Abstract
Quantum sensing exploits the strong sensitivity of quantum systems to measure small external signals. The nitrogenvacancy (NV) center in diamond is one of the most promising platforms for realworld quantum sensing applications, predominantly used as a magnetometer. However, its magnetic field sensitivity vanishes when a bias magnetic field acts perpendicular to the NV axis. Here, we introduce a different sensing strategy assisted by the nitrogen nuclear spin that uses the entanglement between the electron and nuclear spins to restore the magnetic field sensitivity. This, in turn, allows us to detect small changes in the magnetic field angle relative to the NV axis. Furthermore, based on the same underlying principle, we show that the NV coupling strength to magnetic noise, and hence its coherence time, exhibits a strong asymmetric angle dependence. This allows us to uncover the directional properties of the local magnetic environment and to realize maximal decoupling from anisotropic noise.
Introduction
Quantum sensing harnesses the coherence of wellcontrolled quantum systems to detect small signals with high sensitivity^{1,2,3}. Typically, an external signal directly leads to a shift of the quantum sensor’s energy levels. Ancillary sensors, which do not interact with the signal directly, can assist the main sensor by, for example, acting as a longlived quantum memory^{4,5,6,7,8} or providing error correction^{9,10,11,12,13,14}. Solid state spins are promising platforms for quantum sensing techniques and applications, among which nitrogenvacancy (NV) centers in diamond have received the most attention^{2,15,16,17}. The electron spin associated with the negatively charged NV center has long coherence time even at room temperature and is capable of detecting a variety of signals with high sensitivity and nanoscale resolution. These include magnetic^{18,19,20} and electric fields^{21,22,23,24,25,26,27,28}, temperature^{29,30,31,32}, and pressure^{33,34,35}.
For the NV to act as a magnetometer, a bias magnetic field along the NV axis is generally required to put the electron spin (S = 1) in the \(\left{m}_{S}=0,\pm\! 1\right\rangle\) basis, such that the energy levels are firstorder sensitive to magnetic field perturbations^{2}. However, this method fails when the bias field turns toward the direction perpendicular to the NV axis, where the Zeeman interaction no longer induces energy shifts between the levels. To unlock the full potential under this unfavored condition, we introduce an entirely different sensing approach assisted by the ancillary ^{15}N nuclear spin. The essential principle is based on the sensitivity of the hyperfine interaction (between the electron and nuclear spins) to small magnetic signals. By monitoring the entanglement between the two spins using spinecho sequences, we detect small changes in the magnetic field angle. Furthermore, similar to the hyperfine interaction, we show that the coupling between the electron spin and magnetic noise also sensitively depends on the bias field angle, which can be further employed to distinguish and characterize anisotropic noise in the environment. Our exploration extends the capabilities of the versatile sensing toolkit of the NV center, and this sensing strategy based on the interaction between the main sensor and ancillary sensor can be implemented on other quantum sensing platforms as well.
Our experiments were done under ambient conditions, as schematically depicted in Fig. 1a, b. The singlecrystal diamond chip contains individually resolvable ^{15}NVs. A metal stripline fabricated on the diamond surface delivers microwave currents to manipulate the NV spin states. A cylindrical permanent magnet exerts a magnetic field at the NV and provides a coarse control of the field angle (θ_{B}), while a small DC current flowing in the stripline finetunes θ_{B} in the close vicinity of 90°. We define \(\hat{z}\) to be the NV axis and \(\hat{x}\) to be the bias magnetic field direction when it is exactly perpendicular (Fig. 1a).
The paper is structured as follows. We begin by analyzing the NV electron energy eigenstates as the magnetic field angle θ_{B} varies around 90°. In particular, we study how the angle modulates the electron spin operator expectation values 〈S〉 at each state, and consequently affects its interaction with the ^{15}N nuclear spin. Next, we demonstrate magnetic field angle sensing by using spinecho interferometry to measure the angledependent hyperfine interaction. Lastly, we show that the NV coherence exhibits an asymmetric angle dependence, which originates from anisotropic noise in the environment.
Results
Electron spin eigenstate properties
The NV spin ground state Hamiltonian H_{gs} can be written as:
where H_{e} and H_{n} denote the Hamiltonians associated with the electron spin (S = 1) and ^{15}N nuclear spin (\(I=\frac{1}{2}\)), respectively. D_{gs} ≈ 2.87 GHz is the zerofield splitting, γ_{B} ≈ 2.87 MHz/G is the electron spin gyromagnetic ratio, γ_{N} ≈ 0.4316 kHz/G is the ^{15}N nuclear spin gyromagnetic ratio, and A is the hyperfine tensor with only diagonal elements: A_{xx} = A_{yy} ≈ 3.65 MHz and A_{zz} ≈ 3.03 MHz^{36,37,38}. S_{x,y,z} and I_{x,y,z} are the spin1 and spin\(\frac{1}{2}\) Pauli matrices, respectively. We applied the bias field ∣B∣ > 60 G, such that H_{e} always dominates over H_{n} (at any θ_{B}). The electron eigenstates, denoted by \(\left0,\pm\! \right\rangle\) throughout the paper, are thus mainly determined by H_{e}, and H_{n} splits each state into two nuclear spin sublevels (see Supplementary Note 2B).
Under a bias magnetic field along \(\hat{z}\) (θ_{B} = 0), i.e., the conventional magnetometry condition, the electron eigenstates are \(\left{m}_{S}=0,\pm\! 1\right\rangle\) (Fig. 1c). As the bias field direction rotates (θ_{B} ≠ 0), the eigenbasis changes. At θ_{B} = 90° (B_{z} = 0, B_{x} > 0), the eigenstates are as follows: \(\left0\right\rangle \approx \left{m}_{S}=0\right\rangle ,\left\right\rangle =\frac{1}{\sqrt{2}}(\left{m}_{S}=+1\right\rangle \left{m}_{S}=1\right\rangle ),\left+\right\rangle \approx \frac{1}{\sqrt{2}}(\left{m}_{S}=+1\right\rangle +\left{m}_{S}=1\right\rangle )\), and the states \(\left\pm \!\right\rangle\) split in energy by \(\approx \frac{{\gamma }_{B}^{2}{B}_{x}^{2}}{{D}_{\rm{gs}}}\) (Fig. 1c). Under large B_{x}, \(\left{m}_{S}=0\right\rangle\) and \(\frac{1}{\sqrt{2}}(\left{m}_{S}=+1\right\rangle +\left{m}_{S}=1\right\rangle )\) are slightly hybridized in composing \(\left0\right\rangle\) and \(\left+\right\rangle\), hence the approximate equality in the above expressions. This hybridization results in finite 〈S_{x}〉 values for \(\left0\right\rangle\) and \(\left+\right\rangle\): \({\langle {S}_{x}\rangle }_{0}\,<\,0,{\langle {S}_{x}\rangle }_{+}\,>\,0\) (see Supplementary Note 2A).
The states \(\left\pm \!\right\rangle\) are equal superpositions of \(\left{m}_{S}=\pm\! 1\right\rangle\) at θ_{B} = 90°, hence \({\langle {S}_{z}\rangle }_{\pm }=0.\) However, as θ_{B} deviates from 90°, the \(\left{m}_{S}=\pm\! 1\right\rangle\) amplitudes are no longer equal. Due to the large zerofield splitting, the imbalance grows rapidly with the offangle Δθ_{B} ≡ θ_{B} − 90°, and consequently, \({\langle {S}_{z}\rangle }_{\pm }\) acquire finite values (Fig. 2a). On the other hand, 〈S_{x}〉 barely changes (Fig. 2b). The change in 〈S_{z}〉 dramatically affects the electron spin interaction with the nuclear spin, thus providing a way to sense Δθ_{B}.
Angledependent hyperfine interaction
A given electron state exerts effective hyperfine fields at the nuclear spin, determined by its spin operator expectation values. Specifically, the nuclear spin Hamiltonian is \({H}_{\rm{n}}={A}_{  }{I}_{x}{\langle {S}_{x}\rangle }_{\left{\psi }_{e}\right\rangle }+{A}_{\perp }{I}_{z}{\langle {S}_{z}\rangle }_{\left{\psi }_{e}\right\rangle }+{\gamma }_{N}\left({B}_{x}{I}_{x}+{B}_{z}{I}_{z}\right)\), where \(\left{\psi }_{e}\right\rangle =\left0,\pm\! \right\rangle\). H_{n} splits each electron state (\(\left0,\pm\! \right\rangle\)) into two nuclear sublevels and the splitting energy (ћω) can be obtained by diagonalizing H_{n}. Figure 2c plots ω as a function of θ_{B} calculated under ∣B∣ = 65 G and Fig. 2d zooms in on a small angle range centered at 90°. The \(\left\right\rangle\) state splitting (ω_{−}) is especially interesting: it grows linearly with Δθ_{B}. The slope \(\frac{d{\omega }_{}}{d{\theta }_{B}}\), as we will see soon, directly determines the angle sensitivity.
Nuclear spin assisted angle sensing
We now demonstrate detection of small angle changes using the anglesensitive hyperfine interaction. Either ω_{−} or ω_{+} can be used, as they both change with θ_{B}. We choose to use ω_{−} since its angle dependence is steeper. To measure this quantity, we performed electron spinecho interferometry, where the spinecho signal is dramatically affected by the hyperfine splitting due to the electronspinechoenvelopemodulation (ESEEM) effect^{39,40,41,42}.
A typical spinecho sequence is shown in Supplementary Fig. 3. The electron is first prepared in a superposition of \(\left0\right\rangle\) and \(\left\right\rangle\), and then accumulates phase during the free evolution time τ between the two π/2 pulses. The π refocusing pulse decouples the fields at frequencies other than 1/τ.
The ESEEM effect occurs when the nuclear spin undergoes Larmor precession, with the frequency (ω_{0} or ω_{−}) conditioned on the electron state (\(\left0\right\rangle\) or \(\left\right\rangle\)). Consequently, the electron and nuclear spins are periodically entangled and disentangled at a rate determined by ω_{0} and ω_{−}. The spinecho amplitude measures the electron coherence, which is directly affected by its entanglement with the nuclear spin, hence exhibiting collapses and revivals. As ω_{−} is highly sensitive to θ_{B}, the spinecho signal shows angledependent modulation patterns. A more detailed analysis is given in Supplementary Note 2C, where the spinecho signal P obtains a simple expression: \(P=1 {\hat{\omega }}_{0}\times {\hat{\omega }}_{}{ }^{2}{\sin }^{2}\left(\frac{{\omega }_{0}\tau }{4}\right){\sin }^{2}\left(\frac{{\omega }_{}\tau }{4}\right)\).
We performed spinecho experiments between \(\left0\right\rangle\) and \(\left\right\rangle\) under ∣B∣ ≈ 65 G, as θ_{B} varied between 89° and 91° (Fig. 3a). It shows good agreement with the above expression for P (Fig. 3b). At fixed τ = 2.2 μs, the spinecho signal can sensitively detect small angle changes at the largest slope (red dashed line in Fig. 3c). The corresponding sensitivity is ~13\(\frac{mdeg}{\sqrt{Hz}}\), provided the single NV fluorescence ~100 kcps and optical contrast ~15% in the experiment (see Supplementary Note 2D). To get an overall picture, Fig. 3d expands on Fig. 3b, showing the spinecho signal P in broader θ_{B} and τ ranges.
Taking the derivative of P with respect to θ_{B}, we obtain an analytical expression of the angle sensitivity η (see Supplementary Note 2D):
where F represents the NV fluorescence, C the optical contrast of different spin states, t_{ini} the spin initialization time and T_{r} the spin state readout time. \({\gamma }_{\theta }\equiv \frac{d{\omega }_{}}{d{\theta }_{B}}\) denotes the slope of the angledependent hyperfine interaction (Fig. 2d). Note that γ_{θ} is playing an analogous role as the gyromagnetic ratio in conventional magnetometry. The denominator in Eq. (1) causes modulation in τ, i.e. the angle sensitivity is periodically lost and regained at different τ times (Fig. 3d), and η^{*} in Eq. (2) is the modulation envelope. To optimize the sensitivity, we need to pick τ that maximizes \( {\sin }^{2}\left(\frac{{\omega }_{0}\tau }{4}\right)\cdot \sin \left(\frac{{\omega }_{}\tau }{2}\right)\). As examples, sensitivities were evaluated at τ = 2.2 μs and 11 μs and plotted in Fig. 3e, represented by the blue and orange solid curves, respectively.
On the other hand, conventional magnetometry also detects magnetic field angle changes via the electron Zeeman interaction. Its angle sensitivity is proportional to the static magnetic field sensitivity along \(\hat{z}\): \({\eta }_{con}=\frac{{\eta }_{Bz}}{ B \sin {\theta }_{B}}\). With a parallel bias magnetic field, η_{Bz} is typically between tens of \(\frac{nT}{\sqrt{Hz}}\) and a few \(\frac{\mu T}{\sqrt{Hz}}\) depending on experimental parameters^{15,16}, and η_{Bz} decreases as the bias field turns toward a perpendicular direction (see Supplementary Note 2E). Figure 3e plots η_{con} as a function of θ_{B} assuming η_{Bz} is originally 300 or \(800\frac{nT}{\sqrt{Hz}}\) under a parallel bias field.
As illustrated in Fig. 3e, our nuclearassisted approach and conventional magnetometry work in complimentary regimes. The conventional method works well until θ_{B} approaches 90° (Fig. 3e inset), when \(\frac{{\gamma }_{B}^{2}{B}_{x}^{2}}{{D}_{gs}}\gtrsim 2{\gamma }_{B}{B}_{z}\) so the electron eigenbasis changes from \(\left{m}_{S}=0,\pm\! 1\right\rangle\) to \(\left0,\pm\! \right\rangle\), and after that the nuclearassisted approach takes over. The sensitivities of both methods are limited by lowfrequency noise, up to different effective coupling constants (see Supplementary Note 2E).
Detection of anisotropic noise
The NV couples to magnetic field noise through its electron spin operators. Similar to the hyperfine interaction, the noise coupling strength, and hence the spin coherence, also exhibits strong angle dependence. We show that this provides a useful way to distinguish and characterize anisotropic noise.
Magnetic field noise δB(t) induces transition energy fluctuations: δE_{±,0}(t) = δB(t) ⋅ (〈S〉_{±} − 〈S〉_{0}), degrading the coherence between the states \(\left0\right\rangle\) and \(\left\pm\! \right\rangle\). It can be shown that the coherence is affected by the variance of the fluctuation 〈δE^{2}〉 (see Supplementary Note 3A). Recalling 〈S_{z}〉 (〈S_{x}〉) in Fig. 2a, b, we get:
The last terms in Eqs. (3) and (4) suggest the coherence is sensitive to the correlation between δB_{x} and δB_{z}, which is nonzero for anisotropic noise. Since \({\langle {S}_{z}\rangle }_{\pm }\) is an odd function of Δθ_{B}, while \({\langle {S}_{x}\rangle }_{0}\) an even function, depending on the sign of 〈δB_{x}δB_{z}〉, the coherence is longer on one side than the other around θ_{B} = 90°, and the states \(\left\pm\! \right\rangle\) show opposite asymmetry. If the noise is isotropic, i.e., 〈δB_{x}δB_{z}〉 = 0, the coherence is then symmetric around 90°. Therefore, by examining the angle dependence of the coherence time, we can distinguish the anisotropic noise and further characterize its direction based on the asymmetry. This directional information, on the other hand, cannot be obtained under a parallel bias magnetic field (θ_{B} ≈ 0°), where the NV only couples to the \(\hat{z}\) component of the noise.
The coherence asymmetry observed in our experiment (Fig. 4a, b) indicates that the noise coupled to the NV is anisotropic. We argue in Supplementary Note 3B that this is likely due to the dipolar interaction with a few nearby randomlyflipping spins (such as P1 centers or ^{13}C nuclear spins). To further illustrate this effect, we performed spinecho simulation under a fully anisotropic noise, where the noise only fluctuates along a straight line at −45°/+135° relative to \(+\hat{x}\) (Fig. 4e), such that δB_{x}(t) + δB_{z}(t) = 0. Under this condition, there exist optimal angles at which the noise coupling is maximally suppressed (white dashed lines in Fig. 4c, d).
Discussion
While conventional NV magnetometry fails when the bias magnetic field orients perpendicular to the NV axis, here we demonstrated a method which uses the electron eigenbasis change for sensing. The achieved sensitivity (η ≈ 13\(\frac{mdeg}{\sqrt{Hz}}\)) can be further improved by using cleaner diamond samples with less noise (e.g., by using ^{12}C enriched diamonds^{43} or chemical termination to reduce surface spins^{44}), increasing photon collection efficiency (e.g. by fabricating microlens^{45,46,47,48,49} or pillars^{50,51,52,53,54}) or using NV ensembles^{55,56,57}. For example, as the sensitivity is improved by the square root of the number of NVs, a typical ensemble density of 10^{12}/cm^{2} can achieve η < 0.3\(\frac{mdeg}{\sqrt{Hz}}\). Our work expands the already remarkable sensing versatility of the NV center. One example of the use of angle sensing is to allow precise tuning of a perpendicular magnetic field to the NV axis for electric field sensing^{21,22,23,24,27,28}.
Our sensing strategy relies on measuring the interaction with an ancillary sensor. Implementing this idea to other quantum sensing platforms requires two key ingredients. First, the angledependent hyperfine interaction is due to the NV eigenbasis change, which occurs when \(\frac{{\gamma }_{B}^{2}{B}_{x}^{2}}{{D}_{gs}} \sim {\gamma }_{B}{B}_{z}\). At θ_{B} ≈ 90°, this condition becomes \(\frac{{\gamma }_{B}^{2} B{ }^{2}}{{D}_{gs}} \sim {\gamma }_{B} B \sin {{\Delta }}{\theta }_{B}\), i.e. \({{\Delta }}{\theta }_{B} \sim \frac{{\gamma }_{B} B }{{D}_{gs}}\), indicating that the high angle sensitivity arises from the large zerofield splitting (D_{gs}). Second, the hyperfine interaction is the dominating term in the nuclear spin Hamiltonian, which directly affects its entanglement with the electron spin and therefore the spinecho signal.
We also demonstrated a method to distinguish and characterize anisotropic noise in the XZ plane (i.e. δB_{x} and δB_{z}) based on the asymmetric angle dependence of coherence time. By further rotating the bias field in the XY plane (perpendicular to the NV axis), we could obtain information along all directions, which can help locate noise sources at the nanoscale. Moreover, we showed that the anisotropic component of the noise can be maximally decoupled by applying a bias field at the optimal angle, which may allow a great improvement of the NV spin coherence.
Methods
Diamond sample
The diamond sample was an electronicgrade CVD diamond (Element Six), with a natural abundance (1.1%) of ^{13}C impurity spins. Negativelycharged NV centers were created by ion implantation (18 keV) followed by vacuum annealing at 800 °C for 2 h. NV experiments were performed on a homebuilt confocal laser scanning microscope. A 532 nm green laser was used to initialize the NV spin to the m_{S} = 0 state and generate spindependent photoluminescence for optical readout.
Spin dynamics simulation
The simulation of NV spinecho decay was done using the QuTip (Quantum Toolbox in Python) software^{58,59}. To simulate the coupling with the anisotropic magnetic noise, collapse operators were defined, and the time evolution was governed by the Lindblad master equation. More details are provided in Supplementary Note 4.
Data availability
The data generated and analyzed during this study are available from the authors upon reasonable request.
Code availability
The code used for simulation presented in this study is available from the authors upon reasonable request.
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Acknowledgements
We gratefully thank Norman Y. Yao for valuable discussions and advice. Z.Q. also thanks Matthew Turner for diamond annealing assistance. This work was primarily supported by ARO Grant No. W911NF1710023. Fabrication of samples was supported by the U.S. Department of Energy, Basic Energy Sciences Office, Division of Materials Sciences and Engineering under award DESC0019300. A.Y. also acknowledges support from the STC Center for Integrated Quantum Materials, NSF Grant No. DMR1231319, and the Aspen Center of Physics supported by NSF grant PHY1607611.
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A.Y. and Z.Q. conceived the project. Z.Q. performed the experiments, analyzed data, and produced figures. Z.Q., U.V., A.H. and A.Y. discussed the results and contributed to the writing of the manuscript. A.Y. supervised the project.
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Qiu, Z., Vool, U., Hamo, A. et al. Nuclear spin assisted magnetic field angle sensing. npj Quantum Inf 7, 39 (2021). https://doi.org/10.1038/s41534021003746
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DOI: https://doi.org/10.1038/s41534021003746
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