Partition of energy for a dissipative quantum oscillator

We reveal a new face of the old cliched 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 = and E-p = , where and 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. P-k(omega) and 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.