Cross-relaxation studies with optically detected magnetic resonances in nitrogen-vacancy centers in diamond in external magnetic field

In this work, cross relaxation between nitrogen-vacancy (NV) centers and substitutional nitrogen in a diamond crystal is investigated. It is demonstrated that optically detected magnetic resonance signals (ODMR) can be used to probe cross relaxation. ODMR is detected at axial magnetic field values around 51.2 mT in a diamond sample with a relatively high (200 ppm) nitrogen concentration. We observe transitions that involve magnetic sublevels that are split by the hyperfine interaction. Microwaves in the frequency ranges from 1.3 GHz to 1.6 GHz ($m_S=0⟶m_S=-1$ NV transitions) and from 4.1 to 4.6 GHz ($m_S=0⟶m_S=+1$ NV transitions) were used. To understand the cross-relaxation process in more detail and, as a result, reproduce measured signals more accurately, a model is developed that describes the microwave-initiated transitions between hyperfine levels of the NV center at anticrossing that are strongly mixed. Additionally, we simulate the extent to which the microwave radiation driving ODMR in the NV center also induces transitions in the substitutional nitrogen via cross relaxation. The improved understanding of the NV processes in the presence of magnetic field will be useful for designing NV-diamond-based devices for a wide range of applications from implementation of q bits to hyperpolarization of large molecules to various quantum technological applications such as field sensors.

Figure 1

Level scheme of an NV center in diamond, $m_S$ is the electron spin-projection quantum number, $D_g$ and $D_e$ are the ground-state 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 ${}^1\mathrm{E}$ to the triplet ground state ${}^3\mathrm{A}_2$ and, ${\gamma }_0^e$ and ${\gamma }_{\pm 1}^e$ are the relaxation rates from the triplet excited state ${}^3\mathrm{E}$ to the singlet state ${}^1\mathrm{A}_1$ [29].

Figure 2

Level diagram for ${}^{14}\mathrm{N}\mathrm{V}$, black lines, and ${}^{14}\mathrm{N}$ (P1) centers, blue lines, with hyperfine interaction in the vicinity of $51.2\mathrm{mT}$. The arrows show the energy-matching transitions between the ${}^{14}\mathrm{N}\mathrm{V}$ center's $|m_S=0,m_I\rangle$ and $|m_S=-1,m_I\rangle$ magnetic sublevels and the ${}^{14}\mathrm{N}$ (P1) center's $|m_S=-\frac{1}{2},m_I\rangle$ and $|m_S=\frac{1}{2},m_I\rangle$ magnetic sublevels. Each of the arrows are split into three by the ${}^{14}\mathrm{N}\mathrm{V}$ center's hyperfine structure, but they are omitted in the figure for better readability.

Figure 3

Rotating-frame Hamiltonian after fast-oscillating terms have been set to zero for the energy matched (only the black squares) and unmatched (black and gray squares indicating the matrix elements related to the MW) cases. The kets surrounding the figure describe the ordering of the $|m_{\text{NV}},m_{\text{P1}}\rangle$ electron-spin basis states.

Figure 4

Expectation value $\langle {\widehat{S}}_{\text{P1}z}\rangle$ as a function of time without microwaves and in the presence of a constant microwave field.

Figure 5

Experimental setup.

Figure 6

NV Fluorescence signal (normalized, arbitrary units) dependence on the magnetic field strength.

Figure 7

Typical measured ODMR signal (normalized, arbitrary units) from the NV centers at around 51.2 mT external magnetic field for the $m_S=0⟶m_S=-1$ transition with unresolved hyperfine structure, showing the fitted full width at half maximum for the three Lorentzians and the contrast of the ODMR signal. The $m_I=0$ and $m_I=-1$ components are assumed to be small in intensity due to the nuclear spin polarization to the $m_I=+1$ state, as can be seen in the figure as well.

Figure 8

ODMR signal width as a function of the applied external magnetic field. The red points are calculated from the experimental data for the $|m_S=0,m_I\rangle ⟶|m_S=-1,m_I\rangle$ NV center transitions. The tables around the graph show the mean magnetic field values (bold) of three possible transitions for an NV center in combination with three transitions for the P1 center (the same transition at slightly different magnetic field values); these values were calculated using the reasoning in Sec. 2b. The NV center and P1 center transitions are written in the $|m_S,m_I\rangle$ bases. Cells and numbers marked in green represent transitions noted to have the total spin projection (magnetic quantum number) change's absolute value $|\mathrm{\Delta }{m}_{\text{Total}}|=|\mathrm{\Delta }{m}_S^{\text{NV}}+\mathrm{\Delta }{m}_I^{\text{NV}}+\mathrm{\Delta }{m}_S^{P1}+\mathrm{\Delta }{m}_I^{P1}|\le 2$ (this is true only when the projections are along the same axis, otherwise this can be taken only as a classification notation). The off-axis transitions are marked red in the tables.

Figure 9

ODMR contrast as a function of magnetic field for (a) NV center transition $|m_S=0,m_I\rangle ⟶|m_S=-1,m_I\rangle$ and (b) $|m_S=0,m_I\rangle ⟶|m_S=+1,m_I\rangle$. The vertical lines indicate cross-relaxation resonance transitions that have the total spin projection (magnetic quantum number) change's absolute value $|\mathrm{\Delta }{m}_{\text{Total}}|\le 2$, see the tables in Fig. 8, and that originate from a populated state; solid lines indicate transitions between the on-axis P1 centers and NV centers, and the dotted vertical lines indicate cross-relaxation resonance positions between off-axis P1 centers and NV centers.

Figure 10

A zoom in on the “A” peak from the contrast image Fig. 9 showing all possible transitions (black vertical lines). The vertical lines in red show the transitions from the polarized state $|m_S=0,m_I=+1\rangle$ and the one transition written in blue text has the total spin projection (magnetic quantum number) change's absolute value $|\mathrm{\Delta }{m}_{\text{Total}}|\le 2$, see the table for peak “A” in Fig. 8. It can be seen that the transition with the blue text lies close to the perceived minimum of the peak. The horizontal and vertical arrows at each red calculated point from the data indicate the standard deviation of the magnetic field calculated form multiple data sets (max 0.02 mT–0.56 MHz). It can be seen that taking into account the nuclear spin polarization of the NV center a more accurate cross-relaxation analysis of the P1 center can be achieved.

Figure 11

ODMR width as a function of magnetic field value for (a) NV center transition $|m_S=0,m_I\rangle ⟶|m_S=-1,m_I\rangle$ and (b) $|m_S=0,m_I\rangle ⟶|m_S=+1,m_I\rangle$. The vertical lines indicate cross-relaxation resonance transitions that have the total spin projection (magnetic quantum number) change's absolute value $|\mathrm{\Delta }{m}_{\text{Total}}|\le 2$, see the tables in Fig. 8, and that originate from a populated state; solid lines indicate transitions between the on-axis P1 centers and NV centers, and the dotted vertical lines indicate cross-relaxation resonance positions between off-axis P1 centers and NV centers.