Resolving Discrepancy between Liouvillian Gap and Relaxation Time in Boundary-Dissipated Quantum Many-Body Systems

TL;DR

This paper reveals that in boundary-dissipated quantum many-body systems, the relaxation time is governed by large expansion coefficients rather than the Liouvillian gap, resolving a long-standing discrepancy in the field.

Contribution

It demonstrates that relaxation time is determined by eigenvector expansion coefficients, not the Liouvillian gap, challenging previous assumptions in quantum dissipative systems.

Findings

Relaxation time is not set by the Liouvillian gap.

Large expansion coefficients dominate relaxation dynamics.

Resolves discrepancy between gap and relaxation time in literature.

Abstract

The gap of the Liouvillian spectrum gives the asymptotic decay rate of a quantum dissipative system, and therefore its inverse has been identified as the slowest relaxation time. In contrary to this common belief, we show that the relaxation time due to diffusive transports in a boundary dissipated many-body quantum system is determined not by the gap or low-lying eigenvalues of the Liouvillian but by superexponentially large expansion coefficients for Liouvillian eigenvectors with non-small eigenvalues at an initial state. This finding resolves an apparent discrepancy reported in the literature between the inverse of the Liouvillian gap and the relaxation time in dissipative many-body quantum systems.