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Physical Review A

Comparison between the classical and quantum theories of optical coherence

B. Picinbono, M. Rousseau

Published Apr 1, 1977• 8 citations
Publisher

Abstract

The comparison between the classical and quantum theories of optical coherence is presented by using an idea used in the problems of modulation of light beams of classical fields. In such problems it is assumed that it is possible to vary the mean light intensity of a beam without changing its statistical properties defined by a set of coherence functions. We generalize this idea and introduce precisely the class of optical fields which are consistent for modulation. It is shown that all the quantum fields of this class have a positive $P$ representation and are strictly equivalent to classical fields. Moreover, when the field is assumed to be stationary, an interpretation of this condition is given which in particular makes precise the relations between photon-counting and light-intensity measurements. Finally it is shown that all the quantum fields without $P$ representation cannot be consistent for modulation, and the condition of consistency for modulation appears as a characteristic property of fields strictly equivalent to classical ones.

DOI
10.1103/physreva.15.1648

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The comparison between the classical and quantum theories of optical coherence is presented by using an idea used in the problems of modulation of light beams of classical fields. In such problems it is assumed that it is possible to vary the mean light intensity of a beam without changing its statistical properties defined by a set of coherence functions. We generalize this idea and introduce precisely the class of optical fields which are consistent for modulation. It is shown that all the quantum fields of this class have a positive PPP representation and are strictly equivalent to classical fields. Moreover, when the field is assumed to be stationary, an interpretation of this condition is given which in particular makes precise the relations between photon-counting and light-intensity measurements. Finally it is shown that all the quantum fields without PPP representation cannot be consistent for modulation, and the condition of consistency for modulation appears as a characteristic property of fields strictly equivalent to classical ones.
Conceptual problems in quantum optics are investigated by a study of the physical significance of various types of description of the field of a single mode. The descriptions are examined through the effect of the field on matter. Both undamped and damped modes are considered, the descriptions being those of various degrees of precision in classical and quantum-mechanical formalisms. The matter consists of a large number of "molecules," and the interaction is studied by means of low-order perturbation theory. Formal expressions are obtained for the energy interchange between field and molecules in terms of the zeroth-order (uncoupled) operators, with the effect of the commutation properties of the field variables explicitly separated from that of the excitation of the field, in second order. The significance of taking expectation values of the operator results is discussed, and it is concluded that this procedure is justified only when the experiment under consideration involves an averaging process; thus, the molecular variables should be averaged, but not necessarily those of the field. Conditions under which the averaging process is justified are considered, and it is reasoned that, when such a process is not justified, a statistical description of the field can result only in a statistical description of the interaction process. Results of various field descriptions are compared, and it is shown that for induced emission and for spontaneous emission from coherently oscillating molecules, classical and quantum-mechanical fields are equivalent. However, for a self-consistent theory of spontaneous emission from molecules that are not oscillating coherently, the quantum-mechanical commutation properties of the field are essential if the molecules are treated quantum-mecahnically, and the field must be treated classically if the molecules are treated classically; the type of description used for the excitation of the field is inconsequential. Photoelectric emission is examined, and, within the approximation framework used, shown to be accurately described by a classical treatment of the field. It is pointed out that there does not necessarily exist a connection between "photon statistics," that is, the statistics associated with an ensemble of fields, and photocurrent fluctuations in one or more detectors coupled to the same field.
Coherence properties of quantum optical solitons in the normalized fundamental soliton states are discussed in terms of the integrated two-point field and intensity autocorrelation functions, which are the relevant quantities in the quantum description of soliton interference experiments and in the Hanbury-Brown and Twiss experiment, respectively. Both the spatial and the temporal correlation properties are considered. The correlation functions are shown to be stationary in general, which demonstrates the solitary behavior of quantum fundamental solitons and indicates their stability in propagation. Explicit evaluation of the correlation functions in some limit cases enables us to make comparisons between the classical and the quantum-mechanical results. Conditions leading to the classical limit are derived and the intrinsic quantum phenomena are identified. Of great importance is the contribution of the momentum superposition, which can make the quantum results differ significantly from the corresponding classical ones.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
Non-photon-number-resolving single-photon-counting modules (SPCM's), also called on-off photon detectors, have been used in quantum optics for investigating the properties of various light fields based on the Hanbury-Brown-Twiss (HBT) experiment. However, for different incident light fields the experimental conditions can affect the measured photon statistics. In this paper, the second-order degree of coherence g(2){g}^{(2)}g(2), known as the important factor for quantitying a single-photon source, is experimentally investigated by means of the HBT scheme consisting of two SPCM's. By comparing the results of coherent field with that of the thermal field, we show that the measured g(2){g}^{(2)}g(2) is affected by the photon-counting rate and the resolution time from pulsed to continuous wave fields. The proper counting rate and resolution time for characterizing the exact photon statistical properties of input fields are determined.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
The paper contains a quantum theoretical analysis of laser beam fluctuations and of the light beat experiments with two lasers. With the help of experimental results on photon counting fluctuations in a single-mode laser field, some correlation properties of the field are derived. It is shown that the correlation equations are satisfied by states of the field which are much more general than "coherent" states. The equations lead directly to the spectral density of the intensity operator in the light beat experiments, which can be obtained from photoelectric measurements. The resulting expression is practically identical to that found by Forrester for light having thermal statistical properties. The reasons for this are discussed by a comparison of the corresponding probability distributions of photon counts and of the classical wave amplitude.