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Journal of Optics B Quantum and Semiclassical Optics

Number-phase properties of correlated chaotic fields

Vlasta Peřinová, A. Lukš, J. Křepelka, Viktor Jelínek

Published Nov 21, 2000• 3 citations
Publisher

Abstract

We study the arrangement of two down-conversion crystals with aligned and partially aligned idler beams in the quasimonochromatic approximation. Treating the signal fields as correlated chaotic modes, we arrive at a detailed description and illustration of their mathematically well defined state in terms of number and phase properties. We show it to be a mixed partial phase-difference state. We derive the closed formulae for the phase-difference distributions in this state derived from the phase-space distributions related to the normal, symmetrical and antinormal orderings of field operators and show that such distributions depend exclusively on measures of correlation which are variants of the modulus of the familiar degree of coherence. Quantum correlation influences fluctuations of photon-number sum, photon-number difference and quantum phase difference. Starting from a quasidistribution which carries the same information as the statistical operator, we arrive at the joint distributions of number sum and phase difference and the quasidistribution of number difference and phase difference.

DOI
10.1088/1464-4266/2/6/306

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We study the arrangement of two down-conversion crystals with aligned and partially aligned idler beams in the quasimonochromatic approximation. Treating the signal fields as correlated chaotic modes, we arrive at a detailed description and illustration of their mathematically well defined state in terms of number and phase properties. We show it to be a mixed partial phase-difference state. We derive the closed formulae for the phase-difference distributions in this state derived from the phase-space distributions related to the normal, symmetrical and antinormal orderings of field operators and show that such distributions depend exclusively on measures of correlation which are variants of the modulus of the familiar degree of coherence. Quantum correlation influences fluctuations of photon-number sum, photon-number difference and quantum phase difference. Starting from a quasidistribution which carries the same information as the statistical operator, we arrive at the joint distributions of number sum and phase difference and the quasidistribution of number difference and phase difference.
Joint signal-idler photoelectron distributions of twin beams containing several tens of photons per mode have been measured recently. Exploiting a microscopic quantum theory for joint quasidistributions in parametric down-conversion developed earlier we characterize properties of twin beams in terms of quasidistributions using experimental data. Negative values as well as oscillating behavior in the quantum region are characteristic for the subsequently determined joint signal-idler quasidistributions of integrated intensities. Also the conditional and difference photon-number distributions are shown to be sub-Poissonian and sub-shot-noise, respectively.
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A continuous photon-number measurement has been assumed in the idler mode output from parametric down-conversion for an initial coherent state. The states generated in the signal mode due to the mode entanglement and the effects of both the destructive and the ideal reduction are described in terms of a quasidistribution of the complex field amplitude as well as of the density-matrix elements. The resulting near-number states in the ideal case are suitable for the study of the dynamics of phase and quadrature distributions rotated by the actual preferred phase. These states exhibit both number and quadrature squeezing. \textcopyright{} 1996 The American Physical Society.
The properties of a photon counter functioning by stimulated emission---rather than by absorption---of photons in an external field are examined. A possible scheme for such a quantum counter is described, and it is shown that correlation measurements performed with a number of quantum counters correspond to antinormally ordered products of field operators. These correlations, unlike the normally ordered ones, are always positive definite and depend explicitly on the number of radiation modes per unit volume to which the counter is coupled. It is shown that the antinormally ordered correlations carry useful information about the field only when the average photon occupation number per mode is large. A general expression for the probability that the quantum counter registers nnn counts in a certain time interval is derived and is shown to be related in an interesting way to the corresponding expression for the photoelectric detector. The variance of the probability distribution is evaluated for some simple states of the field.
Photon statistics and phase properties of two-mode squeezed number states are studied. It is shown that photon number distribution and Pegg-Barnett phase distribution for such states have similar (N + 1)-peak structure for nonzero value of the difference in the number of photons between modes. Exact analytical formulas for phase distributions based on different phase approaches are derived. The Pegg-Barnett phase distribution and the phase quasiprobability distribution associated with the Wigner function are close to each other, while the phase quasiprobability distribution associated with the Q function carries less phase information.
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.