In hybrid quantum systems of micro/nano photonics devices and cold atoms, the fictitious magnetic field generated by the constrained light field for atoms has the advantages such as a high gradient, ease of manipulation, and chirality. Therefore, such systems hold significant applications in atomic manipulation, quantum information, quantum simulation, and precision measurement. The calculation and analysis of fictitious magnetic fields are crucial and indispensable. However, for micro/nano photonics devices with irregular structures, it is impossible to obtain the optical field distribution analytically, making it difficult to directly acquire the fictitious magnetic field. In our study, the finite element method is used to numerically simulate the electric field distributions of the nanofiber waveguide and rectangular waveguide, and the fictitious magnetic field distributions are calculated accordingly. We systematically study the fictitious magnetic field intensity as a function of the size of the two micro/nano waveguides. This work provides theoretical guidance for the experimental study of single-atom traps and internal state manipulation using this micro/nano photonics structure.

We numerically simulate a fictitious field of nanofibers and rectangular waveguides. The material of the waveguides is silica, with a refractive index of 1.4537 (Fig. 1). The optical wavelength of 880 nm is used because the scalar optical frequency shift of the ground state is 0, which indicates that the optical field of the 880 nm laser will not affect the atom trap and only produces a fictitious magnetic field (Fig. 2). The power of the laser is set to 1 mW, and the polarization is in the y direction. The field intensity distributions of the nanofibers and rectangular waveguide are numerically simulated via the finite-difference time-domain method (FDTD, Lumerical solutions) (Figs. 3 and 8). After obtaining the electric field intensity distribution, we calculate the fictitious magnetic field intensity distribution using Eqs. (1)?(8) for the optical frequency shift and the matrix operation method (Figs. 4 and 9).

The fictitious magnetic field obtained through finite element numerical simulation is consistent with the analytical results for the nanofibers, which verifies the feasibility of this method (Fig. 6). To obtain the maximum fictitious magnetic field strength, we optimize the fiber diameter. The diameter of the nanofiber is changed from 0.2 to 1.0 μm. The maximum value of the fictitious magnetic field first increases and then decreases as the nanofiber diameter increases, and it can reach 68.3 G when the nanofiber diameter is 0.475 μm (Fig. 7). To obtain the maximum fictitious magnetic field, we optimize the aspect ratio of the rectangular waveguide. The side lengths of the square rectangular waveguide are changed from 0.1 to 1.0 μm. When the side length is 0.475 μm, the maximum value of the fictitious magnetic field reaches 53.61 G [Fig. 11 (a)]. To determine the maximum of the fictitious magnetic field with a changing aspect ratio, the rectangular waveguide length is kept at 0.475 μm, and the width value is changed from 0.1 to 1.0 μm. The maximum value of the fictitious magnetic field of the rectangular waveguide first increases and then decreases as the width increases. When the aspect ratio is 1 (a square waveguide), the maximum value of the fictitious magnetic field reaches 53.61 G [Fig. 11(b)].

We verify the feasibility of finite element numerical simulations of fictitious magnetic fields and provide a method for calculating the fictitious magnetic field of complex micro/nano structures. A numerical simulation of the nanofiber waveguide and rectangular waveguide is implemented, and the distributions of the electric field and fictitious magnetic field are obtained. For the nanofibers, the maximum value of the fictitious magnetic field first increases and then decreases as the nanofiber diameter increases, and it can reach 68.3 G when the nanofiber diameter is 0.475 μm. For the square waveguide, the maximum value of the fictitious magnetic field first increases and then decreases as the side length increases. When the side length is 0.475 μm, the fictitious magnetic field reaches a maximum of 53.61 G. When it is a square waveguide, the maximum value of the fictitious magnetic field reaches 53.61 G. The above simulation results provide a theoretical basis for determining the relative size of the fictitious magnetic field when the rectangular waveguide is used to capture atoms.