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Entropy

Correspondence Rules for SU(1,1) Quasidistribution Functions and Quantum Dynamics in the Hyperbolic Phase Space

Miguel Baltazar, Ivan F. Valtierra, A. B. Klimov

Published Oct 31, 2022• 3 citations
Publisher

Abstract

We derive the explicit differential form for the action of the generators of the SU(1,1) group on the corresponding s-parametrized symbols. This allows us to obtain evolution equations for the phase-space functions on the upper sheet of the two-sheet hyperboloid and analyze their semiclassical limits. Dynamics of quantum systems with SU(1,1) symmetry governed by compact and non-compact Hamiltonians are discussed in both quantum and semiclassical regimes.

DOI
10.3390/e24111580

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We derive the explicit differential form for the action of the generators of the SU(1,1) group on the corresponding s-parametrized symbols. This allows us to obtain evolution equations for the phase-space functions on the upper sheet of the two-sheet hyperboloid and analyze their semiclassical limits. Dynamics of quantum systems with SU(1,1) symmetry governed by compact and non-compact Hamiltonians are discussed in both quantum and semiclassical regimes.
Motivated by recent advances in quantum dynamics, we investigate the dynamics of the system with SU(1,1)\text{SU}(1,1)SU(1,1) symmetry. Instead of performing the time-ordered integral for the evolution operator of the time-dependent Hamiltonian, we show that the time evolution operator can be expressed as an SU(1,1)\text{SU}(1,1)SU(1,1) group element. Since the SU(1,1)\text{SU}(1,1)SU(1,1) group describes the ``rotation'' on a hyperbolic surface, the dynamics can be visualized on a Poincar\'e disk, a stereographic projection of the upper hyperboloid. As an example, we present the trajectory of the revival of Bose-Einstein condensation and that of the scale-invariant Fermi gas on the Poincar\'e disk. Further considering quantum gases in an oscillating lattice, we also study the dynamics of the system with time-dependent single-particle dispersion.