Estimation of vector static magnetic field by a nitrogen-vacancy center with a single first-shell <sup>13</sup>C nuclear (NV

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Jiang Feng-Jian, Ye Jian-Feng, Jiao Zheng, Huang Zhi-Yong, Lv Hai-Jiang. Estimation of vector static magnetic field by a nitrogen-vacancy center with a single first-shell ¹³C nuclear (NV–¹³C) spin in diamond. Chinese Physics B, 2018, 27(5): 057601 Copy to clipboard

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Estimation of vector static magnetic field by a nitrogen-vacancy center with a single first-shell ¹³C nuclear (NV–¹³C) spin in diamond

Jiang Feng-Jian^†,, Ye Jian-Feng, Jiao Zheng, Huang Zhi-Yong, Lv Hai-Jiang^‡

School of Information Engineering, Huangshan University, Huangshan 245041, China

† Corresponding author. E-mail: jfjiang@mail.ustc.edu.cn

luhj9404@mail.ustc.edu.cn

Abstract

We suggest an experimental scheme that a single nitrogen-vacancy (NV) center coupled to a nearest neighbor ¹³C nucleus as a sensor in diamond can be used to detect a static vector magnetic field. By means of optical detection magnetic resonance (ODMR) technique, both the strength and the direction of the vector field could be determined by relevant resonance frequencies of continuous wave (CW) and Ramsey spectrums. In addition, we give a method that determines the unique one of eight possible hyperfine tensors for an (NV–¹³C) system. Finally, we propose an unambiguous method to exclude the symmetrical solution from eight possible vector fields, which correspond to nearly identical resonance frequencies due to their mirror symmetry about ¹⁴N–Vacancy–¹³C (¹⁴N–V–¹³C) plane.

PACS: 76.70.Hb;76.30.Mi;07.55.Ge

Keyword:color centers;optical detection magnetic resonance (ODMR);magnetometer

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1. Introduction

The negatively charged nitrogen-vacancy (NV) color center^[1,2] in diamond has attracted great interest from various quantum information processing (QIP) applications^[3–8] to high-resolution sensing of magnetic field^[9,10] and imaging in life science^[11] by using ODMR technique,^[12] even at room temperature. Specifically, NV center has become a prominent magnetometer, the central idea of which is detecting the relative energy shift of ground state by an external AC or DC vector magnetic field.^[9,10,13,14] By detecting frequency and relevant shift, the corresponding parameters of external magnetic field could be precisely determined. The information of strength and polar angle of vector magnetic field (relative to NV symmetry axis) could be easily extracted by observing two resonance frequencies of continuous wave (CW) spectrums, but the information of azimuth angle is still missing due to the symmetry of the defect. One solution to this issue is that NVs were employed in a multi-NV vector magnetometer with different axes.^[13,15] However, these NVs are the best to be close to each other and differ by no more than hundreds of nanometers for achieving high spatial resolution. Another alternative strategy, suggested theoretically by Ref. [16], only used a single high-spin system (such as SiC with spin 3/2) as a vector magnetometer, which might further improve the spatial resolution.

In this paper, using a single NV center with a single first-shell ¹³C nuclear spin (NV–¹³C), an executable experiment scheme of detecting a static vector magnetic field is proposed in diamond. More information of vector magnetic field could be revealed by resonance spectrum. The reason is that the symmetry of the NV–¹³C can be reduced from to , a single mirror plane. Moreover, the energy level splitting of ¹³C nuclear spin is sensitive to the changes of external magnetic field in contrast to ¹⁴N nuclear spin, which remains almost unchanged. Due to the distance between ¹³C nucleus and NV electron less than 1 nm, the advantage of our proposal is that high spatial resolution conditioning precise measured hyperfine coupling tensor of NV–¹³C,^[17–21] high contrast of the fluorescent readout, and narrow linewidth of resonance spectrum lines^[22] may be achieved by experimental implementation.

2. A static vector magnetic field detection with an NV–¹³C

2.1. The NV–¹³C system

An NV–¹³C sensor consists of a substitutional ¹⁴N and a nearest neighbor ¹³C nuclear spins adjacent to the vacancy defect. The spin Hamiltonian of the system under an applied static vector magnetic field can be written in a coordinate system, in which the NV axis is aligned to z axis, and x axis lies in the plane constructed by ¹⁴N–V–¹³C (seen in Fig. 1). The combined spin system can be written by Hamiltonian

where vector magnetic field

can be expended to

, θ and ϕ are the polar and the azimuth angles relative to z and x axes in the NV frame of reference, respectively. The gyromagnetic ratios of NV electron, ¹³C and ¹⁴N nuclear spins are

, and

, respectively. Zero-field splitting D and quadrupole splitting P of ¹⁴N correspond to 2.87 GHz and −5 MHz, respectively. The last two terms of Eq. (1) respectively represent the NV electron spin coupled to ¹⁴N and proximal ¹³C nuclear spin. Since the hyperfine splitting of ¹⁴N nuclear spin (spin I = 1) is a constant of −2.16 MHz^[23] and insensitive to changes of applied vector magnetic field, we could only consider the hyperfine structure of NV spin coupled to the ¹³C nucleus written as

where the NV–¹³C hyperfine interaction

, when choosing the x axis such that

without lack of generality. From Eq. (2), eigenvalues

in descending order and corresponding eigenvectors

of ground state can be obtained. For sub-manifold

with the hyperfine splitting of ¹³C nuclear spin, the relevant eigenvectors

could be approximately equal to

, where the ¹³C nuclear spin states

could be approximated as

with

.^[24] The ordering of the corresponding energy level of eigenvectors

could be exchanged by the switching of polar angle θ between

and

For sub-manifold

, its corresponding eigenvectors

could be approximately equal to

in which the nuclear states

of interest depended entirely on applied vector magnetic field and could be written as

Concrete schematic diagram can refer to Appendix A. Specifically, the quantization axis of sub-manifold m_s = 0 and corresponding spin dynamics could be affected by the microwave power strength.

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	Fig. 1. (color online) A static vector magnetic field is applied. NV–¹³C frame of reference is defined by x, y, z axes, where ϕ and θ represent the azimuth and the polar angles with respect to the x and z axes, respectively.

2.2. The determination of eight candidated hyperfine tensors for a single NV–¹³C

In the NV–¹³C frame of reference, the symmetric hyperfine tensor

associated with the nearest ¹³C coupled to NV spin, was determined in Ref. [19], where

for

is the component of eight candidated hyperfine tensors,

. The components are listed in Appendix B, in which the signs of

can be positive or negative, and can be linked by a π rotation transformation around the z axis.

However, when NV–¹³C is used as a vector magnetometer, its hyperfine tensor should be uniquely identified instead of multiple candidates due to mirror symmetry. A simple scheme may be feasible as an initial state of NV electron spin with ¹³C nuclear spin state is polarized^[1,24] and prepared. Under an applied magnetic field aligned along the NV–¹³C axis (z-axis), its total free evolution under the Hamiltonian (2) demonstrates eight different evolution processes conditioned the chosen hyperfine tensor. The evolving states are projected onto the state with evolving populations as shown in Fig. 2. Along the way, in experiment one needs to observe the free induction decay (FID) signal of the NV–¹³C. 3

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Fig. 2. (color online) The applied external magnetic

is parallel to the NV axis. Red and dashed red lines correspond to hyperfine tensor

with

and

, respectively. Blue and dashed blue lines correspond to hyperfine tensor

with

and

, respectively. Green and dashed green lines correspond to hyperfine tensor

with

and

, respectively. Black and dashed black lines correspond to hyperfine tensor

with

and

, respectively.

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Fig. 3. (color online) (a) The energy-level diagram of NV–¹³C center ground state. The hyperfine splitting of Δ and

correspond to m_s = 0 and

ground-state manifolds due to coupled to the nearest ¹³C nuclear spin. (b) Under the low power microwave, six resonance peaks could be observed due to the hyperfine interaction between electron and ¹⁴N nuclear spin. (c) Ramsey sequence for detecting the Larmor splitting Δ. Notations are defined in the main text.

2.3. The spin dynamics of ground state under microwave field

A microwave (MW) field is applied and causes transitions between the electron spin levels, which modulates the fluorescence intensity. At relative low microwave power, the hyperfine splitting of corresponds to two eigenstates of and between which the analytic form

of the effective Larmor splitting obtained by the second order perturbation theory reveals the direction of applied vector field.^[18] Through numerical calculations in moderate magnetic field (

), the above approximate formula of Δ conforms well to the numerical results based on Eq. (2). In the remaining cases of relatively high MW power, the Larmor splitting of sub-manifold m_s = 0 corresponds to two linear superposition states

of eigenstates

and

, whose quantization axis of nuclear spin is realigned to a new axis defined by nuclear spin states

.^[24]

At relatively low MW power, the doublet transitions between and could be observed. Taking into account the hyperfine splitting of the ¹⁴N nucleus, the double peak transitions above could further show six resonance lines as demonstrated schematically in Fig. 4(a). With the increase of MW power, only a single peak transition could be observed and its resonance frequency is almost centred between the doublet transitions of as shown in Fig. 4(b).

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Fig. 4. (color online) As an example, ODMR CW spectra was simulated with

for preset parameters θ = 57°, ϕ = (11°, 169°, 191°, or 349°), and

satisfying mirror symmetry about ¹⁴N–V–¹³C plane. (a) The relatively low MW power reveals doublet transitions of each resonance line in panel (b). The 24 resonance lines resulted from the ¹³C nucleus and the fixed splitting (2.16 MHz) were induced by ¹⁴N nucleus. The asymmetry of 24 resonance lines is due to the different transition probabilities between

and

. (b) At relatively high MW power, the NV–¹³C indicates four resonance spectral lines due to hyperfine coupling with the ¹³C nucleus. Notations are defined in the main text.

2.4. Numerical simulations of a static vector magnetic field detection

2.4.1. Simulation ODMR spectra of the NV–¹³C system

NV center can be optically addressed at room temperature using a confocal microscope combined with a photon-counting detection system.^[12] An external static vector magnetic field whose direction is relative to the NV–¹³C of interest is shown schematically in Fig. 1.

In the following, these frequency hyperfine magnetic resonance lines could be observed by pulsed CW ODMR spectra. The corresponding four resonance frequencies and the Larmor splitting Δ could be obtained as simulated in Fig. 4. For more precise measurement, the Larmor splitting could also be obtained by using Ramsey-type sequence .^[25] The spectrum in Fig. 4(a) shows 24 resonance peaks due to the coupling with the ¹⁴N nucleus. The fluorescence intensities of 24 spectral lines in Fig. 4 are commonly asymmetry. The reason is mainly that the doublet transition probabilities between and are different. Based on the calculations of MW transition matrix elements with j = 1,2,3,4 and i = 5,6, the change of the relative fluorescence intensities for the doublet transition depends on the polar angle θ of the vector field. Qualitative analysis can refer to Appendix C. It should be noted that if the magnetic field is higher than 200 Gs, the fluorescence of the NV center would be dramatically influenced and could not be measured by the CW spectrum.

2.4.2. The fault tolerance analysis of NV–¹³C sensor

The four resonance frequencies of electron spin are sensitive to the changes of polar angle θ, compared to its insensitivity of the changes of azimuth angle ϕ. However, the Larmor splitting Δ induced by ¹³C nuclear spin is sensitive to the changes of ϕ as shown in Fig. 5. Thus, to precisely detect a static vector magnetic field, both the two factors of and Δ should be taken into consideration.

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Fig. 5. (color online) (a) Red and blue lines correspond to the cases of

and

respectively. The four curves of each group (red or blue lines) associated with Larmor splittings Δ correspond to fixed four azimuth angles

, which can be distinguished by changes of Δ (at least ∼100 kHz/0.1 rad) for relatively large misalignment angle θ relative to the z axis. (b) Dashed and solid lines correspond to the cases of

and

respectively. The four resonance frequencies

of each group (dashed or solid lines) are sensitive and insensitive to the changes of θ and ϕ, respectively. In contrast to panel (a), the resonance frequencies

are almost unchanged (at most ∼2 kHz/0.1 rad) for the chosen fixed four different azimuth angles ϕ = 3π/36, 5π/36, 7π/36, and 9π/36. Notations are defined in the main text.

With a method akin to maximum likelihood estimation, our simulations implemented a numerical search procedure to match both the target four measured resonance frequencies and the measured Larmor splitting Δ as shown in Fig. 4 according to the formula similar to^[13]

where the

and Δ have linewidth

and δΔ, respectively. The linewidth δ is fundamentally limited by the imhomogeneous dephasing rate,

of the NV electron spin, which is decided by magnetic dipolar interactions with a bath of nuclear spin in diamond. The pulsed-ESR spectrum linewidth

is inversely proportional to the π-pulse duration limited by

. Furthermore, the linewidth could also be affected by power broadening, which is from the laser used for initializing electron spin and MW field used for spin manipulation. Thus, in experiment, the laser intensity and MW power should be decreased appropriately for achieving a sharpen linewidth.^[22] For Larmor splitting

Ramesy sequence could achieve sharper linewidth

. Therefore, we introduced random measurement errors

and

limited by linewidth

and

. One could pick the parameters

, and B₀ satisfying

that minimizes Eq. (3), compared to other choices of parameters. The search ranges of direction parameters

are set to

and

respectively.

With above method, the simulations of detecting vector direction with fixed are carried out and results are given in Fig. 6, where random measurement errors and are introduced. It should be noted that for large misalignment angle θ relative to the z axis, high resolution of azimuth angle ϕ may be easily achieved compared to those of small misalignment angle θ. At the same time, relatively higher magnetic field strength could improve the angular resolution of detected due to the higher resolution of Larmor splitting Δ for different azimuth angles ϕ as shown in Figs. 5(a) and 6(c).

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Fig. 6. (color online) (a) The detected magnetic field strength is set to 10 Gs. Each simulated azimuth angle ϕ (5°–85° with 10° intervals) with error bar is obtained from corresponding formula

with repetition cycles 10. The mean value of each cycle was obtained from 60 groups of random numbers

and

, which are limited by linewidths

and δΔ = 26 kHz, respectively. (b) As an example, the statistical distributions of simulated measurement of ϕ are demonstrated with θ = 60°±0.2°. (c) With the increasing magnetic field strength, relatively higher angle resolution of ϕ could be achieved. The searched target azimuth and polar angles are set to ϕ = 65° and θ = 20°, respectively. Notations are defined in the main text.

Next, taking into account the errors of hyperfine components referred in Appendix A, we found that for large misalignment angle θ relative to the NV–¹³C axis, simulated field strength and direction angle (θ, ϕ) were robust against these errors compared to those of small ones. Detailed numerical comparison would not be listed here.

2.5. The determination of vector magnetic field direction

Due to the mirror symmetry of NV–¹³C, eight possible combinations of polar angles and azimuth angles (ϕ,π±ϕ,2π-ϕ) correspond to nearly identical and Δ. However, for a definite NV–¹³C hyperfine tensor, the polar angle θ could be firstly decided by observing relative fluorescence intensity of CW spectrums as shown in Fig. 4(a). More details refer to Appendix C. The eight possible directions could be narrowed down to four ones as illustrated in Fig. 7.

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	Fig. 7. (color online) Four possible magnetic field directions satisfying mirror symmetry about the x–z plane correspond to the same CW resonance spectrum and Larmor splitting.

Finally, we propose a scheme that the authentic one of the four possible magnetic field directions could be singled out by observing their free dynamical evolutions of initial states under the Hamiltonian (2). The initial electron spin state could be prepared in and initial ¹³C nuclear spin state or , respectively. Finally, is projected into ground state with . The resulting population evolutions p(τ) under different initial ¹³C nuclear spin state could exclude the symmetrical solutions of four azimuth angles ϕ as shown in Fig. 8. The envelopes contain rapid oscillations, which are from the term of zero-field splitting D in the laboratory frame. In experiment, FID signal could be measured.

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Fig. 8. (color online) As an example, both the vector magnetic field strength and the four possible directions have preset values

, θ = 3π/16, and four possible azimuth angle ϕ = 3π/8, π ±3π/8, or 2π − 3π/8 satisfying mirror symmetry about ¹⁴N–V–¹³C plane. The four possible angles could be clearly discriminated by the evolutionary processes of preparing two different initial nuclear spin state

. The probability evolution is shown by the envelope. (a1)–(a4), the initial state

(b1)–(b4), the initial state

. Notations are defined in the main text.

3. Conclusion

In this paper, we proposed an experimental scheme implementable with current technology, which could detect a static vector magnetic field including both strength and vector directions by employing the NV center with a first-shell ¹³C nuclear spin. Based on the hyperfine tensors between an electron and a single nearest-neighbor ¹³C nucleus, we could acquire the desired information of the vector field by observing the CW ODMR spectrums of hyperfine splitting induced by both ¹⁴N and ¹³C nucleus. Compared with the previous methods of multi-NV magnetometer, our scheme provides a potential alternative method and may enhance the angular resolution for the inhomogeneous magnetic field in space.

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