Dynamical properties of the soft-wall elliptical billiard

Abstract

Physical systems such as optical traps and microwave cavities are realistically modeled by billiards with soft walls. In order to investigate the influence of the wall softness on the billiard dynamics, we study numerically a smooth two-dimensional potential well that has the elliptical (hard-wall) billiard as a limiting case. Considering two parameters, the eccentricity of the elliptical equipotential curves and the wall hardness, which defines the steepness of the well, we show that (1) whereas the hard-wall limit is integrable and thus completely regular, the soft wall elliptical billiard exhibits chaos, (2) the chaotic fraction of the phase space depends nonmonotonically on the hardness of the wall, and (3) the effect of the hardness on the dynamics depends strongly on the eccentricity of the billiard. We further show that the limaçon billiard can exhibit enhanced chaos induced by wall softness, which suggests that our findings generalize to quasi-integrable systems.

Effects of scarring on quantum chaos in disordered quantum wells

“…Our results assist to develop the theory of PI quantum scarring further as well as to understand the connection between PI scars and related phenomena such as conventional scarring [6,7] and branched flow [40]. Moreover, for classical billiards it has been shown that soft boundaries can bring chaos [36]. In addition, an external magnetic field can cause chaotic behavior [37].…”

“…Moreover, for classical billiards it has been shown that soft boundaries can bring chaos. 36 In addition, an external magnetic field can cause chaotic behavior. 37 It has been hypothesized 41 that the softness of the wall combined with a magnetic field causes fractal behavior of the magnetocondutance observed in semiconductor quantum cavities.…”

Solitary wave billiards

“…A less studied billiard is the "soft" version, where the boundary of the billiard table Q is defined by a finite potential wall V , with V (x, y) = 0 for (x, y) ∈ int(Q) and V (x, y) ∈ (0, ∞) for (x, y) ∈ ∂Q. In such a setting, for a Sinai (dispersing) billiard [12,17], the existence of a stable island around a periodic orbit which is tangent to the billiard's wall (or near the corner) was found numerically in [22]; see also [23,24] for other examples of such soft billiards.…”