Spectral density, memory function, and mean relaxation time for resonant subsystem-reservoir interactions

Abstract

We show that the projection operators P usually employed to derive the Nakajima–Zwanzig master equation are non-Hermitian. As a consequence the operator QLQ which governs the dynamics of subsystem-reservoir interactions (here Q=1−P and L is the Liouville operator) is also non-Hermitian and possesses a complex temperature-dependent spectrum of resonances. We use the self-consistent Born approximation to derive a formula for the spectral density of QLQ. From this spectral density we calculate the memory function and mean relaxation time of subsystem-reservoir correlations. These formulas are compared with numerical results obtained from the spin–spin-bath model to test their accuracy. The memory function and mean relaxation time are essential ingredients for the SRA (statistical resonance approximation) theory of microscopic dissipation and decoherence. We show that the SRA master equation preserves positivity when the memory function derived in this paper is employed.