Spin decoherence and electron spin bath noise of a nitrogen-vacancy center in diamond

We theoretically investigate spin decoherence of a single nitrogen-vacancy (NV) center in diamond. Using the spin coherent-state $P$-representation method, we simulate coherence evolution of the NV center coupled to surrounding nitrogen electron (N) spins. In the system, the strength of N-N coupling is the same order as that of NV-N coupling (the strong intrabath coupling regime). We find that spin decoherence time as well as free-induction decay of the NV center depend on the spatial configuration of N spins. Both the spin decoherence rate ($1/T_2$) and dephasing rate ($1/T_2^{*})$ of the NV center increase linearly with the concentration of the N spins. Using the $P$-representation method, we also demonstrate extracting the noise spectrum of the N spin bath. The capability to calculate the noise spectrum will provide promising pathways to suppress decoherence of spin systems in the strong intrabath coupling regime.

Figure 1

(a) Schematic of a spatial configuration for NV and N spins. The cube consists of a diamond lattice. A NV center (large red sphere) is located at $(0,0,0)$, and N spins (small blue spheres) are located randomly on diamond lattice sites. (b) Simulated FID signals of a NV center with three different N spin bath configurations. The concentration of N spins is $f=10$ ppm for all cases. Blue crosses, green squares, and red circles are simulated results, and lines are fit to $\exp [-{(t/T_2^{*})}^2]$. The obtained values $T_2^{*}=0.97$, 0.69, and 0.61 $\mu$s from the fit are in good agreement with calculated values directly from the N spin bath configuration, where $T_2^{*}=\sqrt{2}/b$ are $0.95$, $0.67$, and 0.62 $\mu$s, respectively. (c) Simulated FID for ensemble of NV spins with $f=10$ ppm of the N spin concentration (red dots). The black line represents a fit to a single exponential function giving $T_2^{*}=1/\Gamma =0.5$ $\mu$s. The inset shows a histogram of $b$ for the 90 instances with $f=10$ ppm. The black solid line represents the expected probability distribution of $b$ [Eq. (5)].

Figure 2

(a) Simulated SE and FID signals with $f=10$ ppm. Circles and crosses are simulation results, and the red solid line shows a fit to Eq. (6). With $b=1.44$ $\mu$s${}^{-1}$ from the FID data, we obtained ${\tau }_C=2.78$ $\mu$s. (b) Simulated SE signals for 90 instances of the N spin bath configuration with $f=10$ ppm. Blue circles are simulation results, and green lines are fits to Eq. (6). The inset shows a histogram of $\alpha$ for the 90 instances when SE signals are fitted by $\exp [-{(t/T_2)}^{\alpha }]$.

Figure 3

(a) $1/T_2^{*}$ as a function of the N spin concentration. Each square is the mean value of all instances. The blue solid line is a fit to a linear function. The inset shows the mean value of $T_2^{*}$ with the standard deviation as an error bar. The blue solid line is a fit to a linear function. (b) $1/T_2$ as a function of the N spin bath concentration. Each blue circle is the mean value of all instances. The blue solid line is a fit to a linear function. The red triangle is the result with the HF interaction between N electron and ${}^{14}$N nuclear spins taken into account. The difference in simulated $1/T_2$ between cases with and without the HF coupling is smaller than the deviation. The inset shows the $f$ linear dependence of $T_2$ on the standard deviation as an error bar. The mean values of $T_2$ are 2.03 and 0.2 $\mu$s for $f$ $=$ 10 and 100 ppm, respectively. The blue solid line is a fit to a linear function.

Figure 4

Simulated SE signals with $f=50$ ppm of the N spin concentration. (a) and (b) show SE signals from different spin bath configurations. Simulated results without (with) the HF coupling are shown by solid red circles (blue triangles). Solid red (dashed blue) lines are corresponding fits for the case without (with) the HF coupling.

Figure 5

Noise spectrum of N spins with $f=10$ ppm. Green circles are the simulated noise spectrum. The black solid line shows the noise spectrum of the O-U process expressed by the Lorentzian function with $1/{\tau }_C=0.134$ $\mu$s${}^{-1}$, where ${\tau }_C$ is extracted by fitting the SE signal. Red dotted and blue dashed lines show the power spectra of FID and SE pulse sequences, respectively. $|\tilde{f}{\left(\omega t\right)|}^2$ equals ${|\sin (\omega t/2)/(\omega /2)|}^2$ with $t=T_2^{*}=$ 0.82 $\mu$s for FID and equals $|{\sin }^2{(\omega t/4)/(\omega /4)|}^2$ with $t=T_2=$ 3.28 $\mu$s for SE.