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Physical Review

Quantum Statistics of Nonlinear Optics

Y. R. Shen

Published Mar 15, 1967• 315 citations
Publisher

Abstract

Nonlinear interaction of light with matter is described from the quantum-statistical point of view. The cases of two-photon absorption, Raman transition, sum-frequency generation, parametric amplification, and incoherent scattering are discussed. It is shown that the nonlinear optical effects depend strongly on the statistical properties of the light fields. The rate of nonlinear absorption, generation, and amplification is higher for chaotic than for coherent, and also higher for multimode than for single-mode pump fields. Measurements of the statistics of the output fields may yield information about the statistics of the input fields and the properties of the medium.

DOI
10.1103/physrev.155.921

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 14, 1999•0 citations
Nonlinear interaction of light with matter is described from the quantum-statistical point of view. The cases of two-photon absorption, Raman transition, sum-frequency generation, parametric amplification, and incoherent scattering are discussed. It is shown that the nonlinear optical effects depend strongly on the statistical properties of the light fields. The rate of nonlinear absorption, generation, and amplification is higher for chaotic than for coherent, and also higher for multimode than for single-mode pump fields. Measurements of the statistics of the output fields may yield information about the statistics of the input fields and the properties of the medium.
The nonlinear interaction of light with matter is described from a quantum-statistical point of view. The phenomena of two-photon emission and two-photon absorption including both the single- and two-mode cases and the Raman effect are discussed in detail. A master equation for the density operator of the light fields alone is derived. This operator equation is converted to a c number equation and analytic solutions are obtained for the diagonal matrix elements of the density operator in the Fock representation. No linearizing approximation is introduced. These solutions allow one to compute the moments of the photon distribution for the above nonlinear processes.
Partial table of contents: PART I: GENERAL CONCEPTS AND METHODS OF NONLINEAR OPTICS. Electromagnetic Fields. Classical Description. The Quantized Free Radiation Field. Interaction Between Radiation and Matter. Semiclassical Description of Nonlinear Optics. Statistical and Coherence Properties of the Radiation Field and Their Measurement. Nonstationary Processes. PART II: EFFECTS AND PROCESSES OF NONLINEAR OPTICS. Nonlinear One--photon Processes in Lasers. Nonlinearities in Transient One--photon Processes. Nonlinearities and Qunatum Phenomena in Transient One--photon Processes. Multiphoton Absorption and Emission. Generation of Harmonics and Sum and Difference Frequencies. Parametric Amplification and Oscillation. Stimulated Raman Scattering. Optical Bistability. APPENDIX A: Compilation of Quantum--Theoretical Definitions and Relations. General References. Index.
A theory of a quantum amplifier based on two-photon transitions is presented. The model consists of an assemblage of atoms interacting with a single mode of the light field via two-photon absorption and emission. The treatment is quantum mechanical, and the emphasis is on the photon-statistical aspects of the problem. Difference-differential equations for the density matrix of the field mode are derived. The equations are solved for a special case. Differential equations for the expectation vaues of the number of photons and of the field amplitudes, and for their higher statistical moments, are also obtained. Approximate solutions for the average number of photons are found, and the effect of the amplification process on the statistical fluctuations of the number of photons and the field amplitudes is discussed. The results of this study suggest that such an amplifier does not have the coherence properties of a laser amplifier because it tends to increase the statistical fluctuations of the field.
An analysis is made of ω 1 +ω 2 =ω 3 sum-frequency generation in a nonlinear crystal and also of twophoton absorption under the action of "biphotons," i.e., correlated two-photon fields formed as a result of spontaneous parametric scattering of the initial pump field in another crystal. It is shown that at low pump powers, the biphoton efficiency in two-photon processes is considerably higher than the efficiency of classically correlated fields of the same intensity. It is found that in the case of coherent pumping, the regenerated sum-frequency field is also coherent.
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.
Quantum statistical properties of light in nonlinear coupler in which Raman or Brillouin scattering are in operation are investigated. Main attention is given to possibility of generation of nonclassical states of light. Phase mismatches and losses are included and their influence is discussed. The second part of the paper is devoted to analysis of stationary point. Coherent pumping of optical modes is assumed and stationary solution for c-number amplitudes is found and small linear operator corrections are calculated. Stability of the stationary point is discussed and it is shown that phase uncertainty induces instability.