Torus Bifurcation of a Dissipative Time Crystal

Using a quantum gas setup consisting of a Bose-Einstein condensate strongly coupled to a high-finesse optical cavity by a transverse pump laser, we experimentally observe an instability of a dissipative continuous time crystal toward a time crystalline state exhibiting two prominent oscillation frequencies. Applying a mean-field approximation model and a Floquet analysis, we theoretically confirm that this transition is a manifestation in a many-body system of a torus bifurcation between a limit cycle (LC) and a limit torus (LT). We theoretically illustrate the LC and LT attractors using the minimal model and experimentally reconstruct them using Takens' embedding theorem applied to the nondestructively measured intracavity photon dynamics.