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Two Coupled Harmonic Oscillators on Non-commutative Plane

Jellal, A, El Hassan El Kinani, Schreiber, M

Published Sep 10, 2003• 22 citations
OpenAlex

Abstract

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.

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arXiv:hep-th/0304136•Journal of Physics A Mathematical and General•Apr 2, 2003•25 citations
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.
We investigate a system of two coupled harmonic oscillators on the noncommutative plane [Formula: see text] 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 α. The obtained eigenstates as well as the eigenvalues depend on the noncommutativity parameter θ. Focusing on the ground state wave function before the transformation, we calculate the density matrix ρ0(θ) and find that its traces Tr (ρ0(θ)) and [Formula: see text] are not affected by the noncommutativity. Evaluating the Wigner function on [Formula: see text] confirms this. The uncertainty relation is explicitly determined and found to depend on θ. For small values of θ, the relation is shifted by a θ2 term, which can be interpreted as a noncommutativity correction. The calculated entropy does not change with respect to the normal case. We consider the limits α = 0 and α = π/2. In the first case, by identifying θ to the squared magnetic length, one can recover basic features of the Hall system.
The thermodynamical properties of a system of two coupled harmonic oscillators in the presence of a uniform magnetic field B are investigated. Using an unitary transformation, we show that the system can be diagonalized in a simple way and then obtain the energy spectrum solutions. These are then used to determine the thermodynamical potential in terms of different physical parameters like the coupling parameter α. This allows us to give a generalization of significant work already published and obtain different results, which can be used to discuss the magnetism of the system. Different limiting cases, in terms of α and B, are discussed. In fact, quantum corrections to the Landau diamagnetism and orbital paramagnetism are found.
We develop an approach to study the entanglement in two coupled harmonic oscillators. We start by introducing an unitary transformation to end up with the solutions of the energy spectrum. These are used to construct the corresponding coherent states through the standard way. To evaluate the degree of the entanglement between the obtained states, we calculate the purity function in terms of the coherent and number states, separately. The result is yielded to two parameters dependance of the purity, which can be controlled easily. Interesting results are derived by fixing the mixing angle of such transformation as π/2. We compare our results with already published work and point out the relevance of these findings to a systematic formulation of the entanglement effect in two coupled harmonic oscillators.
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.
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.
The properties of a non-canonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl (1∣3), are further investigated. Within each state space W ( p ), p = 1, 2, ..., the energy E q , q = 0, 1, 2, 3, takes no more than four different values. If the oscillator is in a stationary state ψ q ∊ W ( p ) then measurements of the non-commuting Cartesian coordinates of the particle are such that their allowed values are consistent with it being found at a finite number of sites, called 'nests'. These lie on a sphere centred on the origin of fixed, finite radius ϱ q . The nests themselves are at the vertices of a rectangular parallelepiped. In the typical cases ( p > 2) the number of nests is 8 for q = 0 and 3, and varies from 8 to 24, depending on the state, for q = 1 and 2. The number of nests is less in the atypical cases ( p = 1, 2), but it is never less than 2. In certain states in W (2) (respectively in W (1)) the oscillator is 'polarized' so that all the nests lie on a plane (respectively on a line). The particle cannot be localized in any one of the available nests alone since the coordinates do not commute. The probabilities of measuring particular values of the coordinates are discussed. The mean trajectories and the standard deviations of the coordinates and momenta are computed, and conclusions are drawn about uncertainty relations.