Stochastic optics: A reaffirmation of the wave nature of light

Foundations o f Physics, Vol. 18, No. 2, 1988

Stochastic Optics: A Reaffirmation of the Wave Nature of Light

Trevor MarshalP and Emilio Santos 1

Received March 25, 1987

Quantum optics does not give a local explanation of the coincidence counts in spatially separated photodetectors. This is the case ./'or a wide variety o f phenomena, including the anticorrelated counting rates in the two channels o f a beam splitter, the coincident counting rates o f the two "photons" in an atomic cascade, and the "antibunching" observed in resonance fluorescence.

We propose a local realist theory that explains all o f these data in a consistent manner. The theory uses a completely classical description o f the electromagnetic field, but with boundary" conditions o f the far field that are equivalent to assuming a real fluctuating, zero-point field. It is related to stochastic electrodynamics similarly to the way classical optics is related to classical electromagnetic theory.

The quantitative aspects o f the theory are developed sufficiently to show that there is agreement with all experiments performed till now.

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

Erwin Schr6dinger must be given a substantial part of the credit for today's renaissance of foundational studies in physics. His famous Cat article (1t was, of course, published in the same year as the Einstein-Podolsky- Rosen (2) article. While we believe that the EPR paper has given, and will continue to give, greater stimulation to experimental work than Schr6dinger's Cat, it is possible to see, in Schr6dinger's criticism of quantum mechanics, a single consistent theme from 1925 until his death in 1962. This theme is the insistence on realism; if a theory denies microscopic realism, it must also deny macroscopic realism. This is the conclusion of the Cat argument and, by a different route, of the EPR argument also. We

Departamento de Fisica Te6rica, Universidad de Cantabria, Santander, 39005 Spain.

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0015-9018/88/0200-0185506,00/0 @ 1988 Plenum Publishing Corporation

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186 Marshall and Santos

use "realism" in the technical sense, as in the hypotheses which form the starting point for modern discussions related to the Bell inequalities. (3'4) There is a great deal of evidence (5) to indicate that, in the wider philosophical sense, Schr6dinger was an idealist rather than a realist, especially in the latest period of his life. Nevertheless, as his correspondence with Einstein confirms, (6) he strenuously opposed the Copenhagen notion of observer-created reality. It is now normal to refer to models based on the above hypotheses as "local realist," so we would claim that such models are compatible with quite a wide range of philosophical positions.

There are two main differences between Schr6dinger's and Einstein's approaches to the anti-realist features, as they saw them, of quantum mechanics. Firstly, Schr6dinger never opposed quantum theory from a deterministic point of view; he never said "God does not play dice." Indeed, according to Forman's (v) study, Schr6dinger was one of the first German- speaking physicists, during the Weimar period, to renounce "causality" (which Forman sometimes equates with determinism). Einstein himself later saw that quantum theory, through its nonlocal character, fails to provide us even with a statistical form of causality. (8) It is clear from the intervention by Pauli in the Einstein-Born correspondence (9) that, after 1935, Einstein's principal criticism of quantum theory effectively coincided with Schr6dinger's. It is to the credit of Schr6dinger that he was the first to draw attention to the misuse, as he saw it, of probabilistic concepts in quantum theory. (1°) In this respect, Schr6dinger anticipated Feynman ~H) in recognizing probability amplitudes as a novel kind of probability theory, but the language he used to describe this novelty ("comical") shows his deep philosophical aversion to it. He clearly considered from the beginning the Born interpretation, of the Schr6dinger wave function as the square root of a probability, to be a kind of personal insult:

What seems most questionable to me in Born's probability interpretation is that, when it is carried out in more detail (by its adherents), the most remarkable things come forth naturally: the probabilities of events that a naive interpretation would consider to be independent do not simply multiply when combined, but instead the "probability amplitudes interfere" in a completely mysterious way (namely, just like my wave amplitudes, of course). In a brand new article by Heisenberg even my much smiled at wave packets are said to have finally found their interpretation as "probability packets." The first is especially comical. It can be expressed this way: the Born probability (more correctly, its square ro, ot) is a two-dimensional vector; its addition is to be carried out vectorially. The multiplication is still more complicated, I believe.

This brings us to his second principal criticism of quantum theory. Schr6dinger, like de Broglie (12) and unlike Einstein, considered that the ~, function must be an objectively existing field in ordinary space, pretty much like the electromagnetic field of Maxwell. Hence the Schr6dinger

Stochastic Optics 187

equation for an atom in an electromgnetic field was viewed by him as a coupling of two classical fields. Indeed Milonni, (~3) in his review of semiclassical radiation theories, recognizes Schr6dinger as the pioneer in this area. Schr6dinger was forced to admit the inadequacy of this idea, in its primitive form, when he realized that the ~ function for a two-electron system was over a six-dimensional configuration space rather than the ordinary three-dimensional one. However, he never ceased to insist on the necessity of real fields, and must, therefore, have been even more estranged by second quantization, where both the ~ field and the Maxwell field are represented by operators in Hilbert space.

In the present article we shall argue that Schr6dinger and Einstein were both correct in their criticism of the universal use of probability amplitudes, and that Schr6dinger, while probably incorrect in insisting on the reality of the ~ field, was correct in treating the electromagnetic field classically. For us the Hilbert-space representation of the electromagnetic field is just a formal way of describing a random field, which we should try to understand classically. We hope that a correct understanding will remove the undesirable features of nonlocality at the macroscopic level that appear in the standard interpretation of quantum electrodynamics. This means that the present article is an attempt to rehabilitate semiclassical radiation theory. We shall show that the form of this theory proposed by Jaynes ~14) is not adequate for describing correlations in "photon" counts, because it treats the electromagnetic field as a deterministic process. If it is treated instead as a stochastic process, we have to regard the zero-point field, which is now a feature of many quantum-optical discussions, (~5) as a real object in the classical sense. As a consequence it becomes necessary to distinguish between field intensities and "photon" densities. Indeed, "photons," according to semiclassical theory, are theoretical artefacts. The objects that really exist are the photoelectrons that are emitted when the electromagnetic field is incident on a photocell. We propose that, instead of a linear relation between the field intensity and the emission probability, there should be a threshold intensity, corresponding to that of the zero- point, below which no emission can occur. With these amendments we believe that the resulting theory is capable of explaining all those "nonclassical" phenomena (16'17) which exclude semiclassical theories of the Jaynes type. This will include those phenomena that have been incorrectly interpreted (ls'19) as evidence of "quantum nonlocality."

We consider stochastic optics to be "semiclassical" in a rather different sense from the Jaynes theory. We think that quantum electrodynamics, despite its formal success, should be regarded as not only incomplete (because of the renormalization difficulties), but also incorrect (because of its nonlocality). What we propose to substitute is more correct (because it


is local) but tess complete than quantum electrodynamics. It is tess complete because it must accept as given, for the moment, the energy-level structure of the nonrelativistic Schr6dinger equation. We would then propose, as does Jaynes, to interpret the transition dipole moments as real, but random, quantities describing the instantaneous state of the atom. This means that the system of transition dipole moments plus Maxwell field replaces the Schr6dinger system of ~ field plus Maxwell field. But the nonrelativistic Schr6dinger equation is itself nonlocal. This is most clearly seen by writing it in the form of a classical Hamilton-Jacobi equation with a quantum potential. (2°) So we are accepting, for the moment, a nonlocal theory to describe the processes occurring inside the atom while insisting that, in optical processes at least, the atom communicates with macroscopic instruments only through the Maxwell field, which is local. The residual nonlocality, which we consider a feature of the approximate theory rather than of the physical world, will disappear only in a con- sistently relativistic field theory which, as Dirac 121) has emphasized, does not exist even in the quantum theory.

2. L I G H T - - W A V E S OR PARTICLES?

Nowadays it has become standard to assume that light consists of particles, because of the discreteness of the detection process; light produces black spots on a photographic plate; a photodetector always records an integral number of counts. This is a naive idea. It is equivalent to assuming that the wind is made of particles because it causes an integral number of trees to fall in the forest. A photographic plate contains a set of grains of silver bromide in a metastable state. An external perturbation, such as a light beam, produces a random blackening of the grains, whose density is more or less proportional to the intensity of the beam.

Thus the existence of the photoelectric effect does not, by itself, give evidence against a purely wavelike description of light. In advocating such a description, we believe we are following in the tradition of Max Planck (22) who set out to describe the emission and absorption of light in the "most classical" (that is realist) way possible.

Even since Thomas Young's experiment of 1808 we have known that light is wavelike. Young's experiment has been performed using an ultraweak source. (23~ The two slits of Young have also been replaced by a beam splitter (24) and by a pair of independently tuned laser sources. (25)

For someone like Schr6dinger, or the present authors, all this evidence is unremarkable. It confirms that light is wavelike. No amount of hyperbole (26) about the interference between two images of a distant star


should persuade us that there is anything quantum like about this simple phenomenon. It is precisely what Thomas Young would have expected.

What is remarkable, from the point of view of Young, Schr6dinger, and the present authors, is the experiment, described in many books, (z7'~8) in which we place two photomultiplier detectors behind the two slits of Young's experiment and observe no coincidences. What is not explained in these books is that the latter experiment has never been done. It is a thought experiment, and until it has been realized we are all free to speculate on what result will be obtained.

To Schr6dinger, however, belongs the credit for suggesting that we might learn something by studying the correlations in photoelectron counts in the two channels of a beam splitter, for he suggested the first experiment of this type, performed by Adam, Janossy, and Varga. (z91 These authors found very few coincidences, and they thought that these few could be explained as "accidents" arising because two different "photons" happened to pass into the two channels at the same time. They therefore thought they had proved that a "photon" could not be divided.

We now know that this conclusion was incorrect. A thermal source, no matter how attenuated, may be described as a classical Gaussian stochastic process (no photons!). The number of coincident counts in the two channels of a beam splitter is actually greater for such a source than it would be for a source of constant intensity, a phenomenon which was correctly predicted classically, derived by the leading quantum mechanics of the day, t3°~ and duly observed by Brown and Twiss (3j~ in the year following the experiment of Adam, Janossy, and Varga.

It has been acknowledged in quantum optics since 1974 (32,337 that no conclusions about the alleged particle nature of light may be drawn from experiments using thermal light. This explains why so much effort in recent years has gone into the study of cascades (16) and of resonance-fluorescence sources. (17) We shall discuss these nonthermal sources in a later section.

It is amazing, in view of all this experience, that the error of Adam, Janossy, and Varga is now being repeated. In the famous delayed-choice experiment of Alley et at. (34) the laser (that is, constant intensity) source is chopped into 100 picosecond instalements. The reason given for the chopping was to manufacture "single photons," but it is known from the experimental work of de Martini (35) that its effects is to convert the source into thermal light. On analyzing the coincidences, it is found that they can all be explained (36) as arising from the fairly rare "two-photon states" coming from the source. Precisely! The frequency of such states is given by the Bose-Einstein distribution, which, as Heitler told us more than 30 years ago, ~37) is the rather clumsy way a classical Gaussian process is represented in quantum field theory. We can be reasonably confident that,


if the intensity correlation is compared with the separate intensities in the two channels, it will be found to satisfy the reltion predicted by Brown and Twiss, which is nowadays treated in standard texts on classical coherence. (38) If Professor Wheeler manages to persuade anyone to do the same experiment with the light from a gravitational lens, (26) we expect the same outcome. It looks as if our delayed choice is one between wave behavior in a pair of channels and wave behavior in a single recombined channel. Where are the particles?

The moral of all this is that no experiment with thermal light will be able to show any nonclassical features. Therefore we will not need to explain those experiments in this paper. We will concentrate, instead, on phenomena dealing with so called "one-photon" or "few-photon" states of light. The most remarkable among these experiments have been those showing anticorrelated counts in the channels of a beam splitter, (3~'33) the atomic cascade tests of the Bell inequalities, (3'39'4°) and the demonstration of "antibunching" in resonance fluorescence. (41) Though only the second type was specifically designed to test for nonlocality, it has been pointed out (42~ that all these bodies of data seem to imply that this highly implausible feature of all existing quantum theories is a real physical phenomenon.

3. SEMICLASSICAL AND STOCHASTIC RADIATION THEORIES

It is well known that semiclassical radiation theory, in which the Maxwell field is not quantized, gives a satisfactory treatment of "single- photon" processes of absorption and stimulated emission/43) If we introduce into the nonrelativistic Schr6dinger equation of the atom a term representing the radiative reaction, the semiclassical theory may be extended to treat spontaneous emission. This much is conceded in standard reviews of the theory (~3'16) and also in textbooks of quantum optics. (44)

Semiclassical radiation theory of the Jaynes type has two points of weakness. First, as pointed out by Milonni, (13) it seeks to couple a deterministic process (the Maxwell field) to a stochastic process (the Schr6dinger field). This it does by considering the expectation current, rather than the instantaneous current, as the source of the electromagnetic field. Mitonni argues, and we agree, that this moves the Jaynes theory towards a Copenhagen-type interpretation rather than a statistical interpretation of quantum mechanics; if stochasticity is denied to the electromagnetic field, the process of detection by a photocell is like the collapse of a wave function. Second, as stated by Clauser and Home, (45) a deter-


ministic radiation field always becomes less intense on passing through a linear polarizer, and therefore satisfies the condition of no enhancement. This means that it must satisfy certain homogeneous Bell inequalities (con- necting coincidence photon-counting rates with each other, see Section 6) that are known to be violated experimentally. ~3'39"4°~ Because the deterministic form of semiclassical theory is the best known, it is now widely, but incorrectly, assumed that all semiclassical theories are excluded by this experimental evidence; one of our principal aims in the present article is to show, by explicit example, that a stochastic classical treatment of the Maxwell field resolves both of the above difficulties in the Jaynes theory.

The treatment of the electromagnetic field as a stochastic process goes back at least to Planck's second theory of 1911. (22~ This theory was derived specifically as an alternative to Einstein's theory of light quanta(46); Indeed, when the current controversy over quantum nonlocality has been resolved, we think that Max Planck will be honored not only as the founder but also as the first opponent of quantum theory. Certainly we can already claim that the zero-point field, now widely recognized as important in quantum electrodynamics, was discovered by Planck using mainly classical ideas. The classical electromagnetic field was also treated stochastically, though without taking into account its zero-point part, by Brown and Twiss (38~ in their successful prediction and analysis of super-Poisson coincidence counts of photoelectrons in the two channels of a beam splitter (see previous section).

The missing ingredient, both in Jaynes' deterministic theory and in Brown and Twiss' stochastic theory, is precisely the zero-point field of Planck. This was largely ignored between 1917 and 1955, though it was used by Welton/47~ to give a semiclassical treatment of the Lamb shift. Since i955, it has received something of a revival under the title of random or stochastic electrodynamics, and has been used to explain classically a number of phenomena thought to belong to the exclusive preserve of quantum theory. There are several review articles of this subject (48-51) and one excellent popular account. (s2) Nevertheless the theory is not widely known, so we begin by summarizing it.

The classical zero-point field has a vector potential represented by

A(r, t) = (2V)-1/2 ~ ~ c(h/~o)l/2[a(k, 2) ~(k, 2) e ik - r - i e J t

k 2

+ a*(k, )~) ~;(k, 2) e - i k r + i°~'] (l)

where V is an (arbitrary) normalization volume, c the speed of light, h a constant, ~(k,)~) a (polarization) unit vector, 2 = 1, 2, and a(k, 2) are


complex, Gaussian, independent random variables, each with zero mean and random phase, the variance being

(a(k, 2)a*(k, 2)) = 1 (2)

Such a field has an energy density per unit frequency interval

p(O9 ) = hoo3/( 27z2c3) (3)

This field has been shown to be (a) Lorentz invariant, (b) spatially isotropic in all Lorentz frames, (c) invariant under adiabatic compression, and (d) invariant under scattering by a dipole oscillator moving with arbitrary constant velocity. It is the only field posessing all of these properties, except that the constant h is arbitrary. Several quantum properties manifest themselves when certain material objects are immersed in this field, provided h is given its normal quantum theoretic value (Planck's constant divided by 27r).

The objects most extensively studied in stochastic electrodynamics have been the charged harmonic oscillator, the free particle, and macroscopic dielectrics. The first of them is by far the simplest, because it interacts with a narrow band of frequencies of the zero-point field. One finds that an equilibrium is established, in which the radiative loss due to the oscillating particle's acceleration is balanced by the energy it picks up from the zero-point field. The joint position-momentum probability density is precisely the Wigner distribution for a quantum harmonic oscillator in its ground state. Furthermore, if we suppose that the zero-point field in a cavity at temperature T has a spectrum that is the sum of (3) plus the Planek spectrum, we obtain the Wigner distribution for the appropriate mixture of excited states.

Although both the position and momentum distributions for the above systems are therefore apparently "correct" according to quantum mechanics, the same cannot be said for any of the other common dynamical variables. For example, while the energy has its "correct" mean value, at zero temperature, of ½hog, this quantity has a nonzero dispersion. Similarly, while all the components of angular momentum have zero mean, their mean squares, and hence the expectation value of the quantity L z, are not zero. This is therefore no possibility of establishing complete equivalence with the quantum formalism in which the ground state has zero angular momentum. Nevertheless, such "incorrect" results may help us to improve our understanding of the system. For example, if we add a uniform magnetic field, we find that the contribution to the magnetic moment arising from the nonzero value of L 2 gives precisely the diamagnetic susceptibility of a harmonic oscillator. This solves a rather old problem in classical statistical mechanics, known as the Bohr-van Leeuwen


theorem, according to which diamagnetism is impossible. Through the study of the free particle in stochastic electrodynamics, it has also been possible to obtain the correct expression for the Landau diamagnetism of free particles.

The other body of results in stochastic electrodynamics has resulted from a study of macroscopic dielectrics. The simplest system of this type is a pair of parallel conducting plates. Because of the new boundary conditions on the zero-point field imposed by the presence of the plates, there is a change in the total field energy, resulting in an attraction between the plates. This is known as the Casimir effect. It may be regarded as a particular example of a long-range van der Waals force, and a similar treatment may be carried out using plates of other shapes and dielectric properties. Although the calculations resemble closely the corresponding calculations based on quantum electrodynamics, it is fair to say that many workers in this area recognize the extra depth of understanding provided by the stochastic interpretation of the formalism.

We can point then to some insights provided by the assumption of a real zero-point field. However, it must be admitted that, compared with some expectations among the originators of stochastic electrodynamics, including ourselves, the theory has failed. It gives a qualitative explanation for the stability of atoms, and indeed it is possible to show that the hydrogen atom has more or tess the correct size in the theory. But it gives no explanation for the sharp line spectra. Similarly while there exist several arguments for the Planck-plus-zero-point spectrum in a cavity at nonzero temperature, the problem of the equilibrium between matter and radiation still retains some anomalous features known from the earliest days of quantum theory.

The success of stochastic electrodynamics in the interpretation of the Casimir effect, and its failure in the hydrogen atom, suggest that the theory of emission and absorption of radiation needs some modification, but that the interaction of radiation with macroscopic bodies is the same as predicted in classical electrodynamics, except for the presence of the real zero- point field. This has led us to develop stochastic optics: the theory of phenomena involving electromagnetic radiation as it interacts with macroscopic bodies, without attempting to give a microscopic theory of these interactions, but inclluding the zero-point field. We shall not include phenomena involving radiation of long wavelength simply because this can be studied by purely classical theory [the zero-point field is negligible according to (3) in this case]. We also do not include radiation with short wavelength because it cannot be considered to interact collectively with macroscopic bodies. Then we are left with visible, near infrared, and near ultraviolet light, which is just the domain of optics.


The basic idea of stochastic optics is to treat the transmission of light (through lenses, mirrors, polarizers, etc.) exactly as in classical optics, but including the existence of a zero-point radiation present everywhere. The assumption is made that the zero-point radiation has the same nature as ordinary light but, for reasons to be discussed later, it cannot be detected directly. In consequence, we assumed for the transmission of light--including the zero point--the same laws as in classical optics. On the other hand, we shall use ad hoc assumptions for the emission and absorption of radiation. We hope that these rules will one day be derived from more basic (classical, although stochastic) principles.

After this general idea of what stochastic optics is, we must consider two problems before developing it. The first one is how to separate, from the extremely energetic sea of zero-point radiation, the part which is relevant in a given phenomenon. The second problem, which is related, is to explain why the zero-point radiation alone does not activate "photon" detectors. These problems are dealt with in the following section.

4. THE DETECTION OF LIGHT SIGNALS

The zero-point field, as given by Eq. (1), has a very high energy. Indeed it is infinite if some high-frequency cut-off is not considered. Even with a cut-off, the contribution of the short wavelengths would be very large. Therefore the question arises: How a light signal--sometimes a weak light signal--may give rise to observable phenomena while the strong zero- point field is not observed? In order to get an answer, we must take into account, in the first place, that in stochastic electrodynamics it is assumed that the zero-point field is one of the causes of the difference between classical and quantum physics, so in that sense it is frequently observed. In the second place, we must remember that the (Maxwell) equations of the electromagnetic field are linear, which means that each part of the radiation evolves independently. This implies that, in stochastic optics, we may forget about the zero-point radiation with frequencies outside the visibe or the near infrared or ultraviolet.

Even if we consider only this range of frequencies, however, the following problem appears. Any sample of radiation, e.g., a light signal, is just a set of values for the variables a(k, 2) of Eq. (1). But any set of values is possible for the zero-point field alone according to our assumptions in Section 3. In consequence, it is impossible in principle to distinguish any light signal from a fluctuation of the zero-point field. In order to solve this problem we must assume that the set of values of the variables a(k, ,~) corresponding to a light signal is extremely improbable for the zero-point


field alone. However, this implies that we cannot exclude the possibility that some accidental fluctuation of the zero-point field may appear as signal (e.g., giving rise to activation of of a photon detector).

We may consider many kinds of light (coherent, chaotic, etc.), but in this paper we shall be mainly concerned with light signals which in quantum optics are said to correspond to "one-photon states." As explained in Section 2, "nonclassical features" in quantum optics have been shown with this type of light. It is reasonable to assume, in a purely classical theory, that "one-photon states" correspond to wave packets of minimum (or almost minimum) uncertainty. This means that

Ax Akx ~- Ay Aky ~- Az Ak z ~- 2~ (4)

Then, if the electric field of the signal is analyzed as in (1), the wave vectors involved [i.e., those such that a(k, 2) departs significantly from the average (2)] are inside a volume in k-space that is of order 8~ 3 divided by the volume occupied by the light signal in ordinary space.

Our idea about the wave packets corresponding to one-photon states is similar to Einstein's original proposal (53) of needle radiation (Nadelstrahlung) interpreted literally as a strongly directed pencil of radiation. Nowadays we have at least one macroscopic example of such a pencil in synchrotron radiation, where the directed character is a consequence of the ultrarelativistic motion of the charged particle in its orbit. We propose that, for reasons not yet understood, the light radiation from an atom is of this same character.

We see that, for the purposes of stochastic optics, it is convenient to analyze the full radiation field in terms of cells of a six-dimensional space consisting of the product of k-space by ordinary space. Each one of these cells has a volume 8~ 3. From (3) it can be derived that the energy of the zero-point field in a unit cell is he) on the average, with ½he) associated with each polarization. In fact, by writing the energy per unit volume in six- dimensional space as f ( k ) , we must identify

f ( k ) d3k = p(e)) de) = he)3/(2~2c 3) de) (5)

Now, as the distribution is isotropic, we have

which leads to

k = e)c, d3k = 4nkZdk (6)

f ( k ) = he)/(8n 3) (7)

A more detailed calculation from (1) shows that the amount of energy


within each cell having a given polarization (i.e., 2 = 1 or 2 = 2 ) is a random variable, 11, with a probability density given approximately by

Pt(I1) ~ 2(ho9) -1 e - 21~/h,~ (8)

The approximate nature of (8) is due to the fact that the exact distribution depends on the specific form of the wave-packet representing the signal. If we consider the total energy, I0, associated to both polarizations, we have, by the standard rule of addition of random variables,

io P(Io) = fO p '(11 ) P l ( I° - 11) dI1 = 4Io(hco)-2 e-2I°/t'~° (9)

The energy Io of (9) is what must be compared with a "one-photon" signal. We shall call it the "relevant" part of the zero-point field for a given signal. Note that cells in six-dimensional space must be chosen with reference to a given signal, and that the phase of the zero-point radiation is well defined over a unit cell, but changes from celt to cell.

In order to describe the state of polarization of a "one-photon" signal we need four parameters and not just one (the energy) as in (9). These could be the amplitudes and phases of the two components of the electric field vector (the component along the direction of motion is zero, of course). For later convenience it is better to use the four parameters {E, ~, ~9, Z} related to the electric field by

E = R e ¢~t + iz (E cos ~, E sin ~b e i~') (10)

where R means real part and

0 ~< ~b ~< n/2, 0 ~< t#, )~ ~< 2~c (11)

A similar description can be apptied to the zero-point radiation. From the assumptions of stochastic electrodynamics [see Eq. (1)] can be derived the probability density of these variables for the zero-point radiation within a cell of six-dimensional space. It is

E 3 sin 2~ exp[- -E2/(2a)] (12) p(E, ~, ~O, X)= 8n2a2

where, assuming that all parameters are constant within the cell, we have

a = lWo/4v

v being the volume of the cell in ordinary space. It is not difficult to check that ( t2) is rotationally invariant and that it is consistent with (9).


In stochastic optics we do not assume the existence of photons. (In certain cases, we may consider instead needle wave packets carrying an energy of the order he) as stated above). Therefore "photon counters" are simply macroscopic devices that can be activated when some amount of light enters into them. For clarity we shall consider phototubes, but the analysis of other "photon detectors" (e.g., photographic plates) would be similar. A phototube is constructed so that it can be put in some metastable state (ready to count) which may go to the stable state under the action of light, this process giving rise to a count. Formulating a purely classical theory of the activation of a phototube (or the blackening of a grain in a photographic plate) is not easy. In fact, in normal classical optics, it is assumed that the energy of a light beam is distributed more or less uniformly over the wave front. Then, if we assume that the activation starts with the detachment of an electron from an atom, the time needed before enough energy arrives at the atom is of the order

t ~ E / I a (13)

where I is the incoming intensity, cr the cross section of the atom, and E the ionization energy. In consequence, it may be quite large for weak light, which does not agree with observations. This has been one of the most widely used arguments for the existence of photons from the early days of quantum theory.

The situation is different in stochastic optics. The existence of the background noise (or zero-point field) gives rise to intensity fluctuations of any light beam, both in time and space. As a consequence, the power arriving at the relevant part of the phototube may have also fluctuations, and Eq. (13) no longer holds. The probability of activation of the detector between t and t + dt will depend in a complex form on the power (and its distribution in frequencies) arriving at the detector at times earlier than t. On the other hand, it is plausible to assume that the activation of the detector at any time modifies the probability of activation at later times, thus explaining--a least qualitatively--the sub-Poissonian statistics of the photon counting with weak light signals in some conditions. (54"s5) We leave for later work a more detailed discussion of this point, and we shall consider here only the case of a "one-photon" signal, that we assume to have a small enough size to either enter completely in the detector or not enter at all (this fits with the idea of needle radiation). We also assume that the time duration of the signal is much shorter than the detection window, as it is usually chosen. Finally~ we consider only the linear regime where the probability of a detection event during the window is much smaller than unity. In these conditions we propose for the probability the expression

P = c o n s t ( I - Ith)+ (14)


where I is the integrated energy in the optical frequency domain entering the detector during the window and /th is a threshold. The notation ( )+ means putting zero if the bracket is negative. The intensity I will include the zero point, because we postulated that there is no difference in nature between the signal and the zero-point radiation. In agreement with our discussion in Section 3, Eq. (14) applies to every unit cell of six-dimensional space within the detector at time t.

It is obvious that Ith must be above the average intensity of the zero- point radiation so that P in (14) is zero when there is no signal. However, the threshold assumption alone is not enough to prevent activation of detectors by the zero-point radiation. In fact, the zero-point radiation fluctuates in intensity according to (9), and a too large probability exists that the intensity exceeds the threshold for any reasonable choice of I~h. The problem of the dark background is a serious one to which we intend to return in the future.

In the present article we shelve the dark background problem by making a modification of the model. Our modification consists in replacing (12) by the density

sin 2~ p(E, ~, 4', Z) =---4U-~ ~(E- Eo) (15)

E o being chosen such as to make the corresponding intensity equal to he). With this modification I is never greater than Ith, provided that Ith is taken to the greater than he) unless there is a signal present. We may therefore use (14) unamended.

One could argue that the replacement of (12) by (15), making possible the use of (14), means that stochastic optics is not really derived from stochastic electrodynamics. That is not our view. We believe that the actual zero-point field is given by (12). It could be that the problem of the dark background posed by (12) will be resolved only when we have a complete stochastic theory covering all the processes of emission and absorption by atoms. What we claim in the later sections of this article is a great deal more modest, but substantial nevertheless: that by considering the angular dependence of (12) in isolation we can explain a large number of the "nonclassical" results of quantum optics. Our model of the zero-point field differs somewhat from that used in explaining, for example, the Casimir effect in stochastic electrodynamics. Nevertheless, we think that the two models are sufficiently closely related to regard both as having been derived from a single theory. For more details on the description and detection of light signals, the reader should consult Ref. 56.


5. THE ACTION OF A BEAM SPLITTER

A "single-photon" signal from an atomic decay is represented, according to the model of the previous section, by the set of auxiliary variables

2 = ( ¢ , ~ , Z ) , 0<<.¢<.n/2, 0 < ~ , X~<2n (16)

related to the electric field components as in (10), i.e.,

E = R e i°~' + iX(E cos ¢, E sin ¢ e i~') (17)

Actually these auxiliary variables may be supposed to vary over a time of the order of the natural lifetime of the atom, typically about 5 ns. Two methods are commonly used for analyzing the coincidences between the two channels of a beam splitter. In the antibunching observations on resonance-fluorescent signals, ~41t the coincidences are classified according to their relative arrival times in the two channels, the time difference being anything between zero and 100 ns. We shall refer to this as the "time- dependent analysis," and postpone discussion of it till Section 8. The other type of analysis, which we study in this section, consists of counting, for various signal densities, the number of joint events in the two channels, occurring within a fixed common "window" (see, for example, Ref. 33) of about 10 ns. This we shall refer to as a "time-independent analysis," and will show that it is possible to understand experimental data of this type using the above model and ignoring completely the time variation of phase and polarization. Thus, until Section 8, the only adjustable parameter in the signal is its intensity, while the distribution of 2 in the signals emitted by an atom will be assumed to be similar to the zero-point distribution (15), i.e.,

P(¢, 6, Z)= ( 47~2)-1 sin 2¢ (18)

According to the ideas of deterministic classical optics, the beam splitter simply produces identical signals in the two channels, with intensities equal to one-half of the incident intensity, that is,

E~,= E~ = 2 1/2E (19)

In stochastic optics, however, we have to take into account the interference of these waves with the relevant part of the zero-point field. The meaning of "relevant" was explained in the previous section; we consider only those components, in each channel, having wave vectors in the same unit cell of


six-dimensional space as Es, and Est. Reference to Fig. t will show that these originate in a set of modes represented by

Eo = R e TM + iZ°( E o cos ~bo, E0 sin ~o ei~'° ) (20)

The set 20 = (¢o, ~o, )~o) has the same distribution as 2, but the two sets of random variables are independent of each other. The effect of the beam splitter on E o is to produce noise in both the t- and r-channels

Eo, = - E o r = 2 - 1 / 2 E o (21)

and we may suppose that the total of signal plus noise in the two channels is the superposition of (19) and (21)

E, = 2-I/2(E + Eo) , E,. = 2-I/2(E - Eo) (22)

The description we have given of the beam splitter has a great deal in common with that in quantum optics, where the role of the zero-point field is also recognized and, indeed, where a diagram similar to Fig. I has been discussed. (tS) However, in stochastic optics the zero-point field is con-

E

E o

~E~ EO)/Vr2

Fig. 1. The action of an ideal beam splitter in stochastic optics. The incoming signal E is mixed with the zero-point component Eo and split to produce two signals whose total intensity is E 2 + E~. The relative intensities of the two components depend on the phase of Eo relative to E.


sidered to be just as real as that of the signal. The only feature of the signal, as opposed to the noise alone, which causes detection events to occur in the photomultiplier is its greater intensity. As discussed in the previous section, we propose to accommodate this idea by supposing that the detection probability P in a given channel of a signal is (11), which we rewrite as

P = ~ (E-"~ - Ith ) + (23)

where ~ is a constant representing the efficiency of the detector and E z denotes the integral of E 2 over the volume occupied by the signal in ordinary space. This formula differs from previous semiclassical treatments (a4) only in the assumption of a nonzero value for Ith. We now define the normalized counting rate

n = P(~ E 2) - ' (24)

together with the two dimensionless parameters

= E/Eo, 7 = I ,h(E~)-1 (25)

The normalized counting rate in the undivided channel is now

n = f12 _ y (26)

while those in the two divided channels are

nt.r = [½(/~2 + 1 ) - 7

+ fl{cos ~ cos ~o cos(z - Zo)

+ sin ~b sin ~bo cos(tp - ~Po + Z - ;go)}] + (27)

Experimentally, "single-photon" signals are best investigated by means of atomic cascades; a detection event triggered by one of the "photons" opens an observation window in both channels of the beam splitter used to analyze its cascade partner (see Fig. 2). The quantities measured are the rate of triggers N1, the counting rates N t and N~ in the two channels during the periods when the window is open, and the coincidence rate N~. for these two channels, again during the open-window periods.

According to the standard analysis of quantum optics,/33~ the parameter

= NI N~. / (NrN,) (28)

should tend ro zero as the cascade rate tends to zero. This is because we are then in the limit where at most one "photon" is said to be passing

825/[8/2-7

202 Ma~hall and Santos

-( PM 1

I I

/•PM r

i _ / " i S t

w

Fig. 2. Triggered experiment using an atomic cascade. The detection of a "photon" by PM1 produces a gate w during which the photomultipliers PMt and PM r are active.

through the beam splitter at any time, and in this limit there are no coincidences.

According to stochastic optics, the corresponding value of ~ is determined by the parameter

0~0=.

where the ensemble average is

(nr)(nt)

( n r ) = ( r / t ) = f fir(J- )-o) P()o) P()~O) d2 d2 o

(29)

(30)

and (nrnt) is defined similarly. The apparently formidable sixfold integration is carried out by making a transformation which effectively converts the elliptical polarization (¢o, ~ko, go) to a plane polarization having ¢o = )~o = 0. (56~ We then obtain

(n r ) = ( n , ) = rc -1 fo r dz ~o~/2 sin 2¢ d¢[½/32 + ½ - 7 +/3 cos ~b cos Z] + (31)

fo fo (nrn,)=rc -1 d)~ sin 2q~ dq~[(½/~2 -t-- ½ -- y)2 -- f12 cos2 q~ cos2 )~] + (32)

In Table I we give the values of c%, calculated from (29), (31), and (32), for various values of the parameter ft. Rather than treating the second parameter 7 as adjustable, we have preferred to fix its its value so that, for each/~,

(nr)=½n (33)


Table 1. Values of the Anticorrelation Parameter eo and Threshold Parameter 7 as a Function of the Signal-to-Noise Ratio f12

for a Perfect Beam Splitter ~

f12 NO 7

2.283 0.000 1.642 2.402 0.072 1.375 2.465 0.111 1.327 2.560 0.164 t,275

It is assumed that the photomultiplier detectors are operated with an anode bias which produces a counting-rate reduction of exactly 50 % in each channel of the beam splitter. The quantum value of e0 is zero.

This means that the "photon" counting rate is reduced by one-half when the beam splitter is inserted, or, equivalently, that the sum of the "photons" in the two channels of the beam splitter is the same as in the undivided channel. Actually the threshold is an experimentally variable quantity, determined by the bias voltage on the anode of the photodetector. Our theory would therefore predict a dependence of n - l ( n r ) on the bias voltage, but as far as we know there are no experimental data available for testing this.

F rom Table I we note that, for the limiting value f12= 2.283, c% takes its quantum-theoretic value. It is also fairly easy to see that, as/32 increases above the last value given in Table I, C~o tends rapidly to its deterministic value of 1. Though entirely classical, the model violates the inequality

c~ ~> 1 (violated) (34)

Previous discussions (17'44) have cited data violating (34) as evidence for the quantum nature of light, and such data have recently been said to exhibit quantum nonlocality directlyJ 42) The model we have developed shows that, within a realist analysis of the experimental data, violation of (34) should be taken rather as evidence for the existence of a real zero-point field together with an intensity threshold in all photodetectors.

In order to determine what value of f12 is appropriate, we must examine the experimental data. The ideal experiment is one in which there are no accidental coincidences; the number of cascading atoms in the observation area during a window w is not greater than one. In the Orsay experiment, (33) the data for such an ideal situation may only be inferred by extrapolating the data from finite values of an experimental parameter,

825/18/2-7"


which those authors designate Nw, the average number of cascading atoms in the observat ion are a during w. Indeed, defining

x = - ( l + N w ) -2 (35)

the parameter c~, defined by (28), takes the value (56)

o~(x) = 1 - x + o:oxl/2/p (36)

where eo is given by (19) and p is the solid angle subtended at the cascade source by the collecting lenses.

We have made an empirical fit (33} to the data f rom the Orsay experiment, using the function (see Fig. 3)

c~(x) = 1 - x + yox (empirical fit) (37)

and we have obtained the 90 % confidence interval

o~

0 ~< 7 o < 0.17 (90 % confidence)

tics

Quantum 0 p t ~

EO =0.17

(38)

X= (1 +NW)-2

Fig. 3. Observed values of the parameter c~, defined in (28) for various values of N, the number of cascade events per second, and w, the observation window. The diagonal straight line gives the quantum optics prediction, while the shaded area shows the 90% confidence interval of the parameter Yo defined by (37). A similar fit of the function (36) would give the estimate (39) of/~2, the only free parameter of stochastic optics. The observed frequencies and error bars are obtained from Fig. 2 of Grangier, Roger, and Aspect. ~33)


A corresponding fit using the expression (36) gives

0 ~< ~o < 0.01

giving for f12 the interval

2.28 < flz < 2.31 (39)

In the following sections we shall show that this same range of f12 gives good agreement also with the data obtained in photon correlation experiments.

Grangier, Roger, and Aspect, by recombining the signals in the r- and t-channels, also exhibited the wavelike property of light, obtaining a very high fringe visibility (98%). For the interpretation of the recombination experiment, we refer to Fig. 4. When the signals I~ and Ia are recombined, the result depends on the relative path lengths in the two channels. The maximum and minimum of the fringes in any given channel will occur when this difference is an even or odd number of half wavelengths respectively. Since our theory is a purely wave one, we naturally have no dif- ficulty explaining these features. What is, however, remarkable is that the fringe visibility in our ideal model is actually 100 %. At first glance it might seem that each intervention of the zero-point field at a beam splitter must cause some fuzziness in the pattern. Indeed, a nonzero value of the

la " , , B S !

\ B S 2 \

\ \

\

\ \

\ \

l!b Fig. 4. Recombination experiment. From an incoming signal (1~) and the relevant part of the zero-point field (lb), two waves are produced (1~ and Id) in a beam splitter (BSI). Another beam splitter (BS2) recombines the waves, reproducing exactly the initial intensities if the waves arrive in phase.


parameter c~ at N w = 0 may be said to manifest this fuzziness. Reference to Fig. 4, however, shows that as long as the signal and zero-point fields retain their phases, the incoming intensities of Ia and I b are perfectly reproduced in the outgoing channels of the second beam splitter. This is bacause all the "relevant" modes of the zero-point field are contained within I~ and Id. No new noise can affect the registration of the photodetectors in the two channels of the second beam splitter. The reconstruction of the original signal in the second beam splitter may be regarded as a consequence of the time reversal invariance of Maxwell's equations.

We have therefore explained both the "wavelike" property of recombination and the "partMelike" property of anticorrelation using a purely wave model in both cases. The zero-point, or subthreshold, field plays a crucial role in the explanation of anticorrelation. Now we see, through the analysis of recombination, that, although one of the signals I~, Id is, with high probability, subthreshold, it nevertheless carries "latent order, ''(58) which becomes manifest when the two signals are recombined at the second beam splitter. A term used by some advocates of wave-particle duality (s9) for the subthreshold signal is that of "empty wave," but we believe that our understanding of the phenomenon is greatly improved by the knowledge that this wave is not empty; it carries energy and does not differ qualitatively from the wave said to be carrying the "photon" in the other channel.

6. THE MEASUREMENT PROCESS IN "PHOTON" CASCADES

In this section we shall analyze whether the measurement of the polarization correlation between the two "photons" emitted in an atomic cascade can provide tests of quantum theory against local realism. (3) It is well known that the relevance of correlated pairs of signals was first shown by Einstein, Podolsky, and Rosen (2) and studied quantitatively after by J. S. Bell. (6°) Here we shall discuss the subject in a heuristic manner, just to see the motivation for the stochastic optics calculations reported in the next section. For more detailed treatments of the EPR argument and the Bell inequalities, we refer to the literature. (3'4'45)

Quantum theory consists of a general formalism, equations of motion, and a theory of measurement. It is the latter which gives rise to all problems of interpretation. For instance, in quantum electrodynamics the Hilbert space formalism could be interpreted as a means of dealing formally with sets of random variables (61) and the equations of motion are just Maxwell equations. A classical interpretation of these two ingredients


gives rise to stochastic electrodynamics. Whenever the measurement of microsystems is not involved, no problem arises for a clear physical picture. The typical example is the Casimir effect, where the measurement (of the force between two conducting plates) is macroscopic, and, as stated in Section 3, a classical transparent interpretation is possibleJ 62)

In constrast, when a measurement on a microsystem is involved, the problems of interpretation appear. For instance, the polarization of two "photons" emitted by an atom in a J = 0 ~ 1 --, 0 cascade can be represented in quantum optics by

~t = 2-1/2(XlX2 + y~ Y2) (40)

where x~ means that the polarization of the "photon" 1 (going to the left) is linear in the z x plane, Y2 that "photon" 2 (going to the right) is linearly polarized in the z y plane, etc.). The wave function 0 of (40) possesses the strange feature of "entanglement," which means that neither the left nor the right "photon" has a well-defined polarization but, nevertheless, they are strictly correlated in polarization. According to quantum theory, this correlation implies, for instance, that if the left "photon" is observed to be polarized in the x z plane, then the right "photon" also becomes polarized with certainty in the x z plane. In order to interpret empirical evidence, quantum theoreticians have been forced to introduce the "quantum theory of measurement," which is independent of (and, to some extent, incom- patible with (63)) the equations of motion. A part of that theory of measurement is the assumption that the wave function "collapses" as a consequence of the measurement. For instance, if we put a polarizer orien- ted in the x z plane and a detector on the left beam, the detection event of one "photon" in the detector produces a "collapse" of the wave function

~ x2 (41)

This means that the right "photon," which did not have a definite polarization prior to the measurement, is changed to have a linear polarization in the xz plane by a measurement performed.., on the other photon! This is the well-known nonlocality of the quantum collapse.

The difficulties for an understanding of the quantum theory of measurement are shown by the fact that the subject has given rise to endless discussions for more than 60 years. Thus, the physics journals provide a continuous flow of new papers, each one containing a "trivial" interpretation of the measurement problem which, however, do not challenge objections already raised in the twenties (two recent examples are Refs. 64 and 65). We shall show in the following that stochastic optics does

offer an interpretation of the collapse in the case of an entangled pair of


photons. The interpretation is purely classical, but stochastic, and so it departs from all interpretations within orthodox quantum theory. The stochastic optics interpretation, however, fits perfectly in the line of the EPR (2) and Bell (6°) papers.

EPR analyzed an ideal experiment with two particles correlated in position and momentum, but their argument applies equally well to a "photon" pair wth polarization correlation. (66) The EPR conclusion is that for a reasonable understanding of the nonlocal collapse it is necessary to assume that (40) is not a pure state but a mixture. That is, (40) represents a statistical ensemble of correlated pairs, and the detection process in the left beam selects a subensemble of the "photons" in the right beam, namely the subensemble of the partners of those "photons" of the lft beam that are actually detected. This subensemble is represented by (41). In this way any nonlocality disappears and the quantum "collapse" becomes the standard one of classical theories, i.e., the fact that the probability attached to an event must be changed when new information is available. The EPR argument implies that quantum mechanics is not complete and auxiliary (so-called hidden) variables should be introduced, beside the wave function, in order to specify completely the state of the system. The development of this idea led Bell ~6°) to write an equation for the joint probability of detecting one "photon" on the left and another one on the right after crossing one polarizer each, at angles a and b. With some modification the equation is (45)

p12(a, b) = f pi2(21, )~2) P1(21, a) P~()~2, b) d)~ 1 d22 (42)

Here 21(22) represents the set of auxiliary variables specifying the state of the left (right) "photon," these variables having a probability density 012(21, 22); P1(21, a) is the probability of a detection event in the left detector, and similarly for P2(22, b). (In most papers on the subjec t (3'45) a

single label, 2, is used for all the variables which, for later convenience, we represent here by ,~1,22). These functions must fulfill the obvious requirements

p12>~0, O<~P1, P2<~ I (43)

The condition of locality is contained in formula (42) because neither PI depends on b nor P2 on a, nor Pt2 on either a or b. As the formula derives from the realist analysis by EPR of an experiment with a correlated pair of signals, it is appropriate to take it as the starting point for any local realist model of the "photon" correlation polarization experiment.

Now we are able to study the collapse from a local realist point of view, consistent with Eqs. (42) and (43). The state represented quantum-


mechanically by (40) is represented by the local realist probability density P12(21,22) and the quantum state (41) becomes the ensemble with probability density

G2()'2) = pI(X) -1 f P12(~I, ~2) P1(21, x) d21 (44)

where

Pl(X) = ; P12(21,22) P1(21, x) d21 d22 (45)

is a normalization factor with the physical meaning of the probability of detecting a "photon" on the left beam after crossing a polarizer in the plane xz. Equation (45) can be easily seen to represent the ensemble of partners of the "photons" detected on the left.

A part of the confusion arising in the interpretation of the quantum collapse is caused by the existence of different kinds of collapse. In a system of "photon" pairs coming from an atomic source, we have considered the nonlocal collapse produced in the "photon" by a measurement performed on its partner. It is possible also to have a local collapse if one "photon" beam, say of the right, passes through a polarizer in the xz plane. In this situation, according to quantum mechanics, some of the "photons" are absorbed, but those that pass change from the initial state (40) of undefined polarization to a state of polarization in the xz plane. Therefore, this collapse can be also represented by (41). In contrast to the nonlocal collapse previously discussed, now there is a physical interaction between an apparatus (the polarizer) and the physical system (the "photon" beam on the right). Of course, there is also nothing remarkable in this kind of collapse according to a local realist analysis. However, in quantum mechanics, people tend to believe that the collapse is always a physical interaction (67) or always a change in the available information of the observer. (64) We see that both are possible.

Some confusion also arises due to misunderstandings about what is a measurement in quantum mechanics. The most frequent definition involves the existence of an irreversible act of amplification. ~26) With this definition the local collapse just discussed does not correspond to an actual measurement. The phenomenon has been called a haunted measurement (58) because it allows a reconstruction of the initial state. For instance, in the case of a two-channel polarizer, a recombination in phase of the ordinary and extraordinary rays should reproduce the initial beam. A similar situation has been discussed at the end of Section 5 in relation to the beam splitter.


Now we are in a position to see how the incompatibility between quantum mechanics and local realism arises. To do that we must compare the quantum wave function (41) with the local realist ensemble (43) in the ideal case.

Quantum mechanics predicts that the beam of "photons" represented by x2 (41) contains half the initial "photons," i.e.,

pl(x) =1 (46)

and also that, when the beam crosses a polarizer at angle ~b with the xz plane, it is attenauted by the Malus' law factor cos 2 ~b. On the other hand, the local realist prediction for the attenuation can be calculated from (44) to be precisely twice the quantity given by (42) with a - b = ~b. That is, we must have

p12(a, b) = i cos2(a _ b) (47)

The incompatibility between quantum mechanics and local realism can be easily seen by deriving from (42)-(45) the Bell inequality (45)

p l ( a ) + p z ( b ' ) > ~ p l z ( a , b ) - p 1 2 ( a ' , b ) + p ~ 2 ( a , b ' ) + P 1 2 ( a ' , b ' ) (48)

where Pl, Pz are defined by expressions similar to (45) with x ~ a or x ~ b'. Then, it is easy to see that the quantum predictions (46) and (47) are not compatible with the local realist inequality (48).

It has been claimed (3'~9) that local realism has been refuted and the quantum predictions have been fairly well confirmed. The second part is true (although one of the experiments (3) reported a disagreement), but the first part is not. In fact, any experiment is far from ideal and many correc- tions are necessarily made to take acount of nonideal behavior. We shall discuss here only one of them, the low efficiency (about 10-20 %) of the available photon counters. If this efficiency is ~/, and the experiment is ideal in all other respects, the quantum-mechanical prediction is, instead of (46) and (47),

pl (a ) = ½~1, pie(a, b) = ½q2 cos2(a _ b) (49)

and these are what have been confirmed by experiments. But the result is perfectly compatible with the Bell inequalities (48). Indeed it has been shown that the incompatibility occurs only if t /> 0.82. (68)

The current interpretation of the experiments performed is that the low efficiency of the "photon" detectors is a minor technical problem and, consequently, the quantum predictions (49), confirmed by the experiments with low q, would surely be confirmed with high q, thus violating (48) and


refuting local realism. In order to state this conjecture in a more scientific fashion, new inequalities have been derived which, being independent of the detector efficiency, can be violated empirically. One such inequality is [compare with (48)]

pi2(a, 00) + p~2(oo, b') >~ p12(a, b) - p12(a', b) + plz(a, b') + p~2(a', b') (50)

where P12(a, ~ ) (P12(oo, b')) is the probability of the joint detection of a "photon" pair when a polarizer is inserted in the left (right) beam, but no polarizer is put in the right (left) one. The local realist prediction for p,2(a, 00) and plz(oo, b') involves the introduction of the probabilities P~(21, 00) and P2(22, 00) for the detection of a "photon" having auxiliary variables 21(22) when no polarizer is inserted. Hence, Pt2(a, o0) and P12(oO, b') are calculated through

p,2(a, oo)=fP12()ol,)~2)Pl(zl,a)P2(]42, ~ ) d2~ d22 (51)

and similarly for p2(oO, b'). Then, the derivation of (50) follows from the "no enhancement" assumption

P1(2~, a)~< ~ PI(,)~I, 00), /02(.)~2, hi) ~ P2(/]~2, (30) (52)

considered as "highly plausible". (3) The quantum predictions for p~2(a, co) and Pl2(~, b') are

plz(a, oo) = P12(~, b') = ½tl 2 (53)

so that (50) or related inequalities can be violated even with low-efficiency detectors, and empitical violations have been actually reported.

Our basic criticism to that interpretation of the experiments performed is that local realism is by far more plausible than any particular assumption, like (52) or the extrapolation of (49) to higher efficiencies. The "homogeneous" no-enhancement conditions (52) are much weaker than the "inhomogeneous" last inequality (43) and, consequently, the "homogeneous" inequalities (50) are also weaker than the "inhomogeneous," genuine, Bell inequalities (48). In fact, (50) should be considered as tests of the no-enhancement hypothesis rather than tests of local realism. However, it is not the purpose of this paper to discuss these general epistemological questions, but to show that stochastic optics predicts in a natural way:

(1) the violation of the no-enhancement assumptions (52)

(2) the violation of the "homogeneous" inequalities (50)


(3) an approximate agreement with the quantum predictions (49) for low-efficiency detectors

(4) a practical upper limit to detector efficiencies

(5) a nonlinear dependence of Pl and P12 on detector efficiency, t/, with the corresponding departure from (49) when r/increases.

The first four points are discussed in detail in the next section. Here we comment on the last one, which is more general. The idea that the probability of any detection event increases in proportion to the detector efficiency, which leads to (49), is quite natural for an experiment dealing with identical noninteracting particles. However, it is not at all natural in dealing with a classical wave theory, like stochastic optics. Furthermore, expressions like "100% efficient" are nonsense for waves. It is well known in semiclassical optics that the probability of a detection event does not increase linearly with the incoming intensity/, but according to the taw

P = 1 - exp(-r/I) (54)

which prevents P from becoming greater than unity. In stochastic optics the situation is more complex due to the zero point and the threshold, but certainly a nonlinear dependence of P on r/should be expected, which will produce a smaller violation of the "homogeneous" Bell inequality (50) with increasing r/. In fact, if some enhancement exists such that, for given 2 and a, we have (see (52))

PI(21, a)/Pi(21, o e ) = e > 1 (55)

the parameter e must necessarily decrease with increasing t/, because P1 ~< 1, thus destroying enhancement. As enhancement is a necessary condition for the violation of the homogeneous inequality (50), the degree of the violation wilt decrease with increasing detector efficiency. In summary, the violation of (50) with low-efficiency detectors does not guarantee a violation of the Bell inequality (48) when the detector efficiency increases.

7. THE ANALYSIS OF ATOMIC CASCADES

In this section we shall show that pairs of atomic-cascade "photons," described by the auxiliary variables 2--(¢, ~, )~) with the distribution (18), give an adequate local realist explanation of all the optical measurements made on cascades. In Section 5 we have already seen how such an explanation is achieved in the case of measurements involving beam splitters; we shall now show that essentially the same explanation holds


well for polarizers, with the one adjustable parameter fl, which gives the signal-to-noise ratio, keeping a common value.

We showed in the previous section that a local realist theory giving a satisfactory description of the cascade coincidences must have the property of enhancement, that is,

P(2, ~ ) < P(2, O) (56)

for some values of 2 and O, the latter being the orientation of the polarizer

0~<0~<~ (57)

We therefore consider it an achievement of stochastic optics that it is the first coherent optical theory to give a satisfactory explanation of this phenomenon.

It is a fairly straightforward matter, using the description of the detection process given in Sections 4 and 5, to show how (56) is satisfied. The quantity P(2, Go) is given by (23), and is actually independent of 2;

e ( ; . 2 (58)

To calculate P(2, 0) we simply need to know the action of a (ideal) linear polarizer on the elliptically polarized wave (17). This is a matter of elemen- tary optics; the incident intensity of f12E2 is reduced to

E 2 (after polarizer)

=! t~zPZ[ l+cos20cos2~b+s in20s in2~bcos~]2e ~o (59)

But this is not the total intensity transmitted by the polarizer. A polarizer is like a beam splitter (see Fig. 1) in that it divides modes of the incoming field into one of two channels. This means that, as in the case of the beam splitter, there is a contribution to both channels (see Fig. 5) coming from the relevant part of the zero-point field. The main difference between the two devices is that, for an (ideal) polarizer, the contribution from the zero- point field is polarized perpendicularly to E s, so that we add intensities rather than amplitudes. We therefore obtain

E-~ = E~ + E~ sin 2 ~bo (60)

(Here we have exploited the isotropy of the distribution (15) by choosing the "x" axis of polarization of the noise to coincide with the axis of the linear polarizer.) Now, substituting in (23) and integrating over the noise variables (~o, @o, Zo), we obtain

P(2, 0) = ~oo f [½/~2{ 1 + cos 20 cos 2~b + sin 20 sin 2q~ cos ~}

+ sin 2 ~bo - ?] + p(20) d2o (61)


(E x,Ey)

(Eox, Eoy)

(E x,EOy)

(Eox, Ey)

Fig. 5. The action of an ideal two-channel linear polarizer in stochastic optics. The incoming signal E is mixed with the zero-point component E0 and split to produce two signals whose total intensity is E z + Eo 2. The axis of the polarizer is taken to be in the "x" direction. For a general orientation the action of the polarizer is given by (59) and (60).

where p(20) is given by (15). The integrat ion gives

where

( ( f + ½), f>~O

P(2, 0) = CE--~ j i ( f + 1)2, - 1 ~<f~<0

~0, f~< - 1

(62)

It is now clear tha t there are values of 2 = (~, ~9, )~), for example ~b = 0 and t# = 0, for which (56) is satisfied, tha t is, enhancement occurs.

Before going on to discuss the coincidence rates resulting f rom (60) and (62), let us see, qualitatively, how enhancement is possible in stochastic optics. We remark first tha t the set of noise variables 2o cor responds to what Sh imony (69) has called hidden variables of environmen- tal type. However , it is now evident tha t this au thor was not correct in claiming tha t no theories of this type can satisfy (56). Enhancemen t occurs, for certain 2, because a s ingle-atom light signal is superpolarized. This

f(2,0)=½fl2[l +cos2Ocos20+sin2Osin2~cosO]-~ (63)


means that, to extend a concept of Dirac, the signal has the same polarization both above and below the "sea" of zero-point radiation. On transmission through a linear polarizer the original intensity of flZE2 iS reduced according to Malus' formula, that is, Eq. (59), but, referring to Fig. 5, it is also enhanced by an intensity of ½E 2 coming from the zero point. For a certain range of values of 2, this enhancement effect is greater than the effect of the Malus' formula reduction.

Note that we speak here of Malus' formula rather than Malus' law. This is because we are supposing that the modification by polarizers of the intensities of atomic light signals is the same as of macroscopic light signals. However, because of the intervention of the zero-point field, and the associated assumption of a threshold intensity in the detectors, we expect that certain deviations from Malus' law will be found for "single-. photon" signals. We discuss this matter in more detail in another publication. (56) We remark here only that, as with the beam-splitter evidence discussed in Sections 2 and 5, investigations of Malus' law based on attenuated thermal signals (7°) are of no use for testing the predictions which we make for single-atom light signals.

The singles rate in one channel of our (ideal) linear polarizer is

p(O) = f P(2, O) p(2) d2 (64)

where again p(2) is given by (18). Actually this quantity is independent of 0;

p(O)= 1 2 ~ E ° fl2[(/~2 _ 7)2 + f12_ 7 + 1] (65)

We have adopted the same procedure for fixing the threshold parameter as we used for the beam splitter in Section 5; for each value of fl, V is chosen so that the singles rate in one channel of the polarizer is one-half of the singles rate in the undivided beam.

The calculation of the coincidence rate in an EPR-type experiment becomes possible once we have fixed the joint distribution, designated p12(21, 22), in Eq. (42), of the auxiliary variables of the "photon" pair. The only obvious constraint on this distribution arises from the requirement that our description of "photon" pairs should be consistent with our description of single "photons," that is,

f p12(~-l, 22) d22 -= p(21)

The procedure we have adopted is to put, for the most widely studied case of the 0-1-0 cascade,

p12(21, 22)= (2/~) -1 p(2, ) c](q~ 1 --q~2) c$(~,- ~t2) (67)


This, taken literally, implies a perfect correlation between ~b1(01) and ~b2(~2 ) and statistical independence for )~1 and )%. However, we stress that we have chosen this particular joint distribution solely because it leads to a simplification in the subsequent calculation. We do not believe that such a perfect correlation is a necessary feature of an auxiliary-variable theory.

For the same reasons of simplicity we have assumed that the detection probabilities, Pl(a, 21) and P2(b, 22), appearing in Eq. (42) are both given by Eq. (62) with the same value of fl (and hence also of 7).

With these simplifications the calculation of the right-hand side of (42) reduces to a double integration which we have performed numerically. The results, for various values of the parameter fl, are shown in Table II. Since our model is a fairly crude one, which in any case has not yet been extended to imperfect polarizers, we draw only two broad qualitative conclusions. First, the quantity

Pt2(a, a + O) r(O) = (68) P12(00, 0(3)

violates a homogeneous Bell inequality, because the Freedman parameter, (3) r(Tr/8)- r(3rt/8), is greater than 0.25 for all values of/~2 listed in Table II. Second, comparison of the values of r(O) with the quantum values for an ideal polarizer

r~(O) = ½ cos 2 0 (69)

seems to suggest values of ]~2 in the range

2.28 < ~2 <~ 2.56 (70)

Table 11, Values of the Normalized Coincidence Rate r(O) Defined by Equation (7.13), for Varions Values of the Signal-to-Noise Ratio fi2~

x 3x

2.155 0.593 0.469 0,225 0.057 0.012 2.283 0.499 0.417 0.241 0.093 0.039 2.402 0.474 0.402 0.244 0.105 0.052 3.465 0.464 0.396 0.245 0.1 t0 0.064 2.560 0.452 0.389 0.246 0,! 16 0.064 rQ(0) 0.500 0.427 0.250 0.073 0.0

The quantum value rQ(0) is given on the bottom line for comparison. Both r(O) and rQ(0) are calculated for the perfect-polarizer case.


We have seen in Section 5 (see Table I) that this range of values of f12 is also compatible with the beam-splitter data.

We have therefore established both the importance and the explanation of what we consider to be the new physical phenomenon of enhancement. Some authors have considered it a weakness of the enhancement proposal that it predicts different behavior for imperfect "photon" detectors than perfect ones, suggesting that there is no reason, apart from cost, which prevents us from performing an experiment with the latter. We indicated, at the end of the previous section, one reason why we think this reflects an incorrect understanding of the detection process. The model considered in this section helps us to understand further the limitations of real detectors. Returning to Eq.(58), the quantity P(2, oo) is the probability that a given atomic decay results in a photomultiplier detection event. This quantity contains a geometrical factor f, which is the probability that the light signal ("photon") enters the lens-collecting system of the photomultiplier; the quantity r/, or "quantum efficiency," is then

= f - l < P ( 2 , oo));.

and takes values in the range

0.1 < rl < 0.2

(71)

(72)

Naturally, r/is a function of the experimentally variable quantity y, which is related to the anode bias of the photomultiplier. Since the noise level corresponds to 7 = 1, it seems to us reasonable to consider "perfect" detection conditions as the case y = 1, within our model. Then an order-of- magnitude estimate of the maximum of ~ would be

f12 ~, "max ~ ~ 2 _ 1 (73)

Putting f12__ 2.28, together with its associated value (see Table II) of 7, we obtain

r/ma x "~ 0.7 (74)

There are good experimental reasons for not going to the "perfect" value of 7, and these reasons are easy to understand in terms of our model. If ? is decreased toward 1, the signals will all be submerged in noise, which we understand as excitation of the photomultiplier by the zero-point radiation.

Our argument indicates that the limit, discovered by Mermin, (68) of 82% photomultiplier efficiency cannot be realized, and that it is not possible to produce those experimental conditions, using atomic-cascade


"photons," in which quantum theory predicts a violation of the inhomogeneous Bell inequalities.

For completeness we should mention that the model discussed in this section also explains the "circular polarization" coincidence rates obtained by Clauser/71) This is because an (ideal) quarter-wave plate inserted between the source and the linear polarizer does not change the distribution (18). Indeed it was in looking for a distribution with this property that we found (72) (18), though it is now clear that this distribution possesses some very much more general symmetry properties. As shown in Section 4, such symmetry arises naturally in the description of signals according to stochastic electrodynamics.

8. TIME-DEPENDENT PROBLEMS: PHOTON ANTIBUNCHING

Up to this point we have examined phenomena in which the time dependence is irrelevant. In fact, we have considered the various detection probabilities of light signals (after they have passed a number of optical devices) without paying attention to the statistical distribution of the activation times of the detectors. Now, we shall analyze time-dependent problems. These are, of course, more complicated, so that our treatment will be less detailed than in Sections 5 and 7. For the sake of clarity, we shall deal only with two-time correlations in "photon" emissions from a single atom, which have been measured by Dagenais and Mandel. ~41) Other time-dependent problems could be analyzed in a similar form.

The experiment consists of an atomic source, excited by laser light, which emits a light beam. This is sent through a beam splitter, whence the outgoing beams are directed to two phototubes. After each activation of the first phototube, say at time t, we measure the time of activation, if any, of the second phototube. If this time is called t + ~, the relevant result of the experiment is the number of counts, N(~), as a function of ~ (the time interval ~ is discretized in small units). In all cases, N(~) starts increasing from a value N(0) up to a maximum and then decreases (sometimes in an oscillatory way), approaching an asymptote as r --, oe. The increase of N(z) for small z is interpreted as a tendency of the "photons" to appear well separated in time, a phenomenon called "photon antibunching." (In order to maximize the effect, the experiment is performed in such a way that usually there is only one or zero atoms in the source during the observation time). The quantum explanation of the photon antibunching is that an atom that emits a "photon" needs some time to be excited again before being able to emit a second "photon."

As a result of the experiment it is claimed that no classical electro-


magnetic field is able to exhibit this behavior. This follows from the assumption that, in any classical theory, N(v) should be proportional to the intensity autocorrelation of the intensity emitted by the source, i.e,

(I(t) I(t + r) ) N(z) oc gd(z)= (75)

( I ( t ) ) ( I ( t + z))

Actually, this is not possible because the function gc~(v) so obtained violates the inequality

g(O)~g(z), t > 0 (76)

which must be fulfilled by the autocorrelation of any stochastic process representing the light intensity.

In stochastic optics, however, the conclusion is not correct for two reasons. In the first place, we should consider not the total intensity, I(t), but only the intensity above threshold, Ith, as capable of producing detection events [-see Eq. (11)]. In the second place we must consider the stochastic action of the beam splitter, thus distinguishing between the reflected and the transmitted beams. This means that N(r) is not given by (75) but, using a notation borrowed from Section 5, by

(nr(t) nt(t + r) ) N(z) oc g~(z)= (77)

(nr(t) ) (nt(t + z) )

where nr (n,) is the probability per unit time of a count in the r- (t-) channel respectively. It can be realized that (76) is just a time-dependent generalization of the anticorrelation parameter (29). Now nr(t) and n,(t) are given by (27) with all variables depending on time except 7. If this dependence is known, gs(z) could be calculated by a procedure similar to, although more complex than, the one used in (31) and (32). However, the time dependence of the parameters fl, ~b, ¢, Z of the light beam emitted by the source is unknown because we do not yet have a theory of resonance fluorescence within stochastic optics. In consequence, we will not use (27) but make the simplifying approximation

which allows writing

g ~ ( ~ ) -

with

nr(t) + n,(t) - n(t) (78)

(n(t) n(t+z) U(t)[1-- U(t+z)]) (n(t) U(t))(n(t + z)[1 -- U(t + z)] )

(79)

U(t) =- nr(t)/n(t) (80)


Also, for simplicity, we may assume that n(t) and U(t) are independent stochastic processes. On the other hand, by definition of ideal beam splitter, we must assume that the average of U(t) is ½. Then, it follows that

g,(r) = f , ( r ) [2 - 4 ( U ( t ) U(t + z ) ) ]

f,(~) - (n(0 n(t + ~)>

(n(t) ) (n( t + z) }

(81)

From Section 5, we see that the stochastic process U(t) can be modelled by a process switching between the levels 0 and 1. For large time intervals the memory of the phases of the noise is lost and this condition implies that the number of switchings follows a Poisson distribution. This would charac- terize the process as a random telegraph signal with autocorretation

( u ( 0 ~7(t+z)> = J ( l + e ~') (82)

Where e/2 is the number of switchings per unit time. However, for small times, which are relevant to our analysis, it is more appropriate to use the autocorrelation(56)

(U(t) U ( t + z ) ) = ¼ ( l + C e ~,2), C < I (83)

Putting (82) into (80) gives

g s ( r ) = ( 1 - C e ~'2) f,(z) (84)

This analysis allows a very simple interpretation of the "photon" antibunching phenomenon. For instance, the raw data of one of the experimental runs by Dagenais and Mandel (see their Fig. 2) shows a function N(z) which increases up to about 3 N(0) in 30ns and then decreases exponentially again to about N(0) for large r [the plot ends at "c ~- 100 ns with N(z) ~ 1.3N(0)]. It is enough to assume N(z) oc g,(z) and

f ,(z) = 1 + A e - ~ (85)

in (84) in order to get a fit to the data. Indeed, an extremely good fit is possible with A = 6.46, C = 0.893, fl = 47.0/zs -1, e = 2030/~s -2. This shows that stochastic optics is able, in principle, to explain the empirical results reported.

We emphasize that our explanation of "antibunching" differs radically from the one currently given in quantum optics. In the latter theory (v3) a quantity equivalent to the intensity autocorrelation is introduced, but this quantity violates (76). The anomalous "autocorrelation" is said to be


directly observed through a negative correlation between the counting rates in the two channels of a beam splitter. In principle, it may also be observed directly, through the counting statistics of a single detector, but, to our knowledge, this has not yet been done, although the theory was developed almost a decade ago. (73)

In stochastic optics, by contrast, the autocorrelation necessarily satisfies (76), and those features of the counting rates which suggest otherwise, either in a one-channel or a two-channel experiment, must be explained in terms of the actions of the beam splitter and the detector. We have undertaken the two-channel analysis first not only because experimental data exist, but also because locality demands that we treat the two detectors as mutually independent. A single detector, according to stochastic optics, can reasonably be expected to show the self-inhibiting effects referred to in Section 4. The analysis of the counting statistics in a single channel is therefore more difficult. It is clear that further experimental and theoretical work in this single-channel area is needed.

9. CONCLUSION

In this work we have tried to indicate that stochastic optics, though clearly at a very immature stage of development compared with quantum optics, is capable, in principle, of explaining the same body of experimental data as the latter theory. It provides a unified description of "photons" from a wide variety of sources: thermal, coherent, single-photon, cascade photons, and resonance-fluorescence. Above all, we claim that our description is free of the interpretational difficulties of quantum optics; it is fully consistent with a local realist view of the natural world.

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Stochastic optics: A reaffirmation of the wave nature of light