Stochastic dynamics in periodic potentials

TLDR: In this paper, the authors describe the dynamics of both electrons and atoms in periodic potentials and explore how such potentials can be used to realize a new type of quantum chaos in which the effective classical Hamiltonian originates from the intrinsically quantum nature of energy bands.

Abstract: This thesis describes the dynamics of both electrons and atoms in periodic potentials. In particular, it explores how such potentials can be used to realise a new type of quantum chaos in which the effective classical Hamiltonian originates from the intrinsically quantum nature of energy bands. Firstly, this study examines electron dynamics in a superlattice with an applied voltage and a tilted magnetic field. This system displays a rare type of chaos known as non-KAM (Kolmogorov-Arnold-Moser) chaos, which switches on abruptly when an applied perturbation reaches certain critical values. The onset of chaos in the system leads to the formation of complex patterns in phase space known as stochastic webs. The electron behaviour under these conditions is analysed both semiclassically and quantum mechanically, and the results compared to experimental studies. We show that the presence of stochastic webs strongly enhances electron transport. We calculate Wigner functions of the electron wavefunction at various times and show that, when compared to the Poincare sections, evidence of stochastic web formation is observed in the quantum mechanical phase space. Two designs of superlattice are studied and we show, in a full quantum mechanical analysis, that the design of the superlattice has a pronounced effect on the probability of inter-miniband tunnelling and hence the calculated and measured transport characteristics. Secondly, we explore the dynamics of an ultra-cold sodium atom falling through an optical lattice whilst confined in a harmonic gutter potential that is tilted at an angle to the lattice axis. We show this system is analogous to the case of an electron in a superlattice, and that the atomic dynamics show similar enhanced transport properties for certain trapping frequencies. We also find that in a full quantum mechanical calculation, the atomic wavepacket tends to fragment as the angle at which the gutter potential is tilted is increased. Finally, we examine the dynamics of a Bose-Einstein condensate falling through an optical lattice whilst confined in a harmonic gutter potential. We vary the strength of the interatomic interaction parameter to investigate the role of interactions in the system and find that, even for small tilt angles, the condensate wavefunction fragments. For large interaction parameters combined with large tilt angles, the wavefunction explodes catastrophically.