Magnetic Field Effect on Dynamics of Entanglement for Time-dependent Harmonic Oscillator

TL;DR

This paper explores how a static magnetic field influences the time-dependent entanglement, uncertainty, and mixedness in a two-dimensional harmonic oscillator with variable frequency and coupling, providing insights into controlling quantum correlations.

Contribution

It introduces a comprehensive analysis of entanglement dynamics under magnetic fields using symplectic and Wigner formalisms, linking entropy measures with quantum uncertainties in a time-dependent setting.

Findings

Magnetic field significantly affects entanglement and uncertainty dynamics.

Explicit relations between von Neumann entropy and linear entropy are established.

Magnetic and coupling parameters can be used to control quantum features.

Abstract

We investigate the dynamics of entanglement, uncertainty and mixedness by solving time dependent Schr\"{o}dinger 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 $S_{L}$ as well as logarithmic negativity $\mathcal{N}$ 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 $S_{von}$ and establish its link with $S_{L}$. We use the Wigner formalism to derive the Heisenberg uncertainties and {show their dependencies on both $S_{L}$ and the coupling parameters $\gamma_{i}$ $ (i=1,2)$ of the quadrature term $x_{i}p_{i}$.} We graphically study the dynamics of the three features (entanglement, uncertainty, mixedness) and present the similar topology with respect to time. We show the effects of the magnetic field and quenched values of $J(t)$ and $\omega_{2}(t)$ on these three dynamics, which lead eventually to control and handle them.