Hyperfine level structure in nitrogen-vacancy centers near the ground-state level anticrossing

Energy levels of nitrogen-vacancy centers in diamond were investigated using optically detected magnetic-resonance spectroscopy near the electronic ground-state level anticrossing (GSLAC) at an axial magnetic field around 102.4 mT in diamond samples with a nitrogen concentration of 1 and 200 ppm. By applying radio waves in the frequency ranges from 0 to 40 MHz and from 5.6 to 5.9 GHz, we observed transitions that involve energy levels mixed by the hyperfine interaction. We developed a theoretical model that describes the level mixing, transition energies, and transition strengths between the ground-state sublevels, including the coupling to the nuclear spin of the NV centers' ${}^{14}\mathrm{N}$ and ${}^{13}\mathrm{C}$ atoms. The calculations were combined with the experimental results by fitting the ODMR spectral lines based on a theoretical model, which yielded information about the polarization of nuclear spins. This study is important for the optimization of experimental conditions in GSLAC-based applications, e.g., microwave-free magnetometry and microwave-free nuclear-magnetic-resonance probes.

Figure 1

(a) Level scheme for an NV center in diamond where $m_S$ is the electron spin projection quantum number, $D_g$ and $D_e$ are the ground- and excited-state zero-magnetic-field splittings, ${\mathrm{\Omega }}_{\text{MW}}$ is the MW Rabi frequency, ${\gamma }_0^g$ and ${\gamma }_{\pm 1}^g$ are the relaxation rates from the singlet state ${}^{1}E$ to the triplet ground-state ${}^{3}A_2$, and ${\gamma }_0^e$ and ${\gamma }_{\pm 1}^e$ are the relaxation rates from the triplet excited state ${}^{3}E$ to the singlet state ${}^{1}A_1$. (b) Levels of the NV centers' electron-spin magnetic sublevels in the ground state. (c) Hyperfine level ($|m_S,m_I\rangle$) anticrossing in the vicinity of the GSLAC. The degree of mixing near the GSLAC (denoted by the dashed ellipses) is indicated by the relative admixture of the colors in each curve; the lines corresponding to unmixed states do not change color.

Figure 2

Experimental setup. (a) High-density sample. The laser light was coupled into an optical fiber that led to a cage-system cross, in which a dichroic mirror (Thorlabs DMLP567R) reflected the green light, which, in turn, was coupled into another fiber, which led to the sample. A small portion of the green light that passed through the dichroic mirror was used to monitor the laser power. The fluorescence from the sample, which was glued to the end of the fiber, was collected with the same fiber, and, after passing through the dichroic mirror and a long-pass filter (Thorlabs FEL0600), was focused onto a photodiode with an amplifier (Thorlabs PDA36A-EC). (b) Low-density sample. Laser light was focused on the sample with a lens, fluorescence was collected and measured with a photodetector (Thorlabs APD410A/M). The diamond sample was located at the center of the magnet bore. The rotation axes of the sample and the electromagnet are depicted with blue arrows.

Figure 3

ODMR signals at high microwave field frequencies. The top row (a)–(c) shows the transitions between respective levels [Eq. (5)]. The transition strength is indicated by the arrow width. In the middle row (d)–(f), the black lines show the experimental signals, and the red lines in (d)–(f) show the results of the theoretical calculations with the parameters from the fitting procedure described in the text for the low-density sample. The bottom row (g)–(i) shows the corresponding results for the high-density sample. The vertical bars in (d)–(i) correspond to the transitions depicted by the arrows in (a)–(c) of the same color, and their length determines the contribution to the overall line shape of that transition, which is proportional to the product of the level population and the transition strength.

Figure 4

ODMR signals at low microwave field frequencies. The top row (a)–(c) shows the transitions between respective levels [Eq. (5)]. The transition strength is indicated by the arrow width. In the middle row (d)–(f), the black lines show the experimental signals and the red lines in (d)–(f) show the results of the theoretical calculations with the parameters from the fitting procedure described in the text for the low-density sample. The bottom row (g)–(i) shows the corresponding results for the high-density sample. The dashed red lines in (e) and (h) show the calculated signal for an angle between the NV axis and the magnetic field $\mathbf{B}$ of $\theta =0.{015}^{\circ }$. In (d), (f), (g), and (i) there is no noticeable difference between calculated signals for $\theta =0^{\circ }$ and $\theta =0.{015}^{\circ }$

Figure 5

Experimental signals (black) obtained on the low-density sample with theoretical calculations (red) for ground-state $m_S=0⟶m_S=-1$ microwave transitions for different magnetic-field values. Experimental signal with the calculated signal for an angle between the NV axis and the magnetic field $\mathbf{B}$ of (a) $\theta =0^{\circ }$ and (b) $\theta =0.{015}^{\circ }$ (transverse magnetic field 0.025 mT). (c) Experimental signal with the calculated signal at $\theta =0^{\circ }$ for ground-state $m_S=0⟶m_S=+1$ microwave transitions for different magnetic-field values. Bars with black dots on one end are placed at the values of the transition frequencies for a specific magnetic-field value, and their length represents the calculated transition probability. For better readability, signals are arranged in order of descending magnetic field, and each curve is normalized separately with its relative intensity depicted at the right side of the graph (see Fig. 7 for details). The gray lines show how the energy and the intensity of the transitions change in the magnetic field.

Figure 6

Experimental signals from the high-density sample (black) with theoretical calculations (red) for ground-state $m_S=0⟶m_S=-1$ microwave transitions for different magnetic-field values. Experimental signal with the calculated signal for an angle between the NV axis and the magnetic field $\mathbf{B}$ of (a) $\theta =0^{\circ }$ and (b) $\theta =0.1^{\circ }$ (transverse magnetic field 0.18 mT). (c) Experimental signal with the calculated signal at $\theta =0^{\circ }$ for ground-state $m_S=0⟶m_S=+1$ microwave transitions for different magnetic-field values. Bars with black dots on one end are placed at the values of the transition frequencies for a specific magnetic-field value, and their length represents the calculated transition probability. For better readability, signals are arranged in order of descending magnetic field, and each curve is normalized separately with its relative intensity depicted at the right side of the graph (see Fig. 7 for details). The gray lines show how the energy and the intensity of the transitions change in the magnetic field.

Figure 7

ODMR signal contrast for different magnetic field values for the low-density sample. The line is drawn using locally weighted scatter plot smoothing. The image shows how the ODMR signal contrast changes close to the GSLAC region, black dots $m_S=0⟶m_S=-1$ transition [Figs. 5 and 5], red squares $m_S=0⟶m_S=+1$ transition [Fig. 5].

Figure 8

Nuclear ${}^{14}\mathrm{N}$ spin orientation (black dots) and alignment (red squares) obtained for the ground-state $m_S=0⟶m_S=+1$ transition from the integral under the fitted transition peaks for the low-density sample. The solid lines are drawn using locally weighted scatter plot smoothing. The gray, shaded region corresponds to the magnetic field range in which we cannot be certain about the polarization value because an additional peak poorly described by the model interferes with the optimization procedure.

Figure 9

Experimental signals (black) for the low-density sample with theoretical calculations including the effects of ${}^{13}\mathrm{C}$ (red) [see (15)] and without the effects of ${}^{13}\mathrm{C}$ (blue) for the ground-state $m_S=0⟶m_S=+1$ microwave transitions for different magnetic-field values. The gray line tracks the position of the nominal $|0,1\rangle ⟶|1,0\rangle$ transition.

Figure 10

Experimental signals (black) for the low-density sample with theoretical calculations including the effects of ${}^{13}\mathrm{C}$ (red) [see (15)] and without the effects of ${}^{13}\mathrm{C}$ (blue) for the ground-state $m_S=0⟶m_S=-1$ microwave transitions for different magnetic-field values. The gray line tracks the position of the nominal $|0,1\rangle ⟶|0,0\rangle$ transition.