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Hall Conductivity, Transition-State Theory and Their Effect on Entanglement and the Energy Density

Ahlem Abidi

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Two isotropic harmonic oscillators in a magnetic field on the non-commutative phase space (NCPS) are studied in this paper.We derive the corresponding entanglement entropies of the vacuum state and we find analytically purity function, Hall conductivity and the energy density.Transition-state theory is also developed to examine the purity function and the energy density.By considering the action of the non-commutativity parameters, entanglement and Hall conductivity present a similar behavior.A large magnetic field acts as a disturbance of the accurate system information and the non-localization of the two harmonic oscillators in the atom by increasing its stored energy.The application of the transition-state theory increases entanglement and accelerates the non-localization phenomenon.

DOI
10.4236/oalib.1113749

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Two isotropic harmonic oscillators in a magnetic field on the non-commutative phase space (NCPS) are studied in this paper.We derive the corresponding entanglement entropies of the vacuum state and we find analytically purity function, Hall conductivity and the energy density.Transition-state theory is also developed to examine the purity function and the energy density.By considering the action of the non-commutativity parameters, entanglement and Hall conductivity present a similar behavior.A large magnetic field acts as a disturbance of the accurate system information and the non-localization of the two harmonic oscillators in the atom by increasing its stored energy.The application of the transition-state theory increases entanglement and accelerates the non-localization phenomenon.
We study the entanglement entropy of harmonic oscillators in non-commutative phase space (NCPS). We propose a new definition of quantum Rényi entropy based on Wigner functions in NCPS. Using the Rényi entropy, we calculate the entanglement entropy of the ground state of the 2D isotropic harmonic oscillators. We find that for some values of the non-commutative parameters, the harmonic oscillators can be entangled in NCPS. This is a new entanglement-like effect caused by the non-commutativity of the phase space.
Abstract In the present paper, we study the influence of non-commutativity on entanglement in a system of two oscillators-modes in interaction with its environment. The considered system is a two-dimensional harmonic oscillator in non-commuting spatial coordinates coupled to its environment. The dynamics of the covariance matrix, the separability criteria for two Gaussian states in non-commutative space coordinates, and the logarithmic negativity are used to evaluate the quantum entanglement in the system, which is compared to the commutative space coordinates case. The result is applied for two initially entangled states, namely the squeezed vacuum and squeezed thermal states. It can be observed that the phenomenon of entanglement sudden death appears more early in the system for the case of squeezed vacuum state than in the case of squeezed thermal state. Thereafter, it is also observed that non-commutativity effects lead to an increasing of entanglement of initially entangled quantum states, and reduce the separability in the open quantum system. It turns out that a separable state in the usual commutative quantum mechanics might be entangled in non-commutative extension.
We investigate a system of two coupled harmonic oscillators on the non-commutative plane \\RR^2_{\\theta} by requiring that the spatial coordinates do not commute. We show that the system can be diagonalized by a suitable transformation, i.e. a rotation with a mixing angle \\alpha. The obtained eigenstates as well as the eigenvalues depend on the non-commutativity parameter \\theta. Focusing on the ground state wave function before the transformation, we calculate the density matrix \\rho_0(\\theta) and find that its traces {\\rm Tr}(\\rho_{0}(\\theta)) and {\\rm Tr}(\\rho_0^2(\\theta)) are not affected by the non-commutativity. Evaluating the Wigner function on \\RR^2_{\\theta} confirms this. The uncertainty relation is explicitly determined and found to depend on \\theta. For small values of \\theta, the relation is shifted by a \\theta^2 term, which can be interpreted as a quantum correction. The calculated entropy does not change with respect to the normal case. We consider the limits \\alpha=1 and \\alpha={\\pi\\over 2}. In first case, by identifying \\theta to the squared magnetic length, one can recover basic features of the Hall system.
Seeing its simplicity of processing, the system of two coupled harmonic oscillators have witnessed a quick growth of research conducted to quantum information. This approach is explicitly investigated in this paper, in particularly entanglement concept is proved to be intimately related to the tunneling effect. We examine analytically entanglement dynamics by introducing the Lewis and Riesenfeld invariant operator in the Heisenberg picture approach to compute the density matrix on the based of an exact treatment. We use Wigner function of the mixed state as an essential tool to move to linear entanglement entropies and we compute the corresponding correction factor. We follow numerically the evolution of entanglement dynamics without and under tunneling through the quantum potential barrier by introducing two particular models between simple and damped coupled harmonic oscillators. Entanglement dynamics is considered to be average without tunneling, it grows upon encountering the potential barrier and remains moderately constant inside barrier. This specificity is extended for both models but with higher values for damped coupled harmonic oscillators consequently the damping effect rapidly increases entanglement. Correction factor is also considered for both models, it show that; low temperature and high potential barriers make the system more disruptive. An increase of the coupling parameter of the system increase correction factor. Damping make the correction factor more important so damping disturbs more the system. Interference effect increase correction factor and it shows an interference between the values of the barrier penetration integral.
We investigate the dynamics of entanglement, uncertainty and mixedness by solving time-dependent Schrödinger equation for two-dimensional harmonic oscillator with time-dependent frequency and coupling parameter subject to a static magnetic field. We compute the purities (global/marginal) and then calculate explicitly the linear entropy [Formula: see text] as well as logarithmic negativity [Formula: see text] using the symplectic parametrization of vacuum state. We introduce the spectral decomposition to diagonalize the marginal state and get the expression of von Neumann entropy [Formula: see text] and establish its link with [Formula: see text]. We use the Wigner formalism to derive the Heisenberg uncertainties and show their dependencies on both [Formula: see text] and the coupling parameters [Formula: see text] [Formula: see text] of the quadrature term [Formula: see text]. We graphically study the dynamics of the three features (entanglement, uncertainty, mixedness) and present a similar topology with respect to time. We show the effects of the magnetic field and quenched values of [Formula: see text] and [Formula: see text] on these three dynamics, which lead eventually to control and handle them.
In noncommutative space, to find the Wigner function, we need to resolve the eigenequation [Formula: see text]. Generally, solving this type of system is very difficult. Deformation quantization is an efficient method applicable to special-form Hamiltonians, and it is limited to two dimensions. In this paper, we apply deformation quantization approach on three isotropic harmonic oscillators to derive the purity function. We numerically analyze our analytical results with respect different system parameters. We show that when the three harmonic oscillators are localized in the volume bounded by the noncommutative parameters, it becomes low entangled.