Center motions of nonoverlapping condensates coupled by long-range dipolar interaction in bilayer and multilayer stacks

We investigate the effect of anisotropic and long-range dipole-dipole interaction (DDI) on the center motions of nonoverlapping Bose-Einstein condensates in bilayer and multilayer stacks. In the bilayer, it is shown analytically that while DDI plays no role in the in-phase modes of center motions of condensates, out-of-phase mode frequency (ω_o) depends crucially on the strength of DDI (a_d). At the small-a_d limit, ω_o²(a_d)-ω_o²(0)∝a_d. In the multilayer stack, transverse modes associated with center motions of coupled condensates are found to be optical-phonon-like. At the long-wavelength limit, phonon velocity is proportional to a_d.