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IEEE Journal of Quantum Electronics

Diagrammatic density matrix theory of transient four-wave mixing and the measurement of transient phenomena

James G. Fujimoto, Ting K. Yee

Published Aug 1, 1986• 22 citations
Publisher

Abstract

A diagrammatic density matrix perturbation technique is applied to describe transient third-order nonlinear optical processes and their applications to the measurement of transient phenomena. Using this approach, pump probe, parametric scattering, and three-pulse scattering are analyzed in a computationally simple and unified manner. The density matrix allows the description of a wide variety of dynamical processes including population and dephasing relaxation as well as orientational randomization and spatial diffusion. In addition, the diagrams provide a physical interpretation for the density matrix terms that they represent, and the effect of varying experimental parameters for the measurement of transient behavior can often be determined directly from the diagrams. Finally, topological similarities between diagrams provide a unified description of several different transient four-wave mixing measurement techniques.

DOI
10.1109/jqe.1986.1073120

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A diagrammatic density matrix perturbation technique is applied to describe transient third-order nonlinear optical processes and their applications to the measurement of transient phenomena. Using this approach, pump probe, parametric scattering, and three-pulse scattering are analyzed in a computationally simple and unified manner. The density matrix allows the description of a wide variety of dynamical processes including population and dephasing relaxation as well as orientational randomization and spatial diffusion. In addition, the diagrams provide a physical interpretation for the density matrix terms that they represent, and the effect of varying experimental parameters for the measurement of transient behavior can often be determined directly from the diagrams. Finally, topological similarities between diagrams provide a unified description of several different transient four-wave mixing measurement techniques.
We apply a diagrammatic density matrix perturbation technique to analyze third order nonlinear optical processes. The diagrammatic approach is shown to yield a significant simplification in the computation of both steady state and transient response while including the effects of damping and linewidth. The topology of the diagrams is stressed in identifying and interpreting their associated perturbation terms as well as predicting frequency resonance and transient behavior. We demonstrate the practical utility of the technique by presenting a detailed analysis and discussion of the processes of stimulated Raman scattering, hot luminescence, and pressure induced resonances in four wave mixing.
In the present paper a diagrammatic analysis of the density operator for the evaluation of nonlinear optical quantities is considered. The present approach extends earlier diagrammatic analysis by treating the time evolution of both the wave function and its complex conjugate. Time-ordered graphs result, each of which corresponds to a term in the density matrix. Examples involving the third-order susceptibility are discussed for both monochromatic and pulse excitation. In particular coherent rotational transient birefringence is discussed. The diagrams provide a convenient means by which nonlinear optical processes can be precisely defined and the susceptibility readily evaluated.
A diagrammatic density matrix technique is developed and applied to study nonlinear optical processes involving responses from weak probe fields interacting with a system under strong excitation. These diagrams consist of lowest-order double-Feynman diagrams which represent the density matrix perturbation terms describing the system response to the weak probe fields1,2 combined with dressed propagators which represent the density matrix elements of the system under strong excitation. The dressed propagators are described by a set of linear equations which, in most cases, may be derived in inspection. The methodology is similar to the bare-atom approach.2 The advantages of using this diagrammatic technique are that (1) it allows one to select and express analytically the dominant and appropriate density matrix elements (diagrams and dressed propagators); (2) analytical expressions may be constructed directly from the diagrams and thus lengthy calculations needed to obtain closed-form solutions are greatly reduced; (3) it provides a physical interpretation for the density matrix solutions and nonlinear optical processes; and (4) it may be generalized to more complicated, multilevel systems. We demonstrate the application of this technique to describe gain saturation, Stark splitting, and Raman scattering.
Chapter 5 begins the systematic application of the density matrix to describe optical interactions. Stationary two-and three-level atoms interacting with a traveling wave are compared to show that the presence of additional levels changes the saturation behavior of atoms. Then motion of the atoms is introduced and the phenomena of spatial and frequency-domain hole-burning are analyzed. Special effects due to standing waves, such as the light shift, are covered in this section. Next, tri-level coherence is considered, to extend the concept of coherences to two-photon transitions or Zeeman interactions in folded 3- or Λ-systems. Higher order interactions are also analyzed. This includes four-wave mixing and pump–probe techniques which form the basis of common nonlinear laser spectroscopies. Quantum interference is covered as a prelude to the topic of electromagnetically induced transparency. Higher order coherent interactions are also treated using a diagrammatic calculation technique based on Feynman diagrams.
An efficient method has been developed for the calculation of third-order time- and frequency-resolved optical signals. To obtain the general four-wave mixing signal, seven auxiliary density matrices have to be propagated in time. For the special cases of two-pulse photon-echo and transient-grating signals, two or three density matrices, respectively, are required. The method is limited to weak laser fields (it is thus valid within the third-order perturbation theory) but allows for any pulse durations and automatically accounts for pulse-overlap effects. To illustrate the method, we present the explicit derivation of the three-pulse photon-echo signal. Any other third-order optical signal can be calculated in the same manner. As an example, two- and three-pulse photon-echo and transient-grating signals for a weakly damped displaced harmonic oscillator have been calculated.
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.