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IEEE Journal of Quantum Electronics

Graphical representation for the density matrix in nonlinear optics

John P. Uyemura

Published Apr 1, 1980• 12 citations
Publisher

Abstract

A technique for describing the time evolution of the density matrix by diagrams is developed. When applied to time harmonic optical mixing problems, the method allows one to write down the required matrix elements by inspection of the graphs. This allows a significant reduction in the calculations in addition to illustrating the physics of the processes.

DOI
10.1109/jqe.1980.1070488

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A technique for describing the time evolution of the density matrix by diagrams is developed. When applied to time harmonic optical mixing problems, the method allows one to write down the required matrix elements by inspection of the graphs. This allows a significant reduction in the calculations in addition to illustrating the physics of the processes.
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 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.
Previous diagrammatic analyses of the density matrix are extended to incorporate quantum-electrodynamic processes fully by inclusion of the photon propagator and its complex conjugate. It is shown that eight fundamental time-ordered diagrams are necessary to describe radiative processes completely. For each of these, inclusion of the photon propagator permits, in addition to polarization, a direct calculation of electric field expectation values. Electric field correlations associated with the fundamental diagrams lead to a diagrammatic (quantum-electrodynamic) interpretation and derivation of the spontaneous emission limited linewidth associated with simulated emission as given by the Townes-Schawlow line narrowing formula. Stimulated Raman scattering, two-wave mixing, and parametric amplification are illustrative of the extension of these diagrammatic techniques to nonlinear optical phenomena.>
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.