Phonon-Induced Spin-Spin Interactions in Diamond Nanostructures: Application to Spin Squeezing

We propose and analyze a novel mechanism for long-range spin-spin interactions in diamond nanostructures. The interactions between electronic spins, associated with nitrogen-vacancy centers in diamond, are mediated by their coupling via strain to the vibrational mode of a diamond mechanical nanoresonator. This coupling results in phonon-mediated effective spin-spin interactions that can be used to generate squeezed states of a spin ensemble. We show that spin dephasing and relaxation can be largely suppressed, allowing for substantial spin squeezing under realistic experimental conditions. Our approach has implications for spin-ensemble magnetometry, as well as phonon-mediated quantum information processing with spin qubits.

Figure 1

(a) All-diamond doubly clamped mechanical resonator with an ensemble of embedded NV centers. (b) Spin triplet states of the NV electronic ground state. Local perpendicular strain induced by beam bending mixes the $|\pm 1⟩$ states. (c) A collection spins in the two-level subspace $\{|+1⟩,|-1⟩\}$ is off-resonantly coupled to a common mechanical mode giving rise to effective spin-spin interactions. (d) Squeezing of the spin uncertainty distribution of an NV ensemble.

Figure 2

(a) Spin squeezing parameter versus scaled precession time with $N=100$ spins. Thick blue (gray) lines show the calculated squeezing parameter for $T_2=10\mathrm{ms}$ and values of ${\overline{n}}_{\mathrm{th}}$ as shown. For each curve, we optimized the detuning $\Delta$ to obtain the optimal squeezing. Dashed lines are calculated from the linearized equations for the spin operator averages. Thin black solid (dashed) line shows exact (linearized) unitary squeezing. (b) Optimal squeezing versus number of spins. Lower (upper) line shows power law fit for ${\overline{n}}_{\mathrm{th}}=1$ (10) and $T_2=1$ (0.01) s. The detuning $\Delta$ is optimized for each point. Other parameters in both plots are ${\omega }_m/2\pi =1\mathrm{GHz}$, $g/2\pi =1\mathrm{kHz}$, $Q={10}^6$.

Figure 3

(a) Number fluctuation spectrum of thermal driven oscillator. Center (blue filled) peak is purely thermal while side (green filled) peaks are due to detuned drive. Solid (dashed) purple line shows filter function $F$ ($K$) for $M=4$ pulses. Inset: corresponding function $f(t)$ for $M=4$. (b) Solid green curves show squeezing parameter versus precession time for ${\overline{n}}_{\mathrm{th}}=10$ and ${\overline{n}}_{\mathrm{dr}}={10}^3$, $5\times {10}^4$, ${10}^6$ (top to bottom). Dashed black line shows unitary squeezing. (c) Minimum squeezing versus drive strength for ${\overline{n}}_{\mathrm{th}}=50$, 10 (top to bottom). Symbols mark corresponding points with (b). Dashed black line shows unitary squeezing. Parameters in (b) and (c) are $M=4$, $g/2\pi =1\mathrm{kHz}$, $T_2=10\mathrm{ms}$, $N=100$, ${\omega }_m/2\pi =1\mathrm{GHz}$, $Q={10}^6$.