Optomechanical systems are a research topic that has been proposed in recent years and has attracted the attention of many researchers. In many optomechanical systems, the radiation pressure-induced optomechanical interactions will lead to phonon modes, which in turn affects the optical properties and then results in remarkable quantum interference effects. Therefore, many important breakthroughs have been achieved in optomechanical systems, such as cooling of mechanical resonators, quantum entanglement, optomechanically induced transparency, optical bistability, four-wave mixing, and so on. The radiation pressure-induced breathing mode oscillations of the boundary of the resonator play a key role in the nonlinear phenomena. However, the possible role of rotation of the resonator itself has not been explored. In a rotating device, an additional phase is accumulated for the propagating light, which is called the optical Sagnac effect. A non-reciprocal optical transmission with an isolation of 99.6% is achieved using a rotating optical cavity in a recent experiment. Subsequently, hybrid spinning optomechanical systems have been extensively studied, and several remarkable phenomena have been founded including nonreciprocal photon blockades, nanoparticle sensing, and slow and fast light. However, optical bistability and four-wave mixing have not yet been explored in hybrid spinning optomechanical systems.

In this paper, a hybrid spinning optomechanical system driven by a probe laser and pump laser is built to study optical bistability and four-wave mixing, and the composition of the system is analyzed and the definition of each parameter is explained. The drive light entering the system enters from the left side of the fiber and travels clockwise through the optical cavity. The clockwise and counterclockwise modes of the optical cavity experience Sagnac-Fizeau shifts. According to the obtained Hamiltonian, the Heisenberg equation of motion, factorization, and other methods are used to solve it, and relational expressions describing optical bistability and four-wave mixing can be established. Finally, the influence of additional phonon pumping on the four-wave mixing of the system is discussed, and it is found that a small external force is applied, and the four-wave mixing spectral line of the system changes significantly.

The study shows that different properties of optical bistability and four-wave mixing can be observed in the hybrid spinning optomechanical system under different parameter mechanisms. When the optical cavity rotates clockwise, the pump power required to observe the optical bistability is relatively large with the increase in rotation rate. In the case of large pump drive power, when the positive rotation rate increases, the upper stable branch of the corresponding bistable curve increases. This result is reversed when the optical cavity rotates counterclockwise (Fig. 3). In case of no external force and no rotation, two symmetrical peaks appear in the four-wave mixing spectrum, and the mode splitting phenomenon occurs at the resonance. When the optical cavity rotates clockwise, the peak value of the four-wave mixing spectrum will increase, and the mode splitting phenomenon will disappear. When the optical cavity rotates counterclockwise, the peak value of the four-wave mixing spectrum will decrease, and the mode splitting phenomenon gets obvious (Fig. 4). With the increase of the rotation rate when the optical cavity rotates clockwise, the peak value of the four-wave mixing spectrum will gradually decrease, while the mode splitting phenomenon will emerge. From here, the critical value for distinguishing whether pattern splitting occurs at resonances when the optical cavity rotates clockwise can be determined. With the increase of the rotation rate when the optical cavity rotates counterclockwise, the peak value of the four-wave mixing spectrum will decrease, and the mode splitting phenomenon gets obvious (Fig. 5). In case of a small external force and rotation rate, two symmetrical peaks become asymmetrical. With the increase of the external force when the optical cavity is at a fixed rotation rate, the peak value of the four-wave mixing spectrum will increase significantly, and there is no pattern splitting at the resonance (Fig. 6). However, whether the optical cavity rotates clockwise or counterclockwise, under the same external force, the peak value of the four-wave mixing spectrum will decrease, and the reduced energy is used for pattern splitting at the resonance (Fig. 7).

In this paper, based on the optomechanical systems, a hybrid spinning optomechanical system is proposed. By controlling the rotation speed and direction of the optomechanical cavity, the frequency shift induced by the rotation and the optical bistable behavior can be controlled effectively. The results indicate that the rotating direction and rotation rate of the optomechanical cavity affect the four-wave mixing intensity of the system. The decrease or increase of the four-wave mixing intensity will enhance or suppress the mode splitting phenomenon in the resonance region of the system. The external force will destroy the symmetry of the four-wave mixing spectrum. Moreover, the increase of external force will significantly increase the intensity of the four-wave mixing at a fixed rotation rate. However, under the same external force, the increase of rotation rate will reduce the four-wave mixing intensity of the system. Although the optical bistability and four-wave mixing phenomena have been studied in some optical systems, these two phenomena have not yet been analyzed in spinning optomechanical systems, and the study of nonlinear optical properties of hybrid spinning optomechanical systems will have potential applications in quantum information networks.