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The Journal of Chemical Physics

Nonperturbative response functions: A tool for the interpretation of four-wave-mixing signals beyond third order

Lipeng Chen, Elisa Palacino‐González, Maxim F. Gelin, Wolfgang Domcke

Published Dec 18, 2017• 19 citations
Publisher

Abstract

Considering an electronic two-level system coupled to vibrational degrees of freedom and driven by short and intense non-overlapping laser pulses, we introduce the concept of nonperturbative response functions. These response functions are expressed in terms of effective electronic transition dipole moments which depend on the strength of the field-matter coupling and on the pulse durations. It is shown that the nonlinear polarization representing four-wave-mixing signals can elegantly be expressed in terms of these nonperturbative response functions to all orders in the field-matter coupling. The nonperturbative response functions generalize the interpretative power of the familiar third-order response functions to four-wave-mixing experiments with intense laser pulses.

DOI
10.1063/1.5004763

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The Journal of Chemical Physics•Jun 16, 2006•74 citations
Considering an electronic two-level system coupled to vibrational degrees of freedom and driven by short and intense non-overlapping laser pulses, we introduce the concept of nonperturbative response functions. These response functions are expressed in terms of effective electronic transition dipole moments which depend on the strength of the field-matter coupling and on the pulse durations. It is shown that the nonlinear polarization representing four-wave-mixing signals can elegantly be expressed in terms of these nonperturbative response functions to all orders in the field-matter coupling. The nonperturbative response functions generalize the interpretative power of the familiar third-order response functions to four-wave-mixing experiments with intense laser pulses.
The present work is devoted to the description of a nonperturbative treatment of anharmonic effect and mode-mode coupling taking place in molecular vibrational systems. The information is obtained from the four-wave-mixing signal detected in the usual photon-echo phase-matched direction, termed 4WM photon-echo, using heterodyne detection to test separately the real and the imaginary parts of the third-order polarization. This approach allows a complete redistribution of the relaxation, dephasing, and transition constants on account of the magnitude of the coupling strength. In addition, our analytical approach accounts for the contributions provided by the various possible chronological orderings of the field interactions. The high sensitivity of the signal with both the energy shifts induced by the internal couplings and off-resonant energy parameter of the laser excitation is properly analyzed in the framework of our analytical description.
We treat the problem of four-wave resonant mixing of ultrashort laser pulses in gaseous media. We show that for some transitions, for example, quadrupole transitions, the polarization effects can be strongly manifested. For four-wave two-photon resonant mixing we find that the signal-wave plane rotation depends on the parameters of the resonant medium and the pulses mixed. We show that for long pump pulses and short signal pulses in the ultrashort-pulse time domain the angle of rotation of the plane of polarization of the signal plane can reach several dozen degrees. We show that the effect of altering the state of polarization of the short signal pulse can be used for measuring its shape and duration. Since the polarization measurements are precise and the proposed regimes of four-wave mixing do not depend on phase and amplitude fluctuations of the measuring pulse, the polarization four-wave mixing technique can be more accurate than the technique widely used for such purposes, measurement of the second-order intensity correlation function. We also discuss new nonlinear spectroscopy for measuring fast resonant susceptibilities (in particular, second-order polarizabilities) of the medium, based on the rotation of the polarization plane or the polarization ellipse of the signal pulse.
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 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 theory for the nonlinear optical response of an assembly of four level atoms representing an s–p transition, with dipole–dipole interactions is developed. Simulation results for two-pulse ultrafast four wave mixing measurements in atomic clusters and in liquids demonstrate that the electronic density of states can be obtained by combining absorption and four wave mixing experiments. We show that a sensitive direct probe for interatomic interactions and correlations is provided by reversing the pulse time ordering in a photon echo sequence. The effect of nuclear motions on the linear absorption spectrum is investigated.
The nonperturbative approach to the calculation of nonlinear optical spectra of Seidner et al. [J. Chem. Phys. 103, 3998 (1995)] is extended to describe four-wave mixing experiments. The system-field interaction is treated nonperturbatively in the semiclassical dipole approximation, enabling a calculation of third order nonlinear spectroscopic signals directly from molecular dynamics and an efficient modeling of multilevel systems exhibiting relaxation and transfer phenomena. The method, coupled with the treatment of dynamics within the Bloch model, is illustrated by calculations of the two-dimensional three-pulse photon echo spectra of a simple model system-a two-electronic-level molecule. The nonperturbative calculations reproduce well-known results obtained by perturbative methods. Technical limitations of the nonperturbative approach in dealing with a dynamic inhomogeneity are discussed, and possible solutions are suggested. An application of the approach to an excitonically coupled dimer system with emphasis on the manifestation of complex exciton dynamics in two-dimensional optical spectra is presented in paper II Pisliakov et al. [J. Chem. Phys. 124, 234505 (2006), following paper].