# Spectral and Steady-State Properties of Random Liouvillians

## TL;DR

This paper analyzes the spectral and steady-state behaviors of random Liouvillian operators in open quantum systems, revealing how dissipation strength influences spectral gaps, distributions, and scaling properties across different regimes.

## Contribution

It introduces a comprehensive analysis of spectral and steady-state properties of stochastic Liouvillians, identifying regimes and scaling laws based on dissipation strength and system size.

## Key findings

- Spectral gap increases with system size for multiple dissipation channels
- Steady-state spectral distribution transitions from Poissonian to random matrix statistics with increasing dissipation
- Different regimes exhibit distinct scaling exponents for spectral features

## Abstract

We study generic open quantum systems with Markovian dissipation, focusing on a class of stochastic Liouvillian operators of Lindblad form with independent random dissipation channels (jump operators) and a random Hamiltonian. We establish that the global spectral features, the spectral gap, and the steady-state properties follow three different regimes as a function of the dissipation strength, whose boundaries depend on the particular quantity. Within each regime, we determine the scaling exponents with the dissipation strength and system size. We find that, for two or more dissipation channels, the spectral gap increases with the system size. The spectral distribution of the steady state is Poissonian at low dissipation strength and conforms to that of a random matrix once the dissipation is sufficiently strong. Our results can help to understand the long-time dynamics and steady-state properties of generic dissipative systems.