Introduction

Bose−Einstein condensates, which are ensembles of atoms in a single quantum state with long coherence time, offer the possibility of studying quantum mechanics in a completely new regime. Ultracold atoms in optical lattices exhibit formation of Bloch energy bands, tunneling effects, Bloch oscillations,Bragg scattering and Josephson effect. Many of these properties are subject to extensive experimental investigations[1]. The fusion of ultracold atoms and the optical lattices has made significant experimental progress in the past years[2−6] by loading the Bose-condensed atoms in the optical lattices and the optical lattices inside the cavities. Therefore, many solid state phenomenon can be explored in these systems as it is possible for the experimentalists to handle both external and internal degrees of freedom of this system. The Hubbard model[7−8] for bosons and fermions have been studied within the framework of ultracold atoms trapped in optical cavities. Ground-breaking experiments[9−10] have also been performed to realize the Bose−Hubbard model in these systems.

Optical potential with and without the cavity are not the same. It is the presence of ultracold atoms inside the cavity which modifies the resonance frequency of the optical cavity. This in turn modifies the amplitude of the field prevailing inside the cavity responsible for the atomic localization. This kind of periodic potential always develops a band structure in the energy spectrum of the condensate. Thus, if the atom−field coupling inside the cavity is strong, this band structure will also lead to band structure of the field inside the cavity. Then this significantly modifies the Bloch energies, effective mass, Bogoliubov excitations and the superfluid fraction of the BEC[11] inside a cavity. Motivated by such interesting developments in the field of cavity opto-mechanics, the work in this chapter emphasizes on the effects of a movable cavity mirror on superfluid properties, tunneling parameters, the Bloch energy spectrum of a BEC confined in an optical cavity.

Optical Lattice

An optical lattice is basically a set of standing wave pattern buildup when two or more counter-propagated laser beams overlap. These OLs can be created in 1D, 2D or 3D depending upon the number of laser beams used.