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Journal of the Optical Society of America B

Nonlinear response function for time-domain and frequency-domain four-wave mixing

Shaul Mukamel, Roger F. Loring

Published Apr 1, 1986• 207 citations
Publisher

Abstract

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.

DOI
10.1364/josab.3.000595

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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 systematic theoretical and computational investigation of the microscopic factors which determine the phase of the signal field in time-resolved quasidegenerate three-pulse scattering experiments is presented. The third-order phase-matched response is obtained by density-matrix perturbation theory using a Green-function formalism for a system composed of two well-separated sets of closely spaced energy levels. Equations for calculating the electric field of four-wave-mixing signals generated by path-length delayed pulses are given. It is found that the phase of the signal field is determined by the excitation pulse phases, the dynamics of the nonlinear polarization decay, the product of four transition dipole matrix elements, and by a pulse-delay-dependent phase modulation at the frequency of the first dipole oscillation in the four-wave-mixing process. Analytic results for a two-level Bloch model show the phase shift from rapid nonlinear polarization decay. The product of dipole matrix elements is real and positive for three-level processes (bleached ground-state absorption and excited-state emission), but can be real and negative for some four-level Raman processes. The pulse-delay-dependent phase modulation treated here is closely related to the interferometric pulse-delay-dependent amplitude modulation observed in some collinear experiments, and plays a role in producing photon echos in inhomogeneously broadened samples. Numerical calculations of phase-resolved electric fields for finite duration pulses using a Brownian oscillator model appropriate for condensed-phase dynamics are presented. The ability of pulse-delay-dependent phase modulation to encode the frequency of the initially excited dipole onto the phase of the signal field can be exploited to examine energy-level connectivity, reveal correlations hidden under the inhomogeneous lineshape, and probe relaxation pathways in multilevel systems.
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.
We present a theory of the four-wave-mixing (FWM) spectroscopy for the multilevel system coupled to the bath with arbitrary time scale, in contrast to most previous work that assumes the fast bath dynamic limit (motionally narrowed limit). Exact quantum mechanical expressions for the relevant nonlinear response tensor and the nonlinear polarization vector are obtained for a model Hamiltonian. Effects of molecular rotation and the polarization directions of the input radiation fields are also fully taken into account. The results show how the cross correlation as well as the autocorrelation functions of the fluctuations in the energy-gaps or transition frequencies between eigenstates contribute to the FWM spectroscopy. The result is applied to three-pulse echo spectroscopy. We examine the results for a variety of model spectral density functions, and recover the results of the previously reported theories as limiting cases.
We will demonstrate that four-wave mixing with linearly chirped (phase-modulated) pulses is a unique tool for obtaining information on the dynamics and level structure of a system. Especially, it will be shown that the transient-grating-scattering type of experiment with chirped pulses provides an immediate answer to the question of whether the dynamics of a system occurs on a fast and/or slow time scale. In addition, we present compelling evidence that chirped four-wave mixing in a molecular system is a viable method for measuring excited-state vibrational frequencies. Double-sided Feynman diagrams are used for a third-order perturbative calculation of two-level four-wave-mixing effects and chirped coherent Raman scattering. The diagrams provide a visual representation of the quantum-mechanical pathways that the system can take as a result of the different field interactions. The number of quantum-mechanical pathways that contribute to the signal is shown to depend on the chirp rate compared to the time scale(s) of the system dynamics. A stochastic model is used to describe the optical dynamics of the system. The resulting expressions for the third-order nonlinear polarization are so complex that numerical calculations are necessary to simulate the time dependence of the optical response. It will also be shown that our theoretical results in the appropriate limits converge to those obtained for impulsive or continuous-wave excitation.
A comprehensive theory of a new type of three-wave optical mixing (${\ensuremath{\omega}}_{3}=2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$) is given for the spectroscopic purpose of determining ultra-short relaxation times (picoseconds or less) associated with excited states of condensed matter. Two incident light frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ are chosen so that they are both resonant with an inhomogeneously broadened optical transition and $|{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}|$ is in the vicinity of inverse relaxation times. The theory is developed for its major application to broad-band electronic transitions. The nonlinear susceptibility ${\ensuremath{\chi}}_{R}^{(3)}$ for this process (named resonant Rayleigh-type mixing) is calculated by a density-matrix formalism with the model of a two-level atomic system incorporating the distribution of resonance frequencies and longitudinal (T1{T}_{1}T1​) and transverse (T2{T}_{2}T2​) relaxation times. The result shows a frequency characteristic depending only on T1{T}_{1}T1​, T2{T}_{2}T2​, and ${\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$ in a broad-band limit, which serves for the determination of T1{T}_{1}T1​ and T2{T}_{2}T2​ in the frequency domain. The analysis is further extended to include the various effects as follows. The calculation of the saturation effect in the lowest order reveals that it modifies the shape of the frequency characteristic of the nonlinear susceptibility but does not affect seriously the determination of relaxation times. The effect of spectral cross relaxation within the inhomogeneous broadening is incorporated by a generalized density-matrix formalism. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ consists of two terms. The dominant term is the same as before except that T1{T}_{1}T1​ is replaced by a combined relaxation time ${T}_{1}^{\ensuremath{'}}={({T}_{1}^{\ensuremath{-}1}+{T}_{3}^{\ensuremath{-}1})}^{\ensuremath{-}1}$, where T3{T}_{3}T3​ is the cross-relaxation time. The other term originates from the inverse spectral-diffusion process. The effect of other energy levels located between the two levels under study is analyzed with a simple three-level model. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ also consists of two terms, the dominant one being the same as that for the two-level model except that ${T}_{1}^{\ensuremath{-}1}$ should be interpreted as total population decay rate of the upper level. For accurate derivation of ${\ensuremath{\chi}}_{R}^{(3)}$ from experiment, we discuss the interference between the resonant and nonresonant susceptibility terms and the light-wave propagation effect. Finally, the close analogy between this type of optical mixing and the photon echo is discussed.
The delayed or time-domain four-wave-mixing experiment is treated in the regime of intense near-resonant pulses. The interaction with the radiation during both pump and probe pulses is considered to all powers of the electric field amplitude. Analytical results are obtained for an effective four-level system. These include the dependence of the coherence amplitudes on the ratio of the pump-field intensities when there is a large vibrational discrepancy between ground and excited electronic states and a general solution for the unitary time development during the probe pulse. For the first time, delayed coherent anti-Stokes Raman scattering is detected from highly dilute (10-ppm) guest molecules. Illustrative examples are presented for the system of pentacene in benzoic acid at low temperature. Vibronic-free induction decay and the effect of field inhomogeneity across the beam profile are found to be essential for understanding the observed intensity and spectral distribution of the signal beam in the region of optimum pulse intensity.