Effective Hamiltonians for periodically driven systems

2003/01/31 by Saar Rahav, Ido Gilary, Shmuel Fishman · 12 citations
Mathematics · Physics and Astronomy · #Atom (system on chip) #Atomic and Molecular Physics #Classical limit #Classical mechanics #Cold Atom Physics and Bose-Einstein Condensates #Hamiltonian (control theory) #Mathematical physics #Mathematics #Omega #Physics #Quantum #Quantum Mechanics and Applications #Quantum mechanics #Quantum system #nlin.CD #quant-ph

paper · pdf · doi:10.1103/physreva.68.013820

published as Phys. Rev A, V68, 013820 (2003) · 22 pages, 3 figures. Revised version. This online version contains some calculations which were ommited in the published version

openalex publication_date 2003/07/28 · arxiv created 2003/08/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The dynamics of classical and quantum systems, which are driven by a high-frequency (\ensuremathω) 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 \ensuremathω^\ensuremath-1 to the order \ensuremathω^\ensuremath-4, and the corresponding time independent Hamiltonian is calculated. Such an effective Hamiltonian for the corresponding quantum problem is computed to the order \ensuremathω^\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.

Citations

• Steady States and Quasienergies of a Quantum-Mechanical System in an Oscillating Field
• Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time
• Evolution of a Model Quantum System¶ Under Time Periodic Forcing:¶Conditions for Complete Ionization
• On the exponential solution of differential equations for a linear operator
• Unitary time-dependent superconvergent technique for pulse-driven quantum dynamics

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