Universal quantum gates between nitrogen-vacancy centers in a levitated nanodiamond

We propose a scheme to realize the controlled-phase gates between nitrogen-vacancy (NV) centers in an optically trapped nanodiamond, through a uniform magnetic field-induced coupling between the NV centers and the torsional mode of the levitated nanodiamond. The gates are insensitive to the thermal noise of the torsional mode. By combining the scheme with the dynamical decoupling, it is found that the high-fidelity universal quantum gates are possible with present experimental technology. The proposed scheme is useful for an NV-center-based quantum network and distributed quantum computation.

Figure 1

(a) An ellipsoid nanodiamond with two NV centers placed at two ends of its long axis is optically trapped in vacuum. (b) There are two NV centers in the levitated nanodiamond. A uniform magnetic field induces the strong coupling between the torsional mode and the NV center electron spin. The rotational direction $\vec{n}$ is perpendicular to both the axis of the NV centers electron spins and the magnetic field $\vec{B}$. The torsional motion of the nanodiamond leads to the change of the angle $\theta$ between the direction of the NV center electron spin1 and $\vec{B}$.

Figure 2

The ratio between the effective coupling strength $g_{\mathrm{eff}}=\sqrt{|g_1{g}_2|}$ and the torsional frequency $\omega$ as a function of angle $\theta$, with the magnetic field $B=0.045\mathrm{T}$ and $\omega =2\pi \mathrm{MHz}$. The magnetic field $B$ is tuned to fulfill $g/\omega =1/4\sqrt{2}$ at the maximum coupling strength $g$. The corresponding size of the nanodiamond is $r_x=r_y=10\mathrm{nm}$, $r_z=150\mathrm{nm}$.

Figure 3

Pulse sequences of five-piece SUPCODE sequences [58]. The time duration ${\tau }_0=(2+\theta /2\pi )t_0$ for a $\theta$ rotation pulse.

Figure 4

Infidelity due to coupling to torsional mode as a function of (a) thermal occupation ${\overline{n}}_{th}$ with Rabi frequency $\mathrm{\Omega }=2\pi \mathrm{MHz}$; (b) Rabi frequency $\mathrm{\Omega }$, with thermal occupation ${\overline{n}}_{th}=5$. The torsional frequency $\omega =2\pi \text{MHz}$ and coupling strength $g/\omega =1/4\sqrt{2}$. Blue circles are numerical results obtained by solving the master equation, and the blue line is the fitted model of Eq. (16) with the fitting parameter $n_0=3.65$.

Figure 5

Errors ($\xi =1-{\mathcal{F}}_{max}$) due to rethermalization of the cavity mode (a) and qubit dephasing (b). (a) Rethermalization-induced error for ${\overline{n}}_{th}=2$ (blue) and ${\overline{n}}_{th}=4$ (orange), with $\mathrm{\Gamma }=0$. (b) Dephasing-induced errors for $g/\omega =1/4\sqrt{2}$ (blue), $g/\omega =1/8$(orange), and $g/\omega =1/4\sqrt{2}$ (green); here $\kappa /\omega ={10}^{-6}$ and ${\overline{n}}_{th}=0.01$. In both cases the linear error scaling is verified.

Figure 6

The effective coupling strength $g_{\mathrm{eff}}$ under different length long axis $r_z$ of the nanodiamond and the external magnetic field $B$. We change the $r_z$ from $100\mathrm{nm}$ to $300\mathrm{nm}$, while keeping the short axis $10\mathrm{nm}$ and the trapping parameters unchanged. The trapping parameters are taken such that the trapping frequency at $r_z=150\mathrm{nm}$ is $2\pi \text{MHz}.$ The coupling strength is optimized along all the relative direction of the spin to the magnetic field, e.g., taking the maximum value in Fig. 2. A shape bend occurs when the Zeeman splitting $g_{NV}{\mu }_{B}B$ equals the zero-field splitting $D$, which corresponds to $B\sim 0.1\mathrm{T}$. Using Eq. (2), it is found that for $B$ larger than $0.1\mathrm{T}$ the spin-torsion coupling strength $g_{1,2}$ saturates.