was proposed that could lead to the identification of certain classical properties of the field, based upon photoelectric counting experiments. Mandel [L.Mandel, Photoelectric counting measurements as a test for the existence of photons, J. Opt. Soc. Am. 67, 1101 (1977)] responded by asserting that "It follows that for any field for which a classical wave description exists, in the sense that the phase-space density φ({v }), and therefore P(u), are non-negative it is impossible to distinguish between photons and light waves in photoelectric counting experiments." This statement is based upon the derivation of similar photoelectron counting equa­tions using quantum and classical techniques. It is the mathematically loose associations and subse­quent rigid physical interpretation implied in relating the two equations that is under contention. It seems to us that the model-computation-effect approach to physics is still solvent. In particular, if we can model an effect as small as the radiation from an individual electron and produce a predictable effect, then the physics underlying the model must be seriously considered.

INTRODUCTION In Ref. 1 a classical model for radiation was proposed by

this author, together with a test which would unequivocally distinguish this model from quantum descriptions. In Ref. 2 an estimate was made of the resources necessary to perform the experiment. In Ref. 3 Mandel took issue with the con­clusions drawn in Refs. 1 and 2 stating "It follows that for any field for which a classical wave description exits, in the sense that the phase-space density φ{u} and therefore P(U) are non-negative, it is impossible to distinguish between photons and light waves in photoelectric counting experiments." It is the purpose of this paper to examine this statement.

Reference 3 assumes the existence of a free, quantized electromagnetic field, in which the distribution of the photon number is shown to be

If the conversion from photon to photoelectron obeys a binary branching process, the photoelecton counting distribution is

Although Mandel has not discussed how one formally arrives at the free, quantized electromagnetic field, we should be able to consider this to be the result of a similar binary branching process between the radiating oscillator and the photon states. As a consequence the discreteness in the process is preserved from generation to reception. The natural discreteness in the process can be obtained by observing the variance in the photoelectron counting distribution resulting from a coherent

state. This can be shown to be equal to the first moment of the photoelectron counting distribution, and for this de­scription of the process is the only discreteness observed.

In contrast, by using the semiclassical approach, one can show that the distribution for photoelectron counting is4

where

Here W is the detected energy in the process obtained by a time-space integral of the intensity of the field. Of course, Pw(W) is always positive and cannot have negative values as is claimed for P(U). For point detectors, the integral over the area can be easily obtained, yielding Eq. (2) of Ref. 3. (For nonpoint detectors, one must be careful in treating the spatial modes and the spatial coherence.) The variance of the pho­toelectron count can now be shown4 to equal

where var(W) is the variance of the integral of the classical intensity and Ek (K) is the first moment of the counting dis­tribution or the shot noise. The analog to a purely coherent state would be a constant-intensity sinusoid for which Pw(W) = δ (W — W0) and var(W) = 0. It is based upon these results and the equivalence of Eqs. (2) and (3) that Mandel states " . . . It follows that for any field for which a classical wave de­scription exists, in the sense that the phase-space density φ({υ}), and therefore P(U), are non-negative, it is impossible to distinguish between photons and light waves in photo­electron counting experiments." Although P(U) may be

1309 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979 0030-3941/79/091309-03$00.50 © 1979 Optical Society of America 1309

normalized to unity, its status as the classical intensity probability density function is more aptly described as5 having " . . . many of the properties of a classical probability density funct ion. . . "

DEPARTURE It is here that differences occur. In the model introduced

in Ref. 1, the field was characterized by a classical ensemble of random-phase wave packets. By a close examination of Pw(W) for such an ensemble, one obtains the exponential distribution in intensity as expected, plus perturbation terms.1

This deviation from the Gaussian case has been noted before and can be identified as radiation shot noise.6 When ob­serving this intensity over a short time T relative to the inverse bandwidth, the resulting photoelectron count variance in Eq. 5 becomes a sum of three terms.1 The first term Ek (k) is the shot-noise term resulting from the detection process as ob­served earlier. The term var(αW) becomes a composition of two parts. One part, E2

k(k), relates to the variance of a Gaussian process, which would occur if this were the true input. The second part is equal to 2(4α)2γ2 where

KT is the average of the observed wave-packet count, r(t) is the envelope of the wave packet, and x and a are random variables. This latter term is the shot noise inherent in the classical radiation field because of its construction from the finite ensemble of wave packets. As observed in Refs. 1 and 2, this term is geometry dependent, with changes predicted by the use of Maxwell's equations. There is no counterpart to this term in the quantum description and this is the basis for the departure. Thus, from the classical model presented we would observe two quantum effects which are separate and distinct, the first occurring upon creation of the field and the second observed when it is detected, thereby decoupling two independent events. In the model proposed, the particle behavior is embodied in the transient superposition among the wave packets, with the average intensity accounting for the average transfer of energy. On an instantaneous basis, however, it is this complex nonstationary interference pattern that would describe the energy transfer. As a result, it is ov-ersimplistic to speak of "(If the incoming light is regarded as) a collection of classical wave packets, each of which is divided into two wave packets at the mirror, (we would expect to find frequent coincidences between pulses emitted by the two photodetectors.)"3 Our experiment attempts to quantita­tively evaluate the difference between the average value of the count variance predicted by the model and that expected from accepted results. This difference would be geometry de­pendent with a classical explanation, and that is where our attention has focused.

DISCUSSION It may well be that the effect we have derived does not

exist.7 However, if it does, then Ref. 3 is incorrect. This re­sult is a measurement of the shot noise inherent in the classical field described and, if it exists, is a direct measurement of the wave packet itself considered as a Maxwellian field. In his own paper Mandel has emphasized that Eqs. (2) and (3) are

identical when P(U) is non-negative definite. Thus the quantum noise explained by him, and introduced in the de­tection process using a classical description are in one-to-one correspondence. On the other hand, his description of the field is made in terms of a complete orthonormal basis, or in the case of the coherent states, an overcomplete basis. Con­sequently, there should be no discrepancy between the field and the expansion, and the quantum noise identified above should be the only source of noise. However, as pointed out by Wiener, it is the time process that corresponds to reality. While we may use alternative domains for convenience of calculation and understanding, expansions are merely rep­resentations. If a molecule radiates during a finite period of time, then something is perturbed during the whole period. This is the time process. A viewer observing this event would be affected by this entire process. A viewer observing many such events would see a time process much like what we have used. In fact, the time process that we have used can be put in one-to-one correspondence, statistically, with any such viewed process.6,8,9 The representation that we have used, however, is not complete. Rather, it has certain statistical convergence properties, analogous to the correspondence principle, which put it in one-to-one correspondence with the finite-energy emissions from a discrete bounded ensemble of radiators. The ensemble can be satisfactorily associated with various temporal forms of radiation. Incoherent sources produce discrete finite ensembles with uniformly distributed phase, while coherent, sources arising from the process of stimulated emission produce phase properties which are correlated across the ensemble.6 It is the discrepancy due to finiteness and discreteness that is isolated by the test that we have proposed. Spatial properties are handled in a standard classical manner. To blindly insert a P(U) function into Eq. (2) that yields the correct answer seems contrary to the basic tenets ofphysics. One could argue with equal justification that it is the non-negative definite nature of Pw(W) which constrains P(U) in a similar manner.

Mandel has cited work by Clauser10 as disproving our pro­posed model. In fact, this work does not preclude our model. Clauser has assumed that the intensities which are trans­mitted and reflected by a beam splitter are proportional. This applies only to the average values. It is necessary that the instantaneous transmitted and reflected intensities sum to the incident intensity to conserve energy at every point in time and space. In fact, the random interference effect, being of a wave origin, can yield correlations between the transmitted and reflected intensities ranging from 1 to —1, with an average value near zero being both plausible and probable. Our measurement, which depends upon the fourth order of the envelope, transcends this anomaly since it can in theory be performed on a single intensity. We have been mathemati­cally precise. We have proposed a well-defined experiment with a quadratic metric. We have performed a detailed analysis of the statistic to be measured, and have predicted an effect that Mandel has yet to reproduce by current theo­ries.

If we assume that the effect will exist, then there are certain insights that can be conjectured. We know that the effect is proportional to the ratio of the absorption cross section to the area of the focused beam. Since the absorption of a quantum of energy would be from a multitude of wave packets, we can

1310 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979 JOSA Letters 1310

see that the more wave packets that are involved, the less discrete is the effect and the less discrete is the appearance of the radiation field. By contrast, as the effect gets larger less wave packets are involved in the absorption and the closer we are to observing the discrete wave-packet nature of the ra­diation field. These conclusions do not depend on the average intensity of the field.

In capsule form, we claim that every discrete process has a quantization noise associated with it. With regard to the ensemble of wave packets that we have analyzed, this quan­tization noise is the familiar shot noise, is a measure of the wave packet itself, and is distinguishable from the quantum effect introduced by the detection process in the classical case or by quantization of the field in the quantum-mechanical case. Consequently, the existence of the effect would estab­lish the wave packet as a measurable field that satisfies Maxwell's equations, produces predictable statistics, and hence precludes the existence of photons.

1S. Karp, "Statistical properties of ensembles of classical wave packets," J. Opt. Soc. Am. 65, 421 (1975).

2S. Karp, "Test for the nonexistence of photons," J. Opt. Soc. Am. 66, 1421 (1976).

3L. Mandel, "Photoelectric counting measurements as a test for the existence of photons," J. Opt. Soc. Am. 67, (1977) 1101.

4R. M. Gagliardi, and S. Karp, Optical Communications (Wiley, New York, 1976), p. 62.

5C. W. Helstrom, J. W. S. Liu, and J. P. Gordon, "Quantum-me­chanical communication theory," Proc. IEEE 58, 1578 (1970).

6S. Karp, R. M. Gagliardi, and I. S. Reed, "Radiation models using discrete radiator ensembles," Proc. IEEE 56, 1704 (1968).

7Since the experiment that we have proposed is difficult, we point out that the measurement time is proportional to the fourth power of the wavelength. Consequently, a reduction in wavelength of only two would decrease the time by sixteen.

8P. Frost, thesis, Stanford University, 1968, University Microfilm order no. 69-221, (unpublished).

9S. Karp and R. M. Gagliardi, "On the representation of a continuous stochastic intensity by poisson shot noise," IEEE Trans. (Inf. Theory) 16, 142 (1970).

10J. F. Clauser, Phys. Rev. D 9, 853 (1974).

1311 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979 0030-3941/79/091311-02$00.50 © 1979 Optical Society of America 1311