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Cold atoms in cavity provides a new platform for exploring exotic many-bodyphases. Here we explore the dynamics of a two-component condensate coupled to afinesse cavity, in which the Raman coupling is mediated by pumping laser andcavity mode. In this model, the energy scale of cavity mode is several order ofmagnitude bigger than that in the condensate, thus the small fluctuations inthe cavity field may have important consequence in the dynamics of condensate.Beyond the steady-state approximaton, we show the cavity can play two differentroles to this dynamics. In the first case, it imprints a gauge potential to thedynamics of condensate, giving rise to zero and $\pi$ Josephson dynamics.Nevertheless, in the other case, it plays the role of non-reciprocialtransportation between the two hyperfine states, in which the stability of thefixed points are tuned from elliptic to stable spiral for one of the trappedphase and unstable spiral for the other trapped phase, thus the oscillatingdynamics will finally ceased. The transition between these dynamics can becontrolled by the parameters of the cavity field and the driving field. Ourresults demonstrate an novel way to engineer the dynamics of condensate bytuning the stability of the fixed points.

Cavity-mediated oscillating and trapping dynamics in a two-component condensate