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Journal of physics. A, Proceedings of the Physical Society. General

On the quantum statistics of the superposition of coherent and chaotic fields

Jan Peřina, R. Horák

Published Nov 1, 1969• 34 citations
Publisher

Abstract

The generating function, the corresponding integrated intensity distribution, and the photon-counting distribution and its fractional moments are derived for the superposition of coherent and chaotic M-mode fields on the basis of a formalism of arbitrary ordering of field operators in quantum optics. An alternative description is also given based on a recent analysis of the superposition of coherent and chaotic fields in terms of the correlation functions which makes it possible to derive a number of results valid for arbitrary mean occupation numbers per mode from which the model considered follows as a special case. Many earlier results obtained for coherent and chaotic fields and their superposition are included as special cases. The present results generally describe a superposition experiment for an N-mode coherent field (e.g. generated by an N-mode laser operating far above threshold) and an M-mode chaotic field (M>or=N) as well as a field generated by an M-mode laser operating above threshold.

DOI
10.1088/0305-4470/2/6/012

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Physical Review•Sep 25, 1969•58 citations
The generating function, the corresponding integrated intensity distribution, and the photon-counting distribution and its fractional moments are derived for the superposition of coherent and chaotic M-mode fields on the basis of a formalism of arbitrary ordering of field operators in quantum optics. An alternative description is also given based on a recent analysis of the superposition of coherent and chaotic fields in terms of the correlation functions which makes it possible to derive a number of results valid for arbitrary mean occupation numbers per mode from which the model considered follows as a special case. Many earlier results obtained for coherent and chaotic fields and their superposition are included as special cases. The present results generally describe a superposition experiment for an N-mode coherent field (e.g. generated by an N-mode laser operating far above threshold) and an M-mode chaotic field (M>or=N) as well as a field generated by an M-mode laser operating above threshold.
Recently derived formulae for the integrated intensity distribution, the photon-counting distribution and its factorial moments in the statistics of the superposition of coherent and chaotic multimode fields are proposed as approximate formulae for light of arbitrary spectrum. It is shown by explicit calculations of the third factorial moment for the superposition of a one-mode coherent field with a Gaussian-Lorentzian field that the proposed formulae hold with good accuracy over a wide range of conditions. An application to the determination of spectral parameters of light is given.
A unified approach to the quantum statistics of single-mode linear processes described by the general Fokker-Planck equation and the generalized initial superposition of coherent and chaotic fields is developed to reveal the common traits in the mathematical models of several quantum optical systems. Processes such as the degenerate parametric amplification and the harmonic oscillator eventually coupled to the thermal or rigged reservoir are involved as special cases. A sufficient condition for the validity of the master equation is formulated as its being formally independent of possible reservoirs. The dependence of non-classical phenomena such as the sub-Poisson photon statistics and squeezing on the drift and diffusion coefficients is discussed.
This paper is concerned with the determination of the quantum statistics of the photoelectric detection, and the E-field detection of multiple-mode superposed chaotic and coherent radiation. The Glauber P-representation is utilized extensively, and results are obtained by a fully quantum-mechanical treatment. The generating function, the mean, and the variance of the photocount statistics are derived for illumination times of arbitrary duration and for modes of arbitrary statistical interdependence. For short illumination times, the complete photocount distribution is obtained. The distribution has the same form as that for a single mode of superposed chaotic and coherent radiation. The multiple-mode distribution, however, is nonstationary. The difference between experimental determinations of the mean and variance which utilize periodic sampling and those which utilize random sampling is derived. A method for determining the P-representation for modulated fields is described, and the detection statistics are derived. The statistics of the E field are found to be Gaussian, and it is found that for multiple-mode classical statistics and quantum statistics differ.
The relation between the two-quantum photocounting distribution p(n, T) and the intensity fluctuations of the radiation incident on the detector is obtained and compared with the one-quantum results. The actual statistics are evaluated for several cases of interest, including a chaotic source, an amplitude-stabilized wave, and the output of a single-mode laser (Van der Pol oscillator). The two-quantum count rate 〈n〉/T is found to depend on the mean-square intensity of the radiation, in contrast to the one-quantum count rate which is proportional to the mean intensity. The effects of photon correlations in the radiation beam become apparent since the two-quantum distributions manifest a distinctly more positive second derivative than the corresponding one-quantum distributions. Both the low-and the high-count probabilities are therefore increased at the expense of counts near the mean. The quantum-theoretical treatment is found to be equivalent to the semiclassical treatment for density operators possessing a positive-definite weight function in the P-representation. Some possible experiments to verify the theory are discussed and shown to be feasible.
Methods are developed for discussing the photon statistics of arbitrary fields in fully quantum-mechanical terms. In order to keep the classical limit of quantum electrodynamics plainly in view, extensive use is made of the coherent states of the field. These states, which reduce the field correlation functions to factorized forms, are shown to offer a convenient basis for the description of fields of all types. Although they are not orthogonal to one another, the coherent states form a complete set. It is shown that any quantum state of the field may be expanded in terms of them in a unique way. Expansions are also developed for arbitrary operators in terms of products of the coherent state vectors. These expansions are discussed as a general method of representing the density operator for the field. A particular form is exhibited for the density operator which makes it possible to carry out many quantum-mechanical calculations by methods resembling those of classical theory. This representation permits clear insights into the essential distinction between the quantum and classical descriptions of the field. It leads, in addition, to a simple formulation of a superposition law for photon fields. Detailed discussions are given of the incoherent fields which are generated by superposing the outputs of many stationary sources. These fields are all shown to have intimately related properties, some of which have been known for the particular case of blackbody radiation.
The quantum theory of the stimulated Raman and Brillouin effects, the parametric amplifier, and the frequency converter is developed, starting with a simple Hamiltonian for a system of three coupled field modes. The intense incident beam is treated classically. The evolution in time of the statistical properties of the quantized Stokes and anti-Stokes waves are analyzed by means of appropriate time-dependent phasespace distributions under various assumptions for the initial fields. Expansions of the field variables in terms of complete sets of orthogonal operators are discussed. Several regimes of operation are considered. It is shown that an initially coherent state develops, in general, into a superposition of a coherent state and a chaotic state. The amplitudes of the coherent components follow the same equations of motion as the mode operators. The chaotic components stem from the amplification of the vacuum fluctuations.