Compared with the traditional method of changing spin freedom through external magnetic fields, the spin-orbit coupling, which utilizes the coupling between the spin freedom and the motion freedom of atoms, is a new method for regulating spin. With the continuous realization of artificial spin-orbit coupling in cold atomic systems in experiments, many novel physical phenomena based on spin-orbit coupling have been widely promoted. In addition, since the realization of the superradiant quantum phase transition in experiments in 2010, the system of coupling ultracold atomic gas and cavity quantum electrodynamics has become an ideal platform for exploring novel many-body physics, which has aroused a research boom among theoretical scientists and experimental scientists. This coupling system couples ultracold atoms into a high-precision optical microcavity. Under specific electromagnetic boundary conditions, light interacts with ultracold atoms and induces novel many-body quantum properties. In this coupling system, one can not only explore the complex quantum behavior induced by the long-range interaction among atoms mediated by cavity photons but also understand the collective dynamical properties of cavity photons and ultracold atoms at the single-photon level. At the same time, the optical microcavity has both driving and dissipation, and it is a natural non-equilibrium system, which allows one to study the non-equilibrium steady-state dynamical properties. However, the time-dependent cavity-assisted spin dynamics has not been considered experimentally and theoretically. On the one hand, the time-dependent Schr?dinger equation is difficult to obtain an exact analytical solution mathematically, and on the other hand, the physical process expressed by the time-dependent Schr?dinger equation involves complex energy changes, time evolution, and interaction problems, which makes it difficult to solve. In view of these problems, we proposed a method for realizing the superradiant phase transition with the assistance of an optical microcavity. This method coupled the optical microcavity system with a Bose-Einstein condensate trapped in a harmonic potential that oscillates with time to obtain a new model, which could be used to study the self-organized phase transition and spin dynamics of Bose-Einstein condensates in microcavities and provide a reference for studying other Bose-Einstein condensates based on spin.

We considered the preparation of Bose-Einstein condensates using a magneto-optical trap and the coupling of these Bose-Einstein condensates bound in an oscillatory harmonic potential field with a high-precision optical microcavity, thereby establishing a one-dimensional coupled system where the Bose-Einstein condensates only moved in the x direction. The atoms we considered were those with four internal energy levels, and under conditions of large detuning, the excited states of the Bose-Einstein condensates were removed adiabatically, and the resulting Hamiltonian was then quantized. Through mean-field calculations, we obtained the coupled mean-field equations, which were then specialized for the time-dependent part, thereby transforming the problem with time dependence into a classification discussion without time dependence.

We studied the steady-state properties of matter, obtained the relationship diagram of the order parameter with the coupling strength and explored the influence of the external magnetic field strength and the harmonic potential field vibration strength on the critical point of the superradiant phase transition. The results show that the effective magnetic field mz experienced by the atoms and the vibration strength ξ0 of the harmonic potential well will affect the phase transition. Specifically, the coupling strength corresponding to the critical point of the superradiant phase transition increases monotonically with the increase in mz. When ξ0/1/πmω≤31.5, the coupling strength corresponding to the critical point of the superradiant phase transition decreases with the increase in ξ0. When ξ0/1/πmω>31.5, the coupling strength corresponding to the critical point of the superradiant phase transition increases with the increase in ξ0(Fig. 4). In addition, we also analyzed the non-trivial spin dynamics induced by the interaction between light and atoms and the influence of the vibration strength of the harmonic potential field on the dynamical properties. It was found that when the system does not undergo superradiance, the oscillation of σxt at zero over time is still symmetric but not smooth, and the value of ξ0/1/πmω affects the atomic spin resonance effect (Fig. 5).

In this study, we propose a feasible method for realizing optical microcavity-assisted superradiant phase transition and spin dynamics and explore the superradiant quantum phase transition and non-trivial spin dynamics that oscillate with time. We adopt the mean-field approximation method for the cavity field and the matter field and treat the time-dependent system, so as to obtain the superradiant phase transition of the system and give the complete phase diagram of the phase transition. On this basis, we study the non-trivial spin dynamics of the system by qualitatively analyzing the average value of the Pauli operator. We find that the coupling strength corresponding to the occurrence of the superradiant phase transition increases with the increase in the external magnetic field, and it decreases first and then increases with the increase in the vibration intensity of the external harmonic potential field. The vibration intensity of the harmonic potential field affects the spin dynamics effect of the system, because the vibration intensity of the harmonic potential field changes the coupling strength corresponding to the critical point of the superradiant phase transition, thus resulting in changes in the spin dynamics effect of the system.