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The Journal of Chemical Physics

Effects of permanent dipole moments in transient four-wave mixing experiments

J. P. Lavoine

Published Sep 7, 2007• 11 citations
Publisher

Abstract

A two-pulse degenerate four-wave mixing experiment is analyzed in the case where the medium under investigation can be modeled by two-level systems having unequal permanent dipole moments. By modeling the light pulses by double exponentials [exp(-Gamma/t/)], we give an analytical expression of the third-order nonlinear polarization of the medium. We apply this result to simulate the measured signal in such experiment. We show that in the case of a two-photon transition, a signal can be detected if the pump pulse interacts with the medium before the probe pulse contrary to what is observed for excitations in the resonance region. An attempt to explain this behavior is made and the detected signal is analyzed in terms of pure coherent processes. This effect appears as a signature of the presence of permanent dipole moments. To test this property on a more realistic system, we then have considered a one-dimensional frequency-selected infrared degenerate four-wave mixing experiment on a molecular anharmonic vibrational mode modeled by a Morse potential and coupled to a dissipative bath of harmonic oscillators. We show that the two-photon transitions allowed by the presence of permanent dipole moments enable to analyze the multilevel system dynamics as if they were the one of a two-level system. Our results can also be extended to the case of inhomogeneous broadening and are of interest to study the infrared photon-echo response of anharmonic vibrational modes.

DOI
10.1063/1.2753472

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A two-pulse degenerate four-wave mixing experiment is analyzed in the case where the medium under investigation can be modeled by two-level systems having unequal permanent dipole moments. By modeling the light pulses by double exponentials [exp(-Gamma/t/)], we give an analytical expression of the third-order nonlinear polarization of the medium. We apply this result to simulate the measured signal in such experiment. We show that in the case of a two-photon transition, a signal can be detected if the pump pulse interacts with the medium before the probe pulse contrary to what is observed for excitations in the resonance region. An attempt to explain this behavior is made and the detected signal is analyzed in terms of pure coherent processes. This effect appears as a signature of the presence of permanent dipole moments. To test this property on a more realistic system, we then have considered a one-dimensional frequency-selected infrared degenerate four-wave mixing experiment on a molecular anharmonic vibrational mode modeled by a Morse potential and coupled to a dissipative bath of harmonic oscillators. We show that the two-photon transitions allowed by the presence of permanent dipole moments enable to analyze the multilevel system dynamics as if they were the one of a two-level system. Our results can also be extended to the case of inhomogeneous broadening and are of interest to study the infrared photon-echo response of anharmonic vibrational modes.
The main goal of this paper is to analyze a two-photon degenerate four-wave-mixing process that occurs in an isotropic media modeled by homogeneously broadened two-level systems having unequal permanent dipole moments and subject to rotational diffusion. Our theoretical treatment is based upon a third-order perturbative solution of the density-matrix equations and does not require the rotating-wave approximation. The medium being excited by three stationary fields, we give an analytical expression of the macroscopic polarization up to the third order, including permanent dipole moments, orientational diffusion, and field polarizations. We apply these general results to the description of a degenerate four-wave-mixing process by selecting, in the expressions, the contributions in the direction ${\mathbf{k}}_{\mathrm{\ensuremath{\alpha}}}$-${\mathbf{k}}_{\mathrm{\ensuremath{\beta}}}$+${\mathbf{k}}_{\ensuremath{\gamma}}$, ${\mathbf{k}}_{\mathrm{\ensuremath{\alpha}}}$, ${\mathbf{k}}_{\mathrm{\ensuremath{\beta}}}$, and ${\mathbf{k}}_{\ensuremath{\gamma}}$ being, respectively, the wave vectors of the exciting fields. We show that the two-photon degenerate four-wave mixing is dependent on the permanent dipole moments and is not present if these dipole moments are equal or zero. We analyze the influence of rotational diffusion. Different cases of field polarizations are also considered, and we discuss a comparison of the well-known one-photon degenerate four-wave-mixing process and this two-photon process. This process is also discussed in the particular case of phase conjugation.
Properties of the resonance peaks associated with the non-degenerate four-wave mixing signal in a two-level molecular system with non-zero permanent dipole moments are studied in the space defined by the pump ( omega 1 ) and probe ( omega 2 ) frequencies. An analytical expression for the Fourier components of the nonlinear homogeneous macroscopic polarization are obtained, where the terms responsible for the oscillation (at frequency omega 3 =2 omega 1 - omega 2 ) of the populations and coherences are also identified. Topological properties of the resulting nonlinear spectra are studied in frequency space using three-dimensional curves and contour plots. We have additionally identified the dominant contributions of the coherence and/or population that determine the intensity and the spectral linewidth. The critical quantities for the present analysis are the transition and permanent dipole moments when the rotating-wave approximation (RWA) is not considered.
We consider the Four-Wave mixing (FWM) signal and the effects of the pump beam on the nonlinear optical responses of a two-level system where intramolecular coupling is modelled by two coupled harmonic electronic potentials, displaced both in their equilibrium minimum position and energy. We investigate the modifications introduced in the FWM signal, considering a simple representation for the vibrational structure and including molecular electronic states with nonzero permanent dipole moments. The critical quantities for this analysis are the transition and permanent dipole moments, which depend strongly on the intramolecular coupling. Our results show how both absorption and dispersion are affected by the vibronic coupling.
Using a perturbation theory based on the density-matrix formulation, we study the nonlinear optical responses of a noninteracting three-level model system to consecutive coherent pulsed excitations, as realized in several recent experiments involving excitonic transitions in quantum wells. The terahertz emission, which is a second-order response, and the four-wave mixing signal, a third-order response, are calculated within the rotating-wave approximation, in the presence of detuning, dephasing, and inhomogeneous broadening. We study the quantum beats in the photon echo from transient four-wave mixing experiments of a three-level system. We find that the temporal profile of the photon echo signal in the four-wave mixing experiments depends very strongly on the amount of inhomogeneous broadening of the energy levels involved. Both the position and the intensity of the peaks exhibit a smooth evolution from strong quantum beats to a conventional photon echo as the inhomogeneous broadening increases. These features from our noninteracting model system are compared with a recent experiment, and found to account for a number of experimental observations.
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.