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Physical Review A

Four-wave mixing using partially coherent fields in systems with spatial correlations

Shaul Mukamel, Eiichi Hanamura

Published Feb 1, 1986• 48 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.33.1099

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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 microscopic correlation-function expression for the photon-echo signal in systems with long-range spatial correlations is developed. In general, the signal is given by an eight-point correlation function of the dipole operator involving two particles. In the absence of long-range spatial correlations, this correlation function may be factorized into a product of two four-point, single-particle correlation functions. Only under these conditions is the signal proportional to the absolute square of the ensemble-averaged nonlinear polarization. We predict that photon echoes may be used to probe the long-range spatial correlations near critical points.
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 quasi-two-level system is a generalization of a two-level system where each of the single levels is replaced by a nearly degenerate multiplet. We develop the theory of the photon echo, the stimulated echo, and fluorescence emission for such systems when they are subjected to pulses of electromagnetic radiation that are short compared to the reciprocal (in frequency units) of the widths of the multiplets. The observable in each of these experiments is related to equilibrium time-correlation functions of the system in the absence of the radiation field. The correlation functions associated with these experiments involve more than two dynamical operators and more than two times, unlike the correlation functions for such processes as low-power optical absorption and neutron scattering. We introduce an exactly solvable factorization model for the transition moments, which includes as special cases the situation where either of the multiplets is a single state. We also consider the unitary model, which was introduced by Mims, and which leads to considerable simplification of the correlation functions. Several examples of these models are discussed to investigate the features of time-dependent fluorescence experiments and coherent optical transient experiments. In general, there is no simple connection between the time dependence of the decay of a photon echo and the Fourier transform of the low-power homogeneous absorption spectrum.
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.
The theoretical basis for new signal transients and spectral features generated in field-correlated four-wave mixing spectroscopies is developed. Special attention is given to those signal responses that are sensitive to phase or amplitude correlation among the input driving fields, and not simply their intensity correlation. Thus the cases of incoherent broadband excitation and of coherent short-pulsed excitation will be discussed and compared. Applications to the coherent Raman spectroscopies, both electronically nonresonant and full resonant, are analyzed. Novel interferometric oscillatory behavior is exposed in terms of radiation-matter detuning beats and matter-matter bilevel and trilevel quantum beats. In addition, new resonances are found that have submaterial linewidths and detuning parameters that lock onto the mode frequency of the driven chromophore. These spectral features are members of a class of bichromophore resonant line shapes arising from nonlinear mixing with correlated driving fields. Analytic results are presented and modeled to anticipate experimental results presented in the following paper [Phys. Rev. A 43, 3922 (1991)].