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Journal of Mathematical Physics

Probability amplitude dynamics for a two-level system

John K. Boyd

Published Jul 1, 2000• 10 citations
Publisher

Abstract

The time-dependent probability amplitudes are determined for a two-level system without invoking the rotating wave approximation. A new analytic solution is obtained in the limit of small ratio of Rabi frequency to driver frequency. An analytic solution is also obtained in the limit of large ratio of Rabi frequency to driver frequency for a restricted range of parameters. The form of this solution guides the selection of parameters that cause substantial changes in the character of the solution. The dependence on parameter values is studied numerically for the transition probability and the coherent spectrum.

DOI
10.1063/1.533345

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The time-dependent probability amplitudes are determined for a two-level system without invoking the rotating wave approximation. A new analytic solution is obtained in the limit of small ratio of Rabi frequency to driver frequency. An analytic solution is also obtained in the limit of large ratio of Rabi frequency to driver frequency for a restricted range of parameters. The form of this solution guides the selection of parameters that cause substantial changes in the character of the solution. The dependence on parameter values is studied numerically for the transition probability and the coherent spectrum.
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The excitation of a two-level system by a chirped laser pulse is analyzed using an analytical approach (in the perturbation-theory limit) that involves the modified rotating-wave approximation and the numerical solution to the Bloch equation. The dependence of the population of the upper energy level on the chirp is studied for various radiation intensities and pulse durations. An exact solution is compared with the result obtained using the modified rotating-wave approximation and the perturbation theory. It is demonstrated that, for a certain range of parameters, the excitation probability of a two-level system can be effectively controlled via a variation in the chirp.
In this paper, the general optical Bloch equation describing interaction between intensive fields and two level system is obtained without using rotating wave approximation. The first-order solution of this equation is obtained by iterative method and analytic formulas of transition probability as a function of time, perturbation frequency and strength are presented. These results are in agreement with those obtained by an iterative step-wise power series and numerical integration solution for the time dependent Schr(?)dinger equation. And these simplified and clear mathmatical formulas facilitate the application of physics.
The problem of calculating transition probabilities in two-level systems is studied in the limit where the detuning is large compared to the inverse duration of the interaction. Coupling potentials whose Fourier transforms $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(\ensuremath{\omega})$ are of the form $f(\ensuremath{\omega}){e}^{\ensuremath{-}(|b\ensuremath{\omega}|)}$ for large frequencies give rise to solutions which may be classified into families according to the form of $f(\ensuremath{\omega})$. Within each family transition probabilities may be calculated from formulas that differ only in the numerical value of a scaling parameter. In cases where the coupling function has a pole in the complex time plane, the families are identified with the order of this singularity. In particular, for poles of first order, a connection with the Rosen-Zener solution can be made. The analysis is performed via high-order perturbation expansions which are shown to always converge for two-level systems driven by coupling potentials of finite pulse area.
A nonperturbative treatment of the dynamical behavior of a vibronic system in a resonant, coherent, driving light field is developed. The potential energy surfaces for the vibrational motion in the electronic ground and excited state are assumed to be displaced and distorted relative to one another. A system of three integrodifferential equations is derived from the infinite set of density-matrix equations. Both approximate analytical and numerical solutions are presented for the intensity and the intensity-correlation function of the scattered light. The results are valid for arbitrary ratios of the Rabi frequency to the relaxation rates. A system with fast vibrational relaxation behaves as a two-level system. For systems with slow vibrational relaxation, the applicability of perturbation theory requires pump fields that are weaker than those of a two-level system. The Rabi oscillations occurring in a strong pump field are damped less than those of a two-level system, and the intensity substantially decreases. The long-time behavior (determined by the vibrational relaxation time) is characterized by the increase of the intensity toward its stationary value.
Abstract We investigate the dynamics of a two-level quantum system driven by a laser pulse characterized by Lorentzian frequency and sub-Lorentzian amplitude modulations. Complete analytical solutions, expressed via confluent Heun functions, are derived. Our analysis reveals that explicit exact analytical solutions exist under infinite sets of specific parameter conditions. The effects of modulation parameters and initial conditions on the final transition probabilities are examined analytically and numerically. Furthermore, the method is demonstrated to be directly applicable to two closely related models with Lorentzian pulses.