On the two-dimensional time-dependent anisotropic harmonic oscillator in a magnetic field

TL;DR

This paper analyzes a two-dimensional anisotropic harmonic oscillator with time-dependent parameters in a magnetic field, constructing invariant operators and solutions to the Schrödinger equation, and examining entanglement properties of bipartite states.

Contribution

It introduces a method to construct invariant operators and solutions for a time-dependent anisotropic oscillator in a magnetic field, including analysis of entanglement criteria.

Findings

Constructed quadratic invariant operators for the system

Derived solutions with geometric and dynamical phases

Applied Peres-Horodecki criterion to bipartite states

Abstract

A Charged harmonic oscillator in a magnetic field, Landau problems, and an oscillator in a noncommutative space, share the same mathematical structure in their Hamiltonians. We have considered a two-dimensional anisotropic harmonic oscillator (AHO) with arbitrarily time-dependent parameters (effective mass and frequencies), placed in an arbitrarily time-dependent magnetic field. A class of quadratic invariant operators (in the sense of Lewis and Riesenfeld) have been constructed. The invariant operators ($\hat{\mathcal{I}}$) have been reduced to a simplified representative form by a linear canonical transformation (the group $Sp(4, \mathbb{R})$). An orthonormal basis of the Hilbert space consisting of the eigenvectors of $\hat{\mathcal{I}}$ is obtained. In order to obtain the solutions of the time-dependent Schr\"{o}dinger equation corresponding to the system, both the geometric and dynamical phase-factors are constructed. Peres-Horodecki Separability Criterion for the bipartite coherent states corresponding to our system has been demonstrated.