Classical and quantum mixed-type lemon billiards without stickiness

Abstract

The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance $2B$ between their centers, as introduced by Heller and Tomsovic in Phys. Today {\bf 46} 38 (1993). This paper is a continuation of our recent paper on classical and quantum ergodic lemon billiard ($B=0.5$) with strong stickiness effects published in Phys. Rev. E {\bf 103} 012204 (2021). Here we study the classical and quantum lemon billiards, for the cases $B=0.42,0.55,0.6$, which are mixed-type billiards without stickiness regions and thus serve as ideal examples of systems with simple divided phase space. The classical phase portraits show the structure of one large chaotic sea with uniform chaoticity (no stickiness regions) surrounding a large regular island with almost no further substructure, being entirely covered by invariant tori. The boundary between the chaotic sea and the regular island is smooth, except for a few points. The classical transport time is estimated to be very short (just a few collisions), therefore the localization of the chaotic eigenstates is rather weak. The quantum states are characterized by the following {\em universal} properties of mixed-type systems without stickiness in the chaotic regions:(i) Using the Poincaré-Husimi (PH) functions the eigenstates are separated to the regular ones and chaotic ones. The regular eigenenergies obey the Poissonian statistics, while the chaotic ones exhibit the Brody distribution with various values of the level repulsion exponent $\beta$, its value depending on the strength of the localization of the chaotic eigenstates. Consequently, the total spectrum is well described by the Berry-Robnik-Brody (BRB) distribution.

arXiv comment13 pages, 22 figures

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Cited by in corpus (5)

• Statistics of phase space localization measures and quantum chaos in the kicked top model
• Phenomenology of quantum eigenstates in mixed-type systems: lemon billiards with complex phase space structure
• Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space
• Further results on the power-law decay of the fraction of the mixed eigenstates in kicked-top model with mixed-type classical phase space
• Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short Correlations in Singlets and Doublets