Atomic clocks based upon coherent population trapping (CPT) offer advantages of low power consumption and compact size^[1–3]. Cold-atom CPT clocks^[4–9] hold the promise of improved accuracy and long-term stability^[10] compared to buffer-gas-filled vapor-cell CPT clocks, which are limited by shifts dependent on pressure and temperature^[11,12]. In cold-atom CPT clocks, a magnetic field on the order of 100 mG is typically required to split the Zeeman levels and establish a quantization axis for laser radiation. While these clocks employ the magneto-insensitive transition (mF=0↔mF=0), the residual second-order Zeeman shift^[13] can still impact long-term stability^[6,14]. For example, in a clock utilizing the |F=1,mF=0⟩↔|F=2,mF=0⟩ transition of Rb87 atoms, a relative fluctuation of 10−4 in the 100 mG magnetic field would limit the long-term stability to the 2×10−13 level. Furthermore, since the atoms are interrogated by CPT light while in free fall, it is crucial to maintain a stable and spatially homogeneous magnetic field. Therefore, ensuring a stable and uniform magnetic field is essential for cold-atom CPT clocks to realize their full potential in terms of long-term stability.

In the majority of setups that utilize magnetic coils to generate the required magnetic field, it is crucial to recognize that fluctuations in coil current are the main contributors to the observed field noise^[15–19]. While excellent power supplies are commercially available, many laboratories rely on homemade low-noise current sources^[20–24] that can be tailored to meet the specific requirements of their setups. In the case of a cold-atom CPT clock employing a pulsed Ramsey-like interrogation scheme^[25], the preparation of cold atoms involves sub-Doppler cooling, necessitating compensation of residual magnetic fields to achieve lower temperatures. However, during the CPT–Ramsey interference process, an additional bias magnetic field is introduced, requiring the current driving the magnetic coils to change abruptly during the clock cycle. As a result, precise control of a static magnetic field driven by low-noise current, while also accommodating the transitions during the clock cycle, is indispensable. Despite the outstanding magnetic field stability achieved in previous studies^[16,20], the rapid and flexible adjustment of the coil’s magnetic field strength needs to be enhanced for cold-atom CPT clocks. By program-controlling the reference voltage in virtue of a field-programmable gate array (FPGA), the diverse current requirements for the bias coils at different stages of the CPT–Ramsey interference process can be satisfied.

In this Letter, we present a magnetic field stabilization system based on an FPGA and a current feedback loop. Our system effectively addresses the challenges associated with varying magnetic field strengths during different phases of the cold-atom CPT clock. We achieve a noise level of 10−5A·Hz−1/2 and a stability of 10−6, comparable to the results reported in previous experiments^[16,22]. To evaluate the applicability of our system in cold-atom CPT clock experiments, we focus on measuring the second-order Zeeman coefficient for the clock transition in Rb87. Through precise measurements, we demonstrate that the second-order Zeeman shift induced by our magnetic field stabilization system is below 10−14, surpassing the long-term stability of current cold-atom CPT clocks. Our approach offers the advantages of noise suppression and flexible control of the magnetic field, making it suitable not only for cold-atom CPT clocks but also for other setups that require low noise and frequent changes in the magnetic field.

To achieve a stable and spatially homogeneous magnetic field with flexible control, we design a magnetic field stabilization system consisting of a pair of Helmholtz coils, a current feedback loop, and an FPGA-based controller, as illustrated in Fig. 1(a). A pair of Helmholtz coils consisting of two square coils with side lengths of 300 mm positioned 167.6 mm apart is used to generate the necessary spatial homogeneity of the magnetic field. Each coil is wound with 6 turns and 11 layers of copper wire with a 0.25 mm radius. Figure 1(b) shows the simulated results for the magnetic field in this square Helmholtz coil configuration. The inset shows that the magnetic field fluctuations remain within a range of 5 µG in a 30 mm region along the axis of the coil’s center. This clearly indicates the capability of coils to provide a high level of magnetic field homogeneity, thus meeting the stringent requirements for atoms’ free fall of 2 mm in a typical CPT–Ramsey interferometry.

Considering requirements of the hopping magnetic field at different stages of the cold-atom CPT clock, we employ an FPGA-controlled digital-to-analog converter (DAC) as the reference voltage. The DAC referenced to a precision DC voltage source with precision of 1 ppm is capable of generating analog voltages ranging from −10V to 10 V via a resolution up to 20 bits and a precision up to 1 ppm. The FPGA communicates with the DAC through the serial peripheral interface (SPI) to output the preset reference voltage UDAC. The FPGA programming in our setup is facilitated using Altera’s Quartus development system. We programmed three modules in Verilog (an asynchronous serial communication module for receiving and decoding commands from the computer), a timing control module that orchestrates the sequence and timing of voltage changes, and an SPI communication module dedicated to operating the DAC. Consequently, these modules allow the DAC to output a series of voltage values in a cyclic manner. It provides flexibility in controlling the DAC output with a swift response time of 20 µs.

The stable magnetic field is realized by a current feedback loop that includes a sampling resistor, an amplifier, a proportional-integral (PI) controller, and an n-channel MOSFET. We use a 0.5-Ω four-wire sense resistor (VPR221S) with a temperature coefficient of resistance (TCR) of ±2ppm/°C and a power rating of 8 W as the sampling resistor. To improve resolution, the magnetic field below 1 mG, corresponding to a coil current variation of approximately 0.3 mA, a low noise instrumentation amplifier (AD8429) with both high common-mode rejection ratio and low noise level is employed to amplify the sampling voltage. By placing a 665-Ω resistor across the gain-setting terminals, a gain factor of 10 is achieved. The precise reference voltage from the DAC and the amplified sampling voltage are fed into the PI controller for a differential computation and filtered by a low pass filter. The error signal is adjusted through a variable gain amplifier with a gain range from −20dB to 40 dB. Then, the PI computation is carried out. The parameters of the PI controller are meticulously fine-tuned to ensure optimal feedback response and stabilization of the bias field. Figure 1(c) shows the response time of the amplified sampling voltage when the current jumps from 100 to 300 mA. Response time begins with the trigger’s rising edge and concludes once the signal voltage achieves 90% of its total change. In the absence of a load coil, the rising time is 60 µs, while the falling time is 40 µs. When the bias coil is connected, the rising time increases to 0.88 ms, and the falling time until stabilization extends to 0.4 ms. This deceleration in response time with the bias coil is attributed to the effects of the inductive load. The n-channel MOSFETs receive signals from the PI controller to regulate and control the current flowing through a pair of Helmholtz coils. It is common practice in practical applications to parallelize two identical MOSFETs to address the issue of heat dissipation. Through this feedback mechanism, we aim to enhance the stability of the coil current.

To characterize the performance of the coil current, we have done two typical tests of the noise level and the current stability. We measure the error signals that represent the difference between the amplified sampling voltage and the DAC output voltage. The performance of the current noise level with and without the feedback loop is shown in Fig. 2. When the feedback loop is locked, the noise in the loop is effectively suppressed to the level of 10−5A·Hz−1/2 below the nominal 10 kHz control bandwidth. For the cold-atom CPT clock, we also focus on the stability of the coil current, since the long-term stability of the magnetic field directly affects the long-term stability of cold-atom clocks. To evaluate the long-term stability of the current, we measure the voltage across a test resistor identical to the one sampled. Employing a multimeter with a resolution of 6.5 digits, we can record the measured voltages at an average interval of 4 s, as Icoil=Utest/Rtest. The stability of the current calculated as the fractional Allan deviation is shown in Fig. 3, which reaches the level of 1×10−6 at 30 s and exhibits superior long-term stability in the locked state compared to the unlocked state. The long-term stability deteriorates to 5×10−6 for averaging times exceeding 1000 s. This degradation is primarily due to the long-term stability of the reference voltage from the DAC (see Fig. 3). When the averaging time reaches 2000 s, the stability of the reference voltage, manifesting at 3×10−6, significantly impacts the current stability in the locked state. In addition, a temperature change of 1°C would result in a relative change in resistance of 2×10−6 with a TCR of 2 ppm/°C.

In order to validate the usefulness of the magnetic field stabilization system, we employ this system to estimate the second-order Zeeman shift in a cold-atom CPT clock. The clock is operated in the pulsed Ramsey-like interrogation scheme, where the free-fall ensemble of Rb87 atoms is coupled by pulsed Raman light. The light contains two CPT frequency components whose frequency difference equals the hyperfine splitting of the ground states. We prepare the ensemble of Rb87 atoms in a three-dimensional magneto-optical trap (3D-MOT) followed by a 5-ms molasses, while simultaneously employing three pairs of Helmholtz coils to cancel the residual magnetic field. One pair of coils generates a magnetic field parallel to the light beam, which also serves as the bias magnetic field. The value of the bias magnetic field is controlled by the FPGA as UDAC. The details of our experimental apparatus can be found in Refs. [26,27].

We calibrate the absolute value of the magnetic field experienced by the atomic ensemble by measuring the first-order Zeeman splitting. For Rb87 atoms, there are three CPT resonances, including the magneto-insensitive transition (|F=1,mF=0⟩↔|F=2,mF=0⟩) and magneto-sensitive transitions (|F=1,mF=±1⟩↔|F=2,mF=±1⟩) in the presence of the bias magnetic field. The splitting of the CPT resonances caused by the Zeeman effect can be approximated to Δ=2Δz=1.402MHz/G, where Δz≡gFµBBz. Hence, we utilize the splitting to measure the bias magnetic field. Similar to the locking of an atomic clock, we measure the splitting Δ via locking the local oscillator to the magneto-sensitive transitions. The magnitude of the bias magnetic field is calculated from the relationship of Bz=Δ/(2gFµB). We increase the applied voltage and measure the coil current from the test resistor; the calibrated bias magnetic field depending on the Icoil is presented in Fig. 4(a). The magnetic field generated by our system is a highly linear function of the coil current Icoil. It should be noted that the direction of the bias magnetic field, in the laboratory coordinate system, is parallel to the geomagnetic field. Therefore, as we gradually increased the strength of the bias magnetic field, the observed CPT resonance transitioned from left-handed circular polarization to right-handed circular polarization. This change indicates a reversal in the direction of the bias magnetic field experienced by the atoms.

We estimate a second-order Zeeman shift based on the calibrated values of the magnetic field. The magneto-insensitive CPT resonance is insensitive to the first-order Zeeman effect, so the second-order Zeeman shift is dominant. The dependence of the second-order Zeeman shift on the absolute value of the magnetic field is measured by locking the cold-atom CPT clock to a 20-ms CPT–Ramsey spectrum. The frequency offset of the clock is obtained by changing the bias magnetic field every 0.5 h; the results are shown in Fig. 4(b). The error bars are computed as the standard deviation of the data points. By a parabolic fitting, the second-order Zeeman coefficient is 574.21±1.36Hz/G2, which is in good agreement with the previous work^[28].

We estimate the effect of our magnetic field stabilization system on the long-term stability of the cold-atom CPT clock. In our practical experiments, the atomic clock in the magnetic field control system unlocked state was found to lose its lock to the local oscillator after merely 2 to 3 h. Therefore, the fractional Allan deviation for the unlocked state is calculated according to data from several discrete atomic clock lockings. By comparing the fractional Allan deviation of the clock under locked and unlocked coil current conditions [see Fig. 4(c)], we can conclude that the magnetic field stabilization system significantly improves the clock’s long-term frequency stability.

To further characterize the limitation of clock stability from the magnetic field control system, we use an error propagation formula to demonstrate that the magnetic field stability of this system is adequate for atomic clock applications^[29]. We have gotten the linear relationship [Eq. (1)] between the coil current (Icoil) and the bias magnetic field and the parabola relationship [Eq. (2)] between the bias magnetic field and the frequency offset of the clock (fclock), Bz=aIcoil+b,(1)fclock=cBz2+d.(2)

Bz is the magnetic field experienced by the atomic ensemble, which includes the space residual magnetic field and the magnetic field Bcoil generated by the current coils. The relative uncertainty of Bcoil is equivalent to that of Icoil as ΔBcoil/Bcoil=ΔIcoil/Icoil. Then we can determine the contribution of the magnetic field effects from the current stability on the total instability of the clock by calculating the error transfer from Icoil to fclock using the error transfer formula, Δfclock=∂fclock∂BzΔBcoil=2cBzBcoilΔIcoilIcoil=2acIcoil(aIcoil+b)ΔIcoilIcoil.(3)

Equation (3) indicates that higher coil currents have a greater impact on the atomic clock instability. However, the magnetic field in the direction of the main quantization axis is nearly parallel to the Earth’s magnetic field in our laboratory coordinate system. We need significantly high coil currents in the direction of the main quantization axis to counteract the Earth’s magnetic field. Therefore, the contribution of coil current in the locked state to the clock’s instability is just at the level of 10−14, as illustrated in Fig. 4(c). When the magnetic field stabilization system is in the unlocked state, the long-term stability of the atomic clock worsens after reaching 5×10−12, which corresponds with calculated contributions of unlocked coil current instabilities to Δfclock. Therefore, we anticipate that if an additional pair of Helmholtz coils or magnetic shielding is employed to compensate for the Earth’s magnetic field, the influence of current stability on clock instability would be diminished by applying a minor current.

In summary, we have successfully demonstrated a magnetic field stabilization system based on an FPGA for actively stabilizing the applied Helmholtz field in a cold-atom CPT clock. The use of an FPGA allows us to adjust the magnetic field’s strength swiftly and flexibly. This system offers both flexible control of the magnetic field and effective mitigation of magnetic noise using the same coils. To assess the practicality of our magnetic field stabilization system, we applied it to investigate the second-order Zeeman shift in a cold-atom CPT clock using Rb87. Through precise measurements of the second-order Zeeman coefficient, we established that the second-order Zeeman shift induced by our magnetic field stabilization system is below 10−14. This level of precision significantly exceeds the long-term stability observed in current cold-atom CPT clocks. The method we have presented here provides notable advantages in noise reduction and flexible magnetic field control. It is well suited not only for cold-atom CPT clocks but also for other experimental setups requiring low noise and frequent adjustments of magnetic field parameters.