Superfluidity and BEC in optical lattices and porous media: a path integral Monte Carlo study

We evaluate the Bose-Einstein condensate density and the superfluid fraction of bosons in a periodic external potential using Path-Integral Monte Carlo (PIMC) methods. The periodic lattice consists of a cubic cell containing a potential well that is replicated along 1D using periodic boundary conditions. The aim is to describe bosons in a 1D optical lattice or helium confined in a periodic porous medium. The One-Body Density Matrix (OBDM) is evaluated and diagonalized numerically to obtain the single boson natural orbitals and the occupation of these orbitals. The condensate fraction is obtained as the fraction of Bosons in the orbital that has the highest occupation. The superfluid density is obtained from the winding number. From the condensate orbital and superfluid fraction we investigate (1) the impact of the periodic external potential on the spatial distribution of the condensate, and (2) the correlation of localizing the condensate into separated parts and the loss of superflow along the lattice. For high density systems, as the well depth increases, the condensate becomes depleted in the wells and confined to the plateaus between successive wells, as in pores between necks in a porous medium. For low density systems, as the well depth increases the BEC is localized at the center of the wells (tight binding) and depleted between the wells. In both cases, the localization of the condensate suppresses superflow leading to a superfluid-insulator cross-over. The impact of the external potential on the temperature dependence of the superfluidity is also investigated. The external potential suppresses the superfluid fraction at all temperatures, with a superfluid fraction significantly less than 1 at low temperature. The addition of an external potential does not, however, significantly reduce the transition temperature.