Detection of Electron Paramagnetic Resonance of Two Electron Spins Using a Single NV Center in Diamond

A nitrogen-vacancy (NV) center is a fluorescent impurity center existing in the diamond lattice.Gruber et al. (1997) The unique electronic structure of the NV center allows the polarization and readout of the spin state through optical excitation and optically detected magnetic resonance (ODMR) technique.Jelezko et al. (2004); Gaebel et al. (2006); Childress et al. (2006a) The NV center also has a long spin coherence time at room temperature.Takahashi et al. (2008); Balasubramanian et al. (2009); de Lange et al. (2010) It has been shown to be a promising quantum sensor, enabling measurement of an extremely small magnetic field,Degen (2008); Balasubramanian et al. (2008); Maze et al. (2008); Taylor et al. (2008) magnetic resonance spectroscopy with single spin sensitivity,Grinolds et al. (2013); Shi et al. (2015); Abeywardana et al. (2016); Fortman et al. (2021); Fortman and Takahashi (2019); Fortman et al. (2020); Li et al. (2021); Ren et al. (2023) and quantum simulations.Cai et al. (2013); Wang et al. (2015); Ju et al. (2014)

A system with interacting spins is a great platform for fundamental physics and applications in quantum information science. For example, an interacting NV spin system is a great candidate to build a quantum sensing network for quantum entanglement sensing that provides enhanced sensitivity.Xie et al. (2021) Interacting spin systems also exhibit novel phases and dynamics, such as spin liquids, time crystals, and non-equilibrium dynamics.Broholm et al. (2020); Choi et al. (2017); González et al. (2022); Bussandri et al. (2024) NV-detected electron paramagnetic resonance (NV-EPR) spectroscopy with a single NV center is a powerful method to study nanoscale environments of interacting electron spins. NV-EPR can determine the number of the detected spins and identify the type of the detected spins. It can also determine the coupling strengths between the NV and each surrounding spin.Abeywardana et al. (2016); Li et al. (2021) NV-EPR has been demonstrated to detect P1 centers, N2 centers and other unidentified spins.Yamamoto et al. (2013); Shi et al. (2013); Cooper et al. (2020); Rosenfeld et al. (2018); Grinolds et al. (2014); Li et al. (2021) Moreover, electron spins on diamond surfaces have been detected.Sushkov et al. (2014) A characterizable and controllable network of spins can be used for entanglement-enhanced surface quantum sensing. However, only a few demonstrations have been performed to determine the number of interacting electron spins and individual coupling strength to NV center.Shi et al. (2015); Sushkov et al. (2014) Identifying and modeling such configuration remains a challenge for single NV-EPR detection.

In this work, we discuss the demonstration of NV-EPR spectroscopy of a few electron spins. We employ a single shallow NV center in diamond as a sensor and utilize a double electron-electron resonance (DEER) technique to show the detection of EPR signals of surrounding spins. The observation of the NV-EPR signal confirms the detection of spins. The type of the detected spin is analyzed from their g-values and hyperfine splitting. Moreover, we present a simple model to determine the number of the detected spins and the strengths of their magnetic dipole interactions. We find a single NV center coupled to two electron spins with dipolar coupling $1.12\pm 0.13$ MHz and $2.24\pm 0.17$ MHz from the analysis of the NV-EPR signal. The presented spin system will be useful for the investigation of quantum effects that require a low number of spinsXiao and Zhao (2016) or open quantum dynamics.Dakir et al. (2023) The experimental methods can be used to realize entangled quantum reporters for quantum sensing.

In this study, we employ single NV centers in a diamond crystal sample purchased from Qnami.Qnami The sample is a (111)-cut, specially designed diamond. Shallow NVs ($\sim$10 nm depth) were created with Nitrogen ion implantation with the energy of 6 keV and subsequent annealing process. The sample has a unique nano-pillar structure that allows easy isolation and identification of single NV centers. We have studied four single NV centers in this sample. In the present paper, our investigation focuses on one of the single NV centers with a unique NV-EPR signal (denoted as NV1). The experimental results on the others (NV2-4) are also presented in the supplementary material. The setup of the experiment is illustrated in Fig. 1. The diamond sample is mounted on an XYZ piezo stage. A permanent disk magnet is placed under the stage to provide a static magnetic field ($B_{0}$) along the [111] direction of the diamond lattice. For ODMR spectroscopy of the NV center, two channels of microwave (MW1 and MW2) excitation are applied using sources (Stanford SG386) and an amplifier (Mini-Circuits ZHL-15W-422-S+). Both MW1 and MW2 excitation are applied with a 20um diameter gold wire attached close to the top of the sample. A diode-pumped solid-state laser (CrystaLaser, 100 mW) and a microscope dry objective (ZEISS, 100x, NA = 0.8) are employed for the laser excitation. Photoluminescence (PL) from the NV center is collected by the same objective and redirected by a dichroic mirror towards an avalanche photon detector (Excelitas, SPCM-AQRH-13-FC). A central computer is used to control the microwave and laser pulses and analyze the photon count signal.

The NV-EPR experiment starts with the characterization of the external magnetic field and the coherent manipulation capability of the NV centers. After identifying single NV centers using the auto-correlation experiment and ODMR spectroscopy,Abeywardana et al. (2016) we focus on our investigation into each single NV and perform a pulsed ODMR measurement to determine the magnetic field strength and tilt angle. Fig. 2(a) shows a PL image of the sample and the studied NV1 is indicated in the image. Next, we perform pulsed ODMR measurements. As shown in Fig. 2(b), the laser pulse with a duration of 5 $\mu$s initialize (Init) the NV state to $\ket{m_{S}=0}$. After a short delay ($\sim$1 $\mu$s), a MW1 $(\pi)_{Y}$ pulse is applied to drive a rotation of the spin in the Bloch’s sphere along the x-axis (see Fig. 2(b)), from brighter $\ket{m_{S}=0}$ to darker $\ket{m_{S}=\pm 1}$ spin states. Afterward a short readout (RO) laser pulse with a duration of 300 ns project NV’s coherent spin state into the population difference, which is measured by the PL intensity, resulting in observation of ODMR signals at the $\ket{m_{S}=0}$ $\leftrightarrow$ $\ket{m_{S}=+1}$ and $\ket{m_{S}=0}$ $\leftrightarrow$ $\ket{m_{S}=-1}$ transition frequencies. Fig. 2(d) shows the PL intensity as a function of the MW1 frequency. Pulse sequence is averaged over 100,000 times in around 7 minutes to obtain the spectrum of each peak. By fitting the two frequencies, $1960.00\pm 6.78$ MHz and $3783.39\pm 3.39$ MHz, to the following NV Hamiltonian, we obtain the magnetic field strength ($B_{0}$) and the tilt angle ($\theta$) of the magnetic field in respect to the NV axis,

where $\gamma_{NV}$ is the gyromagnetic ratio of the NV spin (28.024 GHz/Tesla). $S=(S_{x},S_{y},S_{z})$ are the spin operators of $S=1$. $D$ is the NV zero-field energy splitting (2.87 GHz). As can be seen in Fig. 2(d), the experimental data and fit result are in fair agreement. From the fit result, we obtained the magnetic field strength of $B_{0}=32.59\pm 0.02$ mT and the tilt angle of $\theta=3.5\pm 0.8$ degrees. Next, we perform a measurement of Rabi oscillations. The pulse sequences of the experiment are shown in Fig. 2(c). After the laser pulse initialization, an MW1 $(\pi)_{Y}$ pulse at the NV1 Larmor frequency of 3783.39 MHz (corresponding to the transition $\ket{m_{S}=0}$ $\leftrightarrow$ $\ket{m_{S}=+1}$) with a duration of $T_{swp}$ is applied. Then, the resultant population change is readout with the RO laser pulse. Two additional reference channels (REF1 and REF2) are used for data normalization. REF1 corresponds to the maximum PL detected in Rabi measurement, and REF2 corresponds to the minimum PL. The normalized PL is $(Rabi-REF2)/(REF1-REF2)$. This normalization procedure will reduce degrees of freedom in data fitting. As shown in Fig. 2(e), the normalized PL shows oscillations as the MW1 pulse length is varied. These are the Rabi oscillations that prove the coherent manipulation of the NV spin states. From the period of the oscillations, the duration of the $\pi$, $\pi/2$, and $3\pi/2$ pulses can be determined. In the present case, we obtained the $\pi$, $\pi/2$, and $3\pi/2$ pulse duration to be 46, 92, and 138 ns, respectively.

Next, we characterize the spin decoherence time ($T_{2}$) using a Carr-Purcell-Meiboom-Gill (CPMG) sequence. The particular sequence used here consists of eight $\pi$ pulses, which is referred to later as a CPMG-8 sequence (see Fig. 3(a)). In the CPMG-8 experiment, after the preparation of the superposition state $\frac{1}{\sqrt{2}}(\ket{+1}+\ket{0})$ by the $(\pi/2)_{Y}$ pulse, a chain of $(\pi)_{X}$ pulses are applied at every 2$\tau$ to keep the coherence state for an extended period. The resultant coherent state is mapped into the $\ket{m_{s}=0}$ state with the application of the second $(\pi/2)_{Y}$ pulse (SIG1). In addition, in a different signal channel, we use a $(3\pi/2)_{Y}$ pulse instead of the $(\pi/2)_{Y}$ pulse to map the state into the $\ket{m_{s}=1}$ state (SIG2). Two signal channels are used to improve the signal-to-noise ratio (SNR) by removing systematic noise and doubling signal contrast. PL intensity is normalized using the same method used in the Rabi oscillation experiment. As seen in Fig. 3(b), an oscillating and decaying difference in the normalized PL is observed. These features in the echo signal are attributed to the electron spin echo envelope modulation (ESEEM) effect due to hyperfine interaction with nearby nuclear spins.Childress et al. (2006b) As shown in Fig. 3(c), we found that the simulation of the ESEEM signal can explain the observed signal taken with the CPMG sequence. In the simulation, we consider bath ¹³C nuclear spins and a ¹³C individual spin with the hyperfine couplings of $A=0.314$ MHz and $B=2.827$ MHz. We also obtained the spin decoherence time ($T_{2}$) to be $38\pm 3$ $\mu$s. This detection of the individual ¹³C nuclear spin was observed only with NV1, not with other NVs studied in this work. The details of the simulation are included in the supplementary material.

Finally, we perform NV-detected EPR (NV-EPR) spectroscopy to detect electron spins surrounding NV1. In the NV-EPR experiment, we use a DEER sequence shown in Fig. 4(a). We use two microwaves (MW1 and MW2) to coherently control the NV spin and target environment spins, respectively. The first microwave (MW1) is fixed at NV Lamour frequency. The second microwave (MW2) is swept in frequency, and when the frequency of MW2 is at the Lamour frequency of the target spins, the MW2 pulse flips the target spins. As can be seen from Fig. 4(a), the sequence consists of the CPMG component for the NV spin and the $\pi$ pulses for the target spins (denoted as CPMG-DEER). In the CPMG-DEER experiment, first, the NV’s spin state is prepared into $\ket{\psi_{0}}=\frac{1}{\sqrt{2}}(\ket{+1}+\ket{0})$ state. The spin state is sensitive to small changes in the external magnetic field $B_{0}$, which in our case is the total dipole field $B_{dip}$ from surrounding electron spins. In the present case, with the application of the $\pi$ pulses with MW2 (See Fig. 4(a)), phase shifts due to changes of the magnetic field are accumulated to the NV coherent state, i.e., $\ket{\psi_{0}}\rightarrow\ket{\psi}=\frac{1}{\sqrt{2}}(\ket{+1}+e^{i\delta\phi}\ket{0})$. The phase shifts can be modeled as,

where $\hbar$ is the reduced Planck constant. The pulse length is relatively short compared to $\tau$ time, so the evolution during pulses is neglected in the equation. The spin flip-flop not due to pulse excitation is also neglected.Stepanov and Takahashi (2016) $B_{dip}(t)=\frac{\mu_{0}}{4\pi}\sum_{i}g_{e}\mu_{B}(3cos^{2}\theta_{i}-1)\sigma_{i}/r^{3}_{i}$ is the total magnetic field generated by the target spin(s) at the NV. $g_{e}$ is the $g$-value of the target spin. $\theta_{i}$ and $r_{i}$ are the angle and distance between the NV spin and target spin. $B_{dip}$ is assumed to be constant throughout one evolution period. When the MW2 pulse is not on resonance, it does not drive any state evolution. As a result, the odd and even terms in Eqn. 2 are canceled. On the other hand, when the frequency of MW2 is on resonance, the MW2 pulse flips the target spins. This spin flip changes the sign of $B_{dip}$ in even terms of Eqn.2, i.e., $B_{dip}\rightarrow-B_{dip}$, allowing the phase to accumulate over the total sensing period. Namely, Eqn.2 can be rewritten as,

It is worth noting that even though the bath spins are not polarized to be in the same state in each measurement, we can still observe the NV-EPR signal since it is due to the statistical average of the spin polarization instead of a thermal polarization. In general, the NV-EPR signal due to a phase shift of $\delta\phi$ is given by,

where $\left<...\right>$ represents the statistical average. In the present case, we consider a case where the phase shifts are caused by the flips of target spins with the application of the MW2 pulses ($T_{MW2}$). Then, the NV-EPR signal is given by,Stepanov and Takahashi (2016)

where $\sigma_{j}$ is +1 or -1 representing $\ket{\uparrow}$ or $\ket{\downarrow}$, respectively. $\left<...\right>_{\sigma_{j}}$ here denotes the statistical average of all conditions with different combinations of $\sigma_{j}$ values. $\mu_{0}$ is the vacuum magnetic permeability. $T_{MW2}$ is the MW2 pulse duration. $\omega_{j}$ is the Rabi frequency associated with dipole interaction strength. Similar to the experiment in Fig. 2(d), the Gaussian decay is included to take into account a decay in Rabi oscillationsAbeywardana et al. (2016) where $T_{0}$ is the decay constant. When the NV-EPR signal is from a single spin, the oscillations are given by a single sinusoidal function. When it is from two spins, the oscillations are given by a sum of two sinusoidal functions, etc. Therefore, the analysis of the oscillations can be used to determine the number of spins coupled to the NV center as well as their coupling strength.

Next, we introduce another NV-EPR experiment to determine the number of spins and strength of interactions detected in the NV-EPR signal. The sequence is shown in Fig. 5(a). This sequence is modified from the CPMG-DEER sequence to measure the Rabi oscillations of the NV-EPR signal, so we call it a DEER-Rabi sequence. In the DEER-Rabi sequence, the MW2 frequency is fixed at the EPR signal, and the pulse length ($T_{MW2}$) is varied while keeping the total echo evolution time fixed. Both SIG1 and SIG2 channels are used as well for better SNR. Data is shown in Fig. 5(b). The normalized DEER-Rabi signal (SIG2-SIG1) shows a strong oscillation in the population recovery (shown in Fig. 5(c)). The observed oscillations are due to the change of dipole field from spins felt by the NV coherence state. The MW2 pulses rotate the spins according to the pulse length, changing the strength of the field along the NV axis. The accumulated phase shifts, therefore, undergo oscillations. To analyze the signal (shown in Fig. 5(c)) with Eqn. 5, we consider three situations where the number of target spins ranges from 1 to 3. In the fit, the dipole interaction strength and decay constant are treated as fitting parameters. We set the range for interaction strength to be 2-312.5 MHz. The lower bound is taken to have one full oscillation in the total recorded pulse length, and the higher bound is taken to have one full oscillation for every four data points. This follows from the frequency resolution being 1/(measurement time) and maximum frequency being (sample rate)/2 for a discrete Fourier transform analysis. The fitted lines are plotted as dotted lines in Fig. 5(c). The adjusted R-square value ($1-(1-\frac{\sum(y_{i}-\hat{y}_{i})^{2}}{\sum(y_{i}-\bar{y})^{2}}\frac{n-1}{n-k-1})$) is used, where $k$ = 3. In general the value is between 0 and 1, and the closer case to 1 means better agreement. The two electron spin model has the best fit with the adjusted R-square value of 0.314, while one spin model gives 0.269 and three spin model gives 0.233. From the fitting result, two interaction strengths are $\omega_{1}=2\pi\times(1.12\pm 0.13)$ MHz and $\omega_{2}=2\pi\times(2.24\pm 0.17)$ MHz. The time constant $T_{0}$ is fit to be $0.34\pm 0.06$ $\mu$s. The inset of Fig. 5(c) summarizes the detected two electron spins and one nuclear spin from the present study with NV1.

Finally, we discuss the origin of the NV-EPR signal observed in the present investigation (see Fig. 4(c) and (d)). The signal is a single peak of coherence population difference centered at $914.7\pm 0.9$ MHz with a width of $9\pm 2$ MHz. No signature of the hyperfine splitting is observed. We attribute this signal to an electron spin with $S=1/2$ and $g=2.009\pm 0.003$. Since there are no signatures of hyperfine peaks, the signal is not from P1 centers, which are commonly observed in diamonds with a high nitrogen concentration. The identification of the detected spins remains unclear, but based on the $S$ and $g$-value and lack of the hyperfine splitting, possible candidates of the observed spins are the tri-Nitrogen W21 centerLoubser and van Wyk (1978); Baker (1999) and surface spins.Grinolds et al. (2014); Peng et al. (2019, 2020)

In summary, we demonstrated the detection of EPR from two electron spins using a single NV center in diamond. CPMG-DEER and DEER-Rabi sequences are used to study the spin interactions. The observed NV-EPR signal is in the absence of strong hyperfine peaks, and further measurement of Rabi oscillations of the signal suggests the interaction is of a single NV center with two electron spins. The physical system and method presented in this paper allow easy identification of a single defect center coupled to a small number of spins and characterization of the interaction strength. The analysis of the NV-EPR signal presented can be a method to distinguish the number of interacting spins up to five, which is limited by the detection time window of the oscillations.

The non-equal but measurable interaction strength is favorable in studying non-equilibrium quantum dynamics. Investigation of quantum phase dynamics, like spin liquids or discrete time crystalline (DTC), requires interacting spin systems with disordered structures.Broholm et al. (2020); Bussandri et al. (2024) Cu²⁺ ions formed kagome latticeNorman (2016) and Ru³⁺ ions formed honeycomb latticeTakagi et al. (2019) has been actively researched materials for realization of spin liquid. This multi electron spin system could be a material to look for these emergent quantum effects with the similarity of being a spin 1/2 magnetic disordered type of material. A chain of 9 ¹³C nuclear spins in diamond has been shown to be a programmable spin-based DTC.Randall et al. (2021) The characterizable electron spin system in this study is potentially a smaller-sized system to realize DTC. A good knowledge of the interaction strengths in a spin system will also be significant in better characterization of non-classical correlations, namely quantum entanglement, which can be key in modeling open quantum dynamics and developing entangled quantum sensing. Quantum-enhanced NMR with a 9.4-fold sensitivity increase has been demonstrated by entangling 9 proton spins with a ³¹P spin.Jones et al. (2009) The electron spins detected in this study can also be simultaneously controlled and potentially achieve entanglement enhancement beyond the standard quantum limit. In addition, further study can be done to reveal more information about the nature of the spin species and improve the understanding of the decoherence of shallow NV centers, paving the pathway to higher standard engineering for quantum sensing applications.

The supplementary material includes a discussion and simulation of the ESEEM effect observed in the experiment and additional measurement results on the other NV centers.

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