Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics B

On the collapse revival of entanglement in a two-dimensional noncommutative harmonic oscillator: An algebraic approach

Fethi Madouri, Abdeldjalil Merdaci

Published Jul 14, 2022• 1 citation
Publisher

Abstract

In this paper, we employ an algebraic approach to investigate the dynamics of entanglement in a two-dimensional noncommutative harmonic oscillator. We start by using the so-called the Bopp shift to convert the Hamiltonian describing the system into an equivalent commutative one. This allows us to take advantage of the Schwinger oscillator construction of [Formula: see text] Lie algebra to reveal through the time evolution operator that there is a connection between the noncommutativity and entanglement. The degree of entanglement between the states is evaluated by using the purity function. The remarkable emerging feature is that, as long as the noncommutativity parameter is nonvanishing, the system exhibits the phenomenon of collapse and revival of entanglement.

DOI
10.1142/s021797922250148x

Related papers

Collapse-revival of entanglement in a two-dimensional noncommutative harmonic oscillator: A Schmidt modes analysis

Tarek Sbeouelji, Abdeldjalil Merdaci, Fethi Madouri

In this paper, we study the dynamics of entanglement in a two-dimensional noncommutative harmonic oscillator. With the help of the so-called Bopp shift, we map the Hamiltonian describing the system into an equivalent commutative form. This brings into play a Schwinger realization of the SU(2) Lie algebra. Subsequently, we derive the time-dependent number state wave functions. Then, we use the Schmidt modes decomposition together with the notion of purity to derive explicitly the amount of entanglement encoded in the most general accessible configurations of pairwise states. In doing so, we show that there is an intimate connection between the non-commutativity parameter and the phenomenon of collapse and revival of entanglement exhibited by the system.

Modern Physics Letters A•Jul 4, 2025•0 citations

Collapse-revival of entanglement in a non-commutative harmonic oscillator revealed via coherent states and path integral

Fethi Madouri, Abdeldjalil Merdaci, Tarek Sbeouelji

Abstract We develop an approach using coherent states and path integral to investigate the dynamics of entanglement in a simple two-dimensional non-commutative harmonic oscillator. We start by employing a Bopp shift to convert the Hamiltonian describing the system into a commutative equivalent one. This allows us to construct coherent states and calculate the propagator in standard way. By deriving the explicit expression of the time-dependent coherent states and considering its connection with the number states, we provide exact results for evaluating the degree of entanglement between the ground state and any excited state through the purity function. The interesting emerging result is that, as long as the non-commutativity parameter is non-zero, our system exhibits the phenomenon of collapse and revival of entanglement.

Zeitschrift für Naturforschung A•Nov 18, 2022•1 citation

Emergence of a geometric phase shift in planar noncommutative quantum mechanics

Saptarshi Biswas, Partha Nandi, Biswajit Chakraborty

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.

arXiv:1911.03196•Physical Review A•Aug 28, 2020•20 citations

NONCOMMUTATIVE HARMONIC OSCILLATOR, ENTROPY AND ENTANGLEMENT

P.B. Espinoza, Armando Gallegos

We show that the time evolution of the two dimensional noncommutative harmonic oscillator gives rise to entanglement. This is achieved at the level of the evolution operator by using the algebraic approach. The measure of entropy and the states of maximum entanglement are also considered in this context.

e-Gnosis•Jan 1, 2010•0 citations

NONCOMMUTATIVE HARMONIC OSCILLATOR, ENTROPY AND ENTANGLEMENT OSCILADOR ARMÓNICO NO CONMUTATIVO, ENTROPÍA Y ENREDO

P.B. Espinoza, Armando Gallegos

We show that the time evolution of the two dimensional noncommutative harmonic oscillator gives rise to entanglement. This is achieved at the level of the evolution operator by using the algebraic approach. The measure of entropy and the states of maximum entanglement are also considered in this context.

•Jan 1, 2010•0 citations

On the entanglement induced by the deformation of phase-space

Shilpa Nandi, Shatarupa Maity, Pinaki Patra

Most quantum gravity theories propose that the fundamental concept of space-time is mostly compatible with quantum theory in noncommutative (NC) space. In the present paper, we revisit the notion of entanglement induced by NC deformations of phase space. The positive partial transpose (PPT) criterion for separability of bipartite Gaussian states is extended to a general class of Bopp's shift. In particular, we have considered both the position-position and momentum-momentum noncommutativity, with deformation parameters $θ$ and $η$, respectively. It turns out that $θ$ and $η$ induce the entanglement. We have directly applied the formalism for an anisotropic two-dimensional harmonic oscillator. Peres-Horodecki separability condition leads to a constraint equation for the parameter values of the oscillator in NC space. It turns out that the bipartite Gaussian state is almost always entangled in deformed space. To implement the theoretical idea, we provide an outline for a gedankenexperiment to identify the signature of phase-space noncommutativity, i.e., quantum gravity. In particular, the gedankenexperiment is devised to test the separability of supposedly separable Gaussian states in the usual commutative space, through the covariance matrix, which is constructed via measured output photocurrents after interaction of input Gaussian states and reference states. If the experiment shows that the supposedly separable states are actually entangled, then the entanglement is created through the intermediate background noncommutative space, which is a signature of the quantum nature of gravity.

quant-pharXiv:2606.17587•Jun 17, 2026•0 citations
In this paper, we employ an algebraic approach to investigate the dynamics of entanglement in a two-dimensional noncommutative harmonic oscillator. We start by using the so-called the Bopp shift to convert the Hamiltonian describing the system into an equivalent commutative one. This allows us to take advantage of the Schwinger oscillator construction of [Formula: see text] Lie algebra to reveal through the time evolution operator that there is a connection between the noncommutativity and entanglement. The degree of entanglement between the states is evaluated by using the purity function. The remarkable emerging feature is that, as long as the noncommutativity parameter is nonvanishing, the system exhibits the phenomenon of collapse and revival of entanglement.
In this paper, we study the dynamics of entanglement in a two-dimensional noncommutative harmonic oscillator. With the help of the so-called Bopp shift, we map the Hamiltonian describing the system into an equivalent commutative form. This brings into play a Schwinger realization of the SU(2) Lie algebra. Subsequently, we derive the time-dependent number state wave functions. Then, we use the Schmidt modes decomposition together with the notion of purity to derive explicitly the amount of entanglement encoded in the most general accessible configurations of pairwise states. In doing so, we show that there is an intimate connection between the non-commutativity parameter and the phenomenon of collapse and revival of entanglement exhibited by the system.
Abstract We develop an approach using coherent states and path integral to investigate the dynamics of entanglement in a simple two-dimensional non-commutative harmonic oscillator. We start by employing a Bopp shift to convert the Hamiltonian describing the system into a commutative equivalent one. This allows us to construct coherent states and calculate the propagator in standard way. By deriving the explicit expression of the time-dependent coherent states and considering its connection with the number states, we provide exact results for evaluating the degree of entanglement between the ground state and any excited state through the purity function. The interesting emerging result is that, as long as the non-commutativity parameter is non-zero, our system exhibits the phenomenon of collapse and revival of entanglement.
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.
Most quantum gravity theories propose that the fundamental concept of space-time is mostly compatible with quantum theory in noncommutative (NC) space. In the present paper, we revisit the notion of entanglement induced by NC deformations of phase space. The positive partial transpose (PPT) criterion for separability of bipartite Gaussian states is extended to a general class of Bopp's shift. In particular, we have considered both the position-position and momentum-momentum noncommutativity, with deformation parameters θθθ and ηηη, respectively. It turns out that θθθ and ηηη induce the entanglement. We have directly applied the formalism for an anisotropic two-dimensional harmonic oscillator. Peres-Horodecki separability condition leads to a constraint equation for the parameter values of the oscillator in NC space. It turns out that the bipartite Gaussian state is almost always entangled in deformed space. To implement the theoretical idea, we provide an outline for a gedankenexperiment to identify the signature of phase-space noncommutativity, i.e., quantum gravity. In particular, the gedankenexperiment is devised to test the separability of supposedly separable Gaussian states in the usual commutative space, through the covariance matrix, which is constructed via measured output photocurrents after interaction of input Gaussian states and reference states. If the experiment shows that the supposedly separable states are actually entangled, then the entanglement is created through the intermediate background noncommutative space, which is a signature of the quantum nature of gravity.