Partition of energy for a dissipative quantum oscillator
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 $E_k = \langle \mathcal E_k \rangle$ and $E_p = \langle \mathcal E_p \rangle$, where $\langle \mathcal E_k \rangle$ and $ \langle \mathcal E_p \rangle$ are mean kinetic and potential energies per one degree of freedom of the thermostat which consists of harmonic oscillators too. The symbol $\langle ...\rangle$ denotes two-fold averaging: (i) over the Gibbs canonical state for the thermostat and (ii) over thermostat oscillators frequencies $ω$ which contribute to $E_k$ and $E_p$ according to the probability distribution $\mathbb{P}_k(ω)$ and $\mathbb{P}_p(ω)$, 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.