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

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.