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arXiv:2405.01383 · Physical Review A

Extended degenerate perturbation theory for the Floquet-Hilbert space

Yakov Braver, Egidijus Anisimovas

Published Aug 14, 2024• 0 citations
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Abstract

We consider construction of effective Hamiltonians for periodically driven interacting systems in the case of resonant driving. The standard high-frequency expansion is not expected to converge due to the resonant creation of collective excitations, and one option is to resort to the application of degenerate perturbation theory (DPT) in the Floquet-Hilbert space. We propose an extension of DPT whereby the degenerate subspace includes not only the degenerate levels of interest but rather all levels in a Floquet zone. The resulting approach, which we call extended DPT (EDPT), is shown to resemble a high-frequency expansion, provided the quasienergy matrix is constructed such that each $m\mathrm{th}$ diagonal block contains energies reduced to the $m\mathrm{th}$ Floquet zone. The proposed theory is applied to a driven Bose-Hubbard model and is shown to yield more accurate quasienergy spectra than the conventional DPT. The computational complexity of EDPT is intermediate between DPT and the numerically exact approach, thus providing a practical compromise between accuracy and efficiency.

DOI
10.1103/physreva.110.023315

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We derive a systematic high-frequency expansion for the effective Hamiltonian and the micromotion operator of periodically driven quantum systems.Our approach is based on the block diagonalization of the quasienergy operator in the extended Floquet Hilbert space by means of degenerate perturbation theory.The final results are equivalent to those obtained within a different approach (Rahav et al 2003 Phys.Rev. A 68 013820), (Goldman and Dalibard 2014 Phys.Rev. X 4 031027) and can also be related to the Floquet-Magnus expansion (Casas et al 2001 J. Phys.A 34 3379).We discuss that the dependence on the driving phase, which plagues the latter, can lead to artifactual symmetry breaking.The high-frequency approach is illustrated using the example of a periodically driven Hubbard model.Moreover, we discuss the nature of the approximation and its limitations for systems of many interacting particles.
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We present theoretical methods for studying quantum mechanical systems subjected to fast periodic driving and apply them to model systems with long-range interaction. We provide a comparison between the methods and insight facilitated by these methods. The methods recently occurred in scientific papers, which supports the need for a scrutinized exposition of the theory. One of the main objects of the theory is a so-called Floquet Hamiltonian-an artificial stationary Hamiltonian describing important features of a quantum system. The methods construct Floquet Hamiltonians in the form of series in the powers of the time period. We present the spectra of Floquet Hamiltonians-the so-called quasienergy spectra-computed by the methods and computed numerically (with higher precision). The quasienergy spectra were computed using various approximations of Floquet Hamiltonians and compared. We discuss an interesting topic of the classical limit of an artificial stationary system. We also mention the kicked rotor system and its connection with the kicked top system-one of our model systems. In summary, the method characterized by simultaneous construction of a Floquet Hamiltonian and a so-called kick operator (operator capturing fast changes of the system) was found universal and accurate. The thesis presents an...