Partition of energy for a dissipative quantum oscillator

Abstract

We reveal a new face of the old clichéd system: a dissipative quantum harmonic oscillator. We formulate and study a quantum counterpart of the energy equipartition theorem satisfied for classical systems.Both mean kinetic energy $E_k$ and mean potential energy $E_p$ of the oscillator are expressed as $E_k=⟨{\mathcal{E}}_k⟩$ and $E_p=⟨{\mathcal{E}}_p⟩$, where $⟨{\mathcal{E}}_k⟩$ and $⟨{\mathcal{E}}_p⟩$ are mean kinetic and potential energies per one degree of freedom of the thermostat which consists of harmonic oscillators too. The symbol $⟨...⟩$ denotes two-fold averaging: (i) over the Gibbs canonical state for the thermostat and (ii) over thermostat oscillators frequencies $\omega$ which contribute to $E_k$ and $E_p$ according to the probability distribution ${\mathbb{P}}_k(\omega )$ and ${\mathbb{P}}_p(\omega )$, respectively. The role of the system-thermostat coupling strength and the memory time is analysed for the exponentially decaying memory function (Drude dissipation mechanism) and the algebraically decaying damping kernel.

References in corpus (3)

• Fundamental Aspects of Quantum Brownian Motion
• Heisenberg-Langevin vs. quantum master equation
• Quantum Brownian Motion with Inhomogeneous Damping and Diffusion