Partition of free energy for a Brownian quantum oscillator: Effect of dissipation and magnetic field

Recently, the quantum counterpart of energy equipartition theorem has drawn considerable attention. Motivated by this, we formulate and investigate an analogous statement for the free energy of a quantum oscillator linearly coupled to a passive heat bath consisting of an infinite number of independent harmonic oscillators. We explicitly demonstrate that the free energy of the Brownian oscillator can be expressed in the form $F(T) = \langle f(\omega,T) \rangle $ where $f(\omega,T)$ is the free energy of an individual bath oscillator. The overall averaging process involves two distinct averages: the first one is over the canonical ensemble for the bath oscillators, whereas the second one signifies averaging over the entire bath spectrum of frequencies from zero to infinity. The latter is performed over a relevant probability distribution function $\mathcal{P}(\omega)$ which can be derived from the knowledge of the generalized susceptibility encountered in linear response theory. The effect of different dissipation mechanisms is also exhibited. We find two remarkable consequences of our results. First, the quantum counterpart of energy equipartition theorem follows naturally from our analysis. The second corollary we obtain is a natural derivation of the third law of thermodynamics for open quantum systems. Finally, we generalize the formalism to three spatial dimensions in the presence of an external magnetic field.