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The Journal of Chemical Physics

Coherence vs incoherence in stepwise laser excitation

Bruce W. Shore, Michael Austin Johnson

Published Jun 15, 1978• 27 citations
Publisher

Abstract

We demonstrate the manner in which solutions of the (coherent) Schrödinger equation for multistep excitation approach those of an (incoherent) rate equation. For a two-level system, coherence produces oscillations about the mean rate-equation values. In more general N-level sequences the two types of solutions agree in the limit described by Ackerhalt and Eberly. We point out instances in which a simple rate equation cannot duplicate results of the Schrödinger equation.

DOI
10.1063/1.435694

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Previously published papers have noted the existence of two-level periodicities in idealized theoretical models of monochromatic excitation of multilevel systems. The present paper provides a theoretical derivation of such two-level dynamics from the more general $N$-level rotating-wave-approximation Schr\"odinger equation and applies the resulting formulas to different extremes of excitation in a nondegenerate system of sequential (ladderlike) excitation: first, when intermediate levels have much larger cumulative detunings than Rabi frequencies, and second, when intermediate Rabi frequencies are much larger than detunings. In each case one has simple analytic expressions for the effective two-level Rabi frequencies and detunings.

Physical Review A•Sep 1, 1981•34 citations
We demonstrate the manner in which solutions of the (coherent) Schrödinger equation for multistep excitation approach those of an (incoherent) rate equation. For a two-level system, coherence produces oscillations about the mean rate-equation values. In more general N-level sequences the two types of solutions agree in the limit described by Ackerhalt and Eberly. We point out instances in which a simple rate equation cannot duplicate results of the Schrödinger equation.
Description of detailed temporal excitation dyanmics for coherent excitation, such as is produced by idealized laser radiation, contrasts with evaluation of rate coefficients by means of generalized Golden Rule procedures; it requires an appropriate time-dependent Schroedinger equation. When the atom undergoing excitation is also affected by incoherent processes, such as collisions, this equation no longer suffices. The Heisenberg equations, or equivalent density-matrix equations, permit treatment in which coherence and incoherence play comparable roles in the excitation dynamics. Unlike rate equations, such equations must incorporate complexities that originate in the orientation degeneracy expressed by magnetic quantum numbers. In simple cases of coherent excitation, both for single-photon and multiphoton excitation, the sublevels merely require an average of 2J+1 independent Schroedinger equations. Relaxation couples the independent equations. It has been known for some time that appropriate state-multipole operators can simplify the description of many phenomena connected with optical pumping. This memo discusses application of these multipole operators to the description of Raman (or more general multiphoton) coherent excitation. In some simple limiting cases the equations simplify, but in general one has a hierarchy of coupled multipole polarizations and coherences in place of the populations and coherences that occur as variables in nondegenerate systems. 28 refs., 4 figs.
We consider a gas of molecules absorbing light from a monochromatic laser source and also undergoing both phase randomizing and thermalizing collisions with a heat bath of inert molecules. The generalized master equation formalism, used previously to study a system of this type with a two-level absorber, is extended to an absorber with N equally spaced levels. The possibility of unimolecular decay from any or all of the N levels is included in the formalism. The well-known relaxation model of Rubin and Shuler and the chemical reaction model of Montroll and Shuler are built into the formalism as special cases when the light source is turned off. Specific calculations for an eight-level model are presented and compared with results expected for an incoherent light source. A coherent effect which increases the upper level populations at steady state is found to exist. A corresponding increase in the macroscopic chemical rate constant is calculated for the coherent model when unimolecular decay from the top level is included. A particularly simple and general result is found for the power absorbed by a harmonic oscillator. It is also shown that, for the N-level model, one does not necessarily find equal populations in the limit of infinite light field intensity.
Excitation dynamics for pulsed optical excitation are described with the density-matrix equations and the rate equations for a two-level system. A critical comparison of the two descriptions is made with complete and consistent formalisms that are amenable to the modeling of applied laser-diagnostic techniques. General solutions, resulting from numerical integration of the differential equations describing the excitation process, are compared for collisional conditions that range from the completely coherent limit to the steady-state limit, for which the two formalisms are identical. This analysis demonstrates the failure of the rate equations to correctly describe the transient details of the excitation process outside the steady-state limit. However, reasonable estimates of the resultant population are obtained for nonsaturating (linear) excitation. This comparison provides the laser diagnostician with the means to evaluate the appropriate model for excitation through a simple picture of the breakdown of the rate-equation validity.
For the pair of 238U^{238}\mathrm{U}238U levels at 3868 and 26 558 ${\mathrm{cm}}^{\ensuremath{-}1}$0, the upper-state population was measured as a function of the irradiance of a resonant exciting pulse of fixed duration. The observed damped Rabi flopping showed that the two-level system was partially coherently excited. The results were compared with the prediction of an exact solution of the time-dependent Schr\"odinger equation for two-level systems excited by a detuned rectangular pulse. The Schr\"odinger equation solution, after averaging over laser irradiance fluctuations, agrees well with the data.
Previously published papers have noted the existence of two-level periodicities in idealized theoretical models of monochromatic excitation of multilevel systems. The present paper provides a theoretical derivation of such two-level dynamics from the more general NNN-level rotating-wave-approximation Schr\"odinger equation and applies the resulting formulas to different extremes of excitation in a nondegenerate system of sequential (ladderlike) excitation: first, when intermediate levels have much larger cumulative detunings than Rabi frequencies, and second, when intermediate Rabi frequencies are much larger than detunings. In each case one has simple analytic expressions for the effective two-level Rabi frequencies and detunings.