The adiabatic and non-adiabatic behavior of a particle in optical lattices
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The cold atom experiments provide a clean and controlled environment for realizing many body systems. Recent realizations of artificial gauge fields and adjustable optical lattices paved the way for the study of effectively charged particles with neutral atoms in various lattice and continuum systems. Moreover, it is possible to precisely control the external system parameters, i.e. the artificial gauge fields much faster or slower than the time scales associated with atomic motion in the lattice. It still needs further analysis to fully understand how the adiabatic and non-adiabatic changes affect the stationary and dynamical behavior of the system. We first investigate the effect of the adiabatic changes in the artificial gauge fields, and focus on the famous problem: A charged particle in a periodic potential under magnetic field. This simple system leads a complicated and involved selfsimilar energy spectrum, the Hofstadter butterfly. The whole structure of this energy spectrum is determined by the lattice geometry as well as the external field. In this regard, we consider all possible Bravais lattices in two dimensions and investigate the structure of the Hofstadter butterfly as the different point symmetry groups of the lattices are adiabatically deformed from one into another. We find that each 2D Bravais lattice is uniquely mapped to a fractal energy spectrum and it is possible to understand the interplay between the point symmetry groups and the energy spectrum. This beautiful spectrum, in addition, consists of infinitely many topologically distinct regions as a function of magnetic flux and gap number. The topological character of energy bands are determined through their Chern numbers. We calculate the Chern numbers of the major gaps and Chern number transfer between bands during the topological transitions. In the second part, we investigate the dramatic effect of the non-adiabatic changes in the artificial gauge fields. In a synthetic lattice, the precise control over the hopping matrix elements makes it possible to change this artificial magnetic field non-adiabatically even in the quench limit. We consider such a magneticflux quench scenario in synthetic dimensions. Sudden changes have not been considered for real magnetic fields as such changes in a conducting system would result in large induced currents. Hence we first study the difference between a time varying real magnetic field and an artificial magnetic field using a minimal six-site model which leads to gauge dependent results. This model proves the relation between the gauge dependant dynamics and the absence of scalar potential terms connecting different gauge potentials. In this context, we secondly search for clear indication of the gauge dependent dynamics through magnetic flux quenches of wave packets in two- and three-leg synthetic ladders. We show that the choice of gauge potentials have tremendous effect on the post-quench dynamics of wave packets. Even trivially distinct two vector potentials by an additive constant can produce observable effects, we investigate the effects on the Landau levels and the Laughlin wave function for a filling factor ν = 1/q. We also show that edge solutions in a wide synthetic ladder are protected under a flux quench only if there is another edge state solution in the quenched Hamiltonian.