| Summary: | Ultracold atoms have emerged as an indispensable setting to study quantum many-body systems. Recent experimental and theoretical work has explored the curious phases and novel properties of Bose-Einstein Condensate with optical lattices, and Bose-Einstein Condensate with light-induced artificial spin-orbit coupling. In this thesis, we report our research on these two types of boson systems.
In the first topic, in contrast with calculations of bosons in optical lattices that focus on the tight-binding regime, we note that the single-particle states of bosons in a periodic potential generally satisfy the Mathieu equation, and have developed a formalism for studying bosons in an optical lattice using the Mathieu equation. Moreover, based on this formalism, we have proposed a self-consistent scheme for describing interacting bosons in an optical lattice using Hartree Fock approximation. We apply this scheme to quantify the effects of inter-atomic interactions on the properties of bosons in an optical lattice, as exhibited in the comparison between observables of non-interacting and interacting systems, such as the superfluid transition temperature and momentum distribution as probed in time-of-flight expansion.
In the second topic, the phases of a Bose-Einstein condensate with light-induced spin-orbit coupling are studied within the mean-field approximation. We obtain the phase diagram at fixed chemical potential and at
fixed density for bosons with spin-orbit coupling, finding a regime
of phase separation and a regime in which the bosons condensed into
a mixed phase. We determine how this phase
diagram evolves as a function of the atom interaction parameters
and as a function of the strength of light-atom coupling. The mixed phase is found to be stable for sufficiently small
light-atom coupling. Specifically, we show that the structure of the phase
diagram at fixed chemical potential suggests an unusual density
dependence for the mixed phase in a harmonic trapping
potential, in which the density of
one spin increases with increasing radius, suggesting a unique experimental signature of this state. The collective Bogoliubov sound mode is shown to also provide a signature of the mixed phase, vanishing as the boundary to the regime of phase separation is approached.
Together, in these two topics we address the need to enhance the understanding of the unconventional physical properties of Bose-Einstein Condensate in a controlled electromagnetic environment (optical lattices, Raman lasers, etc.), and provide predictions for possible experimental findings.
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