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arXiv:1303.0685 · Physical Review A

Probing phase-space noncommutativity through quantum beating, missing information, and the thermodynamic limit

Alex E. Bernardini, Orfeu Bertolami A

Published Jul 1, 2013• 42 citations
arXivHTMLPDF

Abstract

In this work we examine the effect of phase-space noncommutativity on some typically quantum properties such as quantum beating, quantum information, and decoherence. To exemplify these issues we consider the two-dimensional noncommutative quantum harmonic oscillator whose component behavior we monitor in time. This procedure allows us to determine how the noncommutative parameters are related to the missing information quantified by the linear quantum entropy and by the mutual information between the relevant Hilbert space coordinates. Particular questions concerning the thermodynamic limit of some relevant properties are also discussed in order to evidence the effects of noncommutativity. Finally, through an analogy with the Zeeman effect, we identify how some aspects of the axial symmetry of the problem suggest the possibility of decoupling the noncommutative quantum perturbations from unperturbed commutative well-known solutions.

DOI
10.1103/physreva.88.012101

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In this work we examine the effect of phase-space noncommutativity on some typically quantum properties such as quantum beating, quantum information, and decoherence. To exemplify these issues we consider the two-dimensional noncommutative quantum harmonic oscillator whose component behavior we monitor in time. This procedure allows us to determine how the noncommutative parameters are related to the missing information quantified by the linear quantum entropy and by the mutual information between the relevant Hilbert space coordinates. Particular questions concerning the thermodynamic limit of some relevant properties are also discussed in order to evidence the effects of noncommutativity. Finally, through an analogy with the Zeeman effect, we identify how some aspects of the axial symmetry of the problem suggest the possibility of decoupling the noncommutative quantum perturbations from unperturbed commutative well-known solutions.
The effects of general noncommutativity of operators on producing deformed coherent squeezed states is examined in phase space. A two-dimensional noncommutative (NC) quantum system supported by a deformed mathematical structure, similar to that of Hadamard billiard, is obtained and the components behaviour is monitored in time. It is assumed that the independent degrees of freedom are two free 1D harmonic oscillators (HOs), so the system Hamiltonian does not contain interaction terms. Through the NC deformation parameterized by a Seiberg–Witten transform on the original canonical variables, one gets the standard commutation relations for the new ones, such that the obtained, new, Hamiltonian represents two interacting 1D HOs. By admitting that one HO is inverted relatively to the other, we show that their effective interaction induces a squeezing dynamics for initial coherent states imaged in the phase space. A suitable pattern of logarithmic spirals is obtained and some relevant properties are discussed in terms of Wigner functions, which are essential to put in evidence the effects of the noncommutativity.
In this paper, we study the two-dimensional Schrodinger oscillator within a dynamical noncommutative (DNC) space. By leveraging perturbation theory, we derive the energy eigenvalues and eigenvectors and systematically analyze the effects of both dynamical and non-dynamical noncommutative settings. First-order corrections to the eigensystem are obtained, revealing that the energy shift explicitly depends on the DNC parameter tau. Furthermore, we explore the thermal properties of the system by employing the partition function. Numerical results are presented to provide a comprehensive analysis of the system behavior under the considered effects. Notably, in the DNC framework, the commutation relations and the deformation parameter are position-dependent. Using the two-dimensional Bopp-shift, we effectively map the noncommutative problem to its commutative counterpart.
Appearance of adiabatic geometric phase shift in the context of noncommutative quantum mechanics is studied using an exactly solvable model of a two-dimensional simple harmonic oscillator in the Moyal plane, where momentum noncommutativity are also considered along with spatial noncommutativity. For that we introduce a modified form of Bopp's shift, that bridges the noncommutative phase-space operators with their effective commutative counterparts, having their dependence on the noncommutative parameters, and study the adiabatic evolution in the Heisenberg picture. An explicit expression for the geometric phase shift under adiabatic approximation is then found without using any perturbative technique. Lastly, this phase is found to be related to Hannay's angle of a classically analogous system, by studying the evolution of the coherent state of this system.
This work investigates a quantum system described by a Hamiltonian operator in a two dimensional noncommutative space. The system consists of an electron subjected to a perpendicular magnetic field B\mathbf{B}B, coupled to a harmonic potential and an external electric field E\mathbf{E}E, within the context of non-extensive statistical thermodynamics. The noncommutative geometry introduces a fundamental minimal length that modifies the phase space structure. The thermodynamics of this quantum system is developed within the framework of Tsallis statistics through the derivation of qqq-generalized versions of the partition function, magnetization, and magnetic susceptibility, following the application of a generalized Hilhorst transformation adapted to non-commutative geometry. The combined effects of the non-extensivity parameter qqq and the noncommutativity parameter θθθ are analyzed by considering the limit q→1q \rightarrow 1q→1, revealing new thermodynamic regimes and anomalous electromagnetic properties specific to quantum systems in non-commutative geometry.
We discuss the main features of noncommutative quantum mechanics, a version of nonrelativistic quantum mechanics that involves noncommuting coordinates. After finding a representation for the algebra obeyed by the coordinates and momenta, we analyze the changes due to the noncommutative nature of the coordinates. The noncommutative two-dimensional harmonic oscillator is discussed in detail. Under certain restrictions, the effect of the noncommutativity is found to be equivalent to a Landau interaction. The modifications produced by the noncommutativity on the thermodynamic functions of the oscillator also are studied.
In this paper, decoherence of a damped anisotropic harmonic oscillator in the presence of a magnetic field is studied in the framework of the Lindblad theory of open quantum systems in noncommutative phase-space. General fundamental conditions that should follow our quantum mechanical diffusion coefficients appearing in the master equation are kindly derived. From the master equation, the expressions of density operator, the Wigner distribution function, the expectation and variance with respect to coordinates and momenta are obtained. Based on these quantities, the total energy of the system is evaluated and simulations show its dependency to phase-space structure and its improvement due to noncommutativity effects and the environmental temperature as well. In addition, we also evaluate the decoherence time scale and show that it increases with noncommutativity phase-space effects as compared to the commutative case. It turns out from simulations that this time scale is significantly improved under magnetic field effects.