Manifestation of Arnol’d Diffusion in Quantum Systems

Abstract

We study an analog of the classical Arnol'd diffusion in a quantum system of two coupled nonlinear oscillators one of which is governed by an external periodic force with two frequencies. In a classical model this very weak diffusion happens in a narrow stochastic layer along the coupling resonance and leads to an increase of the total energy of the system. We show that quantum dynamics of wave packets mimics, up to some extent, global properties of the classical Arnol'd diffusion. This specific diffusion represents a new type of quantum dynamics and may be observed, for example, in 2D semiconductor structures (quantum billiards) perturbed by time-periodic external fields.

Weak chaos in the disordered nonlinear Schrödinger chain: Destruction of Anderson localization by Arnold diffusion

“…The quantum version of Arnold diffusion for a few oscillators was considered in Refs. [75,76]. The minimal number of oscillators giving the Arnold diffusion in each model was taken, so the quantum system was effectively one-dimensional (moreover, of a finite length, since the number of oscillators was finite).…”

Pumped double quantum dot with spin-orbit coupling

“…Since the motion is periodic, here we use the Floquet method [13,19,20] based on the exact calculation at the first period and then transformed into the integer number of periods. Figure 2 demonstrates the interplay between the tunneling and the spin-flip process.…”

Superfluidity in Bose-Hubbard circuits

“…This classification is strictly well-defined for an M =3 ring, as discussed in 19 , because such d = 2 degree of freedom system has a mixed phase-space, where different regions (sea, islands, ponds) are separated by KAM tori. But in the present work we deal with high dimensional chaos (d > 2) for which the distinction between categories ii-iii-iv is blurred, because always in principle there is an escape option via Arnold diffusion 23,[43][44][45] along the web of non-linear resonances. Still this is a very slow process and we have established that a secular approximation is capable of reproducing the essential physics of the superfluidity regime diagram.…”