Coarse-Grained Finite-Temperature Theory for the Bose Condensate in Optical Lattices

In this paper, we derive a coarse-grained finite-temperature theory for a Bose condensate in a one-dimensional optical lattice, in addition to a confining harmonic trap potential. We start with a two-particle irreducible (2PI) effective action on the Schwinger-Keldysh closed-time contour path. In principle, this action involves all information of equilibrium and non-equilibrium properties of the condensate and noncondensate atoms. In constructing a theory for the condensate and noncondensate in a periodic lattice potential, a difficulty arises from the rapid spatial variation due to a lattice potential, compared to the length scale of the harmonic potential. We employ a coarse-graining procedure to eliminate this rapid variation. By introducing a variational ansatz for the condensate order parameter in an effective action, we derive a coarse-grained effective action, which describes the dynamics on the length scale much longer than a lattice constant. Using the variational principle, coarse-grained equations of motion for condensate variables are obtained. These equations include a dissipative term due to collisions between condensate and noncondensate atoms, as well as noncondensate mean-field. As a result of a coarse-graining procedure, the effects of a lattice potential are incorporated into equations of motion for the condensate by an effective mass, a renormalized coupling constant, and an umklapp scattering process. To illustrate the usefulness of our formalism, we discuss a Landau instability of the condensate moving in optical lattices by using the coarse-grained generalized Gross-Pitaevskii hydrodynamics. We find that the collisional damping rate due to collisions between the condensate and noncondensate atoms changes its sign when the condensate velocity exceeds a renormalized sound velocity, leading to a Landau instability consistent with the Landau criterion. Our results in this work give an insight into the microscopic origin of the Landau instability.