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Europhysics Letters (EPL)

Localization in a Driven Two-Level Dynamics

Frank Großmann, Peter Hänggi A

Published Apr 1, 1992• 305 citations
Publisher

Abstract

A periodically driven two-state dynamics, being analysed within the Floquet formalism, exhibits localization of the amplitude dynamics in an infinite frequency range, extending from the bare tunnel splitting up to infinity. In contrast, the suppression of tunnelling in a driven symmetric double well is restricted to a limited frequency regime, extending from the bare tunnel splitting up to the first resonance frequency with higher-lying states. With the amplitude dynamics of a periodically driven two-level system not being restricted to describe coherent tunnelling transport only, the localization phenomenon within the infinite frequency range does allow for novel applications for systems in strong laser fields.

DOI
10.1209/0295-5075/18/7/001

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A periodically driven two-state dynamics, being analysed within the Floquet formalism, exhibits localization of the amplitude dynamics in an infinite frequency range, extending from the bare tunnel splitting up to infinity. In contrast, the suppression of tunnelling in a driven symmetric double well is restricted to a limited frequency regime, extending from the bare tunnel splitting up to the first resonance frequency with higher-lying states. With the amplitude dynamics of a periodically driven two-level system not being restricted to describe coherent tunnelling transport only, the localization phenomenon within the infinite frequency range does allow for novel applications for systems in strong laser fields.
The quantum dynamics of a quartic double well, subjected to a harmonically oscillating field, is studied in the framework of the Floquet formalism. The modifications of the familiar tunneling process due to driving and dissipation are investigated numerically and explained in terms of the quasienergy spectrum. In absence of dissipation, there is a onedimensional manifold in the parameter space spanned by amplitude and frequency of the driving force, where tunneling is almost completely suppressed by the coherent driving. The influence of dissipation is described on basis of the reduced density matrix in the Floquet representation. In particular, we consider the effect of weak Ohmic damping. In the classical limit, this system corresponds to a damped bistable Duffing oscillator
The effect of a periodic driving force on coherent tunneling of a driven double square-well system, especially when its central potential barrier is very tall and very thin, is investigated. The suppression or enhancement of the tunneling of a wave packet properly chosen in the double-well system may be achieved by a judicious choice of the period of a driving force, i.e., by a particular choice of the generating condition of a nonlinear resonance exhibiting a desirable property. We analyze this mechanism by exploiting both the properties of a dynamical symmetry of the Hamiltonian and the localized properties of Floquet states induced by the nonlinear resonance. \textcopyright{} 1996 The American Physical Society.
The dynamic effect of particles in a periodically driven two-level system is studied with the help of SU(2) symmetry. General closed-form solutions for the quasienergy and the Floquet states are obtained, from which it is found that the avoided quasienergy crossing is confined in the range (−ω/2, ω/2), where ω is the frequency of the driving fields. Of particular interest are high-frequency driving fields, of which it is demonstrated that the exact quasienergy crossings occur at certain values of amplitude and frequency of driving fields, and this phenomenon of dynamic localization should be determined by experiments.
We develop a low-frequency perturbation theory in the extended Floquet Hilbert space of a periodically driven quantum systems, which puts the high- and low-frequency approximations to the Floquet theory on the same footing. It captures adiabatic perturbation theories recently discussed in the literature as well as diabatic deviation due to Floquet resonances. For illustration, we apply our Floquet perturbation theory to a driven two-level system as in the Schwinger–Rabi and the Landau-Zener–Stückelberg–Majorana models. We reproduce some known expressions for transition probabilities in a simple and systematic way and clarify and extend their regime of applicability. We then apply the theory to a periodically-driven system of fermions on the lattice and obtain the spectral properties and the low-frequency dynamics of the system.
We develop a low-frequency perturbation theory in the extended Floquet Hilbert space of a periodically driven quantum systems, which puts the high- and low-frequency approximations to the Floquet theory on the same footing. It captures adiabatic perturbation theories recently discussed in the literature as well as diabatic deviation due to Floquet resonances. For illustration, we apply our Floquet perturbation theory to a driven two-level system as in the Schwinger-Rabi and the Landau-Zener-Stuckelberg-Majorana models. Were produce some known expressions for transition probabilities in a simple and systematic way and clarify and extend their regime of applicability. We then apply the theory to a periodically-driven system of fermions on the lattice and obtain the spectral properties and the low-frequency dynamics of the system.