Frequency-tunable magnetic field sensing using continuous-wave optically detected magnetic resonance with nitrogen-vacancy centers in diamond

In natural science, quantum sensing has received considerable attention because it can facilitate the investigation of physical properties on the nanometer scale and the evaluation of miniaturized devices. In recent years, the nitrogen-vacancy (NV) center in diamond has emerged as a promising candidate for magnetometry with high spatial resolution. The significant characteristic of the NV center is a long coherence time, even at room temperature and atmospheric pressure,^1,2 unlike other quantum sensors such as superconducting quantum interference devices (SQUIDs).^3,4 In addition, the electron spin states can be initialized by irradiation with a green laser⁵ and readout through the spin-dependent photoluminescence.^6,7 Moreover, we can manipulate spin states in the NV center using a microwave (MW) field.⁸ For practical magnetic field sensing using NV centers, many methods such as wide-field imaging,^9–11 AFM-based methods,^12–15 and vector magnetic field sensing^9,16–20 have been proposed and demonstrated.

To realize sensitive AC magnetic field detection with NV centers, pulsed measurement such as spin echo is typically used.^21–26 While such pulsed measurement is advantageous for improving sensitivity because of the extended coherence time (⁠ $T 2$⁠),^27,28 it requires a sophisticated setup to generate pulse sequences with appropriate synchronization. In addition, when the pulsed-measurement is used in CCD-based wide-field imaging, we must deal with significantly different time scales for pulse sequence and imaging frame rates. It is also vital to improve the pulse precision, as it is challenging to apply spatially uniform and strong MW over a large area.

To avoid such difficulties, some of the present authors proposed and demonstrated sensing AC fields with MHz frequencies (RF fields) via continuous-wave optically detected magnetic resonance (CW-ODMR).^29,30 This technique utilizes not only a MW but also the RF field, which is the target field. Due to the coherent interaction between the electron spin states and the driving fields, RF-dressed states are created. Similar research is reported in Ref. 31. As a result, we observe a change in the CW-ODMR signal. In this technique, CW-ODMR is used to measure the spectrum of the RF-dressed states created by the coherent interaction of the electron spin with the target RF magnetic field. The RF field strength can be estimated from the resulting ODMR spectrum. In this approach, a sophisticated setup and strong MW power are not required, unlike the conventional pulsed measurement. However, the target frequency is fixed at the non-adjustable physical properties of the sensor, and any detuning from this value rapidly reduces the sensitivity. This narrow band property limits the broad application of this ambitious technique.

Here, we propose and demonstrate a frequency-tunable version of the CW-ODMR-based AC magnetic field sensing by irradiation with RF fields at two different frequencies. One is the control RF (⁠ $R F control$⁠) field that generates the RF-dressed states of the NV centers in the diamond. The other is the target RF (⁠ $R F target$⁠) field, which is the AC magnetic field whose amplitude is to be estimated by our method. The coherent interaction between the RF-dressed states and the $R F target$ field generates RF double-dressed states and induces measurable signals during CW-ODMR measurement. In addition, since the energy levels of the RF-dressed states can be controlled by changing the $R F control$ field amplitude, the detectable frequency can be tuned. Our proof of principle experiments prove that the detection bandwidth can be increased compared with those of the previously reported methods. Our results open the door to realizing an AC magnetic field sensor with a large bandwidth by using CW-ODMR on the NV centers.

We describe the theory underlying our proposed method. We apply a DC magnetic field perpendicular to a specific axis of the NV center and define this direction as the $x$ axis (see the inset of Fig. 1). In this case, the eigenstates of the NV centers are approximately described by the so-called bright state or dark state. Importantly, the RF field can induce transitions between these states. In addition, by applying the perpendicular DC magnetic field, we can suppress magnetic field noise.³² 

These analytical solutions indicate that the resonant frequency of CW-ODMR changes linearly with the $R F target$ field amplitude when the frequency of the $R F target$ field is equal to $2 E x ′± γ e B RFc$⁠, as shown in Fig. 1. In addition, the amplitude of the $R F target$ field can be measured from the change in the resonant frequency of CW-ODMR. Since we can measure the $R F target$ field at the frequency of $2 E x ′± γ e B RFc$⁠, we can determine the detectable frequency by changing the $R F control$ field amplitude. Although we assume that we drive the NV centers using the MW along the $x$ direction, we can perform similar calculations with the MW aligned along the $y$ direction and obtain the same number of resonances.

We describe the setup of our experiments. We used the same diamond sample used in Ref. 34. The ensemble of NV centers has one axis oriented along the (111) direction. The measurement setup was basically the same as that in Refs. 30 and 34, except that we applied the second RF field to the NV centers. The sample of ensemble NV centers was placed on a MW antenna.³⁹ A perpendicular DC magnetic field was applied by a permanent magnet. The $R F control$ field and the $R F target$ field were irradiated from a single copper wire that was closely attached to the sample surface. In our experiments, we performed a careful calibration for the $R F control$ field and the $R F target$ field. The two RF fields were generated from two channels in a function generator (NF Corp., WF1974), respectively. In addition, each RF field was combined using a function called external addition input in the function generator. To investigate the frequency response, we input these signals into an oscilloscope (Rohde&Schwarz, RTO1004). According to the fitting of the recorded signals, we found the measured $R F control$ amplitude was about 93.5% value of the set one. On the other hand, the measured $R F target$ amplitude had a wide range from 65.5% to 89.7%, depending on the $R F target$ frequencies. In the experiments, we conducted the above calibration for all frequencies which is depicted in Fig. 5. These calibration results allowed us to evaluate the amplitude of the two RF waves impartially.

Before demonstrating CW-ODMR for AC magnetic field sensing, we needed to perform CW-ODMR for calibration. First, we performed the CW-ODMR measurement by sweeping the MW frequency under the perpendicular magnetic field, as shown in Fig. 2(a). We observed two dips, as shown by the black line in Fig. 2(a). One of them corresponds to the transition from $|0⟩$ to $|B⟩$⁠, while the other corresponds to the transition from $|0⟩$ to $|D⟩$⁠. We found that the resonant frequencies of $|B⟩$ and $|D⟩$ were 2.878 and 2.886 GHz, respectively. Their difference of 8.47 MHz corresponds to the resonance frequency of $R F control$ for creating the RF-dressed states.

Second, we performed the CW-ODMR measurement under irradiation of the $R F control$ field with a frequency of 8.47 MHz by sweeping the MW frequency. The results are shown by the blue line in Fig. 2(a). We observed four dips and confirmed the generation of the RF-dressed states. We also found that the linewidth of the RF-dressed states was narrower than that of the bare states without the RF field [see Eqs. (A4) and (A5)]. This is consistent with the fact that the RF-dressed states are more robust against the electric field noise than the bare states.^30,34

Third, we performed the CW-ODMR measurement under irradiation by the $R F target$ (⁠ $R F control$⁠) field with a frequency of 5.34 (8.47) MHz; this frequency corresponds to $2 E x ′− γ e B R Fc$ (⁠ $2 E x ′$⁠). The results are shown by the green line in Fig. 2(b). We observed eight dips, which come from the RF double-dressed states, as shown in Eqs. (12) and (13).

Fourth, we performed the CW-ODMR measurement by sweeping the amplitude of the $R F control$ field (with a frequency of 8.47 MHz) and MW frequency. The color mapping of the CW-ODMR spectra (four dips) in the RF-dressed states is shown in Fig. 3(a). As in previous studies,^29,30 the splitting became larger as the $R F control$ field amplitude increased. Here, we succeeded in controlling the energy splitting of the RF-dressed states by changing the $R F control$ field amplitude. This means that we can tune the energy splitting to be resonant with the target RF frequency by changing the $R F control$ field amplitude. In addition, the proportionality constant between the $R F control$ field amplitude and energy splitting was approximately 25.8 GHz/T. If we assume that the DC magnetic field was applied perpendicular to the NV axis, the proportionality constant can be calculated as 28 GHz/T, which is different from the observed value. We expect that this difference comes from the misalignment of the DC magnetic field.

Finally, we performed the CW-ODMR with the $R F control$ field at a frequency of 8.47 MHz and an amplitude of 101  $μ T$ while we swept both the MW frequency and the $R F target$ field amplitude with a frequency of 5.34 MHz. The results are shown in Fig. 3(b). Here, we focus on one (labeled “B” in Fig. 1) of the four dips in the RF-dressed states; this dip is split into two dips (labeled “B1” and “B2” in Fig. 1) when we apply the $R F target$ field, as shown in Eq. (13). During CW-ODMR measurements, we observed a splitting due to the RF double-dressed states. This splitting became larger as we increased the $R F target$ field amplitude. In addition, the proportionality constant between the $R F target$ field amplitude and energy splitting was approximately 13.8 GHz/T, smaller than the theoretically predicted value of 14 GHz/T. We expect that this difference also comes from the misalignment of the DC magnetic fields, as discussed above.

We can estimate the amplitude of the $R F target$ from the change in the contrast of the CW-ODMR as follows. When the $R F target$ field is applied with a frequency of $ω RFt=2 E x ′− γ e B RFc$ (⁠ $ω RFt=2 E x ′+ γ e B RFc$⁠), we fix the MW frequency at $D ′+ E x ′− 1 2 γ e B RFc$ (⁠ $ω MW= D ′+ E x ′+ 1 2 γ e B RFc$⁠) and measure the change in the CW-ODMR contrast. Below, we explain the sensitivity and frequency tunability of this method.

We measured the amplitude of CW-ODMR signal when we applied the target field. More specifically, we set the target frequency to 7.14 MHz. The results obtained using our approach and the previous scheme are compared in Fig. 4. The vertical axis $S$ is the change of the signal amplitude from the $B RFt ( c )=0$⁠. In both schemes, they quadratically increase with the amplitude of the target RF field.^29,30 The obtained signal was significantly larger than that of the previous method. This is because in the previous method, the resonant frequency was detuned by 1.33 MHz with respect to the target frequency; in contrast, in the new scheme, we can use the control RF field to tune the resonant frequency to match the target frequency. Here, it is worth mentioning that in the previous scheme where only one RF field is used, we define $B RFc$ as the sensing target.

Finally, we experimentally estimated the sensitivity of our method and compared it with that of the previous method. We review the previous method in  Appendix A. We measured $δS$⁠, the signal fluctuations per second. More specifically, we collected the signal fluctuations over several measurement times using an avalanche photodiode and obtained $δS$ by linear fitting. We also estimated $d S d B RFt$ (⁠ $d S d B RFc$⁠) for the new (previous) method. From these estimates, we obtained the sensitivity as $δ B RFt=δS/| d S d B RFt|$ (⁠ $δ B RFc=δS/| d S d B RFc|$⁠) for our (previous) method. The sensitivity of the previous method was optimized around 8.47 MHz at the bandwidth of 3.8 MHz. However, the sensitivity rapidly decreased when the frequency was detuned from 8.47 MHz. In this case, the bandwidth is limited by the inhomogeneous linewidth. Compared with the previous scheme, our method has an advantage of an extended detectable frequency range. The optimal sensitivity was around 4.3 $μ T/ Hz$ when the frequency was around $7.97$ MHz. As we increased or decreased the frequency, the sensitivity became worse. We define the bandwidth of the frequency as the frequency range where the sensitivity is less than twice that of the optimal sensitivity. The bandwidth was estimated to be 7.5 MHz, twice as large as that of the previous scheme. It is worth mentioning that due to the electric field noise, we could not measure the target field with our method when the frequency was in the range of 7.7–9.1 MHz. If the frequency is in this range, we can use the previous method. Thus, by combining our method and the previous method, we realize an AC magnetic field sensor with a large bandwidth.

We observed that the optimal sensitivity of our scheme was comparable to that of the previous scheme, which can be explained as follows. According to the theoretical equations from Ref. 30 and Eqs. (12)–(15), the value of the energy splitting in the RF double-dressed states is half that in the RF-dressed states. However, the creation of the RF-dressed states suppresses environmental fluctuations such as the electric field noise. As a result, the linewidth of the photoluminescence dip is narrower, as shown in Fig. 2(a). This means that a larger contrast change can be obtained. Thus, this mechanism compensates for the less responsiveness to the target RF field and contributes to the sensitivity shown in Fig. 5.

As we increased or decreased the frequency of the target field from 8.47 MHz, the sensitivity became worse. Here, we discuss possible reasons for this. To tune the frequency, we need to increase the amplitude of the control field. As discussed earlier, this control field can suppress the electric field noise. However, the fluctuations in the control field can also be a source of noise when the amplitude of the control field is large. This mechanism increases the linewidth in CW-ODMR, as shown in Fig. 6. Such fluctuations could cause a decrease in sensitivity when the frequency of the target field is detuned far from 8.47 MHz. A similar effect is discussed in Ref. 40, in which concatenated dynamical decoupling was adopted.

The dependence on the detuning was asymmetric for our scheme as shown in Fig. 5 where the sensitivity with positive detuning was worse than that with negative detuning. This may come from the shape of the ODMR without the RF. We could not perfectly fit the shape by a sum of Lorentzians due to the asymmetric shape where the gradient of the contrast of the positive detuning is different from that of the negative detuning, as shown in Fig. 2(a). Importantly, such an asymmetric shape of the ODMR without applying DC magnetic fields was discussed.³⁸ However, in our case, we apply an orthogonal DC magnetic field, which is slightly different from the setup.³⁸ We leave the detailed study of these for future work.

Finally, we discuss the frequency tunability of our sensing scheme. We tune the detectable frequency of the target field by changing the $R F control$ amplitude (i.e., $ω RFt=2 E x ′± γ e B RFc$⁠). However, a strong $R F control$ amplitude could lead to the violation of the RWA. The RWA will be invalid when the driving amplitude becomes comparable to half of the resonant frequency. Thus, it would be practical to detect the target field between 4.2 MHz (⁠ $≃ E x ′$⁠, lower limit) and 13 MHz (⁠ $≃3 E x ′$⁠, upper limit).^20,41

Based on the above results, in this section, we provide a comprehensive comparison between the pulsed scheme and our scheme based on CW-ODMR. Also, we explain some methods to broaden the bandwidth and specific applications.

First, we explain the sensitivity and detection bandwidth in the pulsed scheme and CW-ODMR. In our experiments, we obtained the sensitivity from 4.3 to 10  $μ$T/ $Hz$⁠. Also, we expect that the shot-noise limited sensitivity reaches 0.73  $μ$T/ $Hz$ (see  Appendix B). Furthermore, the shot-noise limited sensitivity can be further improved by increasing the photon detection efficiency and the number of NV centers. On the other hand, from Ref. 42, the sensitivity of 10.8 pT/ $Hz$ was reported where dynamical decoupling pulse sequence was adopted. Next, we describe the detectable bandwidth. While our method achieves 7.5 MHz as described in Sec. III, the pulsed scheme can realize a much larger bandwidth from about 1 kHz to 10 MHz.⁴³ The lower limit of the detectable frequency of the pulsed scheme is determined by the coherence time (⁠ $T 2$⁠). On the other hand, the upper limit of the detectable frequency is generally determined by the MW pulse power.⁴³ 

Second, we introduce a possible way to broaden the bandwidth of our method. We can apply a DC electric field to increase the energy splitting between $|B⟩$ and $|D⟩$⁠.⁴⁴ In this case, we can use a larger $B RF C$ amplitude without violating the RWA, and we should be able to detect an AC magnetic field with larger or smaller frequencies. The nonaxial electric dipole moment of the NV axis is $d ⊥=(17±3$⁠) Hz cm/V,⁴⁴ and we need to apply the electric field of 30 kV/cm to obtain an additional energy splitting of 1 MHz. Note that we need to apply the electric field in the same direction as the bias magnetic field (defined as the $x$ direction in this paper). We also require the experimental setup introduced in Refs. 45 and 46 for generating the electric field. Alternatively, we can increase the perpendicular DC magnetic field, which also increases the effective strain as Eq. (3). However, as we increase the perpendicular DC magnetic field, the approximation to consider the energy eigenstates as $|B⟩$ and $|D⟩$ becomes invalid. Therefore, further analysis is required to check the validity of our method if we apply the stronger perpendicular DC magnetic field.

Third, we describe the advantages of our method over the pulsed scheme. The sensitivity and bandwidth of the pulsed scheme are much better than that of our scheme. However, the pulsed scheme requires strong MW power, precise timing control, and phase-locking. In contrast, our CW-ODMR method eliminates the need for them. Importantly, for some biological applications, we cannot use strong MW fields. An increase in apoptosis in cells was observed for high-intensity MW fields.⁴⁷ On the other hand, with our CW-ODMR method, the irradiated MW fields can be much weaker than the pulsed scheme, which could be beneficial for some applications.

Fourth, we also discuss the effect of the inhomogeneous broadening. In the pulsed scheme, it is possible to suppress the inhomogeneous broadening. By applying a magnetic field parallel to the NV axis, strain variations can be effectively eliminated.³⁷ Due to the dynamical decoupling, low-frequency noise coming from randomized magnetic fields is also suppressed.⁴⁸ On the other hand, in our CW-ODMR scheme, we also demonstrate insensitivity to inhomogeneous broadening, similar to the pulsed scheme. The bright and dark states are robust against magnetic field noise under the static magnetic field perpendicular to the NV axis. In addition, the control RF field with our method suppresses the inhomogeneous broadening due to the strain variations and electric fields, as observed in Fig. 2 or Ref. 34. However, note that increasing the amplitude of the $R F control$ field introduces noise, observed in Fig. 6. We need to resolve this problem to obtain a higher sensitivity, which is left for future work.

Finally, we consider specific applications using our method. One example is detecting electron spin resonance/nuclear magnetic resonance (ESR/NMR) signals from targets using the NV center. Our method allows signal detection over a wide MHz frequency range, which is approximately twice as large as that of the previous method. This widened range simplifies finding ESR/NMR signals when sweeping a static magnetic field amplitude. Another example is wide-field imaging of an AC magnetic field around micro-sized circuits. Our sensing method enables the detection of a wide-frequency AC field without rapid sensitivity deterioration or disrupting the experimental setup, facilitating non-contact inspection of electronic devices using CW-ODMR.

In conclusion, we proposed and demonstrated frequency-tunable magnetic field sensing based on CW-ODMR using the RF double-dressed states of NV centers in diamond. We can tune the detectable frequency of the magnetic field by adding the control RF field, which is in stark contrast to the previous method where the detectable frequency is fixed. Using our setup, the sensitivity achieved by our scheme was higher than that of the previous scheme for the target AC magnetic fields with frequencies under 7 MHz or above 10 MHz. We also found that a hybrid scheme combining the previous scheme and our scheme could improve the bandwidth by approximately 3.7 MHz. These results contribute to the development of practical AC magnetic field sensors.

We discuss future work. In our experiments, we detected an AC magnetic field amplitude under the assumption that the frequency is known. On the other hand, in principle, we can detect the frequency of the AC magnetic field if its amplitude is known. This allows us to estimate the spectral resolution of our method. However, this is out of the scope of our paper. Therefore, further research is needed to assess its practicality. Measurement of the phase of the AC magnetic field is another possible application of our scheme. While our current approach cannot directly measure the phase, the application of a reference RF field allows us to detect the phase of the target AC magnetic fields through interference. In addition, we explain the possibility of improving the sensitivity using all of the eight dips in the RF double-dressed states. In our experiments, we utilized the four energy levels in the RF-dressed states and $R F target$ field resonant frequencies for both lower and higher transitions. To explore all eight dips, an additional RF field resonant with the RF double-dressed states is needed. Adjusting the $R F control$ and the $R F target$ fields amplitude equalizes energy differences between adjacent levels in the RF double-dressed states, potentially inducing interaction with the new RF wave. We may detect its amplitude with a higher sensitivity.

This work was supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) (Grant No. JPMXS0118067395), MEXT/JSPS Grants-in-Aid for Scientific Research (KAKENHI) (Grant Nos. JP19H05826, JP20H05661, JP22H01558, JP22K03524, JP23H01103, and JP23H04390), and Center for Spintronics Research Network (CSRN), Keio University. This work was also supported by MEXT Leading Initiative for Excellent Young Researchers, JST PRESTO (Grant No. JPMJPR1919), JST Moonshot R&D (Grant No. JPMJMS226C), Kanazawa University CHOZEN Project 2022, JST CREST (Grant Nos. JPMJCR23I2 and JPMJCR23I5), and Kondo Memorial Foundation.

The authors have no conflicts to disclose.

Ryusei Okaniwa: Investigation (lead); Writing – original draft (lead). Takumi Mikawa: Formal analysis (supporting); Investigation (supporting). Yuichiro Matsuzaki: Investigation (supporting); Supervision (equal); Writing – review & editing (lead). Tatsuma Yamaguchi: Investigation (equal); Writing – original draft (supporting). Rui Suzuki: Investigation (supporting); Software (supporting). Norio Tokuda: Investigation (supporting); Resources (supporting); Writing – review & editing (supporting). Hideyuki Watanabe: Resources (supporting); Writing – review & editing (supporting). Norikazu Mizuochi: Conceptualization (supporting); Writing – review & editing (supporting). Kento Sasaki: Investigation (supporting); Writing – review & editing (equal). Kensuke Kobayashi: Investigation (supporting); Writing – review & editing (supporting). Junko Ishi-Hayase: Conceptualization (supporting); Funding acquisition (lead); Investigation (supporting); Project administration (lead); Resources (equal); Supervision (lead); Writing – review & editing (equal).

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

These results indicate that the resonant frequency has a linear dependence on the amplitude of changes linearly according to the amplitude value of the target RF field when $ω RF=2 E x ′$ is satisfied. Based on the above theory, we obtain the contrast change (⁠ $S$⁠) derived from the target RF field amplitude while fixing the MW frequency at $D ′+ 1 2 ω RF$ or $D ′− 1 2 ω RF$ in actual experiments. Therefore, we can measure the AC magnetic field from the change in the contrast.