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arXiv:2001.03785 · Journal of Physics Conference Series

Anharmonic effects on phase-space quantum profiles: an exact approach

Alex E. Bernardini, Caio Fernando e Silva

Published Aug 1, 2020• 1 citation
arXivHTMLPDF

Abstract

Abstract Given its well known spectral decomposition profile, the 1-dim harmonic oscillator potential modified by an inverse square (1-dim angular momentum-like) contribution works as an efficient platform for probing classical and quantum information quantifiers in the context of the phase-space Weyl-Wigner formalism. In particular, the phase-space informational content related to the canonical ensemble driven by such a singular oscillator can be quantified in terms of well established analytical structures. Considering that, on one hand, the singular oscillator produce a spectral decomposition profile equivalent to that one of the unmodified harmonic system – in the sense that they result into identical thermodynamic statistics, even for different statistical mixtures – on the other hand, a more complete scrutinization of their phase-space information content can capture some different aspects of the encoded information for the related quantum ensembles. Besides the identification of decoherence effects, the Wigner flow analysis is presumedly useful in identifying stable quantum configurations, according to finite temperature and interaction parameter values. Unexpectedly, our results show that the equivalence between the statistical (quantum) mechanics of the anharmonic singular oscillator and an ordinary harmonic oscillator can also be extended to the phase-space quantum purity quantifier, which is analytically computed and reproduces exactly the same quantum ensemble statistical mixture profile, which does not depend on interaction parameter values.

DOI
10.1088/1742-6596/1612/1/012002

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Abstract Given its well known spectral decomposition profile, the 1-dim harmonic oscillator potential modified by an inverse square (1-dim angular momentum-like) contribution works as an efficient platform for probing classical and quantum information quantifiers in the context of the phase-space Weyl-Wigner formalism. In particular, the phase-space informational content related to the canonical ensemble driven by such a singular oscillator can be quantified in terms of well established analytical structures. Considering that, on one hand, the singular oscillator produce a spectral decomposition profile equivalent to that one of the unmodified harmonic system – in the sense that they result into identical thermodynamic statistics, even for different statistical mixtures – on the other hand, a more complete scrutinization of their phase-space information content can capture some different aspects of the encoded information for the related quantum ensembles. Besides the identification of decoherence effects, the Wigner flow analysis is presumedly useful in identifying stable quantum configurations, according to finite temperature and interaction parameter values. Unexpectedly, our results show that the equivalence between the statistical (quantum) mechanics of the anharmonic singular oscillator and an ordinary harmonic oscillator can also be extended to the phase-space quantum purity quantifier, which is analytically computed and reproduces exactly the same quantum ensemble statistical mixture profile, which does not depend on interaction parameter values.
Phase-space features of the Wigner flow for an anharmonic quantum system driven by the harmonic oscillator potential modified by the addition of an inverse square (one-dimensional Coulomb-like) contribution are analytically described in terms of Wigner functions and Wigner currents. Reporting about three correlated continuity equations which quantify the flux of quantum information in the phase space, the nonclassicality profile of such an anharmonic system can be consistently obtained in terms of the fluxes of probability, purity, and von Neumann--like entropy. Considering that quantum fluctuations can be identified from distortions over the classical regime, they can be quantified through the above-mentioned information fluxes whenever some classically bounded volume of the phase space is selected. Our results suggest that the Wigner flow approach works as a probe of quantumness and classicality for a large set of anharmonic quantum systems driven by quantum wells.
Abstract Extending the phase-space description of the Weyl-Wigner quantum mechanics to a subset of non-linear Hamiltonians in position and momentum, Gaussian functions are identified as the quantum ground state. Once a Hamiltonian, Hw (q, p), is constrained by the ∂2Hw/∂q∂p = 0 condition, flow properties for generic 1-dim systems can be analytically obtained in terms of Wigner functions and Wigner currents. For Gaussian statistical ensembles, the exact phase-space profile of the quantum fluctuations over the classical trajectories are found, so to interpret them as a suitable Hilbert space state configuration for confronting quantum and classical regimes. In particular, a sort of quantum camouflage where the stationarity of classical statistical ensembles can be camouflaged by the stationarity of Gaussian quantum ensembles is identified. Besides the broadness of the framework worked out in some previous examples, our results provide an encompassing picture of quantum effects on non-linear dynamical systems which can be interpreted as a first step for finding the complete spectrum of non-standard Hamiltonians.
A fluid analog of the information flux in the phase-space associated to purity and von Neumann entropy is identified in the Weyl-Wigner formalism of quantum mechanics. Once constrained by symmetry and positiveness, the encountered continuity equations provide novel quantifiers for non-classicality (non-Liouvillian fluidity) given in terms of quantum decoherence, purity and von Neumann entropy fluxes. Through definitions in the Weyl-Wigner formalism, one can identify the quantum fluctuations that distort the classical-quantum coincidence regime, and the corresponding quantum information profile, whenever some bounded x − p volume of the phase-space is specified. The dynamics of anharmonic systems is investigated in order to illustrate such a novel paradigm for describing quantumness and classicality through the flux of quantum information in the phase-space.
The Wigner function for the pure quantum states is used as an integral kernel of the non-Hermitian operator K, to which the standard singular value decomposition (SVD) is applied. It provides a set of the squared singular values treated as probabilities of the individual phase-space processes, the latter being described by eigenfunctions of KK(+) (for coordinate variables) and K(+)K (for momentum variables). Such a SVD representation is employed to obviate the well-known difficulties in the definition of the phase-space entropy measures in terms of the Wigner function that usually allows negative values. In particular, the new measures of nonclassicality are constructed in the form that automatically satisfies additivity for systems composed of noninteracting parts. Furthermore, the emphasis is given on the geometrical interpretation of the full entropy measure as the effective phase-space volume in the Wigner picture of quantum mechanics. The approach is exemplified by considering some generic vibrational systems. Specifically, for eigenstates of the harmonic oscillator and a superposition of coherent states, the singular value spectrum is evaluated analytically. Numerical computations are given for the nonlinear problems (the Morse and double well oscillators, and the Henon-Heiles system). We also discuss the difficulties in implementation of a similar technique for electronic problems.
Phase-space features of the Wigner flow for generic one-dimensional systems with a Hamiltonian HW(q,p){H}^{W}(q,\phantom{\rule{0.16em}{0ex}}p)HW(q,p) constrained by the ${\ensuremath{\partial}}^{2}{H}^{W}/\ensuremath{\partial}q\ensuremath{\partial}p=0$ condition are analytically obtained in terms of Wigner functions and Wigner currents. Liouvillian and stationary profiles are identified for thermodynamic (TD) and Gaussian quantum ensembles to account for exact corrections due to quantum modifications over a classical phase-space pattern. General results are then specialized to the Harper Hamiltonian system, which, besides working as a feasible test platform for the framework here introduced, admits a statistical description in terms of TD and Gaussian ensembles, where the Wigner flow properties are all obtained through analytical tools. Quantum fluctuations over the classical regime are therefore quantified through probability and information fluxes whenever the classical Hamiltonian background is provided. Besides allowing for a broad range of theoretical applications, our results suggest that such a generalized Wigner approach works as a probe for quantumness and classicality of Harper-like systems in a framework which can be extended to any quantum system described by Hamiltonians in the form of HW(q,p)=K(p)+V(q){H}^{W}(q,\phantom{\rule{0.16em}{0ex}}p)=K(p)+V(q)HW(q,p)=K(p)+V(q).
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.