Quantum manifestations of chaotic scattering in the presence of KAM tori

Abstract

We investigate the semiclassical scattering amplitude for systems, where the classical dynamics is non-hyperbolic, i.e. where islands of KAM trajectories exist in an otherwise chaotic phase space. With the help of semiclassical calculations for the three-disk billiard in an external magnetic field, in which a hyperbolic-non-hyperbolic transition is observed as a function of the field strength, we show that the "stickiness" of the KAM tori leads to a much slower decrease of the survival probability, as compared with the hyperbolic case. This is reflected by a much narrower shape of the energy correlation function. However, we also find that the algebraic asymptotic decay of the survival probability in the non-hyperbolic case is not important for the quantum fluctuations. (2) The conclusion in ref. [9] might have been caused by erroneously identifying the survival probability N II (E, t) with P II (E, t).

Ray-chaotic footprints in deterministic wave dynamics: a test model with coupled Floquet-type and ducted-type mode characteristics

“…The semi-log scale in the plot highlights the departure in the tail from linear exponential behavior. The power-law behavior of the tail is evidenced in the log-log scale inset plot, and was estimated numerically (via curve fitting) to be , which is consistent with observations in the technical literature (see, e.g., [38] and the references therein).…”

“…This behavior can be explained by recalling that regular dynamics here is typically associated with rapidly escaping rays whose dwell times cluster around certain values. From the technical literature (see, e.g., [38] and the references therein), the dwell-time distribution in these mixed cases is expected to exhibit a power-law tail, attributed here to long-trapped marginally-stable trajectories. Such a tail, however, was not clearly observed in the above simulations, which are based on sets of 10 -10 trajectory realizations (compatible with our current computational capabilities).…”

“…As typical in mode-matching-type algorithms, the truncation of the Floquet expansions (15) and (16) was pursued by inspection, via monitoring the truncation-induced mismatch error in the (most critical) boundary conditions (23c) within the unit cell. As a rule of thumb, we found that truncating (15) and (16) at (38) and utilizing the pragmatic truncation criterion stated above for the the summation in (34) usually yields acceptable accuracy. The physical interpretation of the truncation condition in (38) can be gauged from the modal cutoff condition in (37) and consists of neglecting in the transmitted field expansion the ducted modes with cutoff level at sufficient distance from the free-space interface .…”

“…As a rule of thumb, we found that truncating (15) and (16) at (38) and utilizing the pragmatic truncation criterion stated above for the the summation in (34) usually yields acceptable accuracy. The physical interpretation of the truncation condition in (38) can be gauged from the modal cutoff condition in (37) and consists of neglecting in the transmitted field expansion the ducted modes with cutoff level at sufficient distance from the free-space interface . Using the (large order) asymptotic expansion of the Hankel function [42] in (36), one can show that the neglected modes decay exponentially with (39) Inclusion of such modes, besides not necessarily improving the overall accuracy, can even yield the opposite effect by eventually deteriorating the numerical conditioning of the system [45].…”

Anomalous Magnetotransport in Disordered Structures: Classical Edge-State Percolation

“…The field has gained new momentum by recent investigations on disordered graphene [16][17][18] and other topological insulators [19,20]. For many of these phenomena classical magneto-transport constitutes the basis of a semi-classical description [13,[21][22][23]. For instance, the edge states in quantum Hall systems are the quantum analogue of "skipping orbits", trajectories formed by circular arcs bouncing along the edges of a mesoscopic structure.One of the widely investigated classical models for transport in disordered systems is the Lorentz model [24,25].…”