Fig. 1. (a) The structure of the magnetic field stabilization system. The LabVIEW program sends a sequence of ordered voltage values to the FPGA via a universal asynchronous receiver/transmitter (UART). The FPGA then communicates with the DAC through SPI, commanding it to output the preset voltages in an orderly fashion. The DAC is capable of transitioning to the voltage output of the next sequence number within 20 µs after the rising edge of a trigger signal received by the FPGA. Upon receiving a reset pulse, the FPGA resets the order of the voltage values, achieving a cyclic output of static voltages. The bias coil is a pair of Helmholtz coils, used to generate the bias magnetic field B at atoms. (b) The simulated performance of the Helmholtz coils in terms of producing a homogeneous magnetic field; (c) response time of the servo loop. The trigger (blue line) represents a pulse digital signal received by the FPGA. The yellow and gray lines, respectively, correspond to the amplified sampling voltages when the bias coil is under load and without any load.

Fig. 2. The noise spectral density (NSD) of the feedback loop. The red curve is for the PI controller unlocked, and the blue curve is for the locked.

Fig. 3. The stability of the steady-state current and the reference voltage from the DAC at a set point of 0.74 V. It presents the fractional Allan deviation for the current when the PI controller is in the locked (red dots) and unlocked (blue dots) states, as well as for the reference voltage from the DAC (black dots).

Fig. 4. (a) Measurement of the magnetic field produced by our system as a function of coil current using magneto-sensitive transitions. The measured magnetic fields are shown with markers and the linear fit with a blue line. The lower inset displays the fitting residuals for individual measurements. The error bars represent the standard deviation of each set of magnetic field measurement data. (b) Quadratic Zeeman frequency shift as a function of magnetic field strength; (c) contribution to the instability of the clock from the coil current in locked state (blue triangles) and unlocked state (red squares) at U_DAC = 0.74 V. Measurement of the frequency stability was conducted by operating the clock at U_DAC = 0.74 V, with the magnetic field stabilization system in the locked state (black dots) and unlocked state (red dots).