Generalized quantum Onsager coefficients from a symmetrized Campbell-Hausdorff expansion

The analogue of the classical Onsager theory of entropy production is systematically derived for weakly irreversible processes in open quantum systems with finite-dimensional Hilbert space. The dynamics is assumed to be given by a quantum dynamical semigroup with infinitesimal generator of Gorini-Kossakowski-Sudarshan type. The basic Spohn formula for entropy production is used to obtain an expansion in terms of powers of the deviation of the initial state relative to the final stationary state of irreversible dynamics. To this end, an appropriate Lie series is constructed from a particular symmetrization procedure applied to the ordinary Campbell-Hausdorff expansion. In this way, only Hermitian contributions by higher-order commutators are generated, which allow an identification with so-called generalized Onsager coefficients. The explicit derivations concentrate on second-, third- and fourth-order coefficients, whereas complete detailed expressions are worked out for second and third order. In a suitable coherence-vector representation of density matrices the results can be given in terms of the dynamical parameters fixing the infinitesimal semigroup generator and in terms of symmetric and antisymmetric structure constants of the Lie algebra of SU(N). As an illustration, an application to generalized Bloch equations for two-level systems is studied, where the Onsager-like expansion can be compared with exact results for entropy production. We find that convergence is good even for rather large deviations between initial and final state if the calculation includes second- and third-order coefficients only. The formalism presented in this paper generalizes restrictions on admitted final states adopted in much simpler earlier treatments to the most general case of arbitrary unique final states of irreversible processes.