The Interplay of Finite and Infinite Size Stability in Quadratic Bosonic Lindbladians

TL;DR

This paper develops a framework to understand how stability properties of open bosonic systems can differ between finite and infinite sizes, revealing two types of metastability with distinct physical and dynamical characteristics.

Contribution

It introduces a classification of dynamical metastability in quadratic bosonic Lindbladians, linking non-normality to transient dynamics and boundary conditions, and explores their physical implications.

Findings

Identifies two types of dynamical metastability with distinct stability properties.

Shows metastability affects entanglement entropy and super-volume scaling.

Connects spectral properties of infinite systems to finite response functions.

Abstract

We provide a framework for understanding dynamical metastability in open many-body systems of free bosons, whereby the dynamical stability properties of the system in the infinite-size (thermodynamic) limit may sharply differ from those of any finite-size truncation, and anomalous transient dynamics may arise. By leveraging pseudospectral techniques, we trace the discrepancy between asymptotic and transient dynamics to the non-normality of the underlying quadratic bosonic Lindbladian (QBL) generator, and show that two distinct flavors of dynamical metastability can arise. QBLs exhibiting type I dynamical metastability, previously discussed in the context of anomalous transient amplification [Phys. Rev. Lett. 127, 245701 (2021)], are dynamically unstable in the infinite-size limit, yet stable once open boundaries are imposed. Type II-dynamically metastable QBLs, which we uncover in this work, are dynamically stable for infinite size, but become unstable under open boundary conditions for arbitrary finite system size. We exhibit representative models for both types of metastability in the dissipative, as well as the limiting closed-system (Hamiltonian) settings, and analyze distinctive physical behavior they can engender. We show that dynamical metastability manifests itself in the generation of entanglement entropy, by way of a transient which reflects the stability phase of the infinite (rather than the actual finite) system and, as a result, is directly tied to the emergence of super-volume scaling in type I systems. Finally, we demonstrate how, even in Hermitian, and especially in highly non-normal regimes, the spectral properties of an infinite-size QBL are reflected in the linear response functions of the corresponding finite QBLs, by way of resonant pseudospectral modes.