A new design method is proposed for a triaxial magnetic field gradient compensation system based on ferromagnetic boundary for atomic sensors. The target field method and the whale optimization algorithm are combined to design magnetic field coils thus transforming the complex analytical solution process into an intelligent optimization problem, while the boundary mirror method is introduced to decouple the coil from the magnetic shield. Magnetic field gradient coils are designed according to the coil design method and combined with the divergence and rotation theorems to form the triaxial gradients compensation system to suppress the magnetic field inhomogeneity in nine directions. The simulation and experimental results are compared with the results before and after compensation by using the box line diagram representing the statistics of the magnetic field deviation, in which the median value is reduced by 65.87% and 85.16%, the minimum by 99.66% and 77.76%, the 25 th quantile by 97.89% and 79.94%, and the 75th quantile by 91.52% and 85.17%, respectively.

The coupling between the transverse magnetic field and the angular velocity in atomic spin gyroscopes (ASGs) makes it difficult to achieve high-precision in situ closed-loop control, which seriously affects the long-term stability of the gyroscopes. This article innovatively proposes a transverse magnetic field closed-loop control method based on high-frequency modulation and transient response. A steady-state response model for the electron spin–nuclear spin coupling ensemble based on high-frequency modulation is established, enabling high-precision in situ measurement of the transverse magnetic field. A method for decoupling the transverse magnetic field and angular velocity based on transient response is proposed. Based on the above, a transverse magnetic field closed-loop control scheme is designed, achieving high-precision in situ closed-loop control in the ASGs for the first time. Experimental results demonstrate that the sensitivities of in situ transverse magnetic field measurements reach 50 and 110 fT/Hz$\mathbf {^{1/2}}$, respectively. The inertial sensitivity of the ASGs is improved by 23.8$\%$ to $\mathbf {6.30 \times {10^{ - 6}}}^{\circ }$/s/Hz$\mathbf {^{1/2}}$, and the bias instability is reduced by 31.3$\%$ to $\mathbf {9.04 \times {10^{-3}}}^{\circ }$/h. This approach is of great significance for improving the anti-interference capability of ASGs and broadening their application range.

In this paper, three dimensional spherical magnetic field coils are proposed and optimized for atomic sensors that require highly uniform magnetic field coils within a limited volume. The design objective of the uniform coil is transformed into a nonlinear optimization model with constraints, which is solved by combining with the particle swarm optimization algorithm. Compared with the traditional cylindrical tri-axial magnetic field coil system whose volume is limited by the aspect ratio of the coil structure, the three dimensional spherical coils not only effectively reduce the volume but also provide consistent magnetic field performance in the three coordinate axis directions. It is of great significance to the miniaturization of atomic sensors. The theoretical calculation shows that the uniformity of spherical coil is improved by about 2 to 3 orders of magnitude compared with the Lee-Whiting coil and the saddle coil. The measurements show that the magnetic field uniformity of our design reached 1.5×10-4 along the z-axis within a range of ±0.25R. The actual magnetic field distribution is basically consistent with the theory. However, due to the limited processing and experimental conditions, the actual magnetic field uniformity is 1 to 2 orders of magnitude lower than the theory.

The coupling between the coil and the cylindrical magnetic shield (CMS) in atomic sensors not only results in the inconsistency of the coil constant with the theory but also deteriorates the uniformity, which affects the sensor performance. To solve the abovementioned deficiencies, a design method of highly uniform field coils inside the CMS based on the coupling model with finite permeability and thickness and particle swarm optimization (PSO) with dynamic inertia weights is proposed for the first time. The coupling model of the coil inside the CMS is established by Green’s function method (GFM). Then, it is combined with the nonlinear optimization method, and the coil configuration is optimized by the dynamic inertia weight PSO (DW-PSO) to obtain the optimal parameters. It not only reduces the calculation error of the model, but also improves the uniformity. Comparing this work with the Lee-Whiting coil inside a CMS, the theoretical uniformity is further improved by two orders of magnitude, and measured uniformity errors in the target region reach $1.404\times {10^{-4}}$ , and $2.317\times {10^{-4}}$ , respectively. The correctness and effectiveness are proved by finite-element simulation and experimental results. The accuracy and uniformity of coils can be considerably improved, which is beneficial to further improve the measurement sensitivity of instruments related to magnetic field uniformity.