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arXiv:1907.06467 · Physica Scripta

Quasidistribution of phases in Raman process with weak and strong pumps

Kishore Thapliyal, Jan Peřina

Published Oct 31, 2019• 6 citations
arXivHTMLPDF

Abstract

Abstract Nonclassicality is quantified using a quasidistribution of phases for the Raman process under both weak and strong pump conditions. In the former case, the solution is applicable to both resonant and off-resonant Raman processes, while strong classical pump is assumed at resonance. Under weak pump conditions (i.e. in a complete quantum treatment), the phase difference of phases described by single nonclassical modes is required to be filtered to describe a regular distribution function, which is not the case with strong pump. Compound Stokes-phonon mode shows nonclassical features of phases in both weak and strong pumping, which effect is similar to that for compound pump-phonon (Stokes–anti-Stokes) mode with weak (strong) pump. While anti-Stokes-phonon mode is observed to be classical and coherence conserving in strong pump case, pump-Stokes mode shows similar behavior in a special case in quantum treatment.

DOI
10.1088/1402-4896/ab5316

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Abstract Nonclassicality is quantified using a quasidistribution of phases for the Raman process under both weak and strong pump conditions. In the former case, the solution is applicable to both resonant and off-resonant Raman processes, while strong classical pump is assumed at resonance. Under weak pump conditions (i.e. in a complete quantum treatment), the phase difference of phases described by single nonclassical modes is required to be filtered to describe a regular distribution function, which is not the case with strong pump. Compound Stokes-phonon mode shows nonclassical features of phases in both weak and strong pumping, which effect is similar to that for compound pump-phonon (Stokes–anti-Stokes) mode with weak (strong) pump. While anti-Stokes-phonon mode is observed to be classical and coherence conserving in strong pump case, pump-Stokes mode shows similar behavior in a special case in quantum treatment.
We investigate quantum phase properties of two-mode optical fields whose quasidistributions have Gaussian form. We show how to simplify the calculation of the joint phase distribution defined via radial integration of the quasidistribution related to s-ordering of the field operators. We find an analytical formula for the joint phase distribution when coherent components of both modes vanish. The general results are applied to analysis of quantum phase properties of the two-mode Stokes-anti-Stokes field generated by means of Raman scattering with a broad reservoir phonon system and strong coherent laser pumping.
We investigate quantum phase properties of two-mode optical fields whose quasi distributions have Gaussian form. We show how to simplify calculation of the joint phase distribution defined via radial integration of the quasi distribution related to s-ordering of the field operators. Introducing hyper spherical coordinates, we can carry out analytically one of two required integrations. The second integral over a finite interval is then evaluated numerically. Analytical formula for the joint phase distribution can be obtained in special case of vanishing coherent components of both modes. The general results are applied to analysis of quantum phase properties of two-mode Stokes-anti-Stokes field generated by means of Raman scattering with broad reservoir phonon system and strong coherent laser pumping.
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.
A quantum model of the Raman process with the independent Stokes and anti-Stokes nonlinear interactions is developed to study nonclassical correlations between the photons in the Stokes and anti-Stokes fields. The role of the laser pump amplitude, the ratio of the Stokes and anti-Stokes coupling constants, and the population and losses of the vibrational mode in forming the correlations is elucidated. The g(2){g}^{(2)}g(2) intensity cross-correlation function, noise-reduction factor, two-mode principal squeezing variance, logarithmic negativity, nonclassicality depth, steering parameter, and the Bell parameter are analyzed side by side to shed light to the correlations between the Stokes and anti-Stokes fields. Conditions for having the Stokes and anti-Stokes fields composed of only photon pairs, similarly as it occurs in twin beams in parametric down-conversion, are revealed. They allow for nonzero mean thermal phonon numbers.
Nonclassical light is now available in a number oflaboratories, being generated by Frank—Hertz experiment, optical parametric processes, four-wave mixing including optical fibres, by optical system with feedback, etc. Therefore it is senseful to reconsider various nonlinear optical interactions involving initially nonclassical light ,originally solved with coherent (laser) light. Adopting tools of quasidistributions related to antinormal ordering, the photocount distributions, its factorial moments and characteristics of squeezing of vacuum fluctuations can be investigated assuming initially nonclassical (squeezed) light in the framework of the Heisenberg—Langevin of Schrödinger approaches. Photon statistics of stimulated Raman scattering of squeezed light can be derived assuming phonon system in chaotic, coherent or squeezed states and including additive noise. The influence of nonlinear dynamics can be established on initially inclassical light and it can be shown that the initially nonclassical effects are encresed 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 nonclassical features are smoothed out. Also frequency detuning can be optimized to have maximum nonclassical effects, which are missing with increasing phonon number, even if the phonon system is squeezed. Small initial squeezing can by increased by the nonlinear dynamics, whereas strong squeezing can only be diminished.
A general fundamental relation connecting the correlation of Stokes and anti-Stokes modes to the quantum statistical behavior of vibration and pump modes in Raman-active materials is derived. We show that under certain conditions this relation can be used to determine the equilibrium number variance of phonons. Time and temperature ranges for which such conditions can be satisfied are studied and found to be available in today's experimental standards. Furthermore, we examine the results in the presence of multimode pump as well as for the coupling of pump to the many vibration modes and discuss their validity in these cases.