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Modern Physics Letters A

Non-Gaussian entanglement of magnetically coupled modes: A dynamical analysis

Radouan Hab‐arrih, Ahmed Jellal, Abdeldjalil Merdaci

Published Oct 20, 2022• 0 citations
Publisher

Abstract

The quantum entanglement of two connected harmonic oscillators via angular momentum yielding a magnetic coupling [Formula: see text] is discussed in this study. The corresponding Hamiltonian is diagonalized by using three canonical transformations, and then the stationary wave function is obtained. The Schmidt decomposition is used to explicitly determine the modes [Formula: see text], with [Formula: see text], [Formula: see text] and [Formula: see text] being two quantum numbers associated with the two oscillators. We summarize our findings by looking at the effects of anisotropy [Formula: see text], [Formula: see text], asymmetry [Formula: see text], and dynamics on entanglement. (i) With increasing [Formula: see text], the entanglement grows exceedingly large. (ii) The sensitivity to [Formula: see text] is determined by [Formula: see text] and [Formula: see text]. The physical parameters and quantum numbers play a great role in the periodic resuscitation of entanglement.

DOI
10.1142/s0217732322502170

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The quantum entanglement of two connected harmonic oscillators via angular momentum yielding a magnetic coupling [Formula: see text] is discussed in this study. The corresponding Hamiltonian is diagonalized by using three canonical transformations, and then the stationary wave function is obtained. The Schmidt decomposition is used to explicitly determine the modes [Formula: see text], with [Formula: see text], [Formula: see text] and [Formula: see text] being two quantum numbers associated with the two oscillators. We summarize our findings by looking at the effects of anisotropy [Formula: see text], [Formula: see text], asymmetry [Formula: see text], and dynamics on entanglement. (i) With increasing [Formula: see text], the entanglement grows exceedingly large. (ii) The sensitivity to [Formula: see text] is determined by [Formula: see text] and [Formula: see text]. The physical parameters and quantum numbers play a great role in the periodic resuscitation of entanglement.
This paper surveys the quantum entanglement of two coupled harmonic oscillators via angular momentum generating a magnetic coupling ωcω_{c}ωc​. The corresponding Hamiltonian is diagonalized by using three canonical transformations and then the stationary wave function is obtained. Based on the Schmidt decomposition, we explicitly determine the Schmidt modes λkλ_{k}λk​ with k∈{0,1,⋯ ,n+m}k\in\left\lbrace 0,1,\cdots,n+m\right\rbracek∈{0,1,⋯,n+m}, nnn and mmm being two quantum numbers associated to the two oscillators. By studying the effect of the anisotropy $ R=ω_{1}^{2}/ω_{2}^{2} $, ωcω_{c}ωc​, asymmetry $ |n-m| $ and dynamics on the entanglement, we summarize our results as follows. $ (i)- $ The entanglement becomes very large with the increase of $ (n,m) $. $ (ii)- $ The sensistivity to ωcω_cωc​ depends on $ (n,m) $ and RRR. $ (iii)- $ The periodic revival of entanglement strongly depends on the physical parameters and quantum numbers.
A system of two coupled quantum harmonic oscillators with the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}=\frac{1}{2}(\frac{1}{{m}_{1}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{1}^{2}+\frac{1}{{m}_{2}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{2}^{2}+A{x}_{1}^{2}+B{x}_{2}^{2}+C{x}_{1}{x}_{2})$ can be found in many applications of quantum and nonlinear physics, molecular chemistry, and biophysics. The stationary wave function of such a system is known, but its use for the analysis of quantum entanglement is complicated because of the complexity of computing the Schmidt modes. Moreover, there is no exact analytical solution to the nonstationary Schrodinger equation $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\mathrm{\ensuremath{\Psi}}=i\ensuremath{\hbar}\frac{\ensuremath{\partial}\mathrm{\ensuremath{\Psi}}}{\ensuremath{\partial}t}$ and Schmidt modes for such a dynamic system. In this paper we find a solution to the nonstationary Schrodinger equation; we also find in an analytical form a solution to the Schmidt mode for both stationary and dynamic problems. On the basis of the Schmidt modes, the quantum entanglement of the system under consideration is analyzed. It is shown that for certain parameters of the system, quantum entanglement can be very large.
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.
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.
We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy SL=1−PS_L=1-\mathcal{P}SL​=1−P for arbitrary excitation numbers (n,m)(n,m)(n,m), allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters θθθ and ϑ2\vartheta_2ϑ2​, the frequency parameter β0β_0β0​ and the coupling strength εεε: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case ω1=ω2=1ω_1=ω_2=1ω1​=ω2​=1 and strong couplings ε≈0.99ε\approx 0.99ε≈0.99, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.
We examine the entanglement induced by an angular momentum coupling between two harmonic systems. The Hamiltonian corresponds to that of a charged particle in a uniform magnetic field in an anisotropic quadratic potential or, equivalently, to that of a particle in a rotating quadratic potential. We analyze both the vacuum and thermal entanglement, thereby obtaining analytic expressions for the entanglement entropy and negativity through the Gaussian state formalism. It is shown that vacuum entanglement diverges at the edges of the dynamically stable sectors, increasing with the angular momentum and saturating for strong fields, whereas at finite temperature entanglement is nonzero just within a finite field or frequency window and no longer diverges. Moreover, the limit temperature for entanglement is finite in the whole stable domain. The thermal behavior of the Gaussian quantum discord and its difference from the negativity is also discussed.