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Journal of Physics B Atomic and Molecular Physics

An algorithm for the evaluation of induced transition probabilities for non-sinusoidal and sinusoidal applied fields with an application to Gaussian pulses

Gerald F. Thomas, William J. Meath

Published Mar 28, 1983• 46 citations
Publisher

Abstract

An efficient computational algorithm for evaluating the evolution operator, and related induced transition probabilities, for a multilevel system interacting with an amplitude-modulated sinusoidal field of arbitrary strength, modulating envelope, frequency and phase, is developed. The algorithm is based on the Riemann product integral representation of the evolution operator in conjunction with Frazer's method of mean coefficients and avoids the introduction of commonly used perturbative or rotating wave approximations. Model calculations for the interaction of a two-level system with a Gaussian pulse are presented and used to discuss the dependence of time-resolved and steady-state frequency sweep spectra on some of the pulse characteristics.

DOI
10.1088/0022-3700/16/6/012

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An efficient computational algorithm for evaluating the evolution operator, and related induced transition probabilities, for a multilevel system interacting with an amplitude-modulated sinusoidal field of arbitrary strength, modulating envelope, frequency and phase, is developed. The algorithm is based on the Riemann product integral representation of the evolution operator in conjunction with Frazer's method of mean coefficients and avoids the introduction of commonly used perturbative or rotating wave approximations. Model calculations for the interaction of a two-level system with a Gaussian pulse are presented and used to discuss the dependence of time-resolved and steady-state frequency sweep spectra on some of the pulse characteristics.
An algorithm [G. F. Thomas and W. J. Meath, J. Phys. B 16, 951 (1983)] for evaluating the evolution operator of a multilevel system dipole interacting with either a continuous-wave or Gaussian-pulsed source, based on its Riemann product integral representation in conjunction with Frazer's method of mean coefficients, is appraised and extended. Our extension allows for the inclusion of spontaneous decay or a dissociation (or ionization) channel and useful relations are derived for the eigenvalues of the evolution operator for both the continuous wave and the Gaussian pulse. For the latter we now admit an arbitrary phase in the field and we derive a simple relation for cascading the state amplitudes of the system over the duration of an arbitrary number of identical phase-coherent pulses. As part of our appraisal, improvements in the implementation of the algorithm are provided. For a two-level system, the evolution operator is given as the chronologically ordered product of readily computed 2\ifmmode\times\else\texttimes\fi{}2 matrices while for multilevel system it is conveniently evaluated using a resolvent method based on Leverrier's algorithm for computing the Bateman matrices. The asymptotic behavior of the evolution operator and the fluence in the direction of propagation for both the continuous-wave and the Gaussian-pulsed sources are explored. Applications to model two-level systems illustrate the influence of the source's phase and interaction duration on resonant and nonresonant transition probabilities, respectively. The saturation of a two-level system by a train of resonant phase-coherent Gaussian pulses is demonstrated.
An algorithm for calculating the time-resolved evolution operator of a multilevel system dipole interacting with a Gaussian-shaped laser pulse is developed through use of a compound Magnus expansion in compliance with Maricq's convergence criterion. In conjunction with the Leverrier-Bateman resolvent method, the algorithm allows the frequency spectrum for the interaction of the multilevel system with a train of identical phase-coherent Gaussian-shaped laser pulses to be readily evaluated. Furthermore, the algorithm renders possible the expeditious averaging of operator expectation values over the spectral bandwidth of the laser field. A prescription for gauging the accuracy of the algorithm is provided.
An exact method of solution to the time-dependent wave equation for a system interacting with a sinusoidal field, which formally treats time and the phase of the field on an equal footing, is used to discuss (1) the importance of averaging the properties of a system over the phase of the applied field and (2) the nature and behavior of the characteristic exponents derived from the Floquet solution (with particular emphasis on the frequency-sweep experiment). The results for the Zeeman tuning experiment are used to resolve recent discrepancies in the literature which relate to the validity of Shirley's important result for the two-level average induced transition probability involving the derivative of the characteristic exponents with respect to the Zeeman splitting parameter ${\ensuremath{\omega}}_{0}$. The explicit calculations included in this work are for the single-photon two-level problem. Important implications can be inferred from them with respect to both the importance of phase averaging and the usefulness of the characteristic exponents as a quantitative means of obtaining resonance frequency shifts and half-widths for multiphoton and multilevel problems.
The assumed existence of a time-independent Kubo-type operator renders possible an analytic expression for the evolution operator of a quantized system. For a multilevel system dipole interacting with a continuous-wave laser, the realization of a suitable Kubo-type transformation through an algebraic algorithm may require that the Hamiltonian operator of the system be simplified through, say, the imposition of the rotating-wave approximation and/or the appeal to transition dipole selection rules that validate the neglect of unimportant couplings. The resultant sparse Hamiltonian operator is arbitrarily constructed on the basis of the near-resonant laser coupling of the ground state to the excited states, but without direct appeal to the magnitude of the dipole coupling strength parameters. The implementation and associated computational aspects of a resolvent method for evaluating the evolution operator is discussed and illustrated through the calculation of the transition probabilities for the excitation of an anharmonic oscillator model of the CH stretch in CD3{\mathrm{CD}}_{3}CD3​H.
A solution for the time-dependent Schrödinger equation is given for an N-level system interacting with a sinusoidal field of arbitrary amplitude , frequency v and phase δ, in the presence of an applied static field of arbitrary strength . The solution is obtained by a modification of a previous exact method of solution for and retains the essential analytic features of the exact wavefunction for the problem. The Floquet form of the solution yields (a) the location and the widths of allowed transitions and the location of forbidden transitions, by use of characteristic exponent plots, but without evaluating the spectrum, and (b) convenient expressions for the steady-state induced transition probabilities, including the effects of a uniform relaxation mechanism, that only require the solution of the wave equation over the initial period of the hamiltonian. The approach is equally applicable to Stark (or Zeeman) frequency sweep or tuning problems. The ability to incorporate the effects of static fields in the solution is important since a variety of problems (e.g. anti-crossing spectroscopy) involve the application of such fields to a level configuration interacting with a sinusoidal field. As a detailed specific example, the multi-photon steady-state induced transition probabilities and characteristic exponents are studied, as a function of frequency v, for the unperturbed and the perturbed two-level system under saturating conditions. It is also demonstrated, for the coupling strengths of interest in multi-photon calculations, that the method of solution discussed here is a viable unified alternative to dressed-atom techniques.
A previous exact formal and practical solution for the time-dependent Schrödinger equation for a two-level non-degenerate system in an arbitrarily strong sinusoidal field is transformed for an N-level system to a Floquet form. Using the Floquet form of the solution computationally convenient expressions are obtained for the phase and the phase-long time-averaged induced transition probabilities. The method exploits the time periodicity of the Schrödinger equation and the solution over the initial period of the hamiltonian provides all the information needed to obtain both the unaveraged and the averaged transition probabilities for the system. The method is illustrated by specific examples which are used to discuss some of the temporal features of the multi-photon spectra of two-level systems.