Generating spin squeezing states and Greenberger-Horne-Zeilinger entanglement using a hybrid phonon-spin ensemble in diamond

Quantum squeezing and entanglement of spins can be used to improve the sensitivity in quantum metrology. Here we propose a scheme to create collective coupling of an ensemble of spins to a mechanical vibrational mode actuated by an external magnetic field. We find an evolution time where the mechanical motion decouples from the spins, and the accumulated geometric phase yields a squeezing of $5.9\text{dB}$ for 20 spins. We also show the creation of a Greenberger-Horne-Zeilinger spin state for 20 spins with a fidelity of $\sim 0.62$ at cryogenic temperature. The numerical simulations show that the geometric-phase-based scheme is mostly immune to thermal mechanical noise.

Figure 1

Schematic diagrams for system squeezing and entangling an ensemble of electron spins in a nanodiamond. (a) An ensemble of NV centers in a nanodiamond oscillates along the $x$ direction with a resonance frequency ${\omega }_m$. The principal axis of the nanodiamond is cut along the $z$ direction. A magnetic field $B_z$ is parallel to the $z$ axis, while the magnetic field $B_x\left(x\right)$ is applied along the $x$ direction and has a giant gradient along the $x$ direction. (b) A level diagram of the NV centers in a nanodiamond. (c) The Bloch sphere representation of squeezed $N=20$ NV centers. (d) The Bloch sphere representation of the GHZ state of $N=20$ NV centers.

Figure 2

(a) Squeezing parameter ${\xi }_R^2$ for $N=50$ spins as a function of $\theta$ for different thermal noise $\overline{n}/Q_m$. (b) Optimal squeezing parameter ${\xi }_{\text{R}}^2\text{(dB)}$ as a function of number of spins, $N$ at different thermal noises $\overline{n}/Q_m=0,0.01,0.1$. The numerical results are fitted by $1.4N^{-2/3},1.4N^{-0.64}$, and $1.4N^{-0.48}$.

Figure 3

Squeezing parameter ${\xi }_R^2$ for $N=10$ spins at different $m$ but fixed $\theta \left(t_m\right)$. Here $Q_m={10}^6,\overline{n}=10$.

Figure 4

(a) Fidelity of the generated GHZ state as a function of thermal noise, $\overline{n}/Q_m$, for $N=10$ (blue line) and $N=20$ (red line) fitted two-term exponential functions, $a{e}^{b\overline{n}/Q_m}+c{e}^{d\overline{n}/Q_m}$, with $\{a=0.5208,b=-32.58,c=0.4722,d=-4.241\}$ (dashed black lines) and $\{a=0.5182,b=-54.21,c=0.4704,d=-6.016\}$ (dotted-dashed black lines), respectively. Green lines are guide to eyes at $F=0.5,0.9$. (b) The mechanical thermal decoherence limit for achieving the set fidelity ($F=0.5,0.9$) for different number of ${\text{NV}}^{-}\mathrm{s}$. Blue lines are numerical results by solving the master equation. Dashed red lines are the two-term exponential fitting by $a{e}^{bN}+c{e}^{dN}$ with $\{a=0.063,b=-0.229,c=0.058,d=-0.032\}$ for $F=0.5$ and $\{a=9.{248}^{-3},b=-0.277,c=6.{816}^{-3},d=-0.042\}$ for $F=0.9$.

Figure 5

Fidelity of the GHZ state for $N=10$ spins at different $m$ but a fixed $\theta =\pi /2$. Here $Q_m={10}^6,\overline{n}=10$.

Figure 6

Magnetic field $B_x$ generated by a two-dimensional permanent magnetic tip made from SmCo27MGOe (simulated with FEMM 4.2). The coordinates of the four points are $\left\{\right(-1\mu \mathrm{m},1.5\mu \mathrm{m}),(1\mu \mathrm{m},1.5\mu \mathrm{m}),(1\mu \mathrm{m},-1.5\mu \mathrm{m}),(-1\mu \mathrm{m},-1.5\mu \mathrm{m})\}$. A ${30}^{\circ }$ convex arc connects the points $(-1\mu \mathrm{m},1.5\mu \mathrm{m})$ and $(-1\mu \mathrm{m},-1.5\mu \mathrm{m})$. Therefore, the diameter and the width of the tip are $3\mu \mathrm{m}$ and $2\mu \mathrm{m}$, respectively. (a) The density and contour plots of the magnetic field surrounding the magnetic tip. The arrows show the direction of the magnetic field. (b) Zoom-in plot of the magnetic field in the dashed white box in (a). (c) The component $B_x$ along the dashed white line in (b). (d) The component $B_x$ along the dashed green line in (b), $100\mathrm{n}\mathrm{m}$ away from the top of the arc.

Figure 7

Checking the optimal phase ${\theta }_{\text{opt}}$ for achieving the maximal squeezing in the absence of mechanical decay. (a) The optimal phase ${\theta }_{\text{opt}}$ for different spin number $N$. The dashed line is fitted by ${\theta }_{\text{opt}}=6^{-1/6}{N}^{-2/3}$. (b) The maximal squeezing at ${\theta }_{\text{opt}}$ for different spin number $N$. The dashed line is fitted by ${\xi }_s^2=N^{-2/3}$.