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arXiv:2106.05182 · Physica Scripta

Investigation of a harmonic oscillator in a magnetic field with damping and time dependent noncommutativity

Manjari Dutta, Shreemoyee Ganguly, Sunandan Gangopadhyay

Published Sep 29, 2021• 6 citations
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Abstract

This work is devoted to find the eigenstates of a two dimensional damped harmonic oscillator in the presence of an external magnetic field varying with respect to time with a time dependent spatial noncommutativity. It is observed that there are some specific choices of the damping factor, the time dependent frequency of the oscillator and the time dependent external magnetic field for which one can obtain interesting solutions of the time dependent noncommutative parameters. These solutions follow from the non-linear Ermakov-Pinney equation. We then obtain exact analytic forms for the phase which relates the eigenstates of the Hamiltonian with the eigenstates of the Lewis invariant. An important aspect of this study is to see the relationship between the damping and the applied magnetic field on the energy profile of the system. For this we compute the expectation value of the Hamiltonian. These are found to vary with time for different solutions of the Ermakov-Pinney equation corresponding to different choices of the damping factor, the time dependent frequency of the oscillator and the time dependent applied magnetic field. Finally, we compare our results with those in the absence of the magnetic field to determine the effect of the applied field on our system.

DOI
10.1088/1402-4896/ac2b4c

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This work is devoted to find the eigenstates of a two dimensional damped harmonic oscillator in the presence of an external magnetic field varying with respect to time with a time dependent spatial noncommutativity. It is observed that there are some specific choices of the damping factor, the time dependent frequency of the oscillator and the time dependent external magnetic field for which one can obtain interesting solutions of the time dependent noncommutative parameters. These solutions follow from the non-linear Ermakov-Pinney equation. We then obtain exact analytic forms for the phase which relates the eigenstates of the Hamiltonian with the eigenstates of the Lewis invariant. An important aspect of this study is to see the relationship between the damping and the applied magnetic field on the energy profile of the system. For this we compute the expectation value of the Hamiltonian. These are found to vary with time for different solutions of the Ermakov-Pinney equation corresponding to different choices of the damping factor, the time dependent frequency of the oscillator and the time dependent applied magnetic field. Finally, we compare our results with those in the absence of the magnetic field to determine the effect of the applied field on our system.
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 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 (Î) 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 Î is obtained. In order to obtain the solutions of the time-dependent Schrödinger equation corresponding to the system, both the geometric and dynamical phase-factors are constructed. A generalized Peres–Horodecki separability criterion (Simon’s criterion) for the ground state corresponding to our system has been demonstrated.
In this paper, decoherence of a damped anisotropic harmonic oscillator in the presence of a magnetic field is studied in the framework of the Lindblad theory of open quantum systems in noncommutative phase-space. General fundamental conditions that should follow our quantum mechanical diffusion coefficients appearing in the master equation are kindly derived. From the master equation, the expressions of density operator, the Wigner distribution function, the expectation and variance with respect to coordinates and momenta are obtained. Based on these quantities, the total energy of the system is evaluated and simulations show its dependency to phase-space structure and its improvement due to noncommutativity effects and the environmental temperature as well. In addition, we also evaluate the decoherence time scale and show that it increases with noncommutativity phase-space effects as compared to the commutative case. It turns out from simulations that this time scale is significantly improved under magnetic field effects.
Abstract In this paper we look for the existence of Berry phase in time dependent harmonic oscillators in noncommutative space. Two systems are considered in our study in the noncommutative framework. The first one is a system in which a scale invariant term is present in the Hamiltonian from the beginning, and the second one is a system in which a scale invariant term emerges due to a change of variables from the noncommutative to the commutative ones. We first compute the eigenstates of both the systems using the Lewis invariant approach. We then employ the Lewis invariant technique to obtain the geometric phase under adiabatic approximation. We also calculate the Berry phase explicitly by choosing appropriate forms of the time dependent parameters appearing in the Hamiltonian. Our analysis surprisingly reveals that a scale invariant time reversal symmetry breaking term may not always lead to a non trivial Berry phase.