1. INTRODUCTION

Quantum sensing has garnered increasing interest for its potential to realize unprecedented sensitivity [1,2]. Various systems, such as superconducting quantum interference devices (SQUIDs) [3,4], atomic vapor cells [5–7], trapped ions [8,9], and solid-state spins [10–12], have been proposed for quantum sensors. For practical applications under ambient conditions, integrated and robust quantum sensing devices are particularly important [13–22].

With the ability to operate under ambient conditions from liquid helium temperature to 1000 K, the negatively charged nitrogen-vacancy (NV) color center in diamond has become one of the most promising platforms for magnetic field sensing [23–28] with sub-nanotesla sensitivities [29–31] and up to $\sim 100\mathrm{kHz}$ bandwidth [32]. Benefiting from their compactness and technical robustness, fiberized diamond-based sensors have been demonstrated for DC magnetometer in the applications of current measurement, magnetic flux leakage detection, and equipment status sensing [31,33–38]. However, the slow spin polarization and inhomogeneous distribution of the driving field [39–41] limit the spin preparation and sensing protocol optimization for NV center ensemble. Experimentally, the continuous-wave (CW) excitation [29,42,43] combined with lock-in detection methods is used to manipulate and detect the ensemble spins. But it is difficult to achieve $T_2$-limited sensitivity [1,44] due to the low manipulation fidelity. Moreover, the diamond magnetometer usually requires high-power optical excitation to achieve higher sensing performance. But for the CW scheme, a higher pump rate may lead to a worse signal-to-noise ratio (SNR) [45,46], since accelerated spin polarization will reduce the optically detected magnetic resonance (ODMR) signal contrast [29,38,47,48]. These factors pose a challenge for fiberized diamond magnetometers, and other types of integrated sensors with large volumes, to achieve optimal sensitivity and bandwidth when employing multiple pulse protocols.

In this work, we efficiently collected and detected the red fluorescence from the NV center ensemble, and developed a picotesla fiberized diamond AC magnetometer. By common-mode rejection and exponential fitting of spin polarization processes, we propose a fluorescence signal treatment method for NV spin ensemble detection, which can be applied to enhance the SNR. This approach can be easily implemented in pulse sequence processing with the detection bandwidth dependent only on the pulse cycle time. Experimentally, XY8 sequences are applied for AC magnetometry to measure arbitrary AC magnetic fields, which demonstrate an $8\mathrm{pT}/\sqrt{\mathrm{Hz}}$ $T_2$-limited sensitivity and a 90 Hz $T_1$-limited frequency resolution over a wide frequency band from 100 kHz to 3 MHz. Our technique effectively leverages the quantum coherence properties of solid-state spins, surpassing the previously mentioned CW-based fiberized diamond magnetometers. The high measurement sensitivity with a megahertz-scale bandwidth makes it highly suitable for practical applications.

2. RESULTS

A. Design of the Experiment and Fluorescence Signal Treatment

A 532 nm green laser of approximately 500 mW, guided through fiber 1 with a ceramic ferrule size of 1.25 mm, optically excites the NV centers in the diamond [see Fig. 1(a)]. A customized parabolic lens is pasted on the other side of the diamond, collecting the red fluorescence and reducing optical losses from the diamond. For fluorescence detection, another 3 mm diameter bare polymer optical fiber 2 is attached to the customized parabolic lens to collect the fluorescence emission, as shown in Fig. 1(b). The green laser and red fluorescence are detected as differential signals to suppress common-mode noise from laser fluctuations during the measurement (details are shown in Appendix A).

 

Fig. 1. Experimental scheme and fiberized diamond-based AC magnetometer. (a) Sketch of an NV center in diamond. The crystallographic coordinates are (x, y, z) = ([110],[100],[110]), and one of the NV orientations is in the yz plane. (b) Scheme of the fiberized diamond-based AC magnetometer. The diamond containing high-concentration NV centers is clamped by the silica optical fiber and parabolic lens. Copper coils with outer diameters of 2 mm and 10 mm are used to radiate microwaves (1–5 GHz) and AC magnetic fields (1 kHz–10 MHz), respectively. (c) Spin manipulation sequence used for magnetic field detection and fluorescence intensity acquisition. The green time trace ($\mathrm{\Delta }S$) is obtained by subtracting the two readouts ($S_{\mathrm{Rea}}$, $S_{\mathrm{Ref}}$) for common-mode noise suppression. $\mathrm{\Delta }S$ can be well fitted by $A_0{e}^{-\frac{t}{{\tau }_0}}$, and the spin state population is linearly dependent on $A_0$.

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We performed magnetic field sensing by employing the pulsed protocol depicted in Fig. 1(c), which consists of different microwave spin manipulation sequences (MW-Sequ.1 and MW-Sequ.2, where “Sequ.” is the acronym of “sequences”) and spin state optical readouts (Rea.&Pol. and Ref.&Pol., where “Rea.,” “Ref.,” and “Pol” are the acronyms of “readout,” “reference,” and “polarization,” respectively). To suppress common mode noise from the laser and ultimately improve the SNR, we subtract such two fluorescence acquisitions to obtain $\mathrm{\Delta }S=S_{\mathrm{Rea}}-S_{\mathrm{Ref}}$. Figure 1(c) shows fluorescence time traces of pulse ODMR with a readout time of approximately 200 μs. $\mathrm{\Delta }S$ denotes the time domain spin polarization process and can be exponentially fitted in the form of $A_0{e}^{-\frac{t}{{\tau }_0}}$ [48]. $A_0$ represents the final spin state $m_s=0$ probability after MW pulse manipulation. The ${\tau }_0\approx 70\mathrm{\mu s}$ is the parameter of spin polarization time, which is constant when manipulating the NV spin ensemble in our experiment. In this case, we only need to determine ${\tau }_0$ in advance, and then extract $A_0$ as the result for each sequence loop by fitting. As the spin polarization process is deterministic and the fitting treatment takes advantage of the correlations between data, it can filter out more noise in comparison with the integrating of photon emission [29,44], resulting in a higher SNR. The method can be applied to fiberized diamond magnetometers using spin manipulation sequences such as spin echo, XY8-N, and other complex dynamic decoupling protocols. In practice, the fitting is speedy and can be performed in a field-programmable gate array (FPGA) to achieve real-time detection.

B. AC Magnetometry by Spin Echo and XY8 Protocol

The above fitting treatment enables us to perform AC magnetic field sensing using dynamic decoupling detection protocols. Here, we detect the AC magnetic field ${\mathit{B}}_1(\mathit{t})=B_{\mathrm{ac}}·\sin (2\pi f_{\mathrm{ac}}t+\varphi )$ by spin echo and XY8 protocol, where $f_{\mathrm{ac}}$ and $B_{\mathrm{ac}}$ are the known frequency and unknown amplitude. In the laboratory coordinate system shown in Fig. 1(a), a bias magnetic field ${\mathit{B}}_{\mathrm{0}}\approx 394\mathrm{G}$ is applied along the NV axis to polarize the nitrogen ${}^{14}\mathrm{N}$ nuclear spins. The Hamiltonian is given by $H=D_{\mathrm{z}\mathrm{f}\mathrm{s}}{S}_z^2+{\gamma }_e\mathit{S}\mathit{B}$, where the electron gyromagnetic ratio ${\gamma }_e=2.8\mathrm{MHz}/\mathrm{G}$, $\mathit{S}=(S_{\mathit{x}},S_{\mathit{y}},S_{\mathit{z}})$ is the spin vector, $D_{\mathrm{z}\mathrm{f}\mathrm{s}}=2.87\mathrm{GHz}$ is the zero-field splitting, and $\mathit{B}={\mathit{B}}_{\mathrm{0}}+{\mathit{B}}_1(\mathit{t})$ is the total magnetic field. We first performed spin echo sequences to probe the AC field (details are depicted in Appendix D). A 200 μs laser pulse is applied to prepare the NV ensemble into the $m_s=0$ state and followed by a spin echo microwave pulse sequence (MW-Sequ.1: $[{\pi }_{\mathit{x}}/2-{\pi }_{\mathit{y}}-{\pi }_{\mathit{x}}/2]$, MW-Sequ.2: $[{\pi }_{\mathit{x}}/2-{\pi }_{\mathit{y}}-{\pi }_{-\mathit{x}}/2]$) with free precession time $\tau$.

Figure 2(a) shows the characteristic pattern of echo “collapse and revival” due to Larmor precession of neighboring ${}^{13}\mathrm{C}$ spins under the applied bias field. It presents an overall decoherence envelope $\exp (-2\tau /{\mathit{T}}_2)$ with $T_2\approx 41.5(2)\mathrm{\mu s}$. When applying a test $f_{\mathrm{ac}}=1\mathrm{MHz}$ AC magnetic field, the plot exhibits modulation but maintains the original ${}^{13}\mathrm{C}$ interaction-induced envelope. Since the timing of each spin echo experiment is not phase-locked with the test AC field, the signal is averaged of a uniformly distributed initial phase $\varphi$. For the overall probability of the $m_0=0$ state, we obtain [49]

 

Fig. 2. AC magnetic field sensing with spin echo and XY8 protocol. (a) Spin echo experiment at 394 G with and without the 1 MHz test AC magnetic field. The red dashed line denotes the fitting to the decoherence envelope in the form $\exp (-2\tau /{\mathit{T}}_2)$. The inset shows the amplitude-dependent trace with $\tau \approx 382\mathrm{ns}$, which can be well-fitted by Eq. (1). (b) XY8-4 experiment with and without the 1 MHz test AC magnetic field. The decoherence envelope is fitted by the same formula as the spin echo protocol. The inset shows the output as a function of the amplitude of the 1 MHz signal from the AC generator.

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The XY8 pulse sequences are robust for preserving an arbitrary spin state in an NV ensemble system [50,51], which allows applications in qubit-based high-sensitivity detection of various external fields. For the measurement demonstration, we applied the XY8-4 sequence where multiple $\pi$ pulses were repeated four times. The orange curve in Fig. 2(b) shows the XY8-4 signal and the extended coherent time $T_{2,\text{XY}8-4}$ is estimated to be 75(1) μs by exponential fitting. The XY8-4 protocol presents a narrower band filtering, resulting in periodic spikes when applying a 1 MHz signal. Due to overlap with the integer multiple of the half-period of ${}^{13}\mathrm{C}$ nuclear Larmor precession, the spikes at about 120 μs and 340 μs disappear. Additionally, the time traces as a function of the AC amplitude are mapped in Appendix D, where the dashed line at the peak of the first revival is plotted in the inset of Fig. 2(b). The oscillation is significantly faster than that of the spin echo protocol. However, note that the $t_{\pi }\approx 65\mathrm{ns}$ is not short enough in our experiment, resulting in pulse imperfections; thus the amplitude trace cannot be well fitted by Eq. (1).

C. $T_2$-Limited Sensitivity of AC Magnetometry

To determine the performance of AC magnetometry, we investigated the sensitivity as a function of the AC magnetic field frequency $f_{\mathrm{ac}}$ and the number of control pulses $n$. In general, the coherence time exhibits a sublinear power-law dependence of $T_2^{(n)}\propto n^s$ [51]. The scaling parameter $s$ depends non-trivially on the concentration of N and ${}^{13}\mathrm{C}$ impurities. For the ${}^{14}\mathrm{N}$-rich diamond used in our experiment, we observed $s\approx 0.15$ (see Appendix E). The $T_2$-limited sensitivity of AC magnetometry measurements as a function of AC frequency $f_{\mathrm{ac}}$ and number of $n$ is given by [51]

Here $C$ is the measurement SNR, which depends on the optical collection efficiency, the number of NV spins contributing to the measurement, and the fluorescence contrast.

In an AC magnetometry measurement utilizing $n$-pulse dynamical decoupling, Eq. (2) shows the theoretical measurement sensitivity will change with AC magnetic field frequency and the number of control pulses. It can also be used to analyze the conditions required to achieve optimal performance. Experimentally, the sensitivity is often directly obtained by ${\eta }_B=\frac{\sigma }{{|\partial S/\partial B|}_{\max }}\sqrt{T}$. The deviation $\sigma$ is dependent on the system electronics and shot noise and can be considered constant in the measurement. In this case, optimizing the maximum slopes ${|\partial S/\partial B|}_{\max }$ of amplitude traces and single measurement time $T$ can yield the most sensitive measurement of an AC magnetic field. Here, Fig. 3(a) shows the time trace with applying varied amplitude of 0.25 MHz AC magnetic field, and the results with different XY8-N sequences are illustrated in Appendix F. By extracting the maximum slopes, the normalized sensitivities over a wide range of AC magnetic field frequencies are obtained as in Fig. 3(b). We observed that the XY8 protocol much outperforms the spin echo scheme in high-frequency measurement. However, when the detection magnetic field frequency overlaps with ${}^{13}\mathrm{C}$ nuclear spin Larmor precession, the incoherent magnetic noise from the spin bath contributes to the noise floor of the AC magnetic field measurement. At this point, the sensitivity will tend to be worse. Changing the magnitude of the bias magnetic field to shift the Larmor precession frequency can avoid such a detection frequency window. For a magnetic field frequency $f\approx 0.25\mathrm{MHz}$, the optimal number of control pulses $n_{\mathrm{opt}}$ is measured to be about 32. Further increasing the pulse number will decrease the signal contrast $C$, making it no longer practical for magnetometry reasonably.

 

Fig. 3. Sensitivity of the fiberized AC magnetometer. (a) Detection of a test AC signal from an RF loop at 0.25 MHz with an XY8-4 dynamic decoupling sequence. (b) Dependence of the AC magnetic field sensitivity on the frequency and XY8 sequence cycle. The sensitivity tends to be worse when the measured magnetic field frequency overlaps with the Larmor precession. (c) AC magnetic field sensitivity determinations by applying a test field with and without 10 Hz modulation. The inset shows the 0.6 s time traces. (d) Scaling of Allan deviation from two recorded time traces plotted in (c), and both of them show the same optimal averaging measurement time.

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Hereafter, we utilized XY8-4 dynamical decoupling pulses to investigate the performance of the diamond AC magnetometer. We applied a known test modulation field as $B(t)=B_{\mathrm{ac}}\cos (2\pi f_{\mathit{d}}t)\cos (2\pi f_{\mathrm{ac}}t)$, where $B_{\mathrm{ac}}\approx 0.4\mathrm{\mu T}$, and $f_{\mathit{d}}=10\mathrm{Hz}$ and $f_{\mathrm{ac}}=0.25\mathrm{MHz}$ are the amplitude modulation frequency and carrier frequency, respectively. For this procedure, the output of XY8-4 is continuously recorded for 100 s [Fig. 3(c) inset shows 0.6 s of this], where the single sampling period is consistent with the total duration of the sequence. Then, by calculating and averaging the amplitude spectral density (ASD) per second, the AC magnetometer sensitivity can be obtained, as shown in Fig. 3(c). Since the phase between the test AC magnetic field and XY8-4 sequence is random, the AC diamond magnetometer is sensitive to the field power. These resulted in the ASD trace highlighting the detection of 20 Hz instead of 10 Hz. For comparison, both the time and ASD traces without modulation are investigated, as shown with the blue curves. We finally reach a $T_2$-limited sensitivity of $8\mathrm{pT}/\sqrt{\mathrm{Hz}}$. On the other hand, the stability can be calculated by scaling the Allan deviation of the readout signal, as depicted in Fig. 3(d). As a result, the best magnetic field resolution can be realized for a longer measurement time of about 10 s, but no further improvements can be achieved by averaging measurement thereafter due to long-term drift.

D. $T_1$-Limited Frequency Resolution

The above dynamical decoupling sequence realizes the high-precision amplitude measurement of the AC magnetic field. The ability to reconstruct the frequency spectra of time-dependent signals is also necessary for quantum sensing. The spectral resolution of spin echo sequence spectroscopy in Fig. 2(b) is Fourier-limited by NV-$T_2$ relaxation to $\sim {(\pi T_2)}^{-1}$ for the ensemble NV center. Although XY8-N sequences can further extend coherence time and narrow the bandwidth, the worse signal-to-noise ratio will reduce the precision of the frequency estimation. The XY8-based correlation measurements have been demonstrated to further narrow the bandwidth and to perform NV-$T_1$ relaxation limitation frequency resolution [1]. In particular, such correlation spectroscopy has been applied to nuclear spin detection, and spectral resolutions of a few 100 Hz have been demonstrated [52,53]. Here, a correlation-type measurement with MW-Sequ.1 $[(\mathrm{XY}8-{1)}_{\mathit{x}}-{\tau }_{\mathrm{corr}}-{(\mathrm{XY}8-1)}_{\mathit{x}}]$ and MW-Sequ.2 $[(\mathrm{XY}8-{1)}_{\mathit{x}}-{\tau }_{\mathrm{corr}}-{(\mathrm{XY}8-1)}_{-\mathit{x}}]$ is implemented to sense a sinusoidal AC magnetic field, as depicted in Appendix G. In this regime, the XY8-1 multipulse sequences are subdivided into two equal sensing periods $t_{\text{XY}8-1}$ that are separated by a free evolution time ${\tau }_{\mathrm{corr}}$. Furthermore, the $\pi$-pulse spacing ${\tau }_0$ in the XY8-1 sequences is set to satisfy ${\tau }_0=1/(2f_{\mathrm{ac}})$, where $f_{\mathrm{ac}}$ is the frequency of the signal to be measured. In this case, the final probability $A_0(\tau )$ of ${\mathit{m}}_s=0$ is given by $A_0(\tau )\propto \frac{1}{2}(1-\frac{{\mathrm{\Phi }}^2}{2}\cos (2\pi f_{\mathrm{ac}}\tau ))$, where $\mathrm{\Phi }=2{\gamma }_e{B}_{\mathrm{ac}}{t}_{\text{XY}8-1}/\pi$.

We begin by measuring the ${}^{13}\mathrm{C}$ NMR at the peak of the first revival in the spin echo signal shown in Fig. 2(b). The Larmor precession can be reconstructed with a high SNR, as illustrated by the orange curve in Fig. 4. Next, we observe the performance of the diamond quantum spectrum analyzer under an AC magnetic field with a frequency and amplitude of 0.25 MHz and 80 mV, respectively. The sinusoidal oscillating time-domain signal is depicted with the blue curve. Since $T_1\approx 3.56\mathrm{ms}$ is much longer than $T_2$ (see Appendix C for details), the two oscillation curves exhibit no significant decay. After a fast Fourier transform (FFT), the oscillation frequencies are found to be 0.42 MHz and 0.25 MHz, as marked in the inset. The linewidth in the FFT result gives a frequency resolution of 10 kHz approximately for 180 μs free precession time. This is larger than the theoretical $1/(\pi T_1)\approx 90\mathrm{Hz}$, mainly due to the short measurement time. These correlation measurements can be applied to recently developed heterodyne detection methods, which can provide a relaxation time independent spectral resolution [54–56] and ultimately break through the $T_1$-limit. The fiberized diamond-based AC magnetometer demonstrates great feasibility and performance for high-resolution frequency measurements, making it promising for non-destructive spectrum analysis and nuclear spin detection in vitro.

 

Fig. 4. ${}^{13}\mathrm{C}$ nuclear magnetic resonance (NMR) and magnetic field detection. The orange plot demonstrates the ${}^{13}\mathrm{C}$ nuclear Larmor precession oscillation over 180 μs. Additionally, the correlation spectroscopy detection with an AC magnetic field at 0.25 MHz is depicted with the blue curve. Since the magnetic field amplitude is greater than that generated from ${}^{13}\mathrm{C}$ nuclear spin, the detection result presents a sinusoidal curve with a large signal-to-noise ratio. The Fourier transform data of both time series are displayed in the inset.

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E. Applications for AC Field Mapping

With the capability of high-sensitivity AC magnetic field measurement, diamond AC magnetometers can offer remarkable advantages in three-dimensional magnetic imaging. The diamond is located in the middle of the two fibers, and the magnetometers’ spatial resolution is diamond-size limited. Here we demonstrate a compact magnetic field mapping with a spatial resolution of 200 μm.

The experimental setup is depicted in the method, where a copper coil is mounted on the YZ translation stage, and its central axis is parallel to the x-axis. We performed AC magnetic field mapping using the XY8-4 sequences. First, we applied an AC signal with a voltage of 0.1 V and frequency of 0.2 MHz, and the vertical distance of the diamond was approximately 5 mm from the coil. We then continuously ran the XY8-4 sequences while scanning the position of the coil, and averaged the acquired data every 100 sequence cycles. Due to the nonlinear response of the output signal to the amplitude of the AC magnetic field, we calibrated the averaging results using the zeroth-order Bessel function. This can be realized by establishing a mapping relationship lookup table between the magnetic field and $A_0$. Note that shortening the free evolution time can increase the measurement dynamic range at the expense of sensitivity. As a result, we obtained the amplitude of the AC magnetic field in the $2{\mathrm{cm}}^{}\times 3{\mathrm{cm}}^{}$ area, as shown in Fig. 5(a). Additionally, we repeated the process by making the central axis of the copper coil parallel to the z-axis, and performing magnetic field mapping in the same manner, as illustrated in Fig. 5(b).

 

Fig. 5. Mapping the AC magnetic field amplitude induced by a copper coil with an outer diameter of 1 cm. (a), (b) Two mappings with coil axis parallel and perpendicular to the x-axis as depicted in the inset.

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3. DISCUSSION AND CONCLUSION

Quantum sensing based on the diamond NV color centers is rapidly advancing with the use of an abundance of novel technologies. High sensitivity and integration are the current development trends and challenges for these quantum sensors. Typically, optical fiber technology can simplify diamond fluorescence excitation and collection systems, enabling integrated and portable sensing. To further improve the sensing performance, diamonds with longer coherence times [57,58] and higher concentrations [39] of color centers can be used. Generally, increasing the sensing volume is preferred to enhance the SNR and measurement sensitivity. The diamond is currently adhered to the end face of the optical fiber. In the future, it can be melted inside the optical fiber to improve the temperature resistance and stability.

In summary, we have demonstrated a fiberized diamond AC magnetic field sensor. By using the XY8-4 dynamical decoupling sequence, we achieved a $T_2$-limited sensitivity of $8\mathrm{pT}\sqrt{\mathrm{Hz}}$ and a frequency bandwidth of 100 kHz to 3 MHz. We then adopted an XY8-based correlation sequences protocol to obtain a $T_1$-limited frequency resolution. These results demonstrate the great potential of fiberized diamond AC sensors for amplitude-frequency measurements. This sensor can be employed for non-destructive testing such as eddy current testing, and material identification via in vitro nuclear spin detection, which is critical for chemical and biological sciences.

APPENDIX A: EXPERIMENTAL SETUP AND SAMPLE PREPARATION

The NV center in a diamond consists of substitutional nitrogen associated with a vacancy in an adjacent lattice site of the diamond crystal. Our measurements were performed on a ${}^{13}\mathrm{C}$ naturally abundant (1.1%) diamond sample, grown via chemical vapor deposition, having a nitrogen concentration of $\sim 36\mathrm{ppm}$ (parts per million) and an NV concentration of $\sim 3.7\mathrm{ppm}$. The diamond is mechanically polished and cut into a membrane with dimensions $200\mathrm{\mu m}\times 200\mathrm{\mu m}\times 100{\mathrm{\mu m}}^{}$, and attached on the tip of a multi-mode optical fiber (step refractive index, with a core diameter of $200\mathrm{\mu m}$ and a numerical aperture of 0.37).

The scheme of the fiberized diamond-based AC magnetometer experiment setup is shown in Fig. 6. The green pump laser and red fluorescence were detected by the same photodetector (PD, Thorlabs PDA36A). These two outputs are collected by a DAQ card (DAQ, NI PCIe-6351) as differential signals, to realize the suppression of common mode noise. The microwave, generated from an arbitrary waveform generator (AWG, M8195A, Keysight), is delivered to a 0.5 mm diameter wire wound around the fiber, which is used to manipulate the ensemble’s spin state with the $m_s=0\rightleftharpoons m_s=\pm 1$ transition. All pulse sequences are generated by a pulse blaster card (SpinCore PulseBlaster ESR PRO 500 MHz), which is used to both trigger the timing of the data acquisition unit and switch on and off the laser and AWG. A permanent magnet producing a static magnetic field of $\sim 394\mathrm{G}$ along the NV symmetry axis Zeeman-splits the $m_s=\pm 1$ spin sublevels. An RF coil is placed around the diamond with around 20 turns and a 0.5 mm diameter. The current inducing a magnetic field is produced with a function generator (Rigol, DG800), ranging from 1 kHz to 10 MHz. In this study, the two-level splitting of the $m_s=0$ and $m_s=-1$ Zeeman states is 1.765 GHz and we use microwave pulses resonant with this transition to control the populations and coherences for quantum sensing.

 

Fig. 6. (a) Scheme of fiberized diamond-based AC magnetometer experiment setup. The green 532 nm pump light is split, and one of the parts is monitored by a photodetector; another part is modulated by an acousto-optic modulator (AOM) and finally coupled into a multimode silica optical fiber. The excited fluorescence is collected by a polymer optical fiber, and detected by another photodetector. Subtracting these two photoelectric signals can suppress the common mode noise of the light source. (b) Image of the fiberized diamond-based sensor shown in (a). (c) ODMR spectrum under a bias magnetic field of about 408 G.

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APPENDIX B: AMPLITUDE OF THE AC MAGNETIC FIELD AS A FUNCTION OF VOLTAGE

To accurately assess the sensitivity of the system’s AC magnetic field measurement, it is essential to determine the magnitude of the magnetic field generated by the solenoid coil. In the experiment, an alternating voltage of the form $U=U_0\cos (2\pi ft)$ is applied to the solenoid, producing an alternating current that theoretically yields the AC magnetic field through the Biot-Sava-Laplace law. However, this method typically requires precise spatial coordinates of the diamond relative to the coil, which can be almost impossible or very challenging to obtain.

Here, we demonstrate an alternative method for obtaining the AC magnetic field of any frequency and voltage, which can be derived from Fig. 9(b). To achieve this, we treat the entire quantum magnetometer system, including the coil and voltage source, as an RL circuit. Here, R accounts for the total impedance of the signal source and solenoid, while L represents the inductance of the solenoid. Consequently, the magnitude of the current in the circuit and the magnitude of the voltage, resistance, and frequency can be expressed as

(B1)$$I=\frac{U_0}{\sqrt{R^2+{(2\pi Lf)}^2}}\cos (2\pi ft+{\varphi }_0),$$

where ${\varphi }_0=\arctan \left(\frac{2\pi Lf}{R}\right)$. Since the magnitude of the AC magnetic field is linear with the current as $B_{\mathrm{ac}}=k_1{I}_0$ ($k_1$ is constant), bringing it into Eq. (1) and assuming ${\sin }^2(\pi f\tau )\approx 1$, we obtain

(B2)$$P_0(U_0)\propto J_0(k_2{U}_0),$$

with

(B3)$$k_2=\frac{4{\gamma }_e{k}_1}{fR}\frac{1}{\sqrt{1+{\left(\frac{2\pi Lf}{R}\right)}^2}}.$$

When a fixed-frequency voltage is applied, the fluorescence signal variation conforms to the form of the zeroth-order Bessel function with the magnitude of the voltage amplitude. Based on this, $k_2$ values for different frequencies $f$ are obtained by fitting the voltage-trace signal plots with Eq. (4), which are presented in Fig. 7(a). Hereafter, the ratio of resistance $R$ to inductance $L$ can be extracted by fitting with Eq. (5). Note that an accurate measurement of the resistance $R$ enables the determination of the coil’s inductance $L$. This allows for the estimation of the magnetic field amplitude $B_{\mathrm{ac}}$ generated by any AC voltage $U$, as demonstrated in Fig. 7(b). With this method, the sensitivity of the optical fiber diamond AC magnetic field measurement can be precisely calibrated.

 

Fig. 7. (a) Inductance measurement of the RL loop. The parameter $k$ is derived from spin echo detection by zeroth-order Bessel equation fitting, which is a function of the applied AC magnetic field frequency. The inductance can be obtained by fitting the curve with the function shown in the inset. Once obtaining the inductance, the AC magnetic field amplitude as a function of the frequency and voltage can be derived by $B_{\mathrm{ac}}=k_1{I}_0$, as demonstrated in (b).

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APPENDIX C: RABI OSCILLATION AND $T_1$ MEASUREMENT

Rabi oscillation is the basis for realizing the fiber diamond AC magnetometer, and its measurement sequence is shown in Fig. 8(a), where the optical readout and optical detection window time are both 200 μs. Figure 8(c) shows the results of the 4.3 MHz Rabi oscillation, and the times of the $\pi$ pulse and $\pi /2$ can be obtained to be about 120 ns and 62 ns, respectively. In this case, we also measured the longitudinal relaxation times ($T_1$) of the NV center ensembles. As a result, Fig. 8(b) shows the measurement sequence and a $\pi$ pulse is applied in the first half before free precession, and the second is not but rather a reference signal. Figure 8(d) shows the longitudinal relaxation, yielding a $T_1$ time of approximately 3.56(1) ms.

 

Fig. 8. (a), (b) Measurement sequence of Rabi oscillation and longitudinal relaxation, and (c), (d) measurement time traces.

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APPENDIX D: COMPARISON OF THE SPIN ECHO AND XY8-4 PROTOCOLS

The sensitivity of both the spin echo and XY8 detection protocols ultimately relies on the frequency of the AC magnetic field. In this study, we first illustrate the interaction between the ensemble NV color center and the AC magnetic field from both the time and frequency domain perspectives using the spin echo protocol, as depicted in Fig. 9(a). To mitigate common noise, the pulse sequence is executed twice, with the ${\pi }_x/2$ pulse in the first sequence phase-shifted by 180° relative to that in the second. Figure 9(b) demonstrates the time-domain signal mapping as a function of amplitude when a 1 MHz AC magnetic field is applied. The dashed line at the peak of the first revival can be accurately fitted using the zeroth-order Bessel function, as depicted in the inset of Fig. 2(b). Furthermore, amplitude-trace mapping as a function of AC magnetic field frequencies can highlight the frequency response characteristics, as shown in Fig. 9(c). When detecting low-frequency fields, even small changes in voltage amplitude can result in significant fluctuations in the fluorescence signal, but this approach limits the linear dynamic range. On the other hand, higher frequencies yield a larger signal-to-noise ratio but a weaker response of the fluorescence voltage to changes in the magnetic field amplitude. In this case, when employing the spin echo method for magnetic field measurements, the sensitivity diminishes as the frequency increases within the 0.1–0.5 MHz range. Finally, by applying voltages with identical amplitudes but different frequencies, we obtain the time-domain signal mapping, as depicted in Fig. 9(d).

 

Fig. 9. Comparison of the spin echo and XY8 protocols. (a) Spin echo sequence scheme. (b) Time-trace mapping as a function of the amplitude of the applied AC magnetic field. (c) Amplitude-trace mapping as a function of the frequency of the applied AC magnetic field. (d) Time-trace mapping as a function of the frequency of the applied AC magnetic field. (e) XY8-4 dynamical decoupling sequences scheme. (f)–(h) Performing the same experiment as (b)–(d).

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On the other hand, employing dynamical decoupling sequences featuring multiple pulses offers a viable solution for enhancing the time-evolved filter function of qubit superposition states. These sequences prove particularly beneficial in scenarios involving time-varying fields, as they effectively extend the coherence time and boost qubit phase accumulation. Consequently, they can find valuable applications in the realm of qubit-based high-sensitivity detection, especially for various external fields like AC magnetic fields. For such applications, maximizing the phase accumulation of the NV spin due to an AC magnetic field requires synchronizing the period of $\pi$ pulses in the decoupling sequences with the half-time period of the AC field. This synchronization enhances the signal from a specific frequency field while simultaneously suppressing noise stemming from unwanted frequency fields. Essentially, this synchronization acts akin to a lock-in amplifier in the quantum regime. To achieve this synchronization and conduct experiments effectively, we employ the XY8-4 sequence [refer to Fig. 9(e)]. By comparing the results to those presented in Figs. 9(f)–9(h), we observe that the time window of the echo signal becomes narrower, facilitating its responsiveness to even subtle changes in the amplitude of the AC magnetic field.

APPENDIX E: EXTENDING COHERENCE TIME BY DYNAMIC DECOUPLING SEQUENCES

The spin bath’s decoherence can be effectively suppressed through dynamic decoupling sequences, resulting in a notable enhancement of the coherence time. In Fig. 10, the time-domain signals obtained from various XY8-N sequences are presented, and the coherence time $T_2$ is determined by fitting the envelope with an exponential curve. To account for microwave manipulation imperfections leading to a gradual decline in signal contrast with increasing $n$, all the detection results were normalized. The inset graph displays the coherence time extension ($T_{2,n}/T_{\mathrm{2,1}}$) as a function of the number of applied $\pi$ pulses, denoted as $n=8N$. By fitting the formula $T_{2,n}/T_{\mathrm{2,1}}\propto n^s$, we estimate the value of $s$ to be approximately 0.15.

 

Fig. 10. Detection time traces for different XY8-N sequences, and the horizontal axis is the sum of free evolution time. As the number of $\pi$ pulses increases, the coherence time $T_2$ can be extended, but the contrast decreases, so all results in the figure have been normalized.

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APPENDIX F: AMPLITUDE SENSING WITH XY8-N

The XY8 sequence significantly enhances the sensitivity of AC magnetic field detection compared to the spin echo measurement method. The measurement sensitivity is dependent on both the responsivity and the single measurement time. To assess the magnetic field at various frequencies, we employed several XY8-N measurement sequences, and the corresponding results are presented in Figs. 11(a)–11(f). As the sequence cycle number N increased, the NV color center spin demonstrated a heightened response to variations in the AC magnetic field amplitude. However, this came at the cost of reduced signal contrast and longer measurement time, ultimately limiting the sensitivity increase with N. It is worth noting that different N values also affected the laser duty cycles, leading to shifts in diamond temperatures and the resonant frequency of the NV color center spin, resulting in decreased contrast and measurement performance. To mitigate this effect, it is crucial to match the microwave frequency with the probing sequence duration. During our experiments, we repeated the XY8 sequence up to 32 times, but further repetitions were halted due to the rapidly decreasing signal amplitude, which adversely impacted sensitivity.

 

Fig. 11. Dependence of the experimentally measured XY8-N on frequency and amplitude of the AC magnetic field for different N.

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APPENDIX G: FREQUENCY SENSING WITH XY8-N

High-resolution measurements of the frequency of the AC magnetic field can be achieved via correlation spectroscopy. Figure 12(a) shows the sequence diagram. The XY8-1 multipulse sequences are subdivided into two equal sensing periods that are separated by a free evolution time $\tau$. Furthermore, the $\pi$-pulse spacing ${\tau }_0$ in the XY8-1 sequences is set to satisfy ${\tau }_0=1/(2f_{\mathrm{ac}})$, where $f_{\mathrm{ac}}$ is the frequency of the signal to be measured. Note that the sequences are phase sensitive, and constructive or destructive phase build-up occurs when the free evolution period ${\tau }_{\mathrm{corr}}$ is a half multiple or full multiple of the AC signal period $1/f_{\mathrm{ac}}$. In this case, the final probability $P_0(\tau )$ of $m_s=0$ is given by

(G1)$$P_0(\tau )=\frac{1}{2}(1-\sin (\mathrm{\Phi }\cos (\alpha ))\sin (\mathrm{\Phi }\cos (\alpha +2\pi f_{\mathrm{ac}}\tau ))),$$

where $\mathrm{\Phi }=2{\gamma }_e{B}_{\mathrm{ac}}{t}_{\text{XY}8-\mathit{N}}/\pi$ is the maximum phase that can be accumulated during either of the two XY8-N multipulse sequences and $\alpha$ is the initial phase between the AC field and the detection sequence. However, the AC signal is not synchronized in our experiments, and thus $\alpha$ is arbitrary. Assuming a small AC magnetic field amplitude $B_{\mathrm{ac}}$, $P_0(\tau )$ can be obtained by integrating Eq. (G1) over $\alpha$ as

(G2)$$P_0(\tau )\approx \frac{1}{2}(1-\frac{{\mathrm{\Phi }}^2}{2}\cos (2\pi {\mathit{f}}_{\mathrm{ac}}\tau )).$$

Figures 12(b)–12(d) show the time-trace mappings as a function of the AC magnetic field amplitude for the XY8-1, XY8-2, and XY8-4 protocols, respectively. We found that the time-domain signal can almost maintain a sinusoidal waveform below 0.2 V. But as the voltage amplitude continued to increase, the waveform began to deform, which verified the small magnetic field signal assumption that Eq. (G2) was established, as shown in Fig. 12(e).

 

Fig. 12. Correlation spectroscopy for AC magnetic field sensing at 311 G. (a) The correlation spectroscopy pulse sequence consists of two XY8-N sequences with fixed $\tau$ at half of the AC field period. The timing ${\mathit{\tau }}_{\mathrm{corr}}$ is swept, which correlates the phases $\varphi$ of the AC magnetic field and generates oscillations in the readout data at the AC frequency. (b)–(d) Time-trace mapping as a function of the AC magnetic field amplitude for XY8-1, XY8-2, and XY8-4 protocols, respectively. (e) Time traces for XY8-1 when applying the amplitudes of 0 mV, 40 mV, 80 mV, 120 mV, and 300 mV.

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Funding

Fundamental Research Funds for the Central Universities (WK2030000062); Key Research and Development Plan of Jiangsu Province (BE2022066-2); National Natural Science Foundation of China (12005218, 52130510, 62225506, 62305324, 62305324); CAS Project for Young Scientists in Basic Research (YSBR-049); Innovation Program for Quantum Science and Technology (2021ZD0303200).

Acknowledgment

The sample preparation was partially conducted at the USTC Center for Micro and Nanoscale Research and Fabrication.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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