Effective Hamiltonians for periodically driven systems

By Saar Rahav, Ido Gilary, S. Fishman. 2003

205 citations recorded by the source corpus.

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.

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Citations in the corpus

A Rigorous Theory of Many-Body Prethermalization for Periodically Driven and Closed Quantum Systems (2015, background): “High-frequency asymptotics (instead of strict localization) in periodically driven quantum systems have been investigated in [16, 34, 39] by techniques similar to ours, but not applicable to the many-body problem.”
Topological photonics (2018, background): “However, due to the time-periodicity inherent to the system Ĥðtþ TÞ ¼ ĤðtÞ, this time-evolution operator can be factorized, leading to the more suggestive form (Kiss, Janszky, and Adam, 1994; Rahav, Gilary, and Fishman, 2003; Goldman and Dalibard, 2014; Bukov, D’Alessio, and Polkovnikov, 2015) Ûð”
Many-body energy localization transition in periodically driven systems (2012, background): “Here we show that the dynamical stabilization phenomenon can be understood through the Magnus expansion of the quantum evolution operator in powers of the inverse frequency (a relate perturbative analysis in powers of the inverse frequency was studied in [20, 21]) .”
Periodically Driven Quantum Systems: Effective Hamiltonians and Engineered Gauge Fields (2014, background): “[59], which clearly highlights three relevant notions associated with the driving: the initial kick captured by the operator K̂ðtiÞ, the effective Hamiltonian Ĥeff that rules the long-time dynamics, and the micromotion described by K̂ðtfÞ (Sec.”