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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum phase properties of Raman scattering

Jaromı́r Fiurášek, Jan Peřina

Published Mar 7, 2001• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.417852

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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.
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.
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-photon and photon-phonon anticorrelation effects in Raman scattering are studied theoretically. The Heisenberg equations of motion describing Raman scattering are solved in the short-time approximation. The laser, Stokes, anti-Stokes and phonon modes are quantised. The joint normally ordered characteristic functions are evaluated for modes exhibiting anticorrelation effects. The problem of existence of a Glauber-Sudarshan quasi-distribution function related to anticorrelation effects is considered. The magnitude of the anticorrelation effects depends on the initial statistical properties of the photon and phonon modes.
The quantum statistical properties of Brillouin scattering of intense laser light are derived including the coupling of Stokes, anti-Stokes and phonon modes, if the anti-Stokes interaction prevails. Making use of the coherent-state technique, the Heisenberg equations of this process are solved neglecting the loss mechanism, and the normal quantum characteristic function and the normal generating function are derived. The time dependences of the photon distribution and its factorial moments are demonstrated if the phonon, Stokes and anti-Stokes modes are initially in a coherent state and periodical anti-bunching of the field is found when the phases of the incident fields fulfil certain phase conditions; the field can also return to the coherent state again.
A quantum theory of stimulated Raman scattering is presented that takes into account three-dimensional propagation and collisional dephasing, allowing the study of the spatial and temporal coherence properties of the generated Stokes light. Maxwell-Bloch equations for the Stokes field operator and the collective atomic operators are solved analytically under low-signal-gain conditions, where the laser field and the atomic ground states remain undepleted. The intensity and the space-time autocorrelation function of the Stokes field are calculated. The Stokes field is expanded into a set of statistically independent ``coherence modes,'' which are determined explicitly for the case of a cylindrically shaped pumped volume. The Stokes pulse energy W is found to fluctuate from pulse to pulse. The probability distribution function for pulse energies P(W) is calculated for a range of Fresnel numbers of the excited volume and collisional dephasing rates. For small values of Fresnel number and dephasing rate, P(W) is a negative exponential distribution. For large values of either, P(W) narrows and approaches a Gaussian-shaped distribution. This occurs because many independent modes contribute to the Stokes emission, making it spatially and/or temporally incoherent.
In the coherent anti-Stokes Raman scattering process, the spectrum of the generated optical phonon depends on the degree of temporal correlation between the pump laser field and the Stokes field. When the two fields are strongly correlated, such as when the Stokes field is generated with stimulated Raman scattering (SRS), the spectral shape of the optical phonon is found experimentally and theoretically to be the same as the gain-narrowed Raman line shape because the laser phase fluctuations cancel out totally, leaving only the collisional noise in the SRS process. When the two fields are uncorrelated, the shape of the optical-phonon spectrum is found to be the same as the Raman line shape without gain narrowing. When two fields are partially correlated, then the two spectral components appear together. We provide a method to measure the degree of correlation between two optical fields that have different central frequencies. The theory developed to interpret the experimental results is an extension of the quantum theory of SRS to include anti-Stokes scattering. We show that only in the high-gain limit can the quantum fluctuations be thought of as arising from a classical noise process.