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Modern Physics Letters A

Purity function of three isotopic harmonic oscillators

Ahlem Abidi

Published Aug 5, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217732325501378

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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.
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.
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.
Quantum entanglement serves as a key phenomenon in understanding correlations in many-body systems, but analytical results remain scarce for coupled three-body oscillators. In this work, we address this gap by introducing a geometrical diagonalization approach that constrains Euler angles, thereby reducing the degrees of freedom in the entanglement analysis. It consists of deriving analytical expressions for linear entropy and purity under the bipartitions (x∣yz)(x|yz)(x∣yz), (y∣xz)(y|xz)(y∣xz), and (xy∣z)(xy|z)(xy∣z) using the Wigner function framework. Our results indicate that excitations in any oscillator basically enhance the redistribution of correlations across the system. The mixing angle θθθ governs entanglement intensity, ranging from separability to maximal correlation. Moreover, we reveal the symmetry relations, notably SLy[(n,m,l),θ]=SLz[(n,m,l),−θ]S_{Ly}[(n,m,l),θ]=S_{Lz}[(n,m,l),-θ]SLy​[(n,m,l),θ]=SLz​[(n,m,l),−θ] and an intrinsic symmetry within (x∣yz)(x|yz)(x∣yz). Hence, we clarify how excitation levels and mixing angles create and enhance entanglement in the three coupled harmonic oscillators.
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.
This thesis studies deformation quantization and its application to contact interactions and systems with dissipation. We consider the subtleties related to quantization when contact interactions and boundaries are present. We exploit the idea that discontinuous potentials are idealizations that should be realized as limits of smooth potentials. The Wigner functions are found for the Morse potential and in the proper limit they reduce to the Wigner functions for the infinite wall, for the most general (Robin) boundary conditions. This is possible for a very limited subset of the values of the parameters -- so-called fine tuning is necessary. It explains why Dirichlet boundary conditions are used predominantly. Secondly, we consider deformation quantization in relation to dissipative phenomena. For the damped harmonic oscillator we study a method using a modified noncommutative star product. Within this framework we resolve the non-reality problem with the Wigner function and correct the classical limit.