Description of nonlinear optical response using phase space wave packets

Abstract

Closed-form expressions are derived for the nonlinear response function of a polyatomic molecule with two electronic states and linearly displaced harmonic potentials to arbitrary order in the radiation field. The underlying dynamics can be best visualized using wave packets in phase space. Using exact expressions for these wave packets, we discuss the possible factorization of the process into two parts representing a forward and a backward propagating wave packet. This factorization facilitates the description of sequential measurements, such as pumpprobe spectroscopy, in terms of a doorway wave packet prepared by the pump and a window prepared by the probe. We show that once a reduced description is adopted, where we trace over bath degrees of freedom, this factorization is possible only in certain limiting cases. I. Introduction The density matrix offers a natural and convenient means for calculating and interpreting ultrafast optical measurements.14 This is well recognized for studies in the condensed phase, since it allows performing thermal averaging and developing a reduced description in which bath coordinates have been eliminated and replaced by dephasing processes. However, even for isolated molecules in the gas phase, the density matrix formulation has a lot to offer compared with wave function based calculations.~ The density matrix follows naturally the time ordering of the various interactions with the fields. In the Wigner representation, it can be used to construct a phase-space wave packet representation of the nonlinear response which provides a powerful framework for developing classical and semiclassical approximations.2 The possibility of visualizing time domain spectroscopy in terms of wave packets in phase space is particularly valuable for sequential experiments such as pump-probe, transient grating, and impulsive Raman scattering conducted using well-separated laser pulsesbll Sequential experiments can be described in terms of overlap of two wave packets in phase space (thedoorway window picture).2J A full quantum treatment of these wave packets in the complete phase space (when all degrees of freedom are included) has been de~eloped.~ This level of description is restricted to small systems (e.g., pumpprobe spectroscopy of ICN in the gas pha~e).~ For larger systems and for spectroscopy in the condensed phase, we need to develop a reduced description in which only part of the phase space is treated explicitly and the remaining degrees of freedom are eliminated. Using a Langevin equation, Yan and Mukamel have calculated the wave packets for a multimode Brownian oscillator model.2 These calculations are limited by the Langevin equation which provides a semiclassical high-temperature approximation. We recently developed a theory for the nonlinear optical response of a multimode Brownian oscillator model which extends that theory in the following respects: First, by adopting a harmonic model for the bath degrees of freedom, we derived exact quantum expressions for the phase space wave packets which are valid at all temperatures and for the entire range of parameters. The solution was obtained using path integral techniques. Second, we calculated the response function to arbitrary order in the field so that we can describe processes higher than four-wave mixing. Third, we incorporated an arbitrary dependence of the transition dipole on nuclear coordinates (non-Condon effects). In this article, we use the exact solution of the multimode Brownian oscillator model to explore the applicability of the