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.

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).

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.

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] $$A_0(\tau )\propto \frac{1}{2}(1+e^{-\frac{2\tau }{T_2}}{J}_0(\frac{4{\gamma }_e{B}_{\mathrm{ac}}}{f_{\mathrm{ac}}}{\sin }^2(\pi f_{\mathrm{ac}}\tau )\left)\right),$$(1)where $A_0(\tau )$ can be extracted from $\mathrm{\Delta }S$, and $J_0$ is the zeroth-order Bessel function of the first kind. In this case, the time traces as a function of AC amplitude can be mapped (see Appendix D), where the dashed line at the peak of the first revival with $\tau =\frac{1}{2f_{\mathrm{ac}}}-t_{\pi }$ can be well fitted by Eq. (1), as shown in the Fig. 2(a) inset. By fitting the amplitude traces with different frequencies, a resistor–inductor circuit can be constructed to determine the amplitude of the AC magnetic field (details are shown in Appendix B).

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).

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] $${\eta }_{(n,f_{\mathrm{ac}})}\approx \frac{\pi \hslash }{2g{\mu }_B}\frac{1}{C\sqrt{\frac{n}{2f_{\mathrm{ac}}}}}\exp \left(\frac{n^{1-\mathit{s}}}{2f_{\mathrm{ac}}{\mathit{T}}_2}\right).$$s(2)

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.

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.

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.

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).

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.

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