  A weakly interacting, spin-orbit coupled, two-component, ultracold Bose gas
bound to a Bravais lattice is studied. Motivated by recent experimental
advances in the field of synthetically spin-orbit coupled, ultracold, neutral
atomic gases showing Bose-Einstein condensation, an analytic framework with
which to describe such systems in the superfluid regime is presented. This is
applied to a Rashba spin-orbit-coupled Bose gas in a two-dimensional optical
lattice. The exotic nature of Bose-Einstein condensation in the presence of
spin-orbit coupling is an interesting study by itself. Additionally, when the
optical lattice is introduced, the system provides a highly controllable
experimental testing ground for numerous condensed matter physics phenomena.
Five phases of the system are considered, and their excitation spectra,
critical superfluid velocities and free energies are found. In obtaining the
free energy, the effects of terms in the Hamiltonian that are linear in
excitation operators are included, and such terms have not been studied
previously in this context. Minimization of the free energy at zero temperature
is used to confirm the phase diagrams reported in the literature, where it has
usually been obtained by neglecting the effect of excitations. The plane and
stripe wave phases in the phase diagram are bosonic analogues of
Fulde-Ferrell-Larkin-Ovchinnikov states in superconductors involving nonzero
condensate momenta.
