Coherence Estimation Beyond the Liouvillian Gap in a Finite Nonequilibrium System

Abstract

We investigate the estimation of bath-induced coherence in a finite quantum system interacting with thermal reservoirs. Enhancement of coherence estimation is transient and the estimation precision totally disappears at the steady state despite the system retaining finite coherence. By analyzing the full Liouvillian eigenspectrum, we demonstrate that the optimal sensing window emerges from the competition between identifiable contributory modes' temporal relaxation and statistical importance. Neither is the linear inverse scaling of Liouvillian gap with transient optimal time a signature of unimodal contribution to optimal sensing, nor is the existence of multimodal dynamics a signature of nonlinear scaling. The inverse Liouvillian gap does not obey any general scaling with the optimal sensing time of coherence and we prove our numerical results analytically using a general Markovian framework. We further show that coupling the finite system to a quantum cavity and maintaining a thermal bias, transforms the transient metrological optimization into a sustained steady-state resource.