Advances in classical and quantum wave dynamics on quasiperiodic lattices : a dissertation submitted for the degree of Doctor of Philosophy in Physics, Centre for Theoretical Chemistry and Physics, New Zealand Institute for Advanced Study, Massey University, Albany, New Zealand
Lattices and discrete networks are cornerstones of a number of scientific subjects. In condensed
matter, optical lattices allowed the experimental realization of several theoretically
predicted phenomena. Indeed, these structures constitute ideal benchmarks for light and
wave propagation experiments involving interacting particles, such as clouds of ultra-cold
atoms that Bose-Einstein condensate. Moreover, they allow experimental design of particular
lattice topologies, as well as the implementation of several classes of spatial perturbations.
For example, Anderson localization being observed for the first time in atomic
Bose-Einstein condensate experiments and Aubry-André localization discovered with light
propagating through networks of optical waveguide.
This thesis considers different types of lattices in the presence of quasiperiodic modulations,
mainly the celebrated Aubry-André potential. Particular attention will be given to
spectral properties of models, localization features of eigenmodes and the transition from
delocalized (metallic) eigenstates to localized (insulating) ones within the energy spectrum.
We additionally discuss the relation between the model’s properties and the dynamics of
particles hopping along the lattice.
After introducing the linear discrete Schrödinger equation, we first discuss the spectral
properties of the Aubry-André model. We then study the transition between metallic
and insulating regimes of a class of quasiperiodic potentials constructed as an iterative
superposition of periodic potentials with increasing spatial period. Next, we discuss the
Aubry-André perturbation of flat-band topologies, their energy-dependent transition (mobility
edge), which can be expressed in analytical forms in case of specific onsite energy
correlations, highlighting existence of zeroes, singularities and divergences. We then discuss
two cases of driven one-dimensional lattices, namely an Aubry-André chain with a weak
time-space periodic driving and an Anderson chain with a quasiperiodic multi-frequency
driving. We show anaytically and numerically how drivings can lift the respective localization
and generate delocalization by design. Finally we discuss the problem of the possible
generation of correlated metallic states of two interacting particles problem in one dimensional
Aubry-André chains, under a coherent drive of the interaction.