Lower bounds to the spectral gap of Davies generators

TL;DR

This paper derives explicit lower bounds on the spectral gap of Davies generators, which model quantum thermalization, highlighting the importance of considering full dynamics over simplified master equations.

Contribution

It introduces explicit bounds for the spectral gap of Davies generators and challenges the conjecture that the convergence rate depends solely on the Pauli master equation gap.

Findings

Bounds are explicitly calculable with known Hamiltonian spectrum.

Counterexample shows convergence rate isn't always determined by the Pauli master equation.

Application to physical systems demonstrates the bounds' practical relevance.

Abstract

We construct lower bounds to the spectral gap of a family of Lindblad generators known as Davies maps. These maps describe the thermalization of quantum systems weakly coupled to a heat bath. The steady state of these systems is given by the Gibbs distribution with respect to the system Hamiltonian. The bounds can be evaluated explicitly, when the eigenbasis and the spectrum of the Hamiltonian is known. A crucial assumption is that the spectrum of the Hamiltonian is non-degenerate. Furthermore, we provide a counterexample to the conjecture, that the convergence rate is always determined by the gap of the associated Pauli master equation. We conclude, that the full dynamics of the Lindblad generator has to be considered. Finally, we present several physical example systems for which the bound to the spectral gap is evaluated.