Localization and Delocalization of Quantum Trajectories in the Liouvillian Spectrum

TL;DR

This paper introduces a novel method to analyze open quantum system dynamics by examining individual quantum trajectories in the Liouvillian eigenbasis, revealing persistent delocalization effects at late times.

Contribution

It presents a new approach using trajectory-eigenstate overlaps to characterize trajectory localization, challenging the notion that late-time dynamics are solely governed by steady states.

Findings

Trajectories can remain delocalized over bulk eigenstates at late times.

Delocalization correlates with the purity of the steady state.

Numerical simulations confirm persistent trajectory spreading in spin and bosonic systems.

Abstract

We develop an approach for understanding the dynamics of open quantum systems by analyzing individual quantum trajectories in the eigenbasis of the Liouvillian superoperator. From trajectory-eigenstate overlaps, we construct a quasiprobability distribution that characterizes the degree of localization of the trajectories in the Liouvillian eigenbasis. Contrary to the common wisdom that late-time dynamics are governed solely by the steady state and the slowest-decaying modes, we show that trajectories can remain well spread over transient eigenstates deep within the bulk of the Liouvillian spectrum even at late times. We demonstrate this explicitly using numerical simulations of interacting spin chains and bosonic systems. Moreover, we find that the delocalization of the trajectory strongly correlates with the purity of the trajectory-averaged steady state, establishing a further link between the trajectory and ensemble pictures of open quantum dynamics.