We propose a fast and accurate method for the *in situ* calibration of the transverse coils in miniaturized nuclear magnetic resonance gyroscopes based on the Rabi oscillation of hyperpolarized gaseous ^{129}Xe nuclear spins. In contrast to the conventional method based on the free induction decay initial amplitude of different pulse durations, this method circumvents the error introduced by the frequency detuning of the transverse driving field. We experimentally confirm that the accuracy of the calibration is influenced by the longitudinal relaxation time *T*_{1}, the transverse relaxation time *T*_{2}, and the magnitude of the transverse driving field *B*_{1}. Through the numerical simulation of Bloch equations, we show that the behavior of $My\u2032$ and *M*_{z} can be completely specified by two dimensionless quantities *a* and *b*, and we provide a ready-to-use selection criterion of *b* ≥ 24 for choosing the proper *B*_{1}. By switching the embedded magnetometer into the longitudinal mode, we demonstrate the effectiveness and efficiency of our new calibration method. Finally, we examine the effectiveness of the selection criterion with our new calibration method. This method is expected to improve the authenticity of coil calibration and relaxation measurement in a more efficient way.

## I. INTRODUCTION

Nuclear magnetic resonance gyroscopes^{1–5} (NMRGs) utilize the inherently embedded atomic magnetometer and the Larmor precession of the spin-exchange optically pumped^{6–11} noble gas nuclear spins to sense the inertial rotation. Because of its working principle, the NMRG has the characteristics of high precision, extremely stable scale factor, and high miniaturizability and has been a very active research area in recent years.

The working atoms of a NMRG are encapsulated in a millimeter-scale glass vapor cell and manipulated by a transverse oscillating magnetic field generated by the transverse magnetic field coils. A precise knowledge of the coil constant is of great importance not only for obtaining a better signal-to-noise ratio of the gyroscope but also for characterizing the relaxation time of nuclear spins. A simple and convenient method of calibrating the magnetic field coils is using a fluxgate magnetometer.^{12,13} However, there are several disadvantages related to those approaches, such as the accurate placement and orientation of the sensor and the precision of the fluxgate. A more elegant and precise way of calibrating the magnetic coils is to use the atomic sample itself, which is called the *in situ*^{14–17} calibration. Breschi *et al.*^{18} utilized the Larmor frequency of the alkali-metal electron spins to calibrate the constant of three-axis Helmholtz coil sets. Zhang *et al.*^{19} utilized the Larmor frequency of the noble gas nuclear spins. For a miniaturized NMRG, due to the differences in the system structure, rather than using the Larmor frequency of the electronic or nuclear spins to calibrate the transverse coils, the Rabi frequency of the nuclear spins is preferred, which is a conventional *in situ* calibration method that is wildly used to determine the transverse coil constants and the duration of the 90° and 180° pulses.^{20–24} In the conventional method, as the NMRG is intrinsically a transverse magnetometer, the transverse component of the noble gas nuclear spins is naturally detected and used to calibrate the magnetic coils and determine the pulse duration. However, the accuracy of this method is dramatically influenced by the relaxation rate of the nuclear spins and the magnitude of the transverse driving field. Moreover, the measurement error caused by the frequency detuning of the driving field cannot be avoided. In addition, the whole calibration process is time-consuming.

In this work, we study the dynamics of the noble gas nuclear spins by the numerical simulation of the Bloch equations. We introduce two dimensionless quantities (*a* and *b*) to characterize the dynamic features of the nuclear spins. By calculating the deviation between the real Rabi curve and the $My\u2032(t)$ curve, a graphical selection criterion is obtained to guide the determination of the proper *B*_{1} magnitude. Then, we present a new method of calibrating the coils by using the longitudinal magnetometer. We show that this method can completely circumvent the influence of frequency detuning. Finally, we verify our selection criterion with our new calibration method. The structure of the remainder of the paper is as follows: Sec. II explains the dynamics of the nuclear spins and the principle of coil calibration; Sec. III describes the experimental apparatus and introduces the methodology used to detect the nuclear magnetic moment; Sec. IV presents and discusses the simulation and experimental results; and finally, in Sec. V, we draw our conclusions.

## II. PRINCIPLES

When an atomic vapor cell filled with isotopically enriched ^{87}Rb and ^{129}Xe is subjected to an external magnetic field B_{0} along the *z* axis, both the Rb atoms and the Xe atoms will acquire an extra energy and their atomic energy levels will split into several Zeeman sub-levels. To obtain a relatively large resultant magnetic moment of ^{129}Xe, a circularly polarized beam of Rb D1 line is applied along the *z* axis to polarize Rb atoms. Through Rb–Xe spin-exchange collisions, the electron spins of the polarized ^{87}Rb atoms will be transferred to the nuclear spins of ^{129}Xe atoms. Then, the resultant magnetic moment of ^{129}Xe is established.

After the buildup of ^{129}Xe resultant moment, as the moment is along the *z* direction, to obtain a measurable magnetic resonance signal of ^{129}Xe, a transverse RF field along the *x* axis with an amplitude of 2*B*_{1} and a frequency of *ω* is applied to tilt the resultant moment away from the *z* axis. Under the combined influence of the longitudinal stationary field *B*_{0} and the transverse rotating field *B*_{1}, the dynamics of the resultant moment ** M** can be fully described by the phenomenological Bloch equations,

^{25}

where

*γ* is the gyromagnetic ratio of ^{129}Xe nuclear spins, *T*_{1} and *T*_{2} are the longitudinal and transverse relaxation times of ^{129}Xe, and *M*_{0} is the steady state nuclear spins without the transverse perturbation field, which is generally along the *z* axis. After introducing a frame rotating with a frequency *ω*, the Bloch equation can be simplified to

where Δ = *ω*_{L} − *ω* is the detuning of the oscillating field and *ω*_{L} = *γB*_{0} is the Larmor frequency of ^{129}Xe nuclear spins. From Eq. (3), the steady state solutions of the resultant moment in the rotating frame can be easily obtained by setting the left-hand side of the equations to zero,^{26}

where *ω*_{R} = *γB*_{1}, which will be called the real Rabi oscillation frequency in the following contexts. As the NMRG works in the continuous-wave NMR mode, the maximum transverse signal $My\u2032$ is obtained at Δ = 0 and $\omega R=1/T1T2$. However, in the process of calibrating magnetic coils and measuring *T*_{1} and *T*_{2}, as the pulsed NMR technique is used, a much larger *ω*_{R} is needed.

In the pulsed NMR domain, there are two major methods of studying the transient behavior of the resultant moment: the analytical method and the numerical method. In the analytical method, although the general analytical solution of the Bloch equation was given by Torrey,^{27} it is too unwieldy for practical applications. Instead, with the prerequisite that *ω*_{R} ≫ 1/$T1T2$, the effects of the transverse and longitudinal relaxations are usually neglected during the application of the pulsed field. The schematic of this method is shown in Fig. 1. Viewing from the aforementioned rotating frame, the bias filed *B*_{0} is partially eliminated by a fictitious field *B** = *ω*/*γ* and the resultant moment will rotate around an effective filed $B\u0304$ with an effective Rabi oscillation frequency,

When the frequency detuning Δ equals zero, the effective Rabi frequency Ω will be equal to the real Rabi frequency *ω*_{R}. As the resultant moment rotates with a frequency of Ω, the analytical solutions of equation Eq. (3) are simply given by

In the numerical method, with a given initial state of ** M**(

*t*= 0), the dynamics of the resultant moment can be numerically calculated from the Bloch equation point by point. Although the numerical method is time-consuming compared with the analytical method, the influence of the relaxation effect under different values of

*B*

_{1}can be quantitatively analyzed.

As the NMRG is intrinsically a transverse magnetometer, by varying the duration of the transverse pulsed field, the initial amplitude of the free induction decay (FID) signal (namely, $My\u2032(t)$) can be used to calibrate the transverse coil. Figures 2(a)–2(c) show the schematics of the conventional coil calibration process. When the pulsed field is exactly a 90° pulse, namely, the resultant moment rotates exactly 90° in the rotating frame, the largest FID initial amplitude can be obtained. A common misleading picture of the FID process is also plotted using a dashed red line. From Eq. (6c), we can see that, by switching the optical magnetometer from transverse mode to longitudinal mode, the *M*_{z}(*t*) signal can also be used to calibrate the coil. In Sec. IV B, we will show that this method is more simple and straightforward and can easily circumvent the influence of the frequency detuning.

Theoretically, the transverse field *B*_{1} should be as large as possible, but practically, there are several disadvantages when *B*_{1} is too large: a too large *B*_{1} will lead to an extremely short 90° pulse duration, which will increase the requirement of the current source; the response delay of the filed coil is no longer negligible when the pulse duration is extremely short, which will lead to uncertainty in the calibration of the coil constant; and a large *B*_{1} will be accompanied by a large gradient Δ*B*_{1}, which will also reduce the accuracy of the measurement. Therefore, choosing an adequate *B*_{1} is of great importance both for coil calibrations and for relaxation measurements. To provide a selection criterion for *B*_{1}, a series of numerical simulations of the transient process are carried out and the relaxation-induced deviation between simulation results and equation Eq. (6b) is estimated.

## III. SYSTEM SETUP AND EXPERIMENTAL METHODS

Figure 3 shows the schematic of our experimental system. A cubic atomic vapor cell is located in the center of a four-layer magnetic shield and enclosed in a boron-nitride oven. The inner length of the cell is 3 mm, and the outer length is 4 mm. The cell is filled with isotopically enriched ^{87}Rb, 2 Torr of ^{129}Xe, 8 Torr of ^{131}Xe, and 300 Torr of N_{2}. The boron-nitride oven is heated by a non-magnetic heating film. The temperature of the cell is detected by a thermocouple and controlled by a proportional integral derivative (PID) controller. The working temperature of the cell is about 120 °C, and the temperature fluctuation is below 0.002 °C. A Helmholtz coil is used to generate the longitudinal magnetic field. Two sets of saddle coils are used to generate the transverse magnetic field. The saddle coils are fabricated on a multilayer flexible printed circuit board with a width and height of 12.5 and 25 mm, respectively [see Fig. 3(b)]. A circularly polarized beam of the Rb D1 line is used to pump the Rb atoms. The light power is 1 mW, and the beam diameter is 3 mm. The pump laser frequency is tuned to the *F* = 1 → *F*′ = 1, 2 absorption peak. A linearly polarized probe beam of the Rb D2 line is applied to measure the Rb spin polarization through the Faraday rotation effect. The light power is 0.5 mW, and the beam diameter is 3 mm. The probe laser frequency is tuned 20 GHz to the low frequency side of the *F* = 2 → *F*′ = 1, 2, 3 transitions. The D1 and D2 absolute absorption spectra^{28} of the cell are pre-measured and shown in Figs. 3(c) and 3(d). The probe beam transmitted through the cell is split by a polarizing beam splitter (PBS) and detected by a balanced photodiode amplifier. The differential signal is demodulated at the ^{87}Rb Larmor frequency by a dual-phase lock-in amplifier.

In this study, two working modes of the *in situ* atomic magnetometer are used to detect the dynamics of the ^{129}Xe nuclear spins. Here, we briefly introduce the principle of these two working modes. Detailed descriptions can be found in Ref. 2, 16, and 24. In the transverse mode, a bias magnetic field *B*_{0} of 10 *μ*T is applied along the *z* axis. The Larmor frequency of the ^{87}Rb is about 70 kHz. An oscillating magnetic field of the same frequency is applied along the *z* axis to modulate the Rb Larmor precession. The amplitude of the oscillating field *B*_{ac} is about 1.357 times of *B*_{0} to achieve the maximum sensitivity of the magnetometry. The harmonics of the Larmor frequency can then be detected by the transverse probe light. The differential signal detected by the balanced photodiode detector is sent to a dual-phase lock-in amplifier and demodulated at the ^{87}Rb Larmor frequency. By adjusting the demodulation phase of the demodulator,^{29} the fictitious in-phase component X and quadrature component Y of the demodulated signals can be assigned to respond to the residual magnetic field along the realistic *x* and *y* axes. In the longitudinal mode, a bias magnetic field *B*_{0} of 10 *μ*T is also applied along the *z* axis, while the oscillating magnetic field is applied along the *x* axis. The signal of the balanced photodiode detector is also demodulated at the ^{87}Rb Larmor frequency by a dual-phase lock-in amplifier, while the demodulating phase is used to respond to the magnetic field shift along the *z* direction. Due to the linear region of the phase curve being relatively narrow, the measuring range of the longitudinal magnetometer is quite limited, while it is large enough to measure the magnetic field shift caused by the dynamics of ^{129}Xe nuclear spins.

## IV. RESULTS AND DISCUSSION

### A. Selection criterion for transverse field magnitude

The conventional method utilizes the pulse duration dependence of the FID initial amplitude to calibrate the coil constant and determine the duration of the 90° pulse. Figures 2(d) and 2(e) show the experimental results of this method with two different cells. The fitted results are also plotted. A good agreement is obtained for cell 1, which has a *T*_{1} of 36 s and a *T*_{2} of 34 s [see Fig. 2(d)]. A poor agreement is seen for cell 2, which has a *T*_{1} of 36 s and a *T*_{2} of 7 s [see Fig. 2(e)]. It is obvious that cell 2 requires a larger *B*_{1} to achieve an accurate measurement.

To choose an adequate *B*_{1}, the numerical simulation method is used to investigate the behavior of the resultant moment. In order to simplify the simulation, we shall, for the moment, assume that the frequency detuning Δ is always zero. We will show in Sec. IV B that the frequency detuning can be easily circumvented by using a longitudinal magnetometer. By introducing the dimensionless quantities

the number of the simulation parameters is reduced from 3 (*T*_{1}, *T*_{2}, and *ω*_{R}) to 2 (*a* and *b*). Several typical simulations are carried out and plotted in Fig. 4 to illustrate the influence of parameters *a* and *b* on the behavior of the resultant moment. The simulation parameters are listed in Table I. Each simulation result is presented by a time-domain plot, a 3D diagram in the laboratory frame, and a Y′–Z graph of the rotating frame. In the time-domain plot, *M*_{z}(*t*), $My\u2032(t)$, and *M*_{y}(*t*) are plotted with a black, red, and blue solid line, respectively, while a sinusoidal curve with the frequency of *ω*_{R} is plotted with a solid green line. In the 3D diagram, the trajectory of the resultant nuclear magnetic moment is plotted with a gray line and the last Larmor cycle is marked out with a red line. In the Y′–Z graph, the trajectory of the nuclear spins is plotted with a blue line.

Serial . | No. 1 . | No. 2 . | No. 3 . | No. 4 . | No. 5 . | No. 6 . | No. 7 . |
---|---|---|---|---|---|---|---|

a | 1 | 1 | 1 | 1 | 1 | 0.02 | 1 |

b | 1 | 1 | 1 | 11.8 | 118 | 118 | 1180 |

T_{1} (s) | 0.1 | 1 | 10 | 10 | 10 | 10 | 10 |

T_{2} (s) | 0.1 | 1 | 10 | 10 | 10 | 0.2 | 10 |

ω_{R} (Hz) | 10 | 1 | 0.1 | 1.18 | 11.8 | 11.8 | 118 |

ω_{L} (Hz) | 118 | 118 | 118 | 118 | 118 | 118 | 118 |

Δ (Hz) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Serial . | No. 1 . | No. 2 . | No. 3 . | No. 4 . | No. 5 . | No. 6 . | No. 7 . |
---|---|---|---|---|---|---|---|

a | 1 | 1 | 1 | 1 | 1 | 0.02 | 1 |

b | 1 | 1 | 1 | 11.8 | 118 | 118 | 1180 |

T_{1} (s) | 0.1 | 1 | 10 | 10 | 10 | 10 | 10 |

T_{2} (s) | 0.1 | 1 | 10 | 10 | 10 | 0.2 | 10 |

ω_{R} (Hz) | 10 | 1 | 0.1 | 1.18 | 11.8 | 11.8 | 118 |

ω_{L} (Hz) | 118 | 118 | 118 | 118 | 118 | 118 | 118 |

Δ (Hz) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

From Figs. 4(a)–4(c), we note that although the first three simulations have different values of *T*_{1}, *T*_{2}, and *ω*_{R}, the behavior of the *M*_{z} and $My\u2032$ components is exactly the same because of their identical *a* and *b*. Hence, their Y′–Z graphs are totally identical. From Figs. 4(c) and 4(d), we can see that when *B*_{1} is increased, the dynamic process of the nuclear spins will convert from the overdamped state to the oscillating state. Comparing Fig. 4(d) and Fig. 4(e), we note that when *B*_{1} is large enough, all these three curves (black, red, and green) will become almost identical, except for a certain phase difference between each other. Figures 4(e) and 4(f) show that a cell with shorter *T*_{2} requires a larger *B*_{1} to obtain an accurate measurement. Figure 4(g) shows a classical situation where *B*_{1} equals *B*_{0}. An 8-shaped track is seen in the laboratory frame.

To quantitatively determine how large *B*_{1} should be, a massive number of simulations are performed with *a* ranging from 0 to 1 and *b* ranging from 1 to $a\omega LT1$. Without loss of generality, *T*_{1} is set to a default value of 30 s and *ω*_{L} is set to 118 Hz. The deviation calculation scheme is shown in Fig. 5. The first full period of the real Rabi curve and the $My\u2032(t)$ curve are normalized and the absolute values are taken, and then, the root-mean-square deviation (RMSD) between them is calculated for each simulation result.

The final RMSD results of all simulations are shown in Fig. 6. The yellow stripe is the transition zone between the overdamped state and the oscillating state. The pink lines are the upper limit of *b* with *T*_{1} = 30, 3, and 0.3 s and can be regarded as the contour lines of the same *B*_{1} or *ω*_{R}. The cyan dashed-dotted line is the boundary with a RMSD equal to 10%. For a cell with arbitrary *T*_{1} and *T*_{2}, the range of acceptable *ω*_{R} can be inferred from this figure. Taking *T*_{1} = 3 and *T*_{2} = 2.4 as example, as *a* = 0.8, the effective range of *b* with a RMSD less than 10% is marked out with a white dotted line, which is from 23.1 to 317.9, and the corresponding range of *ω*_{R} is from 9.37 to 118 Hz. Moreover, the corresponding points of the experimental results shown in Figs. 2(d) and 2(e) are also marked out in Fig. 6 with a white (*a* = 0.944; *b* = 45.5) and a black (*a* = 0.167; *b* = 19.1) pentagram, respectively. From this figure, we can intuitively understand why a good result is obtained for cell 1 and a poor result is obtained for cell 2 in the aforementioned experiments. As to the nuclear magnetic resonance of gaseous samples, *T*_{2} is usually not much smaller than *T*_{1}. Hence, for simplicity and convenience, a ready-to-use criterion of *b* ≥ 24, namely, $\omega R\u226524/T1T2$ (black dotted line), can be used.

### B. Calibration of the coil constant

As the conventional method is to measure $My\u2032(t)$ indirectly with a transverse magnetometer, a simpler and efficient way of calibrating the *x*-coil is to measure *M*_{z}(*t*) directly with a longitudinal magnetometer. First, we perform a single FID process with the transverse magnetometer to quickly extract an approximate Larmor frequency of about 127.66 Hz. Then, the magnetometer is switched to the longitudinal mode. As shown in Fig. 7(a), when a near resonance driving field with a frequency of 127.66 Hz is applied along the *x* axis, the Rabi oscillation of ^{129}Xe resultant moment is detected by the longitudinal magnetometer. The oscillation will gradually fade away and stabilize at a certain value. When the driving field is withdrawn, the longitudinal relaxation process will begin at this moment.

The effective oscillation frequency Ω can be directly extracted from the data in the yellow region by Fourier transform [see Fig. 7(b)]. To circumvent the influence of a nonzero detuning Δ, a series of experiments were carried out with artificial detuning. The driving field frequency vs the effective Rabi oscillation frequency is plotted in Fig. 7(c). As the first order Taylor expansion of Eq. (5) around Δ = 0 is

the real Rabi frequency *ω*_{R} and the exact Larmor frequency *ω*_{L} can be obtained by fitting a quadratic curve to the experimental data. After determining the real Rabi frequency *ω*_{R}, with the knowledge of the voltage or current applied on the coil, the coil constant can be easily calculated. In this experiment, the peak voltage of the sine wave applied on the transverse coil is 4 V, the corresponding real Rabi frequency is 6.8783 ± 0.0005 Hz, and the coil constant is calculated to be 292.03 ± 0.02 *μ*T/mV.

To verify the effectiveness of our selection criterion, a sequence of experiments were performed for different values of *b* or *ω*_{R} chosen from Fig. 6. Cell 1 is used to carry out these experiments. The selected points are marked out with white points and connected with a solid red line. As all these values of *ω*_{R} are chosen from the acceptance region, satisfactory experimental results could be expected. The experimental and fit results are shown in Fig. 7(d). Note that a perfect linearity is obtained from the fit and the resulting intercept of the fitted curve is almost zero.

## V. CONCLUSION

In conclusion, a fast and accurate method for the *in situ* calibration of the transverse coils in miniaturized NMRGs based on the Rabi oscillation of nuclear spins is proposed. We experimentally show that choosing a proper *B*_{1} is of great significance for precision coil calibration. Through the numerical simulation of Bloch equations, we show that the behavior of $My\u2032$ and *M*_{z} can be completely specified by two dimensionless quantities *a* and *b*. From the simulation results, a graphical selection criterion is obtained, which is further simplified to a ready-to-use criterion of *b* ≥ 24. By switching the inherent magnetometer from the transverse mode to longitudinal mode, the *M*_{z} signal is detected and used to extract the Rabi frequency, which significantly shortens the calibration period from tens of minutes to several minutes. In addition, the influence of the frequency detuning can be easily circumvented by quadratic curve fitting. Finally, we verify the effectiveness of the selection criterion with the use of our new calibration method. Potential applications are foreseen for minimized atomic gyroscopes, as well as magnetometers.

## ACKNOWLEDGMENTS

This work was supported by the National Key Research and Development Program of China (Grant No. 2018YFB2002404) and the Beijing Natural Science Foundation (Grant No. 1222025).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Jinpeng Peng**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). **Zhanchao Liu**: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Writing – review & editing (equal). **Tengyue Wang**: Software (equal). **Yunkai Mao**: Software (equal). **Binquan Zhou**: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.