Towards reconciliation of completely positive open system dynamics with the equilibration postulate

Abstract

Almost every quantum system interacts with a large environment, so the exact quantum mechanical description of its evolution is impossible. One has to resort to approximate description, usually in the form of a master equation. There are at least two basic requirements for such a description: first, it should preserve the positivity of probabilities; second, it should correctly describe the equilibration process for systems coupled to a single thermal bath. Existing two widespread descriptions of evolution fail to satisfy at least one of those conditions. The so-called Davies master equation, while preserving the positivity of probabilities, fails to describe thermalization properly. On the other hand, the Bloch-Redfield master equation violates the first condition, but it correctly describes equilibration, at least for off-diagonal elements for several important scenarios. However, is it possible to have a description of open system dynamics that would share both features? In this paper, we partially resolve this problem in the weak-coupling limit: (i) We provide a general form of the proper thermal equilibrium state (the so-called mean-force state) for an arbitrary open system. (ii) We provide the solution for the steady-state coherences for a whole class of master equations, and in particular, we show that the solution coincides with the mean-force Hamiltonian for the Bloch-Redfield equation. (iii) We consider the cumulant equation, which is explicitly completely positive, and we show that its steady-state coherences are the same as one of the Bloch-Redfield dynamics (and the mean-force state accordingly). (iv) We solve the correction to the diagonal part of the stationary state for a two-level system both for the Bloch-Redfield and cumulant equation, showing that the solution of the cumulant is very close to the mean-force state, whereas the Bloch-Redfield differs significantly.

Perturbative Steady States of Completely Positive Quantum Master Equations

“…It is well known [1] that the Lindblad equation obtained from secular approximation results in the Gibbs state of the system Hamiltonian as a steady state. However, if we accept that the total system, consisting of the system of interest and the bath, thermalizes, for example by satisfying the condition required for the eigenstate thermalization hypothesis [17][18][19][20], the steady state of the reduced density operator will be the so-called mean force Gibbs (MFG) state [21][22][23][24], which is obtained by tracing the Gibbs state for the total Hamiltonian over the bath variables [25][26][27]. Therefore, the Lindblad equation is not consistent with the thermalization property because it neglects the effect of coupling to the bath.…”

Canonically consistent quantum master equation

“…Despite the lack of CP, a stringent restriction for physical maps [23,39], the Redfield equation is able to capture finite system-bath coupling effects for which the Lindblad is insensitive [26,40]. Thus, it is only recently that the Redfield equation has gained popularity as a tool to incorporate finite systembath coupling effects [27,34,35,[41][42][43]. To this goal, we propose below a scheme that uses the generalized Gibbs state from equilibrium statistical mechanics [28,29,33] to correct the Redfield QME, specifically improving on the approximation in Eq.…”