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Journal of Modern Optics

Quantum Statistics of Stimulated Raman and Hyper-Raman Scattering of Squeezed Light with Pump Depletion

Jan Peřina, J. Křepelka

Published May 1, 1992• 8 citations
Publisher

Abstract

Using the approximation of small fluctuations around a stationary solution, photon statistics and quadrature variances are derived in stimulated Raman and hyper-Raman scattering including the coupling of pumping, Stokes, anti-Stokes and phonon modes. The depletion of pump light is included. Compared to Raman scattering, additional regimes to generate anti-bunched, sub-Poisson and squeezed light were found caused by self-interaction of the pump mode. It was possible to include effects of nonlinear dynamics, initial squeezing of radiation and phonon system, external noise and losses.

DOI
10.1080/09500349214551051

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Using the approximation of small fluctuations around a stationary solution, photon statistics and quadrature variances are derived in stimulated Raman and hyper-Raman scattering including the coupling of pumping, Stokes, anti-Stokes and phonon modes. The depletion of pump light is included. Compared to Raman scattering, additional regimes to generate anti-bunched, sub-Poisson and squeezed light were found caused by self-interaction of the pump mode. It was possible to include effects of nonlinear dynamics, initial squeezing of radiation and phonon system, external noise and losses.
Using the approximation of small fluctuations around a stationary point, photon statistics and squeezing of vacuum fluctuations are derived in stimulated Raman scattering including the coupling of photon-photon and photon-phonon modes. The depletion of pump light is described and it is shown that a regime of inversion of the process is related to an increase in quantum fluctuations, which are finally reduced again, more in the anti-Stokes mode than in the Stokes mode. If the anti-Stokes coupling is stronger than the Stokes coupling, then an oscillating behaviour of the statistical characteristics occurs and substantial squeezing can be found in photon-phonon modes.
Quantum statistical properties are derived for stimulated Raman scattering by squeezed phonons including squeezing of vacuum fluctuations in radiation modes and sub-Poisson photon behaviour.Initial radiation can be in squeezed thermalized and/or sub-Poisson states, phonon system can also be squeezed.In particular, effects of initial phase conditions, losses, initial nonclassical behaviour and phonon squeezing are demonstrated in terms of reduced factorial moments up to the fifth order in combined photon-photon and photon-phonon modes.Also photon number distribution, quadrature variances and principał squeeze parameters are shown.
Photon statistics of stimulated Raman scattering of squeezed light are considered including additive noise. The phonon system is assumed to be in chaotic or squeezed chaotic states. The influence of nonlinear dynamics is established on the initial non-classical features of light adopting quasi-distribution related to anti-normal ordering, photocount distribution, its factorial moments and principal squeezing parameter. It is shown that the initial non-classical effects are increased by the nonlinear dynamics provided that the Stokes mode is less amplified than the anti-Stokes mode is damped, whereas in the opposite case the non-classical effects are swiftly smoothed out. The frequency detuning can be optimized to have maximum non-classical effects, which can periodically return. However, above a certain bound the non-classical effects are missing with frequency detuning, and also with increasing phonon number, even if the phonon system is squeezed. Small initial squeezing can be increased by the nonlinear dynamics, whereas strong squeezing can only be diminished.
Raman scattering from a great number of phonon modes is described from a quantum-statistical point of view within the standing-wave model. The master equation for the completely quantum case, including laser pump depletion and stochastic coupling of Stokes and anti-Stokes modes, is derived and converted to classical equations: either into a generalized Fokker-Planck equation and an equation of motion for the characteristic function or into the master equation in Fock representation. These two approaches are developed both in linear and nonlinear regime. A detailed analysis of scattering into Stokes and anti-Stokes modes in linear regime, i.e., under parametric approximation, is presented. The existence of s-parametrized quasiprobability distributions, in particular the Glauber-Sudarshan P-function, is investigated. An analysis of Raman scattering into separate Stokes and anti-Stokes modes in nonlinear regime, thus including pump depletion, is given. The master equation in Fock representation is solved exactly for the complete density matrix using the Laplace transform method. Short-time solutions, steady-state solutions and approximate compact form solutions are obtained. Relations between the quasidistribution approach based on the Fokker-Planck equation and the density matrix approach based on the master equation in Fock representation are presented. The photocount distribution and its factorial moments as well as variances and extremal variances of quadratures are calculated in both approaches giving the basis for the analysis of the quantum properties of radiation such as sub-Poissonian photon-counting statistics and squeezing. Various statistical moments obtained from numerical calculations utilizing our exact solutions, from the approximate relations for short times, and those obtained under parametric approximation are compared.
Photon statistics of stimulated Raman scattering and degenerate hyper-Raman scattering in the cavity are investigated using the method of small operator corrections to a c-number solution. Weak non-classical effects and super-Poissonian behaviour of all modes are derived in the case of Raman scattering in stationary regime. Higher-order nonlinearity of hyper-Raman scattering leads to more significant non-classical effects—considerable quadrature squeezing and sub-Poissonian behaviour of laser and Stokes modes (Fano factor lower than 0·8) were obtained. A negative quantum noise component is manifested in oscillations of photon-number distribution and in negative values of Wigner distribution of the integrated intensity. The influence of external driving amplitude and thermal bath is examined.
We develop the quantum theory of stimulated hyper-Raman scattering (HRS) and discuss the quantum statistics of pump and Stokes fields. We show that the initially coherent pump field acquires a strong squeezing that exceeds the similar squeezing in other nonlinear processes also involving two-photon absorption. Stokes field statistics are analysed in detail as a function of the initial statistics of both modes. Bunching effects are shown to be amplified in the short-time region, while in asymptotics the antibunching of Stokes photons becomes dominant, which in many cases results in sub-Poissonian statistics of Stokes mode. We also find a number of correlation effects between the photons of the two modes.