In this paper, we formulate and study the quantum counterpart of the energy equipartition theorem for a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a passive quantum heat bath through coordinate variables. The bath is modelled as a collection of independent quantum harmonic oscillators. We derive the closed form expressions for the mean kinetic and potential energies of the charged-dissipativemagneto-oscillator in the form E k = E k and Ep = Ep respectively, where E k and Ep denote the average kinetic and potential energies of individual thermostat oscillators. The net averaging is two-fold, the first one being over the Gibbs' canonical state for the thermostat, giving E k and Ep and the second one denoted by . being over the frequencies ω of the bath oscillators which contribute to E k and Ep according to probability distributions P k (ω) and Pp(ω) respectively. The relationship of the present quantum version of the equipartition theorem with that of the fluctuation-dissipation theorem (within the linear-response theory framework) is also explored. Further, we investigate the influence of the external magnetic field and the effect of different dissipation processes through Ohmic, Drude and radiation bath spectral density functions, on the typical properties of P k (ω) and Pp(ω). Finally, the role of system-bath coupling strength and the memory effect is analyzed in the context of average kinetic and potential energies of the dissipative charged magneto-oscillator.

In this brief report, following the recent developments on formulating a quantum analogue of the classical energy equipartition theorem for open systems where the heat bath comprises of independent oscillators, i.e. bosonic degrees of freedom, we present an analogous result for fermionic systems. The most general case where the system is connected to multiple reservoirs is considered and the mean energy in the steady state is expressed as an integral over the reservoir frequencies. Physically this would correspond to summing over the contributions of the bath degrees of freedom to the mean energy of the system over a suitable distribution function ρ(ω) dependent on the system parameters. This result holds for nonequilibrium steady states, even in the nonlinear regime far from equilibrium. We also analyze the zero temperature behaviour and low temperature corrections to the mean energy of the system.

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