Transition from Classical, “Maxwell-Boltzmann” to Quantum, “Bose-Einstein” Partition Statistics by Stochastic Splitting of Degenerate Light

  • F. De Martini
  • S. Di Fonzo
  • R. Tommasini
Conference paper


The concept of “quantum statistics for distinguishable particles” has been introduced by a remarkable theoretical work a few years ago (1). There it is shown that the classical, Maxwell-Boltzmann (MB) partition law for an ensemble (or a stream) of n particles to be distributed in M “boxes” (or scattered over M channels) can be transformed formally into the Bose-Einstein (BE) law (or into the Fermi-Dirac (FD) law) by allowing the scattering probability over each channel, Wi, to be a stochastic variable instead of a “stationary” cross-section. Since in this process no use is made of “indistinguishability”, i.e., of the fundamental property which is generally attributed to all “quantum” particles, the authors of (1) argue that then the statistics cannot be a criterion for that property. On the other hand, according to the authors of (1), this does not contradict the fundamental prescription according to which “classical” particles obey MB Statistics while “real” particles obey BE or FD quantum-Statistics (2,3). Since the world is actually made of real particles, the interesting transformation theorem given by (1) has been generally considered as a mathematical curiosity with no real physical content. In the present paper we give the first experimental demonstration that the statistical behavior of real particles, i.e., the optical photons, is indeed described either by the classical-Statistics or by the appropriate quantum-Statistics depending on the statistical character of W1.


Photon Number Real Particle Distinguishable Particle Partition Statistics Optical Photon 
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  1. 1.
    J.Tersoff and D. Bayer, Phys.Rev.Lett. 50, 553, 1983MathSciNetCrossRefGoogle Scholar
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    F. De Martini, in: “Squeezed and Nonclassical light”, E.Pike, Plenum, N.Y., 1988.Google Scholar
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    Owing to Bohr’s Completamentarity our photon-counting experiment reveals physical effects that involved only the photon particle-like aspect. This argument legitimates the present statistical approach.Google Scholar
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    The M=2, BE partition law given in the text is a particular case of the general BE law: Pnh= n!(M-l)!/(n+M-1)! (4). The verification of the interstatistics transition theory here reported for M=2 has been obtained for M>>2 in a recent scattering experiment in our laboratory.Google Scholar
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    G(1/m), may be taken as the “quantum-signature” of the electromagnetic field as well as of any Bose-field. An account of this novel method of quantum optics with application to gravity-radiation detection is found in: F. De Martini and K.H. Strobl, Optics Comm., Ref.1.Google Scholar
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    For input |α>-states the statistical picture is valid owing to their n- state expansion. Another picture is provided by electrodynamics.Google Scholar
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    The wide-band noise was generated for sets A,B by reverse-biased 2N2369 transistors. The PC were: EOD125 (A), Lasermetrix 1042 (B), Phase- modulator was Lasermetrix 1039B. All electronics was developed in our laboratory. A “uniform” W1-distribution has been achieved by a suitably clipping via an adjustable saturated amplifier of the wide gaussian-distribution of the noise-voltage V feeding the StO-BS.Google Scholar
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    The state if the input-field coherence was determined by Hanbury Brown- Twiss, StA-BS measurement of g1,2 (5).Google Scholar
  13. 11.
    The most obvious one appears to be the need for rejection of the indistinguishability concept (I), which is in contradiction with MB-statistics (4). In other words, thereare no “classical” and “quantum” particles in physics: as far as I is concerned all particles are the same. Their quantum-statistical behaviour should be determined by scattering events involving nonstationary cross-sections determined by particle motions under momentum transfer in collisions and by the relevance taken in their motion by the dynamical implications of the Heisenberg principle. Since the Statistics is not a fixed particle property, as demonstrated in the present work, we should attribute to the particle-spin the quantum-property that, among other effects, determines the quantum-statistical behaviour of a quantum-gas in equilibrium. Then, according to our model and in agreement with quantum theory, particles with integer or half-integer spins (i.e., “bosons” and “fermions”) are respectively the ones for which the dynamical effect of the spin in the collision process (i.e., theoretically, on the related S-operator) leads to the observed BE or FD equilibrium phenomenology for the quantum-gas (B.D’Espagnat, private comm. to F. D.M.). The classical behaviour of photons interacting with optical instruments has been experimentally verified by: F.De Martini (2) and by R Lange, J.Brendel, E.Mohler and W.Martienssen, Europhysics Lett., 5, 619, 1988. The effect discovered can be reproduced in first-order coherence processes and open new trends in interferometry and spectroscopy. The same concepts can be to FD-statistics to investigate the Pauli principle. This is being done in our laboratory.Google Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • F. De Martini
    • 1
  • S. Di Fonzo
    • 1
  • R. Tommasini
    • 1
  1. 1.Dipartimento di Fisicadell‘Universita’ “La Sapienza”RomaItaly

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