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PRSM

Nonperturbative Analysis of the Harmonic Oscillator with a Time-Dependent Frequency

Abdelaziz Fedoul, Az-Eddine L. Marrakchi, Salaheddine Sayouri, Abdelhafid Chatwiti

Published Mar 16, 2011• 0 citations
Publisher

Abstract

We apply the Floquet theorem and the resonating averages method to solve the Schrödinger equation of the harmonic oscillator whose frequency varies periodically with time. We determine its Floquet states and its corresponding wave functions. The formalism provides an efficiency tool for analyzing the various transition probabilities as well as the uncertainty relation of the system. The effect of a periodic external force on the Floquet states and the two-photon algebra realization are shortly discussed.

DOI
10.34874/prsm.mjcm-vol8iss0.176

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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.
We consider here the exact solution of a nonrelativistic quantum system composed of a two-level atom interacting with a laser with arbitrary large frequency and intensity. We use the analogy of this system and a ℏ/2-spin particle interacting with a time-dependent magnetic field. A systematic use of the dynamical symmetry underlying the physical system is made. Actually the Hamiltonian is a Hermitian element of the SU(2) Lie Algebra. The exact Temporal Evolution Operator in terms of a Generalized Displacement Operator of the group is constructed. It is possible to develop a nonperturbative method that allows us to solve exactly the model for any value of the relevant frequencies (Rabi, Laser, and Atom Frequencies) and in so doing the usual Rotating Wave and Small Interaction approximations are unnecessary. The properties of periodicity of this model and the phenomenon of harmonic generation are considered by using Floquet Analysis. We find that in addition to the so far well-known spectrum composed by odd harmonics, this model generates another type of noninteger harmonics whose frequencies and amplitudes are determined for any value of the relevant parameters of the system.