On the second-order corrections to the quantum canonical equilibrium density matrix

Abstract

We consider the equilibrium state of a quantum system weakly coupled to a quantum bath within second order perturbation theory. It was previously shown by Romero-Rochin and Oppenheim [Physica A 155, 52 (1989)] that the equilibrium state deviates from the canonical form, e−βHs/Zs (Hs is the free system Hamiltonian and Zs the canonical partition function). We reproduce this result via a different derivation, starting from the non-Markovian, rather than the Markovian, quantum Master equation. Our derivation sheds new light on the mechanism that stabilizes the deviation from the canonical form and shows that it involves an interplay between a static distortion to the equilibrium state and dynamical system–bath correlations. We show that this deviation is a necessary consequence of translational invariance and vanishes when the rotating-wave-approximation is applied. The deviation is also shown to vanish for a two-level system off-diagonally coupled to a heat bath or when the Lamb shifts are neglected. Two ways for numerically evaluating the second order deviations are described. Finally, the deviations from canonical equilibrium are given an illuminating geometrical interpretation in terms of the phase space Wigner distribution.