Quantum counterpart of energy equipartition theorem for a dissipative charged magneto-oscillator: Effect of dissipation, memory, and magnetic field

TL;DR

This paper extends the energy equipartition theorem to a quantum charged oscillator in a magnetic field with dissipation, deriving explicit energy expressions and analyzing effects of bath spectral densities, magnetic field, and memory.

Contribution

It provides a novel quantum formulation of the energy equipartition theorem for dissipative charged oscillators in magnetic fields, including explicit energy expressions and analysis of environmental effects.

Findings

Derived closed-form expressions for mean kinetic and potential energies.

Analyzed the influence of magnetic field and bath spectral densities.

Explored the effects of system-bath coupling and memory on energies.

Abstract

In this paper, we formulate and study the quantum counterpart of the energy equipartition theorem for a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a passive quantum heat bath through coordinate variables. The bath is modelled as a collection of independent quantum harmonic oscillators. We derive the closed form expressions for the mean kinetic and potential energies of the charged-dissipative-magneto-oscillator in the form $E_k = \langle \mathcal{E}_k \rangle$ and $E_p = \langle \mathcal{E}_p \rangle$ respectively, where $\mathcal{E}_k$ and $\mathcal{E}_p$ denote the average kinetic and potential energies of individual thermostat oscillators. The net averaging is two-fold, the first one being over the Gibbs canonical state for the thermostat, giving $\mathcal{E}_k$ and $\mathcal{E}_p$ and the second one denoted by $\langle . \rangle$ being over the frequencies $\omega$ of the bath oscillators which contribute to $E_k$ and $E_p$ according to probability distributions $\mathcal{P}_k(\omega)$ and $\mathcal{P}_p(\omega)$ respectively. The relationship of the present quantum version of the equipartition theorem with that of the fluctuation-dissipation theorem (within the linear-response theory framework) is also explored. Further, we investigate the influence of the external magnetic field and the effect of different dissipation processes through Gaussian decay, Drude and radiation bath spectral density functions, on the typical properties of $\mathcal{P}_k(\omega)$ and $\mathcal{P}_p(\omega)$. Finally, the role of system-bath coupling strength and the memory effect is analyzed in the context of average kinetic and potential energies of the dissipative charged magneto-oscillator.