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Physical Review A

Nonimpact unified theory of four-wave mixing and two-photon processes

Shaul Mukamel

Published Dec 1, 1983• 142 citations
Publisher

Abstract

A microscopic correlation-function theory for coherent, parametric, four-photon processes (four-wave mixing) and for single-absorber two-photon processes (two-photon absorption, Raman, and fluorescence spectra) is developed. Both types of experimental observables are expressed in terms of four-point correlation functions of the dipole operator. The latter are then evaluated semiclassically or within the factorization approximation in which they are expressed in terms of ordinary single-photon line-shape functions. The common results of the Bloch equations are shown to be the outcome of two successive approximations: the factorization approximation and the impact limit whereby the single-photon line-shape functions are taken to be Lorentzian. Effects of fluctuations in the radiation fields are easily incorporated within the present approach.

DOI
10.1103/physreva.28.3480

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Journal of the Optical Society of America B•Oct 1, 1985•13 citations
A microscopic correlation-function theory for coherent, parametric, four-photon processes (four-wave mixing) and for single-absorber two-photon processes (two-photon absorption, Raman, and fluorescence spectra) is developed. Both types of experimental observables are expressed in terms of four-point correlation functions of the dipole operator. The latter are then evaluated semiclassically or within the factorization approximation in which they are expressed in terms of ordinary single-photon line-shape functions. The common results of the Bloch equations are shown to be the outcome of two successive approximations: the factorization approximation and the impact limit whereby the single-photon line-shape functions are taken to be Lorentzian. Effects of fluctuations in the radiation fields are easily incorporated within the present approach.
A microscopic correlation function expression for the signal in four-wave mixing is developed. Both stationary and transient (i.e., photon-echo) experiments with coherent or incoherent radiation fields are considered. In general, these observables are described by an eight-point correlation function of the dipole operator involving two particles, and the signal cannot be written as an amplitude squared. Conditions under which the correlation function may be factorized into a product of two four-point, single-particle correlation functions are specified. In this case the conventional approach, based on calculating a nonlinear polarization and substituting it as a source in Maxwell's equations, is justified. However, in the presence of long-range spatial correlations in the sample (e.g., near critical points) the conventional formulation breaks down and the present theory should be used.
A unified theory of time-domain and frequency-domain four-wave mixing processes, which is based on the nonlinear response function R(t3, t2, t1), is developed. The response function is expressed in terms of the four-point correlation function of the dipole operator F(τ1, τ2, τ3, τ4) and is evaluated explicitly for a stochastic model of line broadening that holds for any correlation time of the bath. Our results interpolate between the fast-modulation limit, in which the optical Bloch equations are valid, and the static limit of inhomogeneous line broadening. As an example of the relationship between time-domain and frequency-domain four-wave mixing, we compare the capabilities of steady-state and transient coherent anti-Stokes Raman spectroscopy experiments to probe the vibrational dynamics in ground and excited electronic states.
A unified approach is presented for the treatment of stochastic fluctuations in four-wave-mixing processes, including uncorrelated and fully or partially cross-correlated input laser fields. The theory allows for phase as well as amplitude fluctuations and covers Markovian and non-Markovian (non-Lorentzian) input line shapes. The analogy between laser field fluctuations and molecular dephasing processes is discussed, and several types of fluctuation-induced resonances are considered. Uncorrelated input fields give rise to resonances similar to those described previously by Agarwal, while correlated input fields cause extra resonances of the type presented by us in earlier publications. Partial cross-correlations are treated here for the first time, with predictions for new effects: Oscillations in the intensity of the generated four-wave-mixing signal as a function of the degree of cross-correlation. The conditions for the experimental observation of these effects are discussed.
A unified theory of coherent and incoherent resonance scattering of radiation in gases is presented. Expressions for the scattering rates in collisional redistribution and four-wave mixing (FWM) are obtained, starting from a microscopic formulation based on the Liouville--von Neumann equation. The derivation reemphasizes the point that coherent scattering depends on a product of scattering amplitudes at two separate points in the gas. The theory is extended to the nonimpact domain in which frequency detunings exceed the inverse duration of a single collision, taking into account intracollisional effects. Expressions are given in the so-called quasistatic limit (of semiclassical Franck-Condon transitions) for situations in which one or more radiative transitions occur during the collision. An extension of double-sided diagrams to collision-induced processes is introduced, and is used to illustrate the various phenomena described. Applications are suggested to the study of molecular dynamics, making use of the spectral features of FWM outside the impact domain. Investigations of long-range correlation effects, involving laser fluctuations, or cooperative excitations in dense fluids (utilizing the two-point nature of FWM), are also discussed.
In this paper, we calculate the four coefficients for the quantum theory of multiwave mixing including a local-field correction resulting from dipole-dipole interactions. We make contact with the semiclassical calculations of probe absorption and four-wave-mixing coupling coefficients, and illustrate the effects of local field corrections on resonance-fluorescence and coupled-mode-fluorescence spectra. The method uses the hybrid quantum-Langevin-equation master-equation approach of An and Sargent.
Cross correlation refers to the situation in which the fluctuations in the fields of two or more lasers are correlated in some way. Recently, Thomas et al. [ Opt. Lett.6, 298 ( 1981); Phys. Rev. Lett.48, 867 ( 1982)] proposed the generation of cross correlation between two laser fields interacting with a three-level system as a means of reducing noise and went on to exploit this property in the observation of narrow Ramsay fringes. Dalton and Knight [ J. Phy. B15, 3997 ( 1982)] discussed the theoretical aspects of cross correlation and showed that it produced interesting effects in population trapping for example. We have demonstrated that it has significant effects on the optical double-resonance (ODR) spectra [ J. Phys. B17, L389 ( 1984)]. On the other hand, the effects of non-Lorentzian laser line shapes have been shown by various workers to have significant effects on a wide range of phenomena, such as multiphoton ionization, resonance fluorescence, and ODR. In this paper we present a theory of the interactions of lasers whose fluctuations are governed by the phase-diffusion process with atomic systems that includes the effects of cross correlation and non-Lorentzian line shapes. We derive the basic equations (which in general have to be solved numerically) and obtain analytic results in the case in which the line shape is approximately Lorentzian near line center but falls off faster than a Lorentzian in the wings. We believe that this analytic model shows all the essential features of the interplay between cross correlation and non-Lorentzian line shapes. The theory is applied to the ODR experiment. In particular, we find that the asymmetry of the spectra is strongly affected by these two features, and so is the ODR spectral line shape when the saturating laser intensity is close to the threshold for producing two peaks.