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Soviet Journal of Quantum Electronics

Photon statistics in parametric interaction of three waves

Yu A Il'inskiĭ, Давид Николаевич Клышко, Vera M. Petnikova

Published Nov 30, 1975• 6 citations
Publisher

Abstract

The parametric approximation is used in a statistical analysis of the frequency conversion of an optical field by a noncentrosymmetric crystal which couples an input wave of frequency 0 < ω < ωL/2 to output waves of frequency ω+nωL(n = 0, ±1, ±2, ...; ωL is the pump frequency). An expression is obtained for the characteristic function of the output field which is described in terms of a matrix of the coupling coefficients in the case of Gaussian (displaced) statistics of the input field with independent modes. The mutual statistics of the output waves with n ≥ 0 and n' < 0 is not Gaussian but corresponds, in the lowest approximation with respect to the pump wave and in the absence of the input field, to a spontaneous (direct or cascade) decay of pump photons into photon pairs. The intensity of the radiation represented by the n ≥ 0 modes is equal to the intensity of the radiation emerging from a classical converter which receives one photon in each mode with n' < 0. The special case of three interacting modes with n = −1, 0, 1 (upconverter or light modulator) is considered in detail. The noise intensities and correlations between the numbers of photons in two and three modes are found for this case, and the spatial-temporal correlation function of the fields of the Stokes (n' = −1) and anti-Stokes (n = 1) frequencies is determined.

DOI
10.1070/qe1975v005n11abeh012127

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The parametric approximation is used in a statistical analysis of the frequency conversion of an optical field by a noncentrosymmetric crystal which couples an input wave of frequency 0 < ω < ωL/2 to output waves of frequency ω+nωL(n = 0, ±1, ±2, ...; ωL is the pump frequency). An expression is obtained for the characteristic function of the output field which is described in terms of a matrix of the coupling coefficients in the case of Gaussian (displaced) statistics of the input field with independent modes. The mutual statistics of the output waves with n ≥ 0 and n' < 0 is not Gaussian but corresponds, in the lowest approximation with respect to the pump wave and in the absence of the input field, to a spontaneous (direct or cascade) decay of pump photons into photon pairs. The intensity of the radiation represented by the n ≥ 0 modes is equal to the intensity of the radiation emerging from a classical converter which receives one photon in each mode with n' < 0. The special case of three interacting modes with n = −1, 0, 1 (upconverter or light modulator) is considered in detail. The noise intensities and correlations between the numbers of photons in two and three modes are found for this case, and the spatial-temporal correlation function of the fields of the Stokes (n' = −1) and anti-Stokes (n = 1) frequencies is determined.
The variance and average value of photocounts (counts of electrons emitted by the cathode of a photodetector) are calculated allowing for the existence of the following sources of noise in a frequency converter: fluctuations of the pump intensity, conversion of background radiation, thermal fluctuations of fields in the converter crystal, spontaneous generation of the difference frequency, and illumination of the photodetector with the pump radiation transmitted by filters. A function representing the distribution of photocounts is obtained for short time intervals allowing for fluctuations of the pumping intensity in the case of conversion of a random Gaussian signal.
Quantum properties of parametric amplification and frequency conversion of an optical image in coupled parametric interactions are analyzed for the case of a remote object. The coupled-wave interaction consists of a traditional process of parametric amplification in the high-frequency pump field accompanied by the frequency mixing of generated and pump waves. The photon number density and signal-to-noise ratio at the interacting frequencies are studied in the image plane at the output of the nonlinear frequency converter in the fixed pump field approximation. The degree of entanglement of optical images at frequencies above and below the pump frequency is investigated.
Quasi-phase-matched consecutive three-frequency interactions between waves with nonmultiple frequencies are considered. The quantum theory of parametric amplification is developed in the approximation of an undepleted low-frequency pump field in the Heisenberg representation. It is shown that, with a certain choice of quasi-phase-matching orders, the regime of exponential increase in the mean number of photons of interacting waves can be realized. In this case, the photon statistics of generated waves is super-Poissonian one. The correlation of fluctuations of numbers of photons with different frequencies is also considered.
Statistical properties of optical fields in nonlinear couplers composed of two waveguides in which Raman or Brillouin processes (with classical pumping) are in operation and which are mutually connected through the mutual Stokes and/or anti-Stokes linear interactions are investigated within the framework of a generalized superposition of coherent fields and quantum noise. Heisenberg equations describing the couplers are solved both analytically under special conditions and numerically in general cases. Regimes for nonclassical properties of optical fields, such as sub-Poissonian photon-number statistics, negative reduced moments of integrated intensity and squeezing of quadrature fluctuations are discussed for the cases of single and compound fields. General results are compared with those from the short-length approximation.
A quantum statistical analysis of phase-locked coupled light beams is presented with the coupling mechanism between the individual modes provided by the time-modulated dielectric constant of the crystal inserted in the cavity. The intense electric field that drives the modulator is treated classically. A Hamiltonian including the modulator interaction is formulated and the Heisenberg equations of motion of the mode operators are solved utilizing a generating function of the mode operators. The evolution in time of the phase-space distribution functions for single and multiple modes is derived under various initial conditions. Initially coherent oscillating modes are shown to conserve their coherent character. Below threshold, the modes are assumed to be chaotic Gaussian. It is shown that when all modes have the same average number of photons, they remain uncorrelated Gaussian. When the average photon numbers are nonidentical, the field remains a pure noise field, but the various modes are correlated. The analysis of the resonant-case mode of operation, properly interpreted, leads directly to the case of nonresonant modulation.
The influence of disordering of one-dimensional quasi-phase matched structures in nonlinear optical crystals on spectral characteristics of parametric processes is studied. Two different correlation functions are introduced for describing the random change of the sign of the second-order nonlinear susceptibility. The optimal conditions for quasi-phase matching of the interacting waves are found. The second aim of the paper is the investigation of important features of spontaneous parametric down-conversion (SPDC) in a homogeneous nonlinear slabs with emphasis of spatial inhomogeneity of interacting waves (in paraxial approximation). We estimate the absolute values of the emission rate of photon pairs produced by SPDC when all fields are in single Gaussian modes.