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 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 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 universal properties of mixed-type systems without stickiness in the chaotic regions: (i) Using the Poincare-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 β, 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. (ii) The entropy localization measure A (also the normalized inverse participation ratio) has a bimodal universal distribution, where the narrow peak at small A encompasses the regular eigenstates, theoretically well understood, while the peak at larger A comprises the chaotic eigenstates, and is well described by the beta distribution. (iii) Thus the BRB energy level spacing distribution captures two effects: the divided phase space dictated by the classical Berry-Robnik parameter ρ2 measuring the relative size of the largest chaotic region, in agreement with the Berry-Robnik picture, and the localization of chaotic PH functions characterized by the level repulsion (Brody) parameter β. (iv) Examination of the PH functions shows that they are supported either on the classical invariant tori in the regular islands or on the chaotic sea, where they are only weakly localized. With increasing energy the localization of chaotic states decreases, as the PH functions tend towards uniform spreading over the classical chaotic region, and correspondingly β tends to 1.

Fluctuating Number of Energy Levels in Mixed-Type Lemon Billiards

“…The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between the centers, as introduced by Heller and Tomsovic in 1993 [1,2]. The present study represents a continuation of our recent paper [3] on the classical and quantum ergodic billiard (B = 0.5) with strong stickiness effects, from the family of lemon billiards, as well as on three simple mixed-type lemon billiards with only one regular region, surrounded by a uniform chaotic sea without stickiness regions, namely, with the shape parameters B = 0.42, 0.55, 0.6 [4].…”

“…One can conclude that the two cases B = 0.1953, 0.083 are interesting to verify the Berry-Robnik picture of quantum billiards [24], including the possible quantum localization of chaotic eigenstates, leading to the Berry-Robnik-Brody level spacing distribution and the universal statistical properties of the localization measures [4,9], as there are no significant stickiness effects, based on the results of the analysis of the recurrence time statistics in [2], unlike in the ergodic case, B = 0.5, studied in [3].…”

“…The family of lemon billiards was introduced by Heller and Tomsovic in 1993 [1] and has been studied in a number of papers [16][17][18][19][20], most recently, in [2][3][4]21]. and in the recent studies [3,4] by us.…”

“…The family of lemon billiards was introduced by Heller and Tomsovic in 1993 [1] and has been studied in a number of papers [16][17][18][19][20], most recently, in [2][3][4]21]. and in the recent studies [3,4] by us. The lemon billiard boundary is defined by the intersection of two circles of equal unit radius with the distance between the centers, 2B, being less than the diameters and B ∈ (0, 1), and is given by the following implicit equations in Cartesian coordinates:…”

“…This is certainly due to the complexity of the phase portrait shown in Figure 2, where many structural features are quantum mechanically not yet resolved, even at such high lying energies E = k 2 , starting at k 0 = 2880. This will be studied in more detail in the forthcoming paper by us [9], where the Poincaré-Husimi functions in the phase space are analyzed in a similar way as in [4]. The full circles represent the value of P(S = 0).…”