Full spectrum of the Liouvillian of open dissipative quantum systems in the Zeno limit

TL;DR

This paper analyzes the spectral structure of the Liouvillian in open quantum systems under strong local dissipation, deriving effective equations and explicit eigenvalues, with applications to quantum spin chains.

Contribution

It introduces a method to explicitly compute the eigenvalues and eigenstates of the Liouvillian in the Zeno limit, extending understanding of dissipative quantum dynamics.

Findings

Eigenvalues form distinct vertical stripes in the complex plane.

Derived effective Lindblad equations for each spectral stripe.

Explicit eigenvalue expressions valid at order 1/Γ.

Abstract

We consider an open quantum system with dissipation, described by a Lindblad Master equation (LME). For dissipation locally acting and sufficiently strong, a separation of the relaxation time scales occurs, which, in terms of the eigenvalues of the Liouvillian, implies a grouping of the latter in distinct vertical stripes in the complex plane at positions determined by the eigenvalues of the dissipator. We derive effective LME equations describing the modes within each stripe separately, and solve them perturbatively, obtaining for the full set of eigenvalues and eigenstates of the Liouvillian explicit expressions correct at order $1/\Gamma$ included, where $\Gamma$ is the strength of the dissipation. As a example, we apply our general results to quantum $XYZ$ spin chains coupled, at one boundary, to a dissipative bath of polarization.