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Physics Letters A

Effective Hamiltonian approach to periodically perturbed quantum optical systems

Isabel Sainz, A. B. Klimov, C. Saavedra

Published Nov 11, 2005• 7 citations
Publisher
DOI
10.1016/j.physleta.2005.10.074

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We analyze periodically modulated quantum systems with SU(2) and SU(1,1) symmetries. Transforming the Hamiltonian into the Floquet representation we apply the Lie transformation method, which allows us to classify all effective resonant transitions emerging in time-dependent systems. In the case of a single periodically perturbed system, we propose an explicit iterative procedure for the determination of the effective interaction constants corresponding to every resonance both for weak and strong modulation. For coupled quantum systems we determine the efficient resonant transitions appearing as a result of time modulation and intrinsic non-linearities.
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.
The Schrieffer-Wolff transformation (SWT) is a foundational perturbative method for deriving effective Hamiltonians in quantum systems by systematically eliminating couplings between pairs of energy distant subspaces. Despite recent advancements, the implementation of SWTs for sufficiently complex systems remains computationally challenging and often requires extensive calculations that are prone to errors. In this work, we introduce an analytical software tool, SymPT (Symbolic Perturbation Theory), designed to automate the SWT and its extensions. Building on a universal framework developed in recent research, SymPT provides a systematic and generalizable solution for deriving the generator of the transformation, enabling accurate computation of effective Hamiltonians for arbitrary perturbative systems. The tool supports both time-independent and time-periodic Hamiltonians, extending beyond standard SWT to incorporate arbitrary coupling elimination, block-diagonalization and full-diagonalization routines, thus enabling precise handling of systems with intricate energy structures.