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Physical Review A

Semiclassical treatment of multiphoton interactions: Signal-flow-graph approach

Ho Kyoon Chung, Thomas A. DeTemple

Published Dec 1, 1980• 6 citations
Publisher

Abstract

A signal-flow-graph representation of Schr\"odinger's equation appropriate to the analysis of multiphoton interactions in the semiclassical limit is described. Within the rotating-wave and quasistatic approximations the equations of motion for the density-matrix elements can be recast into a system of algebraic equations on the basis of which the signal flow graphs are constructed. By introducing selected graph manipulations, the graph can be simplified and direct extraction of information is facilitated by using standard graph algebra. Illustrations are given for typical cases involving three waves and three- or four-level systems for serial, parallel, and parametric interactions. Also, the construction of an ac Stark graph, which is specifically used for the determination of ac Stark shifts, is illustrated. We show that perturbation results for virtual transitions as well as for real transitions can be obtained directly from the graph.

DOI
10.1103/physreva.22.2647

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A signal-flow-graph representation of Schr"odinger's equation appropriate to the analysis of multiphoton interactions in the semiclassical limit is described. Within the rotating-wave and quasistatic approximations the equations of motion for the density-matrix elements can be recast into a system of algebraic equations on the basis of which the signal flow graphs are constructed. By introducing selected graph manipulations, the graph can be simplified and direct extraction of information is facilitated by using standard graph algebra. Illustrations are given for typical cases involving three waves and three- or four-level systems for serial, parallel, and parametric interactions. Also, the construction of an ac Stark graph, which is specifically used for the determination of ac Stark shifts, is illustrated. We show that perturbation results for virtual transitions as well as for real transitions can be obtained directly from the graph.
The extraction of ac Stark shifts and splittings from a semiclassical treatment of multiple wave-atom interactions is considered. Within the rotating-wave and quasistatic approximations, we show that the information can be extracted from the complex Beer's coefficient sensed by a weak tunable probe wave, whose lower level is fully populated and unconnected to the strong fields. This situation leads to a great simplification of the density-matrix equations of motion, from which algorithms are derived for the algebraic construction of the Beer's coefficients appropriate to other cases not treated. Both level-shifted resonance and line-strength information are contained in the Beer's coefficient, many examples of which are given, some for the first time. Generalized sum rules for the transitions are derived along with a connection of this approach to other approaches including dressing transformations.
A semiclassical density matrix treatment is developed for the parallel near-resonant interaction of three waves in a four-level atomic or molecular system. Explicit solutions are developed and contain implicit AC Stark-shift and splitting information. The role of splitting and AC Stark-shifts is discussed for a system of two competing emitted waves starting from a common optically pumped level; a situation encountered in many optically pumped lasers. Application of the results to a Raman-Raman competition situation in ammonia is illustrated.
Motivated by recent experiments on second-harmonic generation in three-level systems [K. S. Yngvesson and E. L. Kollberg, Appl. Phys. Lett. 36, 104 (1980)], a semiclassical description of resonant three-wave parametric interactions in three nondegenerate levels is considered. Within the limitations of the rotating-wave approximation, analytical nonperturbative solutions for the nonlinear polarization are derived, which are then coupled to Maxwell's equations and integrated numerically. The results of sample calculations for second-harmonic generation show (a) a maximum harmonic conversion occurs on full resonance, (b) the conversion efficiency can be > 50% but requires a strongly saturating pump, (c) the conversion distance increases with pumping intensity, and (d) the inclusion of ac Stark splittings is crucial for the achievement of high conversion efficiency. The role of level splittings is further explored for the case of a resonant parametric oscillator.
The algebraic structure of a truncated semiclassical Hamiltonian describing the translational–vibrational interaction has been determined and exploited to compute the S matrix for the system. The equations of motion have been integrated in a 4×4 faithful nonunitary representation of the group, and the final results lifted to the infinite dimensional unitary representation of the group. Computations of transition probabilities are facilitated by using powerful disentangling theorems to write the S matrix in convenient operator-ordered form. These methods are in quantitative agreement with results of Gazdy and Micha, and generalize easily to the multimode case. The structure of the Ehrenfest equations of motion for the operators of the algebra is identical to the structure of the dynamical algebra itself. The S matrix is also reconstructed from these equations of motion. Finally, a simple geometric picture is presented to describe the two extreme limits of the parametric oscillator problem involving only single quantum transitions or double quantum transitions.
A semiclassical treatment is given for the serial, near-resonant interaction of three waves in seven equivalent four-level atomic or molecular systems. Explicit equations are presented for the complex susceptibilties for the three waves, valid for $t\ensuremath{\gg}{T}_{2}$, which contain implicit ac Stark-shift and -splitting information, and can be separated into one-, two-, and three-photon contributions. For the case of a weak third or terminal wave, equations for the degree of ac Stark shifting and splitting are found and indicate that the three-photon or triple-resonance contribution is immune to the optical Stark effect if the remaining two waves are tuned to at least two-photon resonance conditions. The conditions for Doppler-free triple resonance are outlined, and it is shown that an equivalent ac Stark broadening may exist but may also be minimized by an appropriate choice of field intensities.
The authors present an exact treatment of two- and multiquantum transitions in three-level systems driven by two intense linearly polarised monochromatic fields based on the semiclassical many-mode Floquet theory developed recently. Further they extend the almost degenerate perturbation theory of Salwen (1955-6) to the two-mode Floquet Hamiltonian and obtain approximate analytical formulae for multiphoton transition probabilities, resonance bichromatic shifts and widths, and absorption lineshapes beyond the conventional rotating-wave approximation (RWA). Detailed comparison of the analytical, the RWA, and the exact results is given. Several novel features of multiphoton lineshape characteristics are pointed out.