Dark state polarizing a nuclear spin in the vicinity of a nitrogen-vacancy center

The nuclear spin in the vicinity of a nitrogen-vacancy (NV) center possesses long coherence time and convenient manipulation assisted by the strong hyperfine interaction with the NV center. It is suggested for the subsequent quantum information storage and processing after appropriate initialization. However, current experimental schemes are either sensitive to the inclination and magnitude of the magnetic field or require thousands of repetitions to achieve successful realization. Here, we propose a method to polarize a ${}^{13}\mathrm{C}$ nuclear spin in the vicinity of an NV center via a dark state. We demonstrate theoretically and numerically that it is robust to polarize various nuclear spins with different hyperfine couplings and noise strengths.

Figure 1

Schematic of polarizing a ${}^{13}\mathrm{C}$ nuclear spin in the vicinity of an NV center. (a) The structure of an NV center. (b) Energy levels of the electron and ${}^{13}\mathrm{C}$ nuclear spins under the hyperfine interaction. Two pulses are simultaneously applied to induce the transitions of $|0,\uparrow \rangle \leftrightarrow |-,\uparrow \rangle$ and $|-,\uparrow \rangle \leftrightarrow |-,\downarrow \rangle$ with respectively Rabi frequencies ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$, and one-photon detuning $\delta$, and two-photon detuning $\mathrm{\Delta }$.

Figure 2

The population dynamics of initial state $|0,\uparrow \rangle$ (red), intermediate state $|-,\uparrow \rangle$ (green), and target state $|-,\downarrow \rangle$ (black) by the Schrödinger equation (solid lines) with a non-Hermitian Hamiltonian expressed by Eq. (7) and the exact master equation with the exact Hamiltonian $H_M$ (dashed lines). The population dynamics are drawn with $A_{\parallel }=130$ MHz [49], $\delta =\mathrm{\Delta }=0$, ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=13$ MHz, and $\kappa =1/58$ MHz [50].

Figure 3

Comparison between the probabilities of the nuclear spin in the $|\downarrow \rangle$ vs time by applying two simultaneous pulses (black line) and two separate pulses (red line). In the simultaneous case, ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=13$ MHz, while ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=4.3$ MHz [31] in the separate case. Other parameters are the same in both cases, i.e., $A_{\parallel }=130$ MHz [49], $\delta =\mathrm{\Delta }=0$, and $\kappa =1/58$ MHz [50]. The optimal operation times are labeled by the blue arrows.

Figure 4

The fidelity of nuclear spin in $|\downarrow \rangle$ vs $A_{\parallel }=A_{\perp }$ for different noise strength: (a) $\kappa =1$ MHz, (b) $\kappa =1/5.8$ MHz, and (c) $\kappa =1/58$ MHz [50]. The solid red line is $A_{\parallel }=130$ MHz [49], the dashed blue line is $A_{\parallel }=14.8$ MHz [49], and the dotted green line is $A_{\parallel }=-7.5$ MHz [49].

Figure 5

The transfer fidelity $P_{|\downarrow \rangle }$ vs the one-photon (two-photon) detuning $\delta$ ($\mathrm{\Delta }$) for the first-shell ${}^{13}\mathrm{C}$. The dependence on $\mathrm{\Delta }$ is shown by the red curve with $\delta =0$. The dependence on $\delta$ is shown by the dashed blue curve with $\mathrm{\Delta }=0$. Other parameters are the same as in Fig. 2.

Figure 6

The effect of pulse amplitude fluctuation on the population dynamics for the first-shell ${}^{13}\mathrm{C}$: dash-dotted red curve for $|0,\uparrow \rangle$, dotted blue curve for $|-,\uparrow \rangle$, and dashed green curve for $|-,\downarrow \rangle$. The population dynamics without 1% amplitude fluctuation is shown by the solid curves. Other parameters are the same as in Fig. 2.

Figure 7

The influence of magnetic field strength on the transfer fidelity for the first-shell ${}^{13}\mathrm{C}$: Red curves with stars for $B=0$ G, black curves with circles for $B=200$ G, and green curves with squares for $B=1500$ G. The fidelity dynamics with (without) transverse hyperfine interaction is shown by the lower solid (upper dashed) curves. Other parameters are the same as in Fig. 2.