Quantum dynamics in a tiered non-Markovian environment

We introduce a new analytical method for studying the open quantum systems problem of a discrete system weakly coupled to an environment of harmonic oscillators. Our approach is based on a phase space representation of the density matrix for a system coupled to a two-tiered environment. The dynamics of the system and its immediate environment are resolved in a non-Markovian way, and the environmental modes of the inner environment can themselves be damped by a wider `universe'. Applying our approach to the canonical cases of the Rabi and spin-boson models we obtain new analytical expressions for an effective thermalisation temperature and corrections to the environmental response functions as direct consequences of considering such a tiered environment. A comparison with exact numerical simulations confirms that our approximate expressions are remarkably accurate, while their analytic nature offers the prospect of deeper understanding of the physics which they describe. A unique advantage of our method is that it permits the simultaneous inclusion of a continuous bath as well as discrete environmental modes, leading to wide and versatile applicability.