Kicked Hall Systems: Quantum-Dynamical and Spectral Manifestations of Generic Superweak Chaos

TL;DR

This paper explores the quantum and spectral effects of superweak chaos in kicked Hall systems, revealing phenomena like quantum antiresonance, universal spectral structures, and slow quantum diffusion, with implications for understanding quantum chaos.

Contribution

It introduces a comprehensive analysis of quantum manifestations of superweak chaos in kicked Hall systems, including effective Hamiltonians, quantum antiresonance, and universal spectral features.

Findings

Quantum antiresonance occurs at integer scaled Planck constants.

The quasienergy spectrum exhibits a doubled Hofstadter butterfly structure.

Quantum diffusion shows universal slow behavior in the semiclassical regime.

Abstract

Classical "kicked Hall systems" (KHSs), i.e., periodically kicked charges in the presence of uniform magnetic and electric fields that are perpendicular to each other and to the kicking direction, have been introduced and studied recently. It was shown that KHSs exhibit, under generic conditions, the phenomenon of "superweak chaos" (SWC), i.e., for small kick strength $\kappa$ a KHS behaves as if this strength were effectively $\kappa^2$ rather than $\kappa$. Here we investigate quantum-dynamical and spectral manifestations of this generic SWC. We first derive general expressions for quantum effective Hamiltonians for the KHSs. We then show that the phenomenon of quantum antiresonance (QAR), i.e., "frozen" quantum dynamics with flat quasienergy (QE) bands, takes place for integer values of a scaled Planck constant $\hbar_{\rm s}$ and under the same generic conditions for SWC. This appears to be the most generic occurrence of QAR in quantum systems. The vicinity of QAR is shown to correspond semiclassically to SWC. A global spectral manifestation of SWC is the fact that a scaled QE spectrum as function of $\hbar_{\rm s}$, at fixed small value of $\kappa /\hbar_{\rm s}$, features an approximately "doubled" structure. In the case of standard (cosine) potentials, this structure is that of a universal (parameters-independent) double Hofstadter butterfly. Also, for standard potentials and for small $\hbar_{\rm s}$ (semiclassical regime), the evolution of the kinetic-energy expectation value exhibits a relatively slow quantum-diffusive behavior having universal features. These approximate spectral and quantum-dynamical universalities agree with predictions from the effective Hamiltonian.