Spin?orbit coupling (SOC) links an atom’s spin angular momentum to its orbital angular momentum, leading to various novel physical phenomena. In Bose?Einstein condensate (BEC) systems, SOC modifies the dispersion relation, giving rise to exotic quantum phases in the ground state, which also exhibits rich collective excitation behavior. Floquet engineering is a powerful tool in quantum physics, enabling precise control over system parameters and manipulation of quantum states. Under high-frequency periodic driving, a spin?orbit coupled spinor BEC can be effectively described by a static effective Hamiltonian. In a spin-1 (spin quantum number is 1) system, periodic driving of the quadratic Zeeman field induces unconventional spin-exchange interactions, leading to new stripe phases. In this paper, we investigate the influence of Floquet high-frequency driving on the Rabi frequency by periodically modulating the Raman laser intensity. We explore the ground state phase transitions and collective excitations in this modulated spin-1 BEC system.

To analyze the ground state properties, we employ Floquet theory to derive the system’s effective Hamiltonian. First, we apply a unitary transformation to eliminate the time-dependent terms in the original Hamiltonian. Then, by averaging over one driving period, we obtain a time-independent effective Hamiltonian. To study the collective excitation properties, we use the Bogoliubov?de Gennes (BdG) method. By introducing perturbations to the ground state wave function, we construct the BdG matrix and extract the excitation properties of the system. We further analyze the density response function and static structure factor. In addition, we compute the sound velocity to compare with the ground state behavior.

The obtained effective static Hamiltonian shows that both the SOC strength and the quadratic Zeeman field are modulated by the zero-order Bessel function of the first kind. Notably, the modulation introduces two distinct frequency components and a new spin operator F^y2, which emerge from the periodic driving of F^z2. Unlike quadratic Zeeman field modulation, Rabi frequency modulation does not generate new interaction terms, as rotational symmetry among spin operators in the interaction Hamiltonian remain intact. With the introduction of modulation, the boundaries of quantum phases in the ground state phase diagram shift (Fig. 1). The parameter space of the stripe phase S1 contracts along the Ω0 direction but expands into the positive range of the ε direction. The stripe phase S2 undergoes significant expansion. Given that in the absence of modulation, the stripe phase only appears when ε<0, the application of periodic driving extends the parameter range in which the stripe phase can be realized, particularly with respect to the quadratic Zeeman field. As modulation intensity increases, the plane wave phase contracts, while the zero-momentum phase expands. The effect of modulation on phase transitions is more pronounced when the constant Rabi frequency Ω0 is small (Fig. 2). In the excitation spectrum of the stripe phase, rotons appear when ε>0 (Fig. 3). As modulation intensity increases, the depth of roton minimum in each excitation band gradually decreases and shifts toward the center of the Brillouin zone. The energy gap between two lowest excitation bands widens, making atoms excitation less probable. At specific parameters where Ω0=1ER and ε=-1ER (Fig. 4), when modulation α/ω is weak, the ground state remains in the stripe phase. At the Brillouin zone boundary, both the structure factor and density response function exhibit divergence. Increasing modulation to α/ω=1.3 shifts the ground state to the plane wave phase. In this process, the roton mode in the excitation spectrum gradually vanishes. At roton position, the response function reaches maximum, while at the maxon position, the structure factor peaks. When α/ω=1.6, ground state transitions into the zero-momentum phase, where all excitation spectra exhibit symmetrical structures. Sound velocity is influenced by both modulation intensity and the constant Rabi frequency (Fig. 5). When α/ω is small, velocity remains largely unchanged. However, under strong modulation, velocity can vary significantly, particularly at phase boundaries. The larger the constant Rabi frequency, the more pronounced the velocity changes. Corresponding to the ground state, as modulation intensity changes, sound velocity undergoes phase transitions and exhibits distinct behaviors across different phases.

In this paper, we apply Floquet high-frequency driving on the Rabi frequency in a spin-1 spin?orbit coupled BEC to investigate ground state and collective excitation properties. Through a unitary transformation and averaging over one driving period, we obtain a time-independent effective Hamiltonian, in which both SOC intensity and the quadratic Zeeman field are modulated by the zero-order Bessel function. In addition, due to rotation symmetry, the interaction Hamiltonian remains unaffected by periodic driving. The ground state phase diagram varies with modulation intensity, and as modulation increases, phase transitions occur between the stripe, plane wave, and zero-momentum phases. Notably, the stripe phase extends into the positive region of the quadratic Zeeman field, thus allowing its observation over a broader parameter range in experiments. Through the BdG equation, we further obtain the excitation spectrum, density response function, and static structure factor, which provide key signatures for distinguishing quantum phases. Phase transition can also be characterized by sound velocity, where its continuity at phase boundaries reflects the transition type. In this paper, we demonstrate a novel method for controlling phase transitions through periodic modulation of the Rabi frequency, offering a more flexible approach to investigating the excitations and dynamics of spinor BECs.