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 = 〈ε_k〉 and E_p = 〈ε_p〉, where 〈ε_k〉 and 〈ε_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 ω which contribute to E_k and E_p according to the probability distribution [Formula: see text] and [Formula: see text], 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.