Phonon Networks with Silicon-Vacancy Centers in Diamond Waveguides

We propose and analyze a novel realization of a solid-state quantum network, where separated silicon-vacancy centers are coupled via the phonon modes of a quasi-one-dimensional diamond waveguide. In our approach, quantum states encoded in long-lived electronic spin states can be converted into propagating phonon wave packets and be reabsorbed efficiently by a distant defect center. Our analysis shows that under realistic conditions, this approach enables the implementation of high-fidelity, scalable quantum communication protocols within chip-scale spin-qubit networks. Apart from quantum information processing, this setup constitutes a novel waveguide QED platform, where strong-coupling effects between solid-state defects and individual propagating phonons can be explored at the quantum level.

Figure 1

Setup. An array of SiV defects embedded in a 1D phonon waveguide. The inset shows the level structure of the electronic ground state of the SiV center. A tunable Raman process involving the excited state $|3⟩$ is used to coherently convert the population of the stable state $|2⟩$ into a propagating phonon, which can be reabsorbed by any other selected center along the waveguide. See text for more details.

Figure 2

Phonon waveguide. (a) Acoustic dispersion relation for a triangular waveguide of width $w=130\mathrm{nm}$ and an etch angle $\phi =35°$. Symmetric (solid lines) and antisymmetric (dashed lines) branches with respect to the vertical mirror-symmetry plane are shown. (b) Normalized displacement profiles of the symmetric phonons at 46 GHz. (c) Corresponding emission rates into the symmetric longitudinal (${\mathrm{\Gamma }}_l$) and transverse (${\mathrm{\Gamma }}_t$) polarization for different positions of the defect within the triangular cross section. (d) ${\mathrm{\Gamma }}_l$ and fraction (${\beta }_l$) of spontaneous emission into the longitudinal branch for different positions of the SiV center along the vertical mirror-symmetry axis. In our calculations, the waveguide is oriented along the [110] crystal axis, and the SiV centers are oriented along the [$\overline{1}11$], orthogonal to the waveguide axis. Experimentally, SiVs of this specific orientation can be pre-selected based on their Zeeman splitting ${\omega }_B$, when a static $B$ field is aligned along [$\overline{1}11$].

Figure 3

State transfer protocol. (a) Schematics showing the relevant fields, retardation times, and propagation phases. (b) State transfer fidelity for constant rates ${\gamma }_e(t)={\gamma }_r(t)={\gamma }_{\max }$. The case of a single resonant mode (red dashed line; ${\varphi }_L^t=0$, ${\varphi }_L^l=\pi$) is compared to the off-resonant case (dot-dashed black line; ${\varphi }_L^t={\varphi }_L^l=\pi$) for $L\sim 100\mu \mathrm{m}$ ($\mathrm{\Delta }{\omega }_t/{\gamma }_{\max }=140$). The full green line represents the long-waveguide counterpart of the off-resonant scenario, where $L\sim 1\mathrm{mm}$ ($\mathrm{\Delta }\omega /{\gamma }_{\max }=14$). (c)–(d) Protocol using slowly-varying control pulses ($t_p{\gamma }_{\max }=1$) where ${\mathrm{\Phi }}_{r,t}^{\mathrm{out},L}(t)$ is completely suppressed. The dashed blue line corresponds to the long waveguide counter part of the dashed red line. For (b)-(d), the two defects are equally coupled to both modes, ${\varphi }_e^n={\varphi }_r^n=\pi$ and ${\beta }_e^n={\beta }_r^n=0.5$. (e) Fidelity for varying positions of the receiving SiV center assuming ${\varphi }_L^t={\varphi }_L^l=\pi$ and a maximal transfer time of $12{\gamma }_{\max }^{-1}$. In all plots, we considered defects near the boundaries (${\tau }_e={\tau }_r\approx 0$) with a reflectivity of $R=0.92$, which corresponds to $Q\approx 5\times {10}^4$ in the cavity limit.

Figure 4

Scalability. (a) Quantum connectivity matrix for 49 SiV centers, equally spaced in a $500\mu \mathrm{m}$ long waveguide. The maximal protocol time is fixed to $12{\gamma }_{\max }^{-1}$ and $R=0.92$. (b) State-transfer protocol in an infinite waveguide where the outermost SiV centers act as switchable mirrors. (c) The full black line shows the fidelity as a function of the characteristic time of the emitting pulse, $t_p$, while the dashed red curve shows the total protocol time, $t_{\text{tot}}$. For the considered positions the propagation phases between each center and their neighboring mirror defect is $\pi$ for both phonon branches and $\mathrm{\Delta }{\varphi }_{er}=2\pi \times n$. In all simulations, a constant decay rate for the mirror defects, ${\gamma }_{m_1}(t)={\gamma }_{m_2}(t)={\gamma }_{\max }$, maximizes the fidelity.