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The Journal of Chemical Physics

Optical four wave mixing spectroscopy for multilevel systems coupled to multimode Brownian oscillators

Jaeyoung Sung, R. Silbey

Published Jan 16, 2003• 20 citations
Publisher

Abstract

We generalize the multimode Brownian oscillator model for a two-level system to optical four wave mixing spectroscopy of a multilevel system, based on the recent work of Sung and Silbey [J. Chem. Phys. 115, 9266 (2001)]. For the Ohmic limit where the phenomenological Langevin equation becomes correct, we present exact expressions for energy-gap correlation functions in the strong damping case as well as in the weak damping case.

DOI
10.1063/1.1534103

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We generalize the multimode Brownian oscillator model for a two-level system to optical four wave mixing spectroscopy of a multilevel system, based on the recent work of Sung and Silbey [J. Chem. Phys. 115, 9266 (2001)]. For the Ohmic limit where the phenomenological Langevin equation becomes correct, we present exact expressions for energy-gap correlation functions in the strong damping case as well as in the weak damping case.
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.
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].
This work provides an alternative derivation of third order response functions in four wave mixing spectroscopy of multichromophoric macromolecular systems considering only single exciton states. For the case of harmonic oscillator bath linearly and diagonally coupled to exciton states, closed form expressions showing all the explicit time dependences are derived. These expressions can provide more solid physical basis for understanding 2-dimensional electronic spectroscopy signals. For more general cases of system-bath coupling, the quantum master equation (QME) approach is employed for the derivation of multistep time evolution equations for Green function-like operators. Solution of these equations is feasible at the level of 2nd order non-Markovian QME, and the new approach can account for inter-exciton coupling, dephasing, relaxation, and non-Markovian effects in a consistent manner.
We study theoretically ultrafast modulation spectroscopy in Doppler-broadened cascading three-level and four-level systems. The pump beams consist of two frequency components that drive two-pathway excitation. The four-wave mixing (FWM) signal exhibits damping oscillation as the relative time delay τ between the two pump beams is varied, from which the resonant frequencies between energy levels can be deduced. Furthermore, Doppler-free precision in the measurement of the resonant frequencies can be achieved when the pump beams have either narrow-band or broadband linewidths. We study two-photon resonant FWM induced by time-delayed correlated fluctuating fields in a Doppler-broadened cascading three-level system. It is related intrinsically to the sum-frequency trilevel echo when the pump beams have broadband linewidths. However, unlike the corresponding incoherent-light stimulated photon echo, the maximum of the FWM signal is shifted from τ≃0. If we consider the fluctuations of the pump beams, the FWM signal exhibits damping oscillation when the laser frequency is off resonant from the two-photon transition.
It was studied the solvent influence over a two level system in the four wave mixing spectroscopy. the solvente effect was incorporated to the model in a stochastic form, through the broadening of the upper molecular level. Two intense waves of the same frequency, ω1, and another wave with minor intensity and ω2 frequency, were considered coupled to the nonlinear system, generating a new signal whose frequency is ω3, and where the wave-vectors follow the relation vector k1 + vector k1 = vector k2 + vector k3. Using the dipolar-electric and the rotating wave approximations, it was calculated the average of the coherence at the ω1, ω2 and ω3 frequencies. From that results, analytical expressions to the susceptibilities at the studied frequencies were obtained. Linear, coherent and coupled processes were identified.