Self-organised Limit-Cycles, Chaos and Phase-Slippage with a Superfluid inside an Optical Resonator

We study dynamical phases of a driven Bose-Einstein condensate coupled to the light field of a high-Q optical cavity. For high field seeking atoms at red detuning the system is known to show a transition from a spatially homogeneous steady-state to a self-organized regular lattice exhibiting super-radiant scattering into the cavity. For blue atom pump detuning the particles are repelled from the maxima of the light-induced optical potential suppressing scattering. We show that this generates a new dynamical instability of the self-ordered phase, leading to the appearance of self-ordered stable limit-cycles characterized by large amplitude self-sustained oscillations of both the condensate density and cavity field. The limit-cycles evolve into chaotic behavior by period doubling. Large amplitude oscillations of the condensate are accompanied by phase-slippage through soliton nucleation at a rate which increases by orders of magnitude in the chaotic regime. Different from a superfluid in a closed setup, this driven dissipative superfluid is not destroyed by the proliferation of solitons since kinetic energy is removed through cavity losses.