Theory of the optical spin-polarization loop of the nitrogen-vacancy center in diamond

The nitrogen-vacancy (NV) center in diamond is of high importance in quantum information processing applications. The operation of the NV center relies on the efficient optical polarization of its electron spin. However, the full optical spin-polarization process, which involves the intersystem crossing between the shelving singlet state and the ground-state triplet, is not understood. Here we develop a detailed theory of this process which involves a combination of pseudo- and dynamic Jahn-Teller interactions together with spin-orbit interaction. Our theory provides an explanation for the asymmetry between the observed emission and absorption spectra of the singlet states. We apply density functional theory to calculate the intersystem crossing rates and the optical spectra of the singlets, and we obtain a good agreement with the experimental data. Since the NV center serves as a template for other solid-state-defect quantum bit systems, our theory provides a toolkit to study them that might help optimize their quantum bit operation.

Figure 1

Properties of the NV center in diamond. (a) Geometry structure showing the symmetry axis of the $C_{3v}$ symmetry. The vacancy is the dotted circle, whereas the solid circles depict carbon atoms in the diamond lattice. (b) Calculated HSE06 Kohn-Sham levels in the diamond band gap between the valence (VB) and conduction (CB) bands. (c) Many-electron states are expressed in two-particle Slater determinants in the parentheses [see Eqs. (1, 2, 3, 4, 5, 6)]. The many-electron levels are also depicted with the measured zero-phonon lines (ZPL). The zero-field splittings in the triplet manifolds are artificially scaled up by five orders of magnitude for the sake of clarity. We label the possible intersystem crossing rates ($\mathrm{\Gamma }$'s) with colored dotted arrows that participate in the spin-polarization cycle. We label the radiative transitions in the aforementioned cycle with solid black arrows. We refer to the vibronic states (coupled electron-phonon wave functions) with tilde labels above the many-electron states. The nature of the vibronic ${}^1\tilde{A}_1$ and ${}^1\tilde{E}$ states is explained (blue and red solid curves and text) where $ee$, PJT, and DJT stands for electron-electron, pseudo-Jahn-Teller, and dynamic Jahn-Teller interactions, respectively.

Figure 2

(a) Jahn-Teller nature of the ${|{}^{1}A_1\rangle \oplus |}^{1}E\rangle$ states as can be calculated by means of Kohn-Sham DFT, which corresponds to ${\mathrm{\Lambda }}_e=0\mathrm{eV}$. The resulting Jahn-Teller energy $E_{\text{JT}}=316\mathrm{meV}$. (b) After ${\mathrm{\Lambda }}_e\approx 1.13\mathrm{eV}$ is switched on, the resulting Jahn-Teller energy is $E_{\text{PJT}}\approx 30\mathrm{meV}$ according to the solution of Eq. (17) that results in $\mathrm{\Lambda }=1.19\mathrm{eV}$ ZPL energy. (c) The vibronic levels of ${|}^1\tilde{E}\rangle$ and $|{}^1\tilde{A}_1\rangle$. The selection rules for the photoluminescence spectrum are indicated. Here the ZPL energy of 1.19 eV between ${|}^1\tilde{E}\rangle$ and $|{}^1\tilde{A}_1\rangle$ is not scaled for the sake of clarity. (d) Experimental photoluminescence spectrum of the singlets at low (black solid line) and room (dotted black line) temperatures compared to the simulated spectrum from the ab initio solution (red curve). We note that the experimental spectra show a substantial and minor background at room and cryogenic temperature, respectively. The simulation curve does not include background signal. The ZPL energy is now set to zero in order to easily read out the position of vibration features in the spectrum. We used 2, 5, and 10 meV Gaussian smearing for the linewidth of the ZPL, first, and second vibronic emissions, respectively, where the width of the ZPL and vibration bands are read out from the experimental spectrum recorded at cryogenic temperature.

Figure 3

(a) Calculated APES of singlet states of the NV center including the pseudo- and dynamic Jahn-Teller effects simultaneously but with $Y=0$ constraint. Zero value at the configuration coordinates corresponds to $C_{3v}$ symmetry. The configuration coordinates and distances are given in the unit of the reduced coordinate of the harmonic oscillator. The localization of the wave functions is depicted. We note that the potential energy is not axial symmetric, as explained in the text. (b) Low-temperature experimental absorption spectrum (black solid curve) and the deduced spectral function (black dotted curve) from Ref. [18] are compared with the calculated absorption spectra (solid blue and red curves) and spectral functions (dotted blue and red curves) using the Huang-Rhys theory based on the APES of singlet states. We applied 1.5 meV Gaussian smearing on the ZPL and 7.5 meV Gaussian smearing on the vibronic sideband.

Figure 4

Calculated low-temperature ISC rates $[{\mathrm{\Gamma }}_z$, ${\mathrm{\Gamma }}_{\pm }$, ${\mathrm{\Gamma }}_{\mp }$ from Eqs. (23, 24, 25, 26)] as a function of the energy gap ($\mathrm{\Sigma }$) between the shelving state singlet state and the triplet ground state. Here we applied ${\lambda }_{\perp }=1.2{\lambda }_z$ by following Ref. [19]. Two experimental data about the lifetime of the singlet state ($T_E$) are applied from Ref. [22] with $T_E^{-1}\approx 2.16\mathrm{MHz}$ (yellow horizontal line) and Ref. [12] with $T_E^{-1}\approx 2.70\mathrm{MHz}$ (purple horizontal line). The crossing point between the simulated inverse lifetime (black curve) and the experimental ones is depicted by a circle that results in $\mathrm{\Sigma }=402$ and $\mathrm{\Sigma }=386\mathrm{meV}$, respectively.

Figure 5

${\mathrm{\Gamma }}_z/{\mathrm{\Gamma }}_{\perp }$ is plotted as a function of ${\lambda }_{\perp }/{\lambda }_z$, where ${\lambda }_z=15.78\mathrm{GHz}$ is our accurate DFT value. The ${\lambda }_{\perp }=56.31\mathrm{GHz}$ value approximated from DFT wave functions is an overestimation. ${\lambda }_{\perp }=39.76\mathrm{GHz}$ yields ${\mathrm{\Gamma }}_z/{\mathrm{\Gamma }}_{\perp }=1.2$.

Figure 6

The calculated lifetime of the singlet shelving state is plotted as a function of the temperature with the observed lifetimes for two single NV centers (dot and triangle data points with uncertainties) taken from Ref. [12].