Decoherence of three-level systems: Application to nitrogen-vacancy centers in diamond near a surface

We study the dynamics of quantum three-level systems due to fast fluctuations of their environment (white noise). As an example, we model the decoherence and dephasing of nitrogen-vacancy (NV) centers in diamond that are positioned near a surface that gives rise to quickly fluctuating magnetic field noise, with and without the presence of a rf field that couples levels of the ground electronic state manifold. We use a simple quantum mechanical model that allows for analytical solutions. The model treats the NV three-level ground state in the presence of rapidly fluctuating magnetic fields that arise from the environment and result in decoherence, dephasing, and dissipation. We show that the magnetic field fluctuations can couple all the elements of the density matrix of the three-level system. Hence, all 9 eigenmodes may contribute to both population and coherence dynamics. We also discuss consequences for fitting experiments in which decoherence plays a role.

Figure 1

Population decay for the case with $\epsilon =0$: ${\rho }_{11}\left(t\right)$ (blue dotted curve), ${\rho }_{00}\left(t\right)$ (orange), and ${\rho }_{-1-1}\left(t\right)$ (green dashed curve).

Figure 2

Population dynamics in the presence of a near-resonant rf field, ${\omega }_{\mathrm{rf}}=\mathcal{D}+\mu B_z-0.24$ GHz and $\mathrm{\Omega }=0.2$ GHz: ${\rho }_{11}\left(t\right)$ [blue, ${\rho }_{11}\left(0\right)$ = 0], ${\rho }_{00}\left(t\right)$ [orange, ${\rho }_{00}\left(0\right)=1]$, and ${\rho }_{-1-1}\left(t\right)$ (green), (a) with the RWA and (b) without the RWA.

Figure 3

Distributions of the eigenmodes and fitting for diagonal elements (on resonance, ${\omega }_{\mathrm{rf}}=\mathcal{D}+\mu B_z,\mu Bz=0.2$). The initial state is a pure ground state. (a) Distribution of the eigenmodes 0 through 8 in ${\rho }_{11}\left(t\right)$. (b) Distribution of the eigenmodes 0 through 8 in ${\rho }_{-1-1}\left(t\right)$. (c) Error of the two-time-scale fit for ${\rho }_{\mathrm{fit},3}\left(t\right)-{\rho }_{11}\left(t\right)$ (red) and that of the single-time-scale fit, ${\rho }_{\mathrm{fit},2}\left(t\right)-{\rho }_{11}\left(t\right)$ (blue). The inset shows ${\rho }_{\mathrm{fit},3}\left(t\right)$ (red), ${\rho }_{\mathrm{fit},2}\left(t\right)$ (blue), and ${\rho }_{11}\left(t\right)$ (green dashed), respectively. (d) Error of the two-time-scale fit for ${\rho }_{\mathrm{fit},3}\left(t\right)-{\rho }_{-1-1}\left(t\right)$ (red) and that of single-time-scale fit, ${\rho }_{\mathrm{fit},1}\left(t\right)-{\rho }_{-1-1}\left(t\right)$ (blue). The inset shows the fits ${\rho }_{\mathrm{fit},3}\left(t\right)$ (red), ${\rho }_{\mathrm{fit},2}\left(t\right)$ (blue) for ${\rho }_{-1-1}\left(t\right)$ (green dashed), respectively.

Figure 4

The nine eigenvalues of the Liouvillian for the case with $\mu B_z=0.2$ GHz (blue points), and the three decay modes obtained by fitting. (a) Resonance case, ${\omega }_{\mathrm{rf}}=\mathcal{D}+\mu B_z$. Red points are obtained from the two-time-scale fit of ${\rho }_{-1-1}$. Green points are obtained from the two-time-scale fit of ${\rho }_{11}$ (the modes obtained by the fit of ${\rho }_{00}$ are located in exactly same position as those obtained for the two-time-scale fit for ${\rho }_{11}$, and are therefore not shown in figure). (b) Off-resonant case, ${\omega }_{\mathrm{rf}}=2.83$ GHz. Red points are obtained from the two-time-scale ${\rho }_{11}$ fit, green points are obtained from the two-time-scale ${\rho }_{00}$ fit, and the purple crosses are obtained from the two-time-scale fit of ${\rho }_{-1-1}$.

Figure 5

Relaxation of the coherence ${\rho }_{10}$ in the presence of a rf field. The initial state is a pure state of ${\psi }_{\mathrm{in}}\left(0\right)=\frac{1}{\sqrt{2}}\left(\right|m=0\rangle +|m=1\rangle )$. (a) The dynamics of the off-diagonal density matrix element ${\rho }_{10}\left(t\right)$ in the interaction picture in the presence of a resonant rf field for $B_z=\mathcal{D}+\mu B_z=3.07$ GHz (resonant case). (b) Distribution of the eigenmodes 0 through 8 for ${\rho }_{10}$ shown in panel (a). (c) The dynamics of the off-diagonal density matrix element ${\rho }_{10}\left(t\right)$ in the interaction picture in the presence of a resonant rf field for $B_z=2.83\mathrm{GHz}$ (off-resonant case). (d) Distribution of the eigenmodes 0 through 8 for ${\rho }_{10}$ shown in panel (d).

Figure 6

Nitrogen vacancy dynamics, eigenvalues, and their characteristics for $\mu B_z=0.2$ GHz. (a) Relaxation dynamics of ${\rho }_{10}$ in the presence of a rf field. The initial state is the pure state ${\psi }_{\mathrm{in}}\left(0\right)=\frac{1}{\sqrt{2}}\left(\right|m=0\rangle +|m=1\rangle ).\mathrm{Re}\left[{\rho }_{10}\right]$ (green dashed) and fits with ${\rho }_{\mathrm{fit},3}\left(t\right)$ (red) and ${\rho }_{\mathrm{fit},2}\left(t\right)$ (blue). Off-resonance case, ${\omega }_{\mathrm{rf}}=2.83$ GHz. (b) Eigenvalues of the Liouvillian (blue) and decay modes extracted from the two-time-scale fit of $\mathrm{Re}\left[{\rho }_{10}\right]$ (red), and the single-time-scale fit of $\mathrm{Im}\left[{\rho }_{10}\right]$ (green). Off-resonance case, ${\omega }_{\mathrm{rf}}=2.83$ GHz. (c) ${\omega }_{\mathrm{rf}}$ dependence of $R$ and $|a_{i_1}^{1,0}|/|a_{i_2}^{1,0}|$ calculated for ${\rho }_{10}$. (d) $\gamma$ dependence of $R$ and $|a_{i_1}^{1,0}|/|a_{i_2}^{1,0}|$ calculated for ${\rho }_{10}$. Off-resonance case, ${\omega }_{\mathrm{rf}}=2.83$ GHz.