Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Chemical Physics

Four wave mixing spectroscopy for a multilevel system

Jaeyoung Sung, R. Silbey

Published Nov 22, 2001• 104 citations
Publisher

Abstract

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.

DOI
10.1063/1.1413979

Related papers

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

Shaul Mukamel, Roger F. Loring

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.

Journal of the Optical Society of America B•Apr 1, 1986•207 citations

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

Jaeyoung Sung, R. Silbey

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.

The Journal of Chemical Physics•Jan 16, 2003•20 citations

Chirped four-wave mixing

Koos Duppen, Foppe de Haan, Erik T. J. Nibbering +1 more

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.

Physical Review A•Jun 1, 1993•65 citations

Strong-field theory of four-wave mixing

Alfred M. Levine, N. Chencinski, W. M. Schreiber +2 more

A nonperturbative semiclassical theory is presented for the treatment of four-wave mixing (FWM) in a four-level system. This treatment is based on a transformation to a generalized rotating frame followed by a numerical diagonalization of the resulting time-independent Hamiltonian. For strong fields, all resonant frequencies are shifted from the atomic (molecular) frequencies. Lines are split up in a way similar to the ac Stark effect for a two-level system, and for even stronger fields, the effect of stirring is observed for the first time in four-wave mixing. In addition, a new extra resonance appears which is induced by the saturating field. This field-induced resonance is the new member in the family of extra resonances (pressure induced, or stochastic fluctuation induced), and it should be observed whenever saturating fields interact with a multilevel atomic or molecular system.

Physical Review A•Mar 1, 1987•29 citations

Four-wave mixing spectroscopy of two-level system with intramolecular coupling: the effect of relaxation times

Teresa Cusati, José Luis Paz, Emilio Squitieri +3 more

A study has been made of the effect of relaxation times on the four-wave mixing (FWM) signal response in frequency space for a two-level system where intramolecular coupling is explicitly taken into account within a vibronic model. The model is a generalization of the conventional two-level approach in that it includes a simplified description of the molecular structure and the explicit consideration of permanent dipole moments. Significant changes in both the intensity and shape of the peaks of the FWM spectrum are observed, due mainly to changes in the ratio between the longitudinal and transversal relaxation times, T 1/T 2.

Molecular Physics•May 20, 2002•1 citation

Two-dimensional optical three-pulse photon echo spectroscopy. I. Nonperturbative approach to the calculation of spectra

Tomáš Mančal, Andrei V. Pisliakov, Graham R. Fleming

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].

The Journal of Chemical Physics•Jun 16, 2006•74 citations
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.
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.
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 nonperturbative semiclassical theory is presented for the treatment of four-wave mixing (FWM) in a four-level system. This treatment is based on a transformation to a generalized rotating frame followed by a numerical diagonalization of the resulting time-independent Hamiltonian. For strong fields, all resonant frequencies are shifted from the atomic (molecular) frequencies. Lines are split up in a way similar to the ac Stark effect for a two-level system, and for even stronger fields, the effect of stirring is observed for the first time in four-wave mixing. In addition, a new extra resonance appears which is induced by the saturating field. This field-induced resonance is the new member in the family of extra resonances (pressure induced, or stochastic fluctuation induced), and it should be observed whenever saturating fields interact with a multilevel atomic or molecular system.
A study has been made of the effect of relaxation times on the four-wave mixing (FWM) signal response in frequency space for a two-level system where intramolecular coupling is explicitly taken into account within a vibronic model. The model is a generalization of the conventional two-level approach in that it includes a simplified description of the molecular structure and the explicit consideration of permanent dipole moments. Significant changes in both the intensity and shape of the peaks of the FWM spectrum are observed, due mainly to changes in the ratio between the longitudinal and transversal relaxation times, T 1/T 2.
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].