Multi-stability in cavity QED with spin-orbit coupled Bose-Einstein condensate

We investigate the steady-state multi-stability in a cavity system containing spin-orbit coupled Bose-Einstein condensate and driven by a strong pump laser. The applied magnetic field splits the Bose-Einstein condensate into pseudo-spin states, which then become momentum sensitive with two counter propagating Raman lasers directly interacting with ultra-cold atoms. After governing the steady-state dynamics for all associated subsystems, we show the emergence of multi-stable behavior of cavity photon number, which is unlike previous investigation on cavity-atom systems. However, this multi-stability can be tuned with associated system parameters. Furthermore, we illustrate the occurrence of mixed-stability behavior for atomic population of the pseudo-spin-up arrow and spin-down arrow states, which appear in so-called bi-unstable form. The collective behavior of these atomic number states interestingly possesses a population transitional phase (or population equilibrium intersection) among both of the spin states, which can be enhanced and controlled by spin-orbit coupling and Zeeman field effects. Additionally, we illustrate the emergence of another equilibrium intersection mediated by the increase in mechanical dissipation rate of the pseudo-spin states. These equilibrium intersections or population transitional phase could be caused by the non-trivial behavior of synthetic spin state mediated by cavity. Our findings are not only crucial for the subject of optical switching but also could provide a foundation for future studies on mechanical aspect of synthetic atomic states with cavity quantum electrodynamics.