on the possible linear term in light coincidence experiments

Foundations of Physics, Vol. 19, No. 12, 1989

On the Possible Linear Term in Light Coincidence Experiments

P6ter Varga I

Received August 12, 1988

An attempt is made to clarify the confusion about the interpretation of an early experiment aiming to demonstrate the dual nature of light. While interferometer experiments show that photons interact with both mirrors of a Michelson interferometer, it was verified that a photon interacts with one of the detectors put in place of the mirrors. Any deviation from the effect predicted by QED would lead to a term in the coincidence rate linearly proportional to the number of photons; the absence of this term--for the light source used -is in accordance with QED. It is shown that by appropriate preparation of the light source a linear term can be obtained, but this is" not in contradiction with QED.

1. INTRODUCTION

Our experiment on the coincidence of photons ~1) continues to be misunderstood (2) even 30 years after its publication. The experimental aim discussed here was to verify Dirac's prediction on the indivisibility of photons. If Dirac was not right, a term proportional to the number of photons in the light beam would have been observed in coincidence counts. This supposed effect, the nonexistence of which was demonstrated, should not be confused with the intensity correlation effect discovered by Hanbury Brown and Twiss, ~3) the latter work being performed coincidentally with publication of Ref. 1. But the authors of Ref. 3 were "able to face with confidence objections" concerning the existence of the intensity correlation. ~a~ This effect--as measured--is proportional to the square of the photon number and can be explained classically too, if it is supposed that the probability for the emission of a photoelectron is proportional to the

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instant value of the intensity and the probability of simultaneous counts of two detectors is proportional to the square of the intensity. (The problem of energy conservation is left open, of course.)

On the other hand, if a linear term were observed in a photon correlation experiment, it would not necessarily lead to the rejection of Dirac's prediction. Such a linear effect can be observed using a source containing independently radiating atoms, the light emitted by them differing slightly from the Gaussian (thermal) light.

2. THE B A C K G R O U N D FOR THE E X P E R I M E N T

Let us suppose that photons do exist as energy quanta. (The opponents of QED are asked to accept this supposition as a working hypothesis and consider whether the experiments described actually meet Dirac's requirements.)

In the very first chapter of his famous textbook, (5) Dirac introduces the concept of the "translational state of the photon," represented by a beam of roughly monochromatic light, i.e., the photon has a momentum and is localized within the beam. (It can be regarded as analogous to a classical, slightly diverging wave or to the coherent state of Glauber. (6~) Dirac analyzes two thought experiments (gedanken experiments). In the first, he splits the translational state of a photon into the superposition of two or more translational states by means of an interferometer. The essen- tial part of Dirac's statement is that there exists a state consisting of the superposition of states of spatially separated beams with different directions of momenta. The possibility of this superposition may be verified by perfor- ming a real interferometer experiment. If the assumption about the superposition is valid, after bringing together the two states in the interferometer the residual wave will inherit the states of the split photon and will exhibit interference. Moreover, if the interference phenomenon is independent of the photon number, any interaction between photons can be excluded.

The other thought experiment is that before rejoining the split beams we measure the energy in the beams. The answer is well known: The photon will be found in one of the beams and never in both of them.

These thought experiments are familiar to everybody. While physics is based on experiments that are actually performed, the thought experiments were to be fulfilled precisely following the ideas of Dirac. The programme for the experimental work was outlined by Jfinossy, (7~ who also analyzed the difficulties encountered in understanding the basic phenomena of quantum theory by a mind trained on classical grounds.

It may be questioned whether real experiments do satisfy the demands

The Linear Term in Light Coincidence Experiments 1517

and whether they do prove the statements of the theory. Even if the experiments are interpretable on the ground of another theory too (which is not fully the case here), the legitimacy of QED will not be destroyed. This will happen only if an alternative theory explains facts of experience-- at the very least.

3. T H E I N T E R F E R E N C E E X P E R I M E N T

To demonstrate the superposition of the translational states in the interferometer, the separation of light quanta is required. Though the problem was not precisely formulated before QED, the need to perform interference experiments, using light of weak intensity, arose_ in 1909 shortly (8) after the explanation of the photoelectric effect by light quanta in 1907. The first experiment (9) was repeated in 1927; in both cases the means of recording was a photolayer. In the experiments carried out by J~.nossy and N~ray, (~°) the light was registered by photon counting in the common arm of a Michelson interferometer. The arm length of the interferometer was 0.1 m in the first run and 14 m in the second. The intensity was so low that the sum of the mean energy in the two arms of the interferometer was lO-3hv and lO-lhv, respectively. In both cases, the same interference pattern was obtained when the intensity was increased by a factor of 4 decimal orders. The existence of the interference at long arm length shows that the superposition of the translational states exists at long-lasting and distant separations too.

This experiment is open to criticism: Though the mean energy was weak, the number of photons taking part in the interference was very high; it was only the probability of their occurrence in the input beam of the interferometer that was extremely low. If we use the rigorous quantum theory, then all the photons emitted from the light source in all directions are present in the beam with some very small probability, this being determined by the minute spatial angle covered by the beam-forming optical system and other losses in the optical arrangement.

The interference experiment has been repeated several times by other authors: The last time it was performed was by Franson and Potocki. ( ~ The merit of this work was not only the enlarged arm length (45 m), but the use of the very low energy atomic beam light source. According to the published data, the probability of finding a photon (of the wavelength filtered out for inteference) within a 45-m-thick sphere shell around the light source was 10 1.

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4. THE PHOTON COINCIDENCE EXPERIMENT

This experiment was carried out using the apparatus shown schemati- cally in Fig. I. The light source S after monochromatic filtering by F passed into a Michelson-like interferometer consisting of a semitransparent mirror M and two photomultipliers D1 and D2. The number of pulses Nt and N2 were counted, and a coincidence counter of resolving time 2.5 #s registered the simultaneous counts (C).

Let us note the number of photons falling on the semitransparent mirror by n and suppose they are divided into two beams each of n/2 photons. (The mirror was not completely semitransparent, and there was some loss in it, but for the present situation this is of no consequence.) Let p be the probability of counting a photon, thus the counting rate in each of the channels is

N = pn/2 ( 1 )

Now we assume that, in spite of Dirac's prediction, the photons are split. This assumption will not change the form of (1), but only the probability of finding a photon will be p/2 and the number of half photons is n. The probability of obtaining a simultaneous answer from both of the photomultipliers is (p/2) 2, and the number of coincidence counts is given by

C'= (p/2)2n (2)

not by (pn/2) 2, which is the punctum saliens! Since photons come at random, aside from these hypothetical counts,

the coincidence counter obviously registers all the pulses coming from the

S ®

F I I . M

4"---- I 52 /

I

Fig. 1.


two different multipliers with a time difference of less than the resolving time z. These events are really accidental coincidences and always occur when registering random pulses. If Nz is small enough--and this was the case for the experiment under discussion--the mean number of accidental coincidences is

C~ = 2N2z (3)

Thus the total number of coincidences should be, if the photons are split,

C = C' + Co= ½pN+ 2N2r (4)

The number of spurious coincidences could be measured by illumi- nating the photomultipliers by independent sources $1 and $2 (see Fig. 1). Because the mean counting rate for three independent sources could not be made equal, the resolving time was calculated from the formula C a = 2N~Nzr, where the quantities other than r were measured directly with the help of the auxiliary sources.

Formula (4) differs from that of Hanbury Brown and Twiss describing the number of counts for the intensity correlationJ 3) In this case we have, instead of (4),

K = 2N2`gg + 2Net (5)

where 9̀ is the coherence time (,9 < ~ was assumed), and g is a quantity depending on spatial coherence ( g < l ) . The excess coincidence rate is proportional to the number of counts (i.e., to the light intensity) in (4), and to the square of it in (5).

No excess coincidences were found, either in the linear term (because the photons do not split) or in the quadratic term (because of the long resolving time compared with the coherence time and the poor spatial coherence). To be truthful, we would have been in trouble had we used a resolving time of one nanosecond and happened to find the intensity correlation. The resolving time of 2.5/~s was sufficiently short to make the number of spurious coincidences small enough to find a linear effect. If we had realized that more than one boson can exist in a phase cell, we would have used the long resolving time and poor spatial coherence intentionally to lessen the mean number in one phase cell.

To find some qualitative measure for the level of confidence of the experiment, formula (4) was slightly modified to

C = epN/2 + 2N2r (6)

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and we calculated which value of ~ is allowable in terms of statistical error. From three series of experiments, we obtained

ep = (1.8 + 2.6). 10 -6 (7)

The quantum efficiency p was measured in an independent experiment and was found to be p = 3 - 10-3. Substituting this value into (7), we obtain

e+ 3 A e = 7 . 1 0 -3 .

Thus, even allowing for fluctuations amounting to three times of the standard error, the measurement would be incompatible with an assumption that more than 0.7% of the photons were split. 2

From the viewpoint of quantum theory, the same objection can be raised against this experiment as against the interference test: The photon number was too high, and only the detection probability was small. But, if we accept the hypothesis of needle radiation used in Ref. 2 (the authors used it as a means of explaining some quantum effects without introducing photons), this objection is overcome: The wave packets are localized within a small spatial angle, and enter the optical system untruncated. Of course, I do not misuse this hypothesis, since is well known that wide-angle interference experiments (~2'13) disprove the existence of needle radiat ion--at least for slowly moving emitters. It is quite amazing that even at a Schr6dinger centenary conference his important experimental work was neglected.

5. THE M E A S U R A B L E LINEAR T E R M

The number of coincidences measured in split beams may be less than the number of spurious ones (for squeezed light), or they may be more than (for thermal light); possibly they do not differ (for ideal laser light), but this excess rate always depends on the features of the light source itself. The light inherits its statistical behavior from the source and can change it only in interaction. The excess rate-- i f it exists--may depend on the mean number of photons (or the mean intensity) and on other parameters of the light source in various ways. Now we show two examples when the excess rate is positive but a linear function of the intensity. The first example is well known; the second somehow remained unnoticed, but there is now the technical possibility of measuring it.

2The value of 5 + 3zJ~ given for the level of confidence in Ref. 1 was 0.6%. I was unable to repeat the early calculations, but, on evaluating the data obtained from the series of experiments, this is the value I now find.


The fluctuation of the energy inside a cavity that is in thermal equilibrium consists of two terms which are proportional to the energy belonging to the fluctuations of independent particles and that of waves respectivelyJ 14) Exept for some early experiments described in Ref. 15, no experiments have been carried out to measure the correlation nor the fluctuations of the blackbody radiation with any modern technique. When one performs such an experiment, special care needs to be taken in the choice of the cavity-wall material, because KirchholTs theory of blackbody radiation relates only to the mean value of the energy density and not to its fluctuations. As an extreme example, if we were to observe a cavity whose walls were in the phase transition state, the short-term fluctuations ought to inherit the fluctuations of the medium itself.

Let us now turn our attention to the fluctuations emitted by another source consisting of a finite number of independently radiating atoms. According to J~.nossy's theory, ~6~ the classical correlation between the intensities of two beams of light emitted by a source consisting of atoms exhibiting only spontaneous emission may be expressed as

(J=Ja)-(J=)(Ja)=l(g~E~)12+[(g=ga)12+L (8)

where J is the intensity, E the field strength; ~ and/~ denote the parameters characterizing the observation (i.e., the polarization, the time, and the loca- tion of the observation), and L is the linear term, determined by the fourth- order derivative of the cumulative function. Angle brackets represent the expected value of the variable.

Let MdT be the probability of the emission of a wave train in the time interval between T and T+ dT, M being the mean number of excited atoms per unit time. Let an emission takes place at moment T' at a point S of the source. The wave train at a distant point P: ( j = 1, 2) can be represented as

{ ~ t exp T s - T' { 20 i[~o(Tj-T')+q~,]} if Tj>~T' q=

if T j < T '

(9)

Where A denotes the initial amplitude of the train, depending of course on the distance between source S and observation point Pj, l denotes the polarization ( l= 1, 2), and

sPj 5=tj c

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tj is the moment of observation at point p/. Using the generating function method, we obtain for the mean intensity

(J~} =MO(A 2 } (10)

i.e., the intensity is proportional to the mean number of emissions during the lifetime of the excited state and to a factor depending on the geometry and polarization.

The first term on the right-hand side of (8) is the squared modulus of the correlation function of the field strength (the first-order coherence), which can be factorized under not very strict conditions:

• (AlAre-i(%- ~")) ( exp [ - i(oJ - ~oo) to] )

" ( e x p - k ° ° S P 1 - (11)

to being the mean time difference and o~ o the mean frequency. The first pair of angle brackets on the right-hand side of (11) corresponds to the correlation of the amplitudes, which equals zero for crossed polarization of naturally polarized light. The second pair contains the Doppler term, and the third expresses the effect of the size of the light source•

The second term on the right-hand side of (8) is similar to the first, except that, instead of the exponential term, we have

lfoO (2t+lto[icOot) dt exp 20

If the frequency is much higher than the extinction constant, the second term can be neglected. Finally,

1 2 2

The L term does not vanish even if the first-order correlation cannot be observed; that is the case if the time difference is long compared with the interference length due to the Doppler effect or if the size of the source is too large. The existence of such a term is only too obvious. Two events are independent if and only if all the correlations of all orders are zero. The field strengths at two points in space, when the time difference is not much longer than the time of emission, cannot be fully independent. If the first-


order cor re la t ion term is zero, some of the higher orders will differ from zero.

To see the o rde r of magn i tude of this term, we normal ize L with the square of intensities, so we ob ta in the n u m b e r of excess counts c o m p a r e d with spur ious ones:

L 1 exp ( - I to l /O) ( A 2 A ~ , ) ( J ~ ) ( J [ i ) ~ m o 2 2 (13) ( A , ) ( A r )

The term L is observab le if the n u m b e r of emi t t ing a toms of the source dur ing the exci ta t ion t ime is small. Tha t was not the case for exper iment . (1) I now es t imate tha t it was M > 10 4. However , the L term can be observed using a very weak l ight source, such as that in Ref. 11.

Final ly , L is l inear, i.e., if we doub le the n u m b e r of excited a toms, bo th the in tensi ty and L will be doubled.

The existence of the L te rm con t rad ic t s the view that the thermal l ight is of Gauss i an character , but this can be conc luded only if the n u m b e r of r a n d o m var iables proceeds to infinity, which is no t the case now.

R E F E R E N C E S

1. A. Adfim, L. J/tnossy, and P. Varga, Acta Phys. Hung. 4, 301 (1955); Ann. Phys. (Leipzig) 16, 408 (1955).

2. Trevor Marshall and Emilio Santos, Found. Phys. 18, 185 (1988). 3. R. Hanbury Brown and R. Q. Twiss, Nature (London) 177, 27 (1956). 4. R. Hanbury Brown, The Intensity Interferometer (Taylor & Francis, London, 1974), p. 8. 5. P. A. M. Dirac: The Principles of Quantum Mechanics, 3rd edn. (Clarendon Press, Oxford,

1947). 6. Roy J. Glauber, Phys. Rev. 131, 2766 (1963). 7. L. Jfinossy, Acta Phys. Hung. 1, 423 (1952). 8. G. I. Taylor, Proc. Cambridge Philos. Soc. 15, 114 (1909). 9. A. J. Dempster and J. Batho, Phys. Rev. 30, 644 (1927).

10. L. Jhnossy and Zs. N/tray, Acta Phys. Hung. 7, 403 (1957); Nuovo Cimento 9, 588 (1958). 11. J. D. Franson and K. A. Potocki, Phys. Rev. A37, 2511 (1988). 12. P. Sel~nyi, Math. Naturw. Ber. Ungarn 27, 76. (1909); Ann. Phys. (Leipzig) 35, 444 (1911);

Phys. Rev. 56, 477 (1939). 13. Erwin Schr6dinger, Ann. Phys. 61, 69 (1920). 14. A. Einstein, Phys. Z. 9, 185 (1909). 15. S. I. Vavilov The Microstructure of Light (in Russian) (Publishing House of the Academy

of Science of the USSR, Moscow, 1950). 16. L. J/tnossy, Nuovo Cimento 6, 111 (1957); 12, 369 (1959).

on the possible linear term in light coincidence experiments

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