Identification of quantum scars via phase-space localization measures

TL;DR

This paper introduces a phase-space localization measure based on Rényi occupations of the Husimi function to identify quantum scars and unstable periodic orbits in complex quantum systems like the Dicke model.

Contribution

It develops a new phase-space localization measure using Rényi occupations and demonstrates its effectiveness in detecting quantum scars in the Dicke model.

Findings

Rényi occupations with α>1 effectively reveal quantum scars.

The measure identifies unstable periodic orbits associated with scarred eigenstates.

Application to the Dicke model shows high effectiveness in complex phase spaces.

Abstract

There is no unique way to quantify the degree of delocalization of quantum states in unbounded continuous spaces. In this work, we explore a recently introduced localization measure that quantifies the portion of the classical phase space occupied by a quantum state. The measure is based on the $\alpha$-moments of the Husimi function and is known as the R\'enyi occupation of order $\alpha$. With this quantity and random pure states, we find a general expression to identify states that are maximally delocalized in phase space. Using this expression and the Dicke model, which is an interacting spin-boson model with an unbounded four-dimensional phase space, we show that the R\'enyi occupations with $\alpha>1$ are highly effective at revealing quantum scars. Furthermore, by analyzing the high moments ($\alpha>1$) of the Husimi function, we are able to identify qualitatively and quantitatively the unstable periodic orbits that scar some of the eigenstates of the model.