Optomechanical coupling in a one-dimensional optical lattice

In a recent paper [J. K. Asboth et al., Phys. Rev. Lett. 98, 203008 (2007)] we have shown that traveling density wavelike collective oscillations can arise in an asymmetrically pumped optical lattice, and by increasing the lattice size or pump asymmetry, these waves can destabilize the structure even in the overdamped limit. The long-range interaction giving rise to collective motion stems from the back-action of the atoms on the field creating the lattice. In this paper we present a detailed description of these phenomena. We derive the force on a disk-shaped cloud of trapped particles including the back-action on the trapping light, and analyze its relation to the standard perturbative approach giving the ``dipole force'' and ``radiation pressure.'' We calculate the self-consistent lattice constant for both red and blue detuned lattices and find that it decreases---by the same amount in the two cases---as the pump asymmetry is increased. We present the detailed derivation of the lattice vibration eigenmodes using the transfer matrix method, which reveals that the instability is enhanced resonantly at certain settings of the asymmetry.