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Entropy

Tunneling Currents in the Hyperbolic Phase Space

Ivan F. Valtierra, A. B. Klimov

Published Jul 28, 2024• 2 citations
Publisher

Abstract

We introduce the quantum currents for quantum systems with an SU(1,1) dynamic symmetry group whose evolution is governed by a non-linear Hamiltonian possessing a continuous spectrum and apply them to the analysis of the tunneling dynamics on the hyperbolic phase space.
DOI
10.3390/e26080639

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Quantifiers of stationarity, classicality, purity and vorticity are derived from phase-space differential geometrical structures within the Weyl-Wigner framework, after which they are related to the hyperbolic stability of classical and quantum-modified Hamiltonian (non-linear) equations of motion. By examining the equilibrium regime produced by such an autonomous system of ordinary differential equations, a correspondence between Wigner flow properties and hyperbolic stability boundaries in the phase-space is identified. Explicit analytical expressions for equilibrium-stability parameters are obtained for quantum Gaussian ensembles, wherein information quantifiers driven by Wigner currents are identified. Illustrated by an application to a Harper-like system, the results provide a self-contained analysis for identifying the influence of quantum fluctuations associated to the emergence of phase-space vorticity in order to quantify equilibrium and stability properties of Hamiltonian non-linear dynamics.
We describe a general method to solve a time-dependent Hamiltonian system exhibiting dynamical symmetry under the Lie algebra SU(1,1). The instantaneous eigenstates can be obtained by applying a displacement operator which is constructed explicitly. The construction proves to be equivalent to the solution of the classical equations of motion. One can also obtain the exact solution of the quantum Hamiltonian using this displacement operator. We define and discuss the differences between the cyclic evolution in the space of the physical states and in the parameter space. This leads to a method to define and find the T-cyclic and periodic states and their phases for a system with dynamical symmetry SU(1,1) just by using its time evolution operator. For SU(1,1) one can show that T-cyclic states only exist for some values of T which can be determined through an inequality which is found explicitly and discussed. The example used to check these results has been the degenerate optical parametric oscillator with dynamical symmetry SU(1,1). The system has no periodic states but it does possess T-cyclic states. The characteristic interaction times are distributed in forbidden and allowed regions as in the band theory of particles in a periodic potential. We also define exact phases and adiabatic phases along the lines of the above discussion on T-cyclic and periodic states. For sufficiently low frequencies the initial eigenstates of the system acquire non-trivial Berry phases in each cycle of the evolution of the functional parameters on which H depends. Such a phase can be measured experimentally and from the discussion of its possible values under various circumstances emerges an interesting relationship with the golden mean.