Quantum dynamics in a tiered non-Markovian environment

Abstract

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 thermalization 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.

Photon-assisted tunneling and charge dephasing in a carbon nanotube double quantum dot

“…Appendix C: Approximation of dephasing time Ref. 30 allows us obtain an analytical estimate of the 'spin' coherence time in the presence of one or more oscillator modes. This is more appropriate than solutions to the canonical spin-boson model, as the discrete oscillator environment is too small to assume short environmental correlations times.…”

When do perturbative approaches accurately capture the dynamics of complex quantum systems?

“…We consider two different weak-coupling techniques: time convolutionless (TCL) master equations 4 and a second method based on the phase-space representation of the full density matrix 10 (P-mat). Interestingly, we show that both approaches give rise to exactly the same criterion, despite their rather different nature.…”

“…In the ‘grey area’ where a second-order expansion is no longer strictly justified, we find that the quality of the different approaches differs. Some of us have previously found that the P-mat method outperformed the commonly used secular, second-order Born-Markov master equation 10 . For the examples studied here, we find that a TCL ME gives slightly better short time dynamics than P-mat, whilst the latter frequently performs better at longer times as the system approaches thermalisation.…”

Harnessing non-Markovian quantum memory by environmental coupling

“…A new analytical method based on a phase space representation of the system density matrix has been also proposed to study the dynamics of a discrete system in a two-tiered non-Markovian environment [52].…”