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arXiv:nlin/0302023 · Physical Review Letters

Time Independent Description of Rapidly Oscillating Potentials

Saar Rahav, Ido Gilary, Shmuel Fishman

Published Sep 12, 2003• 94 citations
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Abstract

The classical and quantum dynamics in a high frequency field are found to be described by an effective time independent Hamiltonian. It is calculated in a systematic expansion in the inverse of the frequency (omega) to order omega(-4). The work is an extension of the classical result for the Kapitza pendulum, which was calculated in the past to order omega(-2). The analysis makes use of an implementation of the method of separation of time scales and of a quantum gauge transformation in the framework of Floquet theory. The effective time independent Hamiltonian enables one to explore the dynamics in the presence of rapidly oscillating fields, in the framework of theories that were developed for systems with time independent Hamiltonians. The results are relevant, in particular, for exploring the dynamics of cold atoms.

DOI
10.1103/physrevlett.91.110404

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The classical and quantum dynamics in a high frequency field are found to be described by an effective time independent Hamiltonian. It is calculated in a systematic expansion in the inverse of the frequency (omega) to order omega(-4). The work is an extension of the classical result for the Kapitza pendulum, which was calculated in the past to order omega(-2). The analysis makes use of an implementation of the method of separation of time scales and of a quantum gauge transformation in the framework of Floquet theory. The effective time independent Hamiltonian enables one to explore the dynamics in the presence of rapidly oscillating fields, in the framework of theories that were developed for systems with time independent Hamiltonians. The results are relevant, in particular, for exploring the dynamics of cold atoms.
The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.
The dynamics of classical and quantum systems, which are driven by a high-frequency $(\ensuremath{\omega})$ field, is investigated. For classical systems, the motion is separated into a slow part and a fast part. The motion for the slow part is computed perturbatively in powers of ${\ensuremath{\omega}}^{\ensuremath{-}1}$ to the order ${\ensuremath{\omega}}^{\ensuremath{-}4},$ and the corresponding time independent Hamiltonian is calculated. Such an effective Hamiltonian for the corresponding quantum problem is computed to the order ${\ensuremath{\omega}}^{\ensuremath{-}4}$ in a high-frequency expansion. Its spectrum is the quasienergy spectrum of the time dependent quantum system. The classical limit of this effective Hamiltonian is the classical effective time independent Hamiltonian. It is demonstrated that this effective Hamiltonian gives the exact quasienergies and quasienergy states of some simple examples, as well as the lowest resonance of a nontrivial model for an atom trap. The theory that is developed in this paper is useful for the analysis of atomic motion in atom traps of various shapes.
A quantum system driven by a high-frequency periodic perturbation is studied. By using the asymptotic expansions in terms of inverse powers of the driving frequency, a class of unitary time-dependent canonical transformations is defined which renders the transformed Hamiltonians time independent. One representative of those Hamiltonians is the quasienergy operator, which is explicitly derived up to the fourth order. The classical limit of the theory and the possibility of separating the mean-motion Hamiltonian is discussed. It is shown that this separation can consistently be carried out to higher orders only in the case of a uniform external force. The application of the method to unperturbed systems with regular confining and singular potentials is discussed by considering the examples of harmonic oscillator, particle in the double-well potential, and particle in the Coulomb potential.
We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper–Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case, one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer–Wolff transformation extending the latter to driven systems.