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

Abstract

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 ${\gamma }_e/{\gamma }_n$, and their temperature dependences for both isotopes. The temperature dependences of the nuclear-spin transitions within the $m_s=0$ spin manifold near room temperature are found to be 0.52(1) ppm/K for ¹⁴N-V(|m_I = -1⟩ ↔ |m_I = +1⟩) and -1.1(1) ppm/K for ¹⁵N-V(|m_I = -1/2⟩ ↔ |m_I = +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.

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₋) for ¹⁴N-V and two transitions (f₊, f₋) for ¹⁵N-V. At this field, the nuclear spins are optically polarized to their largest m_I sublevels, m_I = +1 for ¹⁴N-V and m_I = +1/2 for ¹⁵N-V, 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.

FIG. 1.

Energy-level diagrams for the electronic ground states of ¹⁴N-V and ¹⁵N-V. Energy levels are described by electron-spin (m_s) and nuclear-spin (m_I) quantum numbers. The electron-spin transitions used in this experiment are shown with blue arrows, and are labeled as $f_{\pm }^{\left(m_I\right)}$ for $\left|m_s=0\right.⟩\leftrightarrow \left|m_s=\pm 1\right.⟩$, where $m_I$ denotes the nuclear-spin state of the transition. The nuclear-spin transitions are shown with purple arrows, and are labeled f₁ to f₉. The energy-level diagrams are depicted for the magnetic field values shown at the bottom.

FIG. 2.

Ramsey measurements of ground-state transition frequencies. (a) Nuclear-spin transition frequencies f₁ to f₆ (¹⁴N-V) and f₇ to f₉ (¹⁵N-V) 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 $\delta$ for a variable time $\tau$, after which a $\pi /2$ 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 $\delta$ from the transition frequency f₁. (c) Temperature dependence of f₁ to f₆ for sample G2 at B ≈ 470 G. The y-axis range for the 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 ¹⁴N-V (¹⁵N-V 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.

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 both ¹⁴N-V and ¹⁵N-V, using data from all diamond samples. The solid lines are fourth-degree polynomial fits. (f) Fractional temperature dependence of all parameters whose data are presented in panels (a)–(e). In those, ¹⁴N-V and ¹⁵N-V were found to have similar fractional temperature shifts in D and in A_‖.

FIG. 4.

Temperature-insensitive parameters. (a) Difference in the zero-field splitting parameter D between ¹⁴N-V and ¹⁵N-V. (b) Ratio of electron-spin gyromagnetic ratio ${\gamma }_e$ and ¹⁴N-V nuclear-spin gyromagnetic ratio ${{}^{14}\gamma }_n$. (c) Isotopic ratios of ${\gamma }_n$ and of A_‖ for ¹⁴N-V and ¹⁵N-V. Markers are experimental data, and dashed lines are mean values. In panels (a) and (c), the data were measured using sample G3, which has a 50:50 ratio of [¹⁵N-V]:[¹⁴N-V]. The error bars were determined from the reproducibility of the results.

FIG. 5.

Temperature and angular dependences of f_DQ and f₇. (a) The fractional shifts in nuclear transition frequencies f_DQ (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) and (6) at 480 G and 10 G, respectively. (b),(c) The fractional shifts in nuclear transition frequencies f_DQ (red) and f₇ (blue), respectively, are plotted as a function of magnetic field misalignment with respect to the N-V axis for fixed values of B_z. 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 Table II. For f₇ in panel (c), A_⊥ is treated as a free parameter, and fit to the experimental data in order to obtain A_⊥ = 3.68(2) MHz.

FIG. 6.

Anisotropy of the ¹⁴N-V magnetic hyperfine coupling. Temperature dependences of (a) the Fermi-contact (f = A_‖ + 2A_⊥) and dipolar (d = A_‖ − A_⊥) terms, and (b) the N-V orbital hybridization |c_p|²/|c_s|² and spin density $\eta$ obtained from Eqs. (9) and (10). Markers are experimental data, and solid lines are polynomial fits.