Spin dipole oscillation and relaxation of coherently coupled Bose-Einstein condensates

We study the static and the dynamic response of coherently coupled two component Bose-Einstein condensates due to a spin-dipole perturbation. The static dipole susceptibility is determined and it is shown to be a key quantity to identify the second order ferromagnetic transition occurring at large inter-species interaction. The dynamics, which is obtained by quenching the spin-dipole perturbation, is very much affected by the system being paramagnetic or ferromagnetic and by the correlation between the motional and the internal degrees of freedom. In the paramagnetic phase the gas exhibits well defined out-of-phase dipole oscillations, whose frequency can be related to the susceptibility of the system using a sum rule approach. In particular in the interaction SU (2) symmetric case, i.e., all the two-body interactions are the same, the external dipole oscillation coincides with the internal Rabi flipping frequency. In the ferromagnetic case, where linear response theory in not applicable, the system show highly non linear dynamics. In particular we observe phenomena related to ground state selection: the gas, initially trapped in a domain wall configuration, reaches a final state corresponding to the magnetic ground state plus small density ripples. Interestingly the time during which the gas is unable to escape from its initial configuration is found to be proportional to the square root of the wall surface tension.