Identification of quantum scars via phase-space localization measures

Saúl Pilatowsky-Cameo1,2, David Villaseñor1, Miguel A. Bastarrachea-Magnani3, Sergio Lerma-Hernández4, Lea F. Santos5, and Jorge G. Hirsch1

1Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apdo. Postal 70-543, C.P. 04510 CDMX, Mexico
2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, C.P. 09340 CDMX, Mexico
4Facultad de Física, Universidad Veracruzana, Circuito Aguirre Beltrán s/n, C.P. 91000 Xalapa, Veracruz, Mexico
5Department of Physics, Yeshiva University, New York, New York 10016, USA

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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 Rényi 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 Rényi occupations with $\alpha \gt 1$ are highly effective at revealing quantum scars. Furthermore, by analyzing the high moments ($\alpha \gt 1$) 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.

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