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Physical Review A

Spontaneous generation of phase waves and solitons in stimulated Raman scattering. II. Quantum statistics of Raman-soliton generation

John C. Englund, Charles M. Bowden

Published Jul 1, 1992• 25 citations
Publisher

Abstract

The formalism of Englund and Bowden [Phys. Rev. A 42, 2870 (1990)] is used to conduct a detailed investigation into the statistics of spontaneously generated Raman solitons and their origin in phase waves. Two approaches are adopted. In one, a Monte Carlo simulation of the stochastic differential equations describing stimulated Raman scattering is carried out, with quantum noise entering through the Stokes vacuum. The other involves determining analytically the field statistics in the quantum-initiation regime, finding the corresponding phase-wave statistics, and determining numerically the class of phase waves that appear in the nonlinear regime as solitons. Computations involving quasi-Gaussian and rectangular pump-pulse profiles consistently indicate soliton yields of roughly 10% and 4%, respectively; differences in the distributions of soliton delay times and peak intensities are also indicated. The Monte Carlo method is adapted to the experimental parameter values of MacPherson et al. [Phys. Rev. A 40, 6745 (1989)], and gives yields and distribution in good accord with the experiment.

DOI
10.1103/physreva.46.578

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The formalism of Englund and Bowden [Phys. Rev. A 42, 2870 (1990)] is used to conduct a detailed investigation into the statistics of spontaneously generated Raman solitons and their origin in phase waves. Two approaches are adopted. In one, a Monte Carlo simulation of the stochastic differential equations describing stimulated Raman scattering is carried out, with quantum noise entering through the Stokes vacuum. The other involves determining analytically the field statistics in the quantum-initiation regime, finding the corresponding phase-wave statistics, and determining numerically the class of phase waves that appear in the nonlinear regime as solitons. Computations involving quasi-Gaussian and rectangular pump-pulse profiles consistently indicate soliton yields of roughly 10% and 4%, respectively; differences in the distributions of soliton delay times and peak intensities are also indicated. The Monte Carlo method is adapted to the experimental parameter values of MacPherson et al. [Phys. Rev. A 40, 6745 (1989)], and gives yields and distribution in good accord with the experiment.
The statistical properties of spontaneously generated Raman solitons are of significant interest because they reflect the statistics of the photon vacuum. These solitons arise when the fluctuating vacuum nearly vanishes at a space-time point (i.e., its expectation values nearly vanish). At such a point, a near phase reversal takes place, so that molecules that are phased to absorb incident radiation instead emit radiation at that frequency; with subsequent phase disruption caused by molecular collisions, phase locking is reestablished, and the radiation is again absorbed. The sequence of pump-frequency absorption, emission, and absorption produces a peak in the pump signal whose width is approximately the mean collision time. Such peaks are shaped by further propagation and show remarkable stability. Although they eventually decay, many survive into the macroscopic regime and may be detected. Specifically, using parameters consistent with ongoing experiments, we have found that solitons having peak intensities greater than one-tenth the pump-pulse maximum can be detected in ~12% of the pulses sent through the Raman medium. The delay time distribution of the solitons decays quasiexponentially, while the soliton height distribution appears to be bimodal.
Theoretical and experimental details of initiating and amplifying a Stokes pulse using stimulated Raman scattering in H2 are presented.Statistics associated with the spectral fluctuations of a Stokes seed pulse have been measured and theoretically calculated.The spectral fluctuations are shown to result from quantum noise associated with spontaneous initiation.These spectral fluctuations are an example of an easily measurable macroscopic fluctuation of quantum origin.The effects of frequency fluctuations on the generation of solitons in the nonlinear regime of stimulated Raman scattering are calculated and measured.A detailed derivation of the fully quantum mechanical stimulated Raman scattering equations is given along with their solution for various approximations.The driven Maxwell's equation for the Raman system is derived from the system Hamiltonian.For treating the quantum initiation of stimulated Raman scattering, the radiation field is expanded in localized coherent modes.The details of calculating and using these modes to generate an ensemble of Stokes pulses are presented along with specific examples and ensemble average statistics.The use of localized modes for treating the pulsed cavityless Raman laser gives insight into the spontaneous initiation of Stokes pulses.iv ACKNOWLEDGMENTS I feel very fortunate to have had Dr. John Carlsten for my thesis adviser.I would like to thank John for making graduate school an enjoyable part of life and for the many hours o f stimulating discussions on fundamental physics.I am also greatly indebted to my wife, Susie, for filling my big lunch bag every morning and for her many hours spent typing this thesis
We use the dressed-state representation to generate the full quantum-mechanical treatment of stimulated Raman scattering (SRS). Equations of motion and propagation are obtained from the Heisenberg equations of motion in the plane-wave limit and under conditions such that anti-Stokes and higher-order Stokes processes can be neglected. Equations for a stochastic model, based upon antinormal ordering and a quantum-classical correspondence are developed from them, suitable for numerical ensemble calculations for the study of amplified quantum fluctuations and spontaneously generated phase waves and solitons. Several alternative approaches, which have more general validity but present greater difficulty in numerical implementation, are also developed by introduction of quasiprobability functions and their associated Fokker-Planck equations from which corresponding stochastic equations of motion and propagation are derived. The Q representation, whose moments correspond to expectation values of products of antinormally ordered fields, and normally ordered atomic operators yield stochastic equations identical to the equations for the stochastic model developed directly from the Heisenberg equations for the same initial and boundary conditions, giving strong credence to the stochastic model. The Wigner representation, whose moments correspond to expectation values of symmetrically ordered operators, leads to a set of equations that are valid where the stochastic model fails, namely, where conditions are such that quantum fluctuations and nonlinearity are simultaneously important. The stochastic model, which is shown to be, by far, the simplest to implement in numerical calculations, is shown also to be valid for the conditions of SRS.
Manifestations of quantum fluctuations in stimulated Raman scattering ( SRS ) have been intensively studied lately1. Quantum noise present in spontaneous scattering stimulates fluctuations both in phase and in amplitude characteristics of output Stokes emission. At present statistical properties of amplitude fluctuations are studied in detail and it is shown that in linear transient regime of SRS without pump depletion statistical scatter of Stokes energy Es has negative exponential distribution2. In saturation regime of SRS the amplitude fluctuations are almost suppressed simultaneously with pump depletion. The phase fluctuations manifest both in linear and in nonlinear SRS and produce beam pointing fluctuations3, a change in Stokes emission spectra from pulse to pulse4 and cause soliton appearance in Rayleigh component of SRS5. Phase noise is observed in saturation regime due to relation between phases of Stokes and phonon waves which is preserved in processes of stationary and quasistationary SRS amplification. Spectrum phase fluctuations manifest as fine structure of SRS spectrum4, fluctuations of average Stokes frequency and Stokes linewidth fluctuations. The evolution of phase fluctuations in time is determined by the nature of spontaneous scattering and polarization of Raman media and is of great interest as one can follow its development from vacuum modes level to saturation regime. However up to now this problem has not been studied yet.
The π phase jump generated spontaneously in stimulated Raman scattering, which is the origin of Raman soliton formation, has been investigated. The probability of Raman soliton generation is measured as a function of the pressure of the Raman medium and the pump intensity. The relation between the soliton’s shape and the magnitude of the phase jump has been revealed by homodyne phase detection of the Stokes phase. These results are compared with numerical calculation, and qualitative agreement is obtained. The stochastic nature of Raman soliton generation is discussed in terms of the phase jump’s magnitude and evolution through the Raman medium.