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

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 (I(over cap)) have been reduced to a simplified representative form by a linear canonical transformation (the group Sp(4, R). An orthonormal basis of the Hilbert space consisting of the eigenvectors of I(over cap) is obtained. In order to obtain the solutions of the time-dependent Schrodinger 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.