Quantum localization measures in phase space

TL;DR

This paper introduces a general scheme for quantifying quantum state localization in phase space using Rénnyi occupations, and compares two Husimi-based measures in the chaotic regime of the Dicke model.

Contribution

It proposes a unified framework for localization measures in measure spaces and applies it to compare Husimi-based localization measures in a complex quantum system.

Findings

Different Husimi-based measures yield context-dependent localization results.

Maximal delocalization requires a bounded reference subspace in unbounded phase spaces.

The scheme clarifies the origin of differences between localization measures.

Abstract

Measuring the degree of localization of quantum states in phase space is essential for the description of the dynamics and equilibration of quantum systems, but this topic is far from being understood. There is no unique way to measure localization, and individual measures can reflect different aspects of the same quantum state. Here, we present a general scheme to define localization in measure spaces, which is based on what we call R\'enyi occupations, from which any measure of localization can be derived. We apply this scheme to the four-dimensional unbounded phase space of the interacting spin-boson Dicke model. In particular, we make a detailed comparison of two localization measures based on the Husimi function in the regime where the model is chaotic, namely one that projects the Husimi function over the finite phase space of the spin and another that uses the Husimi function defined over classical energy shells. We elucidate the origin of their differences, showing that in unbounded spaces the definition of maximal delocalization requires a bounded reference subspace, with different selections leading to contextual answers.