Bloch-Siegert shift under a bias magnetic field along arbitrary direction

• Propose a method to extract Bloch-Siegert shift by numerical solution. • Observe the Bloch-Siegert shift under different conditions. • First obtain the model of Bloch-Siegert shift under any bias magnetic field. In order to reduce the measuring error of magnetic resonance frequency caused by Bloch-Siegert shift, and improve the measurement accuracy of optical pumped alkali-metal magnetometers, the Bloch-Siegert shift is studied under a bias magnetic field in any direction. In this condition, the evolution equation of atomic spin is deduced. Based on the numerical solution of this equation, a method for extracting the Bloch-Siegert shift is proposed, which avoids the difficulty in achieving the analytical solution of the evolution equation. The numerical and analytical solutions of the evolution equation under special conditions are compared, verifying the rationality and necessity of using the numerical solution instead of approximate analytical solution to study the Bloch-Siegert shift under the bias magnetic field in any direction. By studying the Bloch-Siegert shift under different conditions, the typical conditions are selected and the Bloch-Siegert shift is numerically calculated when the bias magnetic field is in different directions. Through the analysis and verification of the calculation results, a theoretical model of the Bloch-Siegert shift under the bias magnetic field in any direction is obtained. This model is of great significance for correcting and reducing the measurement error of optical pumped alkali-metal magnetometers.