We study the quantum counterpart of the theorem on energy equipartition for classical systems. We consider a free quantum Brownian particle modeled in terms of the Caldeira-Leggett framework: a system plus thermostat consisting of an infinite number of harmonic oscillators. By virtue of the theorem on the averaged kinetic energy $E_k$ of the quantum particle, it is expressed as $E_k=\langle {\mathcal{E}}_k\rangle$, where ${\mathcal{E}}_k$ is the thermal kinetic energy of the thermostat per 1 degree of freedom and $\langle \cdots \rangle$ denotes averaging over the frequencies $\omega$ of thermostat oscillators which contribute to $E_k$ according to the probability distribution $\mathbb{P}\left(\omega \right)$. We explore the impact of various dissipation mechanisms, via the Drude, Gaussian, algebraic, and Debye spectral density functions, on the characteristic features of $\mathbb{P}\left(\omega \right)$. The role of the system-thermostat coupling strength and the memory time on the most probable thermostat oscillator frequency as well as the kinetic energy $E_k$ of the Brownian particle is analyzed.

• Received 19 August 2018

DOI:https://doi.org/10.1103/PhysRevA.98.052107