Identification of quantum scars via phase-space localization measures

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 α-moments of the Husimi function and is known as the Rényi occupation of order α. 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 spinboson model with an unbounded four-dimensional phase space, we show that the Rényi occupations with large α are highly effective at revealing quantum scars. Furthermore, by analyzing the large moments of the Husimi function, we are able to find the unstable periodic orbits that scar some of the eigenstates of the model.

Quantum chaos in triangular billiards

“…Higher order Rényi occupation measures are more sensitive to larger values of the Husimi function and therefore suitable for detecting highly localized or scarred states, as recently demonstrated in Refs. [58,59] for the Dicke model. Analytical calculation based on the random wave approximation (see [59] and references therein) give the estimate l α max ≈ Γ(1 + α) 1/(1−α) for maximally extended pure states.…”

“…[58,59] for the Dicke model. Analytical calculation based on the random wave approximation (see [59] and references therein) give the estimate l α max ≈ Γ(1 + α) 1/(1−α) for maximally extended pure states. This gives us l max = l 1 max ≈ 0.66 and l 2 max ≈ 0.5.…”