Temperature Sensitivity of 14N-V and 15N-V Ground-State Manifolds

We measure electron- and nuclear-spin transition frequencies in the ground state of nitrogen-vacancy (N-V) centers in diamond for two nitrogen isotopes (14N-V and 15N-V) over temperatures ranging from 77 to 400 K. Measurements are performed using Ramsey interferometry and direct optical readout of the nuclear and electron spins. We extract coupling parameters Q (for 14N-V), D, A‖, A⊥, and γe/γn, and their temperature dependences for both isotopes. The temperature dependences of the nuclear-spin transitions within the ms=0 spin manifold near room temperature are found to be 0.52(1) ppm/K for 14N-V(|mI = −1⟩ ↔ |mI = +1⟩) and −1.1(1) ppm/K for 15N-V(|mI = −1/2⟩ ↔ |mI = +1/2⟩). An isotopic shift in the zero-field splitting parameter D between 14N-V and 15N-V is measured to be ~ 120 kHz. Residual transverse magnetic fields are observed to shift the nuclear-spin transition frequencies, especially for 15N-V. We have precisely determined the set of parameters relevant for the development of nuclear-spin-based diamond quantum sensors with greatly reduced sensitivity to environmental factors.


I. INTRODUCTION
In recent years, color centers in diamond, and in particular, the nitrogen-vacancy (NV) center, have emerged as one of the key platforms for quantum-technology applications, particularly in sensing [1].As the technology matures, detailed knowledge of the parameters of the system and their environmental dependence become a prerequisite of development of accurate devices such as magnetometers, gyroscopes, clocks, as well as multisensors.For NV-based rotation sensing [2][3][4], one may use 14 NV and 15 NV centers, where using two isotopes is important for differential measurements to separate rotational, magnetic, and temperature effects [5][6][7][8][9][10].
In this work, which follows the earlier experimental studies of the temperature dependence of the ground state zero-field splitting parameter D [11], the electric quadrupole hyperfine splitting parameter Q [12] and the theoretical analysis [13], we present a complete experimental characterization of temperature dependence of the coupling parameters of NV centers for both nitrogen isotopes ( 14 NV and 15 NV), including the dependence of the magnetic hyperfine coupling parameters A || and A ⊥ .We find that NV parameters are generally temperature dependent, with relative sensitivity ranging from 7 to 90 ppm/K at room temperature, depending on the parameter.
Additional findings of this work include identification of the relatively high sensitivity of 15 NV nuclear spin * slourette@berkeley.edulevels to misalignment of the magnetic field to the NV axis, as well as measurement of the isotopic shift in D at a level of ∼40 ppm.The latter is important for testing the theoretical models of the system in order to attain a level of understanding necessary for accurate modeling of devices.Note that 15 NV nuclear spins were recently explored as a resource for quantum sensing not relying on microwave or radio-frequency fields in [14].

II. THEORETICAL BACKGROUND
The Hamiltonian for the electronic ground state of 14 NV and 15 NV [15] is given by: where D is the ground state zero-field splitting parameter of the NV center, Q is the nuclear electric quadrupole parameter (only for 14 NV), A || and A ⊥ are the longitudinal and transverse magnetic hyperfine coupling parameters (see Table II for parameter values at 297 K), γ e is the gyromagnetic ratio of the NV center (2.8033(3) MHz/G [16]), γ n is the gyromagnetic ratio of the nitrogen nuclear spin ( 14 γ n = 307.59(3)Hz/G, 15 γ n = −431.50(4)Hz/G), B is the external magnetic field applied along the z-axis (NV symmetry axis), and S and I are electron and nuclear spin operators, respectively.The energy level diagrams for the electronic ground states of 14 NV and 15 NV are shown in Fig. 1.The f 6 14 NV 15 NV Energy level diagrams for the electronic ground states of 14 NV and 15 NV. , where m I denotes the nuclear-spin state.The 14 NV and 15 NV nuclear-spin transitions are labeled f 1 to f 6 and f 7 to f 9 , respectively, according to the diagram.The nuclear-spin double quantum transition with frequency f 1 − f 2 is labeled as f DQ and is of particular interest for rotation sensing [17,18] and comagnetometry [19,20].
Nuclear-spin transition frequencies for both 14 NV and 15 NV can be derived using perturbation theory (see Appendix C), and are described to lowest order in A ⊥ /(D ± γ e B) by the following expressions: In this work, the transition frequencies f 1 to f 9 , f (+1) ± , and f (+1/2) ± are measured in the presence of an axial field (B ≈ 470 G) at temperatures ranging from 77 K to 400 K.

III. EXPERIMENTAL METHODS
We used a custom-built epifluorescence microscopy setup to measure optically detected magnetic resonances (ODMRs) in an ensemble of NV centers.Four diamond samples, whose properties are listed in Table I, were used in our experiments: two with natural isotopic ratio of nitrogen and two with enhanced 15 N concentration.
The diamond sample was mounted inside a continuousflow microscopy cryostat (Janis ST-500).A bias magnetic field B (470-480 G) was applied along one of the NV axes using two temperature-compensated samariumcobalt (SmCo) ring magnets, which were arranged in

Polariz.
Readout 2. Ramsey measurements of ground state transition frequencies.(a) Nuclear-spin transition frequencies f1 to f6 ( 14 NV) and f7 to f9 ( 15 NV) as well as electron-spin transition frequencies f+ and f− for both isotopes are measured using Ramsey interferometry.After optical polarization (green) with green light, the nuclear spin is manipulated with a series of RF (purple) and MW (blue) pulses to prepare a superposition of the two relevant energy states.The superposition then precesses at the detuning frequency δ for a variable time τ , after which a π/2 RF pulse converts the acquired phase into a population difference to be read out optically.a Helmholtz-like configuration that minimizes magnetic field gradients across the detected volume.A 0.79-numerical aperture aspheric condenser lens was used to illuminate a spot on the diamond with ∼30 mW of 532 nm laser light and collect fluorescence.The fluorescence was separated from the excitation light by a dichroic mirror, passed through a band-pass filter (650 to 800 nm), and detected by a free-space Si photodiode.Microwave (MW) and radio-frequency (RF) signals were delivered using a 160 µm diameter copper wire placed on the diamond surface next to the optical focus.
The transition frequencies of the NV ground state were measured with Ramsey interferometry using pulse sequences that are shown in Fig. 2(a).For each nuclearspin transition frequency (f ) between a pair of states (|m s , m I and |m s , m ′ I ), the following steps are performed: optical polarization, state preparation, Ramsey interferometery, and optical readout.Polarization is done with a green laser pulse with duration 100-200 µs, which polarizes the electron and nuclear spins into |0, +1 (or |0, +1/2 for 15 NV) [12,[21][22][23].State |m s , m I is prepared by transferring population using a sequence of RF and MW π pulses.Next, a superposition of the states |m s , m I and |m s , m ′ I is created using a Ramsey π/2 pulse with frequency f RF (duration 10-100 µs).The superposition then accumulates a phase at a rate given by the detuning frequency, δ = f RF − f (typically 0.5-10 kHz).After a variable delay time τ , the acquired relative phase is projected into a population difference with a second π/2 pulse and read out optically [12].To determine f , we obtain δ by fitting the Ramsey oscillations with an exponentially decaying sinusoidal function.
Measurements of electron-spin transition frequencies (f + , f − ) were performed in the same way, using MW pulses instead of RF pulses.Figure 2(b) shows an example of a Ramsey measurement for the nuclear transition frequency f 1 .

IV. RESULTS
The transition frequencies of 14 NV (f ) were measured using the previously described Ramsey interferometry technique [Fig.2(a,b)] in the presence of an axial magnetic field B ≈ 470 G for temperatures ranging from 77 K to 400 K using four samples; see Table I.
Figure 2(c,d) shows the temperature dependence of the nuclear-spin and electron-spin transitions in 14 NV for sample G2.Transition frequencies f 1 and f 2 are least sensitive to temperature, varying by 8 kHz across the range, while f 3 , f 6 and f 4 , f 5 vary by ∼40 kHz and ∼50 kHz, respectively.These temperature dependences are largely determined by the temperature dependence of Q and A || , according to Eq. (2). Figure 2(e) shows the temperature dependence of the nuclear-spin transitions in 15 NV for sample M1, which has been isotopically enriched with 15 N. Transition frequencies f 8 and f 9 were measured to vary by ∼70 kHz across the measured range of temperatures, while the temperature dependence of f 7 was found to be three orders of magnitude smaller; see Sec V for detailed analysis.
Using all measured transition frequencies, we apply numerical methods described in Appendix A to extract values for the coupling parameters for both 14 Numerically extracted coupling parameters for 14 NV (D, Q, A || , A ⊥ ) and 15 NV (D, A || ) are plotted as a function of temperature in Fig. 3(a-e).Each parameter's temperature dependence was fitted to a 4th order polynomial.Figure 3(f) shows the fractional shift of each parameter as a function of temperature.We observe 14 NV and 15 NV to have identical fractional dependence for both D and A || .
In the case of 15 NVA ⊥ , the following factors prevented the determination of its temperature dependence: (i) A ⊥ has a weak contribution to the nuclear-spin transition frequencies, (ii) the 15 NV nuclear-spin transition frequencies (f 7 in particular) are sensitive to magnetic field misalignment θ; see Sec V A, (iii) 15 NV only has three nuclear-spin transitions, whose frequencies are determined by four parameters: A || , A ⊥ , 15 γ n B, θ.We were able to overcome these issues at room temperature by scanning the transverse magnetic field with coils [see Fig. 5(c)] and obtained A ⊥ = 3.68(2) MHz, which is in agreement with [16].In future studies, this approach can be used to obtain the temperature dependence of 15 NVA ⊥ .
Table II lists the values and temperature derivatives of each coupling parameter at 297 K for both 14 NV and 15 NV, obtained from the polynomial fits.The room temperature value for each measured parameter is consistent with previously reported values [16,[22][23][24].Temperature derivatives are also consistent with previously reported values for 14 NV parameters D [11,25,26], Q [12,[26][27][28], and A || [26][27][28].Recent theoretical work [29] provided ab initio evaluation of the temperature dependences of several parameters for 14 NV system.In the cases where the same parameters were calculated in [29] and measured here, we find good agreement.To the best of our knowledge, temperature derivatives for 14 NV A ⊥ and 15 NV A || are reported here for the first time.
We observe an isotopic shift of ∼120 kHz (∼40 ppm) in the D parameter, with D larger for 15 NV. Figure 4(a) shows the shift in D across the full range of measured temperatures for the G3 sample, which has a 50 : 50 isotopic ratio.This effect is also clearly visible as a difference in separation of peaks in the ODMR signal (see Appendix B).The relatively small isotopic effect in D is in line with the fact that even a replacement of the species adjacent to the vacancy changes the zero-field splitting only slightly, see recent work [30], where the D values were reported to be 2888 MHz and 2913 MHz for the OV 0 and BV − , respectively.
Figure 4(b) shows the numerically extracted values of γ e / 14 γ n for three different samples.The mean value of γ e / 14 γ n is measured to be 9113.9(1), which is in agreement with [31].Using the literature value for (f) 15 NV  γ e = 2.8033(3) MHz/G [16], this corresponds to 14 γ n = 307.59(3)Hz/G.The value for γ e / 14 γ n can also be approximated without numerical methods from the measured frequencies directly [see Eq. ( 2)]: Figure 4(c) shows the isotopic ratio of γ n 15 γ n / 14 γ n = 1.402 85 (6), in agreement with [32] and the isotopic ratio of A || 15 A || / 14 A || = 1.400 96(1) obtained from measurements using the G3 sample.The difference in these ratios can be used to verify theoretical models.When using the literature value for γ e , we obtain 15   The nuclear-spin transitions f DQ and f 7 within the m s = 0 manifold of 14 NV and 15 NV are of particular interest for sensing applications such as rotation sensing [17,18] and comagnetometry [19,20].Precise knowledge of their transition frequencies and their dependence on environmental factors (temperature, magnetic field) is essential for optimal sensor performance.The transition frequencies f DQ and f 7 can be obtained from Eqs. ( 2),(3) and are described by the following expressions: (5) The frequencies of these transitions do not depend on the coupling parameters Q, A || and are determined primarily by the nuclear Zeeman shift ∆m I γ n B, resulting in a greatly reduced temperature dependence compared to other nuclear spin transitions.Therefore, measurements of the temperature dependences of f DQ and f 7 require more precise control of the bias magnetic field.Over the range of temperatures used in this experiment, the bias magnetic field varied by ∼ 1 G, primarily due to thermal expansion of the sample holder in the presence of magnetic field gradients.This variation in the bias magnetic field was measured using electron-spin transition frequencies and subsequently used to obtain corrected values for f DQ and f 7 corresponding to 480 G.When the temperature is changed from 77K to 400K, the transition frequencies f DQ and f 7 are observed to shift by 140 ppm (44 Hz at 480 G) and −260 ppm (−55 Hz at 480 G), respectively; see Fig. 5(a).This corresponds to fractional temperature derivatives of 0.52(1) ppm/K (0.15 Hz/K) for f DQ and −1.1(1) ppm/K (−0.10 Hz/K) for f 7 at 297K.The temperature dependence of f DQ and f 7 arises from the temperature dependence of A 2 ⊥ /D 2 and is described by Eqs. ( 5), (6).These equations are used with experimentally obtained polynomial fits of A ⊥ and D (see Fig. 3), to generate the solid and dashed lines in Fig. 5(a).For f 7 , A ⊥ is assumed to have the same fractional dependence on temperature as A || .For sufficiently small fields γ e B << D (i.e., B = 10 G), the transition frequencies are approximately linear in the magnetic field, and thus the fractional shift is independent of magnetic field.
Magnetic field misalignment (from the NV axis) is another factor that can significantly shift the transition frequencies of f DQ and f 7 , and must be considered in sensing applications.Angular-dependent shifts in f 7 are significantly stronger than shifts in f DQ because of the lack of a stabilizing coupling parameter (i.e., Q) in the effective nuclear-spin Hamiltonian.The frequency shifts f DQ and f 7 exhibit a quadratic dependence on misalignment angle, and are shown in Figs.5(b) and 5(c), respectively.The alignment of the magnetic field was controlled using two pairs of coils oriented perpendicular to the bias magnetic field B z .The transverse magnetic field B x was precisely determined using the NV electron spin transitions of the three non-axial NV sub-ensembles.We measure a frequency shift of 5.0 Hz in f DQ and of 130 Hz in f 7 when misaligning the magnetic field by θ = 0.1°(B x ≈ 0.8 G) at a field of B z = 480 G. II to obtain theoretical predictions for f DQ and f 7 at 480 G (solid line) and 10 G (dashed line).For f 7 , we fit the theoretical model to the experimental data in order to obtain a more precise value of A ⊥ for 15 NV, which we measure to be 3.68(2) MHz at 297 K.

We use numerical methods (Appendix A) together with experimental values from Table
The frequency shifts in f DQ and f 7 due to magnetic field misalignment can be approximated using perturbation theory; see Appendix C. For f DQ the dominant term is a second-order correction, whose fractional shift is de-  II.For f7 (c), A ⊥ is treated as a free parameter, and fit to the experimental data in order to obtain A ⊥ = 3.68(2) MHz.
scribed by the following expression: where θ is the angle between the NV axis and magnetic field, and β ≈ −9.9 at B z = 480 G, and β ≈ −0.003 at B z = 10 G.While the second-order correction is similar to that of f DQ , for f 7 the fourth-order correction is much larger and is described by the following expression: where β ≈ 460 at B z = 480 G, and β ≈ 280 at B z = 10 G.

B. Anisotropy of the hyperfine coupling
The temperature dependence of the magnetic hyperfine coupling components, A || and A ⊥ , is used to obtain the temperature dependence of the Fermi contact and dipolar terms.These in turn can be expressed in terms of the effective spin density η occupying the atomic orbitals of the nitrogen atom and their effective hybridization ratio |c p | 2 /|c s | 2 via the expressions Figure 6 shows that both the spin density and hybridization ratio are observed to increase with temperature.This is consistent with the ab initio calculations [33], which concluded that with increasing temperature, the spin density diffuses away from the three carbon atoms surrounding the vacancy and the nitrogen atom moves towards the vacancy (away from its nearest-neighbor carbon atoms).Thus, one would expect the spin density to increase at the nitrogen atom as a result of the outward diffusion, and the displacement of nitrogen atom would lead to an increase in the hybridization ratio (as the orbitals connecting the nitrogen to its nearest-neighbours must become more p-like to achieve the new geometry of the bond).

VI. CONCLUSION AND OUTLOOK
We measured the nuclear-spin and electron-spin transition frequencies for NV centers containing 14 N and 15 N, as a function of temperature.To describe the results, we used numerical diagonalization of the Hamiltonian, including the effect of magnetic field misalignment.The model allows us to extract the underlying parameters Q (for 14 NV), D, A || , A ⊥ , γ e /γ n for both isotopes and their temperature dependences (except A ⊥ for 15 NV).The magnitude of each one of these parameters (Q, D, A || , A ⊥ ) decreases with temperature in the range from 77 to 400 K, showing a reduction of ∼0.1 % (in the case of Q) to ∼2 % (in the case of A || ).
Comparison of the determined parameters reveals a difference in D of ∼120 kHz (∼40 ppm) between NV centers containing 14 N and 15 N. To our knowledge, this is the first report of such an isotopic difference for NV centers.We also observe a difference of ∼0.1 % between 15 A || / 14 A || and 15 γ n / 14 γ n .
The temperature dependence of the anisotropy of the hyperfine coupling between electron and nuclear spins (A || , A ⊥ ) in 14 NV can be used to infer the temperature dependence of the Fermi-contact and dipolar interactions, which, in turn, can provide information about the electron spin density and orbital hybridization.
We determined the temperature dependence of f DQ (0.52(1) ppm/K) and f 7 (−1.1(1)ppm/K), which are three orders of magnitude smaller than the other nuclear-spin transition frequencies.Nevertheless, this sensitivity to temperature may limit the performance of nuclearspin-based sensors and should be taken into account.
We found that residual transverse fields should be carefully considered in order to precisely determine the frequencies, especially for f 7 , for which a misalignment of the field by 0.1 • leads to a fractional change of ∼600 ppm.This strong dependence allowed us to measure A ⊥ for 15 NV to be 3.68(2) MHz, which is in agreement with the previously measured value [16].
The combination f 3 − f 6 has negligible dependence on A ⊥ and is described to high precision by f 3 − f 6 = 2γ n B, and therefore, its temperature dependence is predicted to be weak: < 10 ppb/K.
In summary, we have precisely determined the set of parameters relevant for the development of NV-diamond rotation sensors, magnetometers, frequency standards and multisensors, along with the temperature dependence of these parameters.The results indicate a promising path to developing such devices with greatly reduced sensitivity to environmental variations.The general idea is to use multiple transitions with different sensitivity to, for example, temperature, which allows one to isolate the environmental parameter drift from the effect of interest (e.g., inertial rotation).(plus six) energy eigenvalues and therefore the frequencies of the six (plus three) RF transitions and two (plus two) MW transitions of the ground state Hamiltonian of the 14 NV center, denoted as .
Here, we need to solve the inverse problem, starting with experimental values for the transition frequencies, f , and ending with values for the coupling parameters, a.This is done starting with an initial guess for the coupling parameters, a 0 , calculating the transition frequencies associated with this guess f (a 0 ), and an associated weighted error between the calculated and measured transition frequencies S(a) , where δf is the measurement uncertainty associated with f .The fitted values for the coupling parameters correspond to the value of a that minimizes S(a), which was obtained using MATLAB's fminsearch function (Nelder-Mead simplex algorithm).This process was repeated for each temperature to obtain the temperature dependence of each parameter.
Using the fitted temperature dependence of the coupling parameters, we numerically diagonalize the Hamiltonian at and near 297 K for B = 470 G.This allows us to estimate the temperature dependence of the magnetically sensitive nuclear transitions (f 1 to f 9 ), which are shown in Table III.The isotopic shift in D can be measured directly from the pulsed ODMR spectrum, which is shown in Fig. 7. Signals were recorded for the G3 sample at 475 G and 298 K.At this field, the nuclear spins are optically polar-ized to their largest m I sublevels, m I = +1 for 14 NV and m I = +1/2 for 15 NV, due to the opposite signs of 14 γ n and 15 γ n , which allows them to be individually resolved.By extracting D using (f + + f − )/2 for each isotope, we obtain D = 2870.26MHz for 14 NV and D = 2870.38MHz for 15 NV, which corresponds to an isotopic shift in the D parameter of 0.12(1) MHz.

Appendix C: Perturbation Theory
Perturbation theory can be used to describe how both the transverse hyperfine coupling parameter (A ⊥ ) and transverse magnetic fields (B x ) shift the nuclear-spin transition frequencies for both 14 NV and 15 NV.Terms in the Hamiltonian [Eq.(A1)] can be divided into terms that do (H || ) and do not (V ) commute with S z and I z : Here we have omitted the transverse nuclear-spin Zeeman term, which is small compared the the transverse electron-spin Zeeman term.Using H || as the unperturbed Hamiltonian and treating V as a perturbation, the second-order perturbation shift of each unperturbed state is calculated using the following expression where the eigenstates of the unperturbed state are denoted as |m s , m I , and their energies are described as follows For each nuclear-spin transition, an approximate expression for the total energy shift is obtained to second order in 1/F ± , where F ± = D ± γ e B. There are also fourthorder perturbation shifts that produce effects that are of second order in 1/F ± , which appear when m Combining these shifts gives us expressions for the shifted nuclear-spin transition frequencies, both for 14 NV and 15 NV: These expressions can be reduced to obtain simplified expressions for the transition frequencies [see Eqs.(2),(3)], as well as their angular dependences [see Eqs. ( 7), (8)].
For 14 NV, we can linearly combine nuclear-spin transition frequencies in order to obtain simple approximations for 14 γ n B, Q, and A || FIG. 1. Energy level diagrams for the electronic ground states of 14 NV and 15 NV.Energy levels are described by electronic spin (ms) and nuclear spin (mI ) quantum numbers.The electron-spin transitions used in this experiment are shown with blue arrows, and are labeled as f (m I ) ± for |ms = 0 ↔ |ms = ±1 , where mI denotes the nuclear-spin state of the transition.The nuclear-spin transitions are shown with purple arrows and are labeled f1 to f9.
FIG. 2.Ramsey measurements of ground state transition frequencies.(a) Nuclear-spin transition frequencies f1 to f6 ( 14 NV) and f7 to f9 ( 15 NV) as well as electron-spin transition frequencies f+ and f− for both isotopes are measured using Ramsey interferometry.After optical polarization (green) with green light, the nuclear spin is manipulated with a series of RF (purple) and MW (blue) pulses to prepare a superposition of the two relevant energy states.The superposition then precesses at the detuning frequency δ for a variable time τ , after which a π/2 RF pulse converts the acquired phase into a population difference to be read out optically.(b) Example of the nuclear Ramsey interferometry measurement.The oscillation frequency of the Ramsey fringes corresponds to the detuning δ from the transition frequency f1.(c) Temperature dependence of f1 to f6 for sample G2 at B ≈ 470 G.The y-axis range for f1 and f2 subplots has been reduced (70 kHz → 20 kHz) to show the reduced temperature dependence of f1 and f2.(d) Temperature dependence of f+ and f− for 14 NV ( 15 NV not shown) for sample G2 at B ≈ 470 G. (e) Temperature dependence of f7 to f9 for sample M1 at B ≈ 468 G.The y-axis range for the f7 subplot has been reduced (80 kHz → 12 kHz).

FIG. 3 .
FIG. 3. Temperature dependence of coupling parameters.(a)-(e) Coupling parameters D, Q, A || , and A ⊥ were extracted numerically (see Appendix A) from measured nuclear-spin and electron-spin transition frequencies.Coupling parameters are plotted against temperature, for both14 NV and 15 NV, using data from all diamond samples.The solid lines are fourth-degree polynomial fits.(f ) Fractional temperature dependence of all parameters whose data is presented in (a)-(e).14NV and 15 NV were found to have similar fractional temperature shifts in D and in A || .
Figure 4(c)  shows the isotopic ratio of γ n 15 γ n / 14 γ n = 1.402 85(6), in agreement with[32] and the isotopic ratio of A ||15 A || / 14 A || = 1.400 96(1) obtained from measurements using the G3 sample.The difference in these ratios can be used to verify theoretical models.When using the literature value for γ e , we obtain 15 γ n = −431.50(4)Hz/G.

3 FIG. 5 .
FIG.5.Temperature and angular dependence of fDQ and f7.(a) The fractional shifts in nuclear transition frequencies fDQ (red) and f7 (blue) are plotted as a function of temperature.Markers represent experimental data after correcting for variations in the magnetic field between measurements.Solid and dashed lines were obtained from Eqs. (5),(6) at 480 G and 10 G, respectively.(b),(c) The fractional shifts in nuclear transition frequencies fDQ (red) and f7 (blue), respectively, are plotted as a function of magnetic field misalignment with respect to the NV axis for fixed values of Bz.Markers represent experimental data, and solid (480 G) and dashed (10 G) lines were obtained by numerically diagonalizing the Hamiltonian [Eq.(1)] using values from TableII.For f7 (c), A ⊥ is treated as a free parameter, and fit to the experimental data in order to obtain A ⊥ = 3.68(2) MHz.

FIG. 7 .
FIG.7.Measurement of isotopic shift in D using ODMR.The spectrum is obtained from the G3 sample (50 : 50 isotopic ratio) at 475 G and consists of four resonances corresponding to two transitions (f+, f−) for14 NV and two transitions (f+, f−) for 15 NV.At this field, the nuclear spins are optically polarized to their largest mI sublevels, mI = +1 for14 NV and mI = +1/2 for 15 NV, which creates a resolvable splitting.A slight difference is observed between the splitting of f+ (3.76 MHz) and that of f− (3.52 MHz), which corresponds to a difference in D of 0.12(1) MHz.

TABLE I .
Diamond samples.The estimated concentrations of substitutional nitrogen [N], NV centers [NV], and 13 C atoms13C , in addition to the nitrogen isotopic ratio 14 N :15N , electron-spin dephasing time T * 2 , and electron-spin coherence time T2 are listed for each sample used in experiments.Diamonds were grown using chemical vapor deposition and were obtained from Element Six.Electron-spin T * 2 was measured using Ramsey interferometry, and T2 was measured using Hahn echo techniques.

TABLE II .
Experimentally determined coupling parameters at 297 K.The values and temperature derivatives of coupling parameters 14 NV (D, Q, A || , A ⊥ ) and15NV (D, A || ) at 297 K are obtained from the polynomial fits of the temperature dependences shown in Fig.3.15NVA⊥ is obtained by scanning the transverse magnetic field with coils; see Fig.5(c).

TABLE III .
Transition frequencies and temperature derivatives at T = 297 K and B = 470 G. Values are determined by performing numeric diagonalization of the Hamiltonian (Eq. 1) using values and uncertainties of D, Q, A || , and A ⊥ listed in TableII.For 15 NV, it is assumed that A ⊥ has the same fractional temperature dependence as A || within 5 %, or (dA ⊥ /dT )/A ⊥ = 1.00(5) × (dA || /dT )/A || .