State-selective intersystem crossing in nitrogen-vacancy centers

The intersystem crossing (ISC) is an important process in many solid-state atomlike impurities. For example, it allows the electronic spin state of the nitrogen-vacancy (NV) center in diamond to be initialized and read out using optical fields at ambient temperatures. This capability has enabled a wide array of applications in metrology and quantum information science. Here, we develop a microscopic model of the state-selective ISC from the optical excited state manifold of the NV center. By correlating the electron-phonon interactions that mediate the ISC with those that induce population dynamics within the NV center's excited state manifold and those that produce the phonon sidebands of its optical transitions, we quantitatively demonstrate that our model is consistent with recent ISC measurements. Furthermore, our model constrains the unknown energy spacings between the center's spin-singlet and spin-triplet levels. Finally, we discuss prospects to engineer the ISC in order to improve the spin initialization and readout fidelities of NV centers.

Figure 1

Electronic structure of the NV center, including the fine-structure states of the spin-triplet levels, the ZPL transitions of the center's visible and infrared resonances (solid arrows), and the state-selective nonradiative ISCs between the spin-triplet and -singlet levels (dashed arrows). ${\Gamma }_X$ denote the rates of the individual ISC transitions.

Figure 2

Schematic of (left) the phonon-induced mixing within the ${}^{3}E$ manifold and (right) the stages of the ISC from the ${}^{3}E$ manifold to the ${|}^1{A}_1\rangle$ state. The phonon mixing transitions within the ${}^{3}E$ manifold are depicted by solid arrows. ${\Delta }_{xy}$ is the strain-induced splitting of the $|E_{x,y}\rangle$ states. The shaded regions denote the quasicontinua of the vibrational levels of ${}^{3}E$ and ${|}^1{A}_1\rangle$. The stages of the ISC, which are described in the text, are denoted by the arrows labeled (1) and (2). The ISC vibrational overlap function $F\left(\omega \right)$, as approximated by the visible emission PSB, is depicted on the far right. This provides a visual representation of how the ${}^{3}E–{|}^1{A}_1\rangle$ energy separation $\Delta$ influences the ISC rate.

Figure 3

ISC rate ${\tilde{\Gamma }}_{E_{1,2}}$ from $|E_{1,2}\rangle$ as a function of the energy of the mediating $E$-symmetric phonon. Contributions due to phonon absorption (red line), stimulated and spontaneous emission (blue line), and spontaneous emission only (blue dashed line) are shown. At $T=5$ K, $k_{B}T=2\pi \hslash \times 104\mathrm{GHz}=0.43$ meV. We assume that the vibrational overlap function $F(\Delta \pm \omega )$ is flat for the range of $\omega$ (2.6 meV) shown.

Figure 4

Mixing rate ${\tilde{\Gamma }}_{\mathrm{Mix}}$ as a function of the energy of the lower-energy mediating $E$-symmetric phonons. At $T=5$ K, $k_{B}T=2\pi \hslash \times 104\mathrm{GHz}=0.43$ meV. The dashed line shows ${\omega }^2$, properly scaled, for comparison.

Figure 5

ISC rate from $|A_1\rangle$. The values of ${\Gamma }_{A_1}$ calculated using Eq. (6) (blue line) and measured in Ref. [16] (orange line) are shown, with confidence intervals given by the uncertainty bounds on ${\lambda }_{\perp }$ described in the text and the $95\%$ confidence interval given for the measurement. The predicted range of $\Delta$, which is defined by the intersection of the two curves, is shown in black. The points of intersection below 148 meV are excluded by the measured ${\Gamma }_{E_{1,2}}/{\Gamma }_{A_1}$ ratio, as explained in Sec. 3b.

Figure 6

Ratio of the ISC rates from $|E_{1,2}\rangle$ and $|A_1\rangle$. The values of ${\Gamma }_{E_{1,2}}/{\Gamma }_{A_1}$ calculated using Eq. (19) (blue line) and measured in Ref. [16] (orange line, with $95\%$ confidence interval) are shown. The dashed plots represent the results obtained when the ISC process that uses ${|}^1{E}_{1,2}\rangle$ as intermediate states is neglected (see Appendix). In (a), we assume no acoustic phonon cutoff $\left(\Omega \to \infty \right)$, so the blue plot represents an upper bound on ${\Gamma }_{E_{1,2}}/{\Gamma }_{A_1}$. The calculated value's uncertainty is due to the uncertainty in $\eta$. This comparison excludes $\Delta <148$ meV, as indicated by the red region. In (b), we use a range of $\Delta$ that corresponds approximately [35] to the range shown in Fig. 5, and we vary the acoustic cutoff energy $\Omega$. The gray region represents the predicted range of $\Omega$ [36].

Figure 7

Error in ${\Gamma }_{E_{1,2}}/{\Gamma }_{A_1}$ ratio due to assumption of the low-temperature limit. The theoretical values of ${\Gamma }_{E_{1,2}}/{\Gamma }_{A_1}$ shown in Fig. 6 were calculated using Eq. (19), which neglects thermal occupation of the mediating phonon modes. We calculate the error due to working in the low-temperature limit by using the temperature-dependent expression for ${\Gamma }_{E_{1,2}}$ [Eq. (10)] rather than the temperature-independent expression [Eq. (12)]. We perform this calculation for the cases where (a) $\Omega$ is infinite and $\Delta$ is varied, and where (b) $\Delta$ is given by the values extracted from our analysis of ${\Gamma }_{A_1}$ and $\Omega$ is varied. These are the same cases shown in the corresponding subfigures of Fig. 6. The results extracted from Fig. 6, the minimum allowed value of $\Delta$ in (a) and the expected values of $\Omega$ in (b), are shown to emphasize that the conclusions we draw using the low-temperature limit are valid.

Figure 8

Calculated temperature-dependent vibrational overlap functions $F(\omega ,T)$. The measured low-temperature vibrational overlap function $F\left(\omega \right)$ is shown in black [17].

Figure 9

Comparison of the predicted high-temperature fluorescence lifetimes with measured lifetimes. The data from 5 to 26 K are taken from Ref. [16] and the data from 295 to 700 K are taken from Refs. [12] $(\square$ and $\circ )$, [14] $(\diamond )$, and [13] $(▿)$. In (a), the solid green line is the lifetime of the $m_s=0$ states predicted by our model, and the dashed green line includes a fit to the Mott-Seitz model with $95\%$ confidence interval, as described in the text. This decay mechanism is not included in the predicted lifetime of the $|m_s|=1$ states. In (b), we plot the predicted lifetime of the $|m_s|=1$ states with varying degrees of coupling to the high-temperature decay mechanism.