Effective dimensions of infinite-dimensional Hilbert spaces: A phase-space approach

TL;DR

This paper introduces a phase-space method using Husimi distributions to define a finite effective dimension in infinite-dimensional Hilbert spaces, aiding the analysis of quantum phenomena like localization and scarring.

Contribution

It presents a novel approach to quantify effective dimensions in infinite-dimensional systems via phase-space techniques, with applications demonstrated on the spin-boson Dicke model.

Findings

Effective dimension can be finite in unbounded phase spaces.

The method characterizes quantum phenomena such as localization and scarring.

Numerical results validate the phase-space approach.

Abstract

By employing Husimi quasiprobability distributions, we show that a bounded portion of an unbounded phase space induces a finite effective dimension in an infinite dimensional Hilbert space. We compare our general expressions with numerical results for the spin-boson Dicke model in the chaotic energy regime, restricting its unbounded four-dimensional phase space to a classically chaotic energy shell. This effective dimension can be employed to characterize quantum phenomena in infinite dimensional systems, such as localization and scarring.