Decay of the rotary echoes for the spin of a nitrogen-vacancy center in diamond

We study dynamics of the electron spin of a nitrogen-vacancy (NV) center subjected to a strong driving field with periodically reversed direction (train of rotary echoes). We use analytical and numerical tools to analyze in detail the form and time scales of decay of the rotary echo train by modeling the decohering spin environment as a random magnetic field. We demonstrate that the problem can be exactly mapped onto a model of spin 1 coupled to a single bosonic mode with imaginary frequency. This mapping allows comprehensive analytical investigation beyond the standard Bloch-Redfield-type approaches. We explore the decay of the rotary echo train under assumption of strong driving and identify the most important regimes of the decay. The analytical results are compared with the direct numerical simulations to confirm quantitative accuracy of our study. We present the results for realistic environment of substitutional nitrogen atoms (P1 centers) and provide a simplified but accurate description for the decay of the rotary echo train of the NV center's spin. The approach presented here can also be used to study decoherence and longitudinal relaxation of other spin systems under conditions of strong driving.

Figure 1

(a) The quantity $\psi$ is plotted vs $\tau$ with red (dark gray) from Eq. (20) for a slow bath, $\rho =0.01$ and $\omega =10$. Its short-$\tau$ asymptote is plotted with a green (gray) dashed line from Eq. (23). The line through the origin intersects $\psi$ at $\tau ={\tau }_0$. (b) The real and imaginary parts of $q_z$ are plotted vs $\tau$ from Eq. (21) (continuous lines), for the same bath. The large-$\tau$ asymptotes, plotted from Eq. (26), are shown with the dashed lines.

Figure 2

Simulations results for the central spin oscillations in a quasistatic bath, $\rho =5\times {10}^{-5}$. Individual oscillations caused by the periodically reversed driving are shown in gray, as a function of the total time $\tilde{t}=4N\tau$. The envelope of the conventional Rabi oscillations for the same bath is plotted with black. Rotary echoes are well pronounced and decay much slower than the Rabi oscillations.

Figure 3

Numerical simulations for the slow bath, $\omega =20$ and $\rho =0.005$. Horizontal axes represent the total time $\tilde{t}=4N\tau$. Individual oscillations within the rotary echo signal (gray) have very high frequency, and are not well resolved in the picture. Analytical predictions for the rotary echo amplitudes (red diamonds), as obtained from Eq. (22), perfectly match the simulations. The envelope of the conventional Rabi oscillations with the same parameters is shown with the black solid line. (a)–(c) correspond to three different values of $\tau$: (a) $\tau =12$, (b) 30, and (c) 60. (d) Zoom-in of the region marked as a dotted rectangle from the panel (b). Note that the rotary echo protocol is very efficient for short $\tau$, but its efficiently drops, and becomes comparable to a standard Rabi oscillation, as $\tau$ approaches ${\left|P\right|}^{-1}\approx 63.3$.

Figure 4

Simulation results for the fast bath with $\omega =20$ and $\rho =0.3$. Horizontal axes in (a)–(c) represent the total time $\tilde{t}=4N\tau$. Individual oscillations within the rotary echo signal (gray) have very high frequency, and are not well resolved in the picture. Analytical predictions for the rotary echo amplitudes (red diamonds), as obtained from Eq. (32), match the simulations very well. The envelope of the conventional Rabi oscillations with the same parameters is shown with the black solid line. (a) and (b) The short-$\tau$ and long-$\tau$ regimes, respectively; the values of $\tau$ are chosen according to Eq. (31), the values of $\tau$ are shown in the graphs. (c) Zoom-in view of the marked area in (a), demonstrating good agreement between analytics and numerics. (d) $\langle S^z\left(N\right)\rangle$ as a function of $\tau$ for $N=150$. Gray dots are the values obtained from simulations, and red line is obtained analytically from Eq. (32) (short-$\tau$ regime).

Figure 5

Numerical simulations with six noise fields. The rotary echo $\langle S^z\left(N\right)\rangle$ is plotted as a function of total time $t=4NT$ for different values of the quarter-periods $T$ (given in microseconds) of the driving reversal. The four figures correspond to different driving amplitudes, $h=2\pi \times 4.85$ ($\omega =6$), $2\pi \times 12.93$ ($\omega =16$), $2\pi \times 20.2$ ($\omega =25$), and $2\pi \times 29.9$ MHz ($\omega =37$). The bath parameters are given in Table 1. The gray dots denote the numerically obtained $\langle S^z\rangle$, and the red symbols correspond to the analytical results of Eq. (32), where we assumed a single effective noise field with the parameters $R_e$ and $b_e$, as explained in the main text.

Figure 6

Simulation results for the rotary echo decay in the presence of six O-U noise fields and the hyperfine interaction with $A_0=2\pi \times 2.16$ MHz. Parameters of the noise fields are taken from Table 1. The four panels correspond to four different driving amplitudes ($h/2\pi =4.85$, $12.93$, $20.2$, and $29.9$ MHz, shown at the top of each panel). Three curves in each panel represent the results obtained for three different durations $T$ of the protocol quarter periods, the values of $T$ (in microseconds) are shown on the graphs. The simulation results are denoted by gray dots, showing $\langle S_{\lambda }^z\left(N\right)\rangle$ as a function of total time $t=4NT$. Red symbols are the analytical results obtained from Eqs. (38) and (E14).