Introduction

The interaction of light with solid matter via radiation pressure forces is a remarkable phenomenon whose discovery dates back to the 17th century1,2. In recent decades, progress on understanding and engineering this light-matter interaction has produced groundbreaking experiments in cavity optomechanics, including laser feedback cooling3, parametric light-matter processes in kg-scale4 and picogram-scale5,6,7 optomechanical systems, and laser cooling of mechanical modes to their ground state7,8. These quantum optics-like experiments have paved the way for optomechanical devices to be used in quantum transduction9,10,11,12,13 and entanglement14,15.

Solid-state vacancy-defect complexes are a developing technology that is complementary to cavity optomechanics. These complexes are atomic defects in dielectric media, such as diamond, that can be intentionally created in a dielectric lattice16,17. The free electron spin or nuclear spin of the resulting lattice vacancies can be coherently controlled as solid state quantum bits16,18. Additionally, research efforts demonstrating acoustic control of spin centers has opened the door to multi-modality quantum systems, such as spin-optomechanical interfaces19,20,21. These complex coupled systems can potentially allow for dark-state operation of spin centers, optical-to-spin quantum transduction, and new architectures for quantum repeaters in a quantum network.

Here, we propose a spin-optomechanical interface in diamond for strong coupling between the mechanical mode of an optomechanical resonator and an embedded group IV defect-vacancy complex. Our device introduces an optical resonance to ultra-small mechanical cavities previously used for spin interfacing22. By jointly confining the optical (mechanical) mode to a normalized mode volume of (10−3λ3 (\(1{0}^{-5}{\Lambda }_{p}^{3}\)), where λ (Λp) is the optical (longitudinal acoustic) wavelength, we simulate competitive optomechanical coupling rates gom ~800 kHz while achieving spin-mechanical coupling rates gsm ~30 MHz to a spin state of the vacancy complex. Thus, we show that this ultrasmall mode volume device can be used to interact with a vacancy without optically exciting the spin at its native wavelength, operating at the cavity wavelength instead through a optomechanically mediated interaction. Hence, using quantum modeling, we explore utilizing this spin-optomechanical interface for entanglement protocols in quantum networks in a setup depicted in Fig. 1. We use the modeled entanglement rates to feedback on a computer-aided design of an optimized spin-optomechanical interface.

Fig. 1: Depiction of spin entanglement via a spin-optomechanical interface.
Fig. 1: Depiction of spin entanglement via a spin-optomechanical interface.
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Each node contains an optical resonator (orange cavity/operator) coupled to a mechanical resonator (blue spring/operator), with an embedded color center (inset/green operator). A pump (red) is used to induce a two-mode squeezing in the opto-mechanical system. The leakage of an optical photon (orange waveguide) and its detection (orange detector) herald the creation of a single mechanical phonon. A beamsplitter (in gray) can be used to “erase” the knowledge of which is the original source of the photon, leading to the heralding of an entangled state\(| 10\left.\right\rangle \pm | 01\left.\right\rangle\) between two neighboring nodes. The phase depends on which of the two detectors clicked78. The ground state splittingΔgs (black) can be actuated with DC strain to selectively detune ωσ and gsm23 to turn on and off the spin-phonon interaction and swap the entangled phonon to the spin degree of freedom.

Results

Theory of Spin-Optomechanical Coupling

A spin-optomechanical interface accomplishes two effects. First, it couples the photonic mode of a photonic crystal cavity to the phononic modes of the crystal in a pump-driven interaction. Next, it couples the spin transition of a solid-state color center to the same phononic modes. Let us denote the operating frequency of the photonic mode as ωa, the spin transition frequency as ωσ between two states of mixed spin–orbit character to permit strain interactions23, and the pump beam frequency as ωp. Without loss of generality, we assume only a single phononic mode Ω is nearly resonant with the pump detuning, such that Δ = ωp − ωa ≈ Ω. Then we can simplify the system Hamiltonian by considering only a single phononic mode. In this picture, the unperturbed Hamiltonian \({\hat{H}}_{0}\) can be written as

$${\hat{H}}_{0}=\hslash {\omega }_{{\rm{a}}}{\hat{a}}^{\dagger }\hat{a}+\hslash \Omega {\hat{b}}^{\dagger }\hat{b}+\frac{\hslash {\omega }_{\sigma }}{2}{\hat{\sigma }}_{{\rm{z}}}.$$
(1)

Here, \({\hat{a}}^{\dagger },\hat{a}\) and \({\hat{b}}^{\dagger },\hat{b}\) are the ladder operators of the photonic and phononic modes, respectively, and \({\hat{\sigma }}_{j}\) is the spin qubit’s j-Pauli operator.

Additionally, the parametric coupling between the mechanical and optical resonators takes the form \({\hat{H}}_{{\rm{om}}}=\hslash {g}_{{\rm{om}}}{\hat{a}}^{\dagger }\hat{a}({\hat{b}}^{\dagger }+\hat{b})\), i.e., an optical resonance shift dependent on the position of the mechanical resonator. To linearize this interaction, we drive the optical cavity with a pump ωp = ωa + Δ. By applying the rotating wave approximation and rewriting the photon ladder operators around a mean population \(\overline{a}\) as \(\hat{a}\to \overline{a}+\hat{a}\), we arrive at the typical optomechanical interaction Hamiltonian in the blue-detuned regime,

$${\hat{H}}_{{\rm{om-bl}}}=\hslash {g}_{{\rm{om}}}\overline{a}\left({\hat{a}}^{\dagger }{\hat{b}}^{\dagger }+\hat{a}\hat{b}\right).$$
(2)

In the red-detuned regime, we get the following Hamiltonian

$${\hat{H}}_{{\rm{om-rd}}}=\hslash {g}_{{\rm{om}}}\overline{a}\left({\hat{a}}^{\dagger }\hat{b}+\hat{a}{\hat{b}}^{\dagger }\right).$$
(3)

Next we consider the spin–mechanical interaction. In a Group IV diamond defect, spin–strain interactions occur under low static strain and under static magnetic field17,23,24. In this regime, the lower branch of the ground manifold weakly mixes with the upper orbital, strain-sensitive states to produce mixed spin–orbit states that are strain susceptible. In the resulting spin–strain interaction picture, the spin-mechanical interaction is generated by deformation-induced strain perturbing the spin states to first order. When the transverse magnetic field experienced by the Group IV defect is such that ωσ = Ω, this interaction is described by the Hamiltonian

$${\hat{H}}_{{\rm{sm}}}=\hslash {g}_{{\rm{sm}}}\left({\hat{\sigma }}_{+}\hat{b}+{\hat{\sigma }}_{-}{\hat{b}}^{\dagger }\right).$$
(4)

Here, gsm is the strain-induced coupling by the zero-point fluctuation of the mechanical resonator and \(\hat{{\sigma }_{\pm }}=\frac{1}{\sqrt{2}}({\sigma }_{{\rm{x}}}\pm i{\sigma }_{{\rm{y}}})\) (see Supplementary). As such, any phonon excitation will induce zero-point coupling between the spin qubit and resonator phonon and vice versa. In this paper, we assume a static magnetic field around 0.18 T, where the transverse field component relative to the axis of the Group IV defect is sufficiently large such that gsm approaches the strain-orbit coupling term25. We will henceforth ignore the upper orbital branch, as well as the optically excited states of the defect, which are far off resonance from the mechanical and/or optical modes of the 1D OMC.

An efficient spin-optomechanical interface requires tuning of couplings gom and gsm, as well as quality factors Qopt and Qmech for a targeted experiment. We delve into the design considerations that affect these parameters below.

Device Simulations

At the core of our proposal is a strain concentrator embedded in a one-dimensional optomechanical crystal (1D OMC) with rectangular cross-section (Fig. 2a). The 1D OMC consists of a nanobeam with periodically etched ellipses, 2nd of which are adiabatically morphed into a defect cell. We then modify the central defect cell by tapering to a width b using a linear taper. This linear taper simultaneously enhances strain by forcing strain energy through the small cross-sectional area of a sharply angled, narrow bridge–as in previous mechanical resonator proposals22–and concentrates the optical mode profile–shown in previous photonic crystal designs26–in the center of the 1D OMC. In our design, only the central cell is modified with this taper to maximize the interaction of the acoustic phonon with a single spin, as well as the optical field concentration in the same region of concentrating mechanical displacement. We simulate (at b = 60 nm, nd = 6) an optical mode of frequency ωa/(2π) ≈ 200.2 THz and Qopt ≈ 9.6 × 104 (Fig. 2f), which lies in the mirror cells’ 28.7 THz optical bandgap from 175.28 THz to 203.98 THz (Fig. 2b). We predict an acoustic resonance around Ω = 5.39 GHz (Fig. 2e) between the 2.41 GHz acoustic bandgap from 4.96 GHz to 7.37 GHz (Fig. 2d).

Fig. 2: Diamond rectangular 1D nanobeam OMC with embedded concentrator, drawing from previous examples in silicon5,7 and diamond20,47,79 as well as ultrasmall mode volume photonic and phononic crystals22,26.
Fig. 2: Diamond rectangular 1D nanobeam OMC with embedded concentrator, drawing from previous examples in silicon5,7 and diamond20,47,79 as well as ultrasmall mode volume photonic and phononic crystals22,26.
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a Diagram of the nanobeam photonic crystal. Free parameters include taper width b; unit cell period as a function of cell number n, a(n); unit ellipse width hx(n) and height hy(n); and beam width w alongside beam thickness t. b Plot of quadratically varying a(n), hx(n), and hy(n) on either side of the beam center. This characterizes the cavity parametrized by Table 1 later in the text. c optical and (d) mechanical bandstructure for the mirror unit cell of the cavity, providing a 28.7 THz bandgap around and a 2.41 GHz mechanical bandgap. e mechanical displacement and (f) electric field norm profiles of the 5.39 GHz mechanical mode and 200.2 THz optical mode of the cavity. These simulations are for parameters \(\{{h}_{{y}_{d}},{h}_{{x}_{d}},{a}_{d}\}=\{218.2,334.8,456.56\}\) nm, α = 135°, rest of the parameters are same as from Table 1.

We note here that the parameters governing the defect cell {b, ad, \({h}_{{x}_{d}}\), \({h}_{{y}_{d}}\)}, as well as the crystal orientation in the device α, heavily affect {gsm, gom, Qopt} which are critical to the device performance (Qmech is relatively unaffected by the device geometry as long as the phonon frequency Ω lies within the acoustic bandgap, and is rather limited by phonon-thermally and materially-governed phonon–phonon processes27,28,29,30). As such, we explore the behavior of these performance parameters as a function of the defect unit cell parameters in Fig. 3. From these sweeps, we find a mix of simple and non-trivial relations between the defect cell parameters and the performance parameters. Setting \(\alpha =\frac{3\pi }{4}\) rad, for example, maximizes both gom and gsm by simultaneously maximizing the photoelastic contribution to gom–gpe–and the strain along the transverse axis of an emitter in the center of the spin-optomechanical interface.

Fig. 3: Analysis of spin-mechanical coupling profiles and the defect cell parameters that affect this coupling.
Fig. 3: Analysis of spin-mechanical coupling profiles and the defect cell parameters that affect this coupling.
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(a) and (b) shows the 2D and 1D spin-mechanical coupling gsm profile of the spin-optomechanical crystal breathing mode, with expected position of the Group IV spin overlayed. The parameters b, ad, \({h}_{{x}_{d}}\), and \({h}_{{y}_{d}}\) impact the spin-mechanical coupling strength among other variables. The one-dimensional profile in the center of the bridge shows that strain is maximized at the bridge edge but retained in the center away from sidewalls. The crystal axis orientation in the device affects gsm (c) and gom (d, red) by changing the strain elements the spin sees and the photoelastic effect for gpe (d, yellow), which sums with a constant gmb (d, gray). e Sweeping b, ad, \({h}_{{x}_{d}}\), and \({h}_{{y}_{d}}\) about a naive parameter set shows that the parameters governing device performance, namely var ∈ {Qopt, gom, gsm}, are maximized about different variables, motivating protocol-based numerical optimization of the exact design parameters. Qmech was routinely near 106 or greater in the sweeps and not depicted. These sweeps are centered around the following parameters \(\{{h}_{{y}_{d}},{h}_{{x}_{d}},{a}_{d}\}=\{220.5,341.25,456.75\}\) nm, α = 135°; the rest of the parameters are the same as in Table 1.

Additionally, decreasing b, which can be thought of as the spring constant in the central bridge, increases the strain energy density, or equivalently the mechanical mode volume22, in the central bridge of the spin-optomechanical interface. Intuitively, this can be seen as forcing the same amount of strain energy in the central bridge through a smaller bridge volume. We estimate through FEM that \({V}_{{\rm{mech}}}/{\Lambda }_{{\rm{p}}}^{3}\) and \({V}_{{\rm{mech}}}/{\Lambda }_{{\rm{s}}}^{3}\) drop from ~ 10−4 and ~ 10−3, respectively, to ~ 10−6 and ~ 10−5, respectively, as b decreases from 100 nm to 20 nm. Here, Λp and Λs are the longitudinal and shear wavelengths in bulk diamond22. As Vmech decreases, gsm increases, which also increases the “mechanical Purcell enhancement.” Vopt/λ3 or Vopt/(λ/n)3 similarly decrease from ~ 10−2 and ~ 10−1, respectively, to ~ 10−3 and ~ 10−2, respectively, with decreasing b–a beneficial effect for simultaneously concentrating the cavity mechanical and optical modes. Here, λ is the free space cavity wavelength, and n is the refractive index of diamond. Practically, we find that both gsm and gom increase as b is made as small as possible, although for b ≤ 20 nm, the frequency of the mechanical mode (which decreases with decreasing b by the intuitive picture of reducing the 1D OMC spring constant) falls below 4.96 GHz and thus lies outside the acoustic bandgap of the mirror cells. Additionally, fabrication constraints make achieving b < 20 nm otherwise challenging. However, the effects of {ad, \({h}_{{x}_{d}}\), and \({h}_{{y}_{d}}\)} on the performance parameters are less predictable, owing to a possible interplay between these parameters in the defect cell geometry. In previous studies on optomechanical crystals, these parameters are numerically optimized to yield the best performance parameter set7. In this study, which considers a complicated tripartite interface of an optical cavity, a phononic cavity, and an embedded spin system, we would like to motivate the optimization not just by {gsm, gom, Qopt, Qmech} but by a protocol that utilizes this interface. We consider non-classical spin state heralding and opto-mechanically mediated entanglement between two spins in the sections below.

Spin-Mediated Entanglement via DLCZ

The controlled opto-mechanical two-mode squeezing represented by Eq. (2) enables us to herald the creation of single phonons in the mechanical resonator. Such excitations can then be deterministically transferred to the spin via the spin-mechanical interaction in Eq. (3) for long term storage. Crucially, if we employ the Duan, Lukin, Cirac, and Zoller’s31,32 entangling protocol, we can herald an entangled \(| 01\left.\right\rangle \pm | 10\left.\right\rangle\) state in two remote mechanical resonators. Each of the two mechanical resonators can then deterministically swap its content with their embedded spins by DC strain-actuating the optomechanical crystal to detune the spin from the mechanical mode (see Supplementary), leading to two remote entangled long-lived spins for use in quantum networking.

The DLCZ protocol is, at its core, two single-phonon heralding experiments running in parallel as seen in Fig. 1. However, the detector triggering the heralding is placed after a “path erasure apparatus”, e.g., a simple 50–50 beamsplitter. Therefore, when a phonon is heralded by the detection of a photon, the phonon is in equal superposition of being in the left or in the right node. This results in the two mechanical resonators being in the state \(| 01\left.\right\rangle \pm | 10\left.\right\rangle\), with the phase depending on which detector clicked. For details on this path erasure consult31,33.

Hence, we develop a protocol for single-spin heralding by first initializing our spin-phonon system in the ground state via optomechanical cooling and then performing single-phonon heralding and swapping using optomechanical squeezing and spin-swapping. The protocol is shown in Fig. 4a, b, where we (1) cool, (2) herald on the creation of an entangled phonon, and (3) conditionally swap to the vacancy spin state.

Fig. 4: Diagram of the full heralding protocol and timing.
Fig. 4: Diagram of the full heralding protocol and timing.
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Schematic of the full protocol depending on the (a) absence or (b) presence of a photon click on the photo-detectors at t = tC. (c) Map of all possible quantum trajectories.We start with N trajectories. Out of them, there is a photon click in x1 of them. After getting a click, there are three possibilities, where events a, b and b1 correspond to the three collapse operators described in section 2.3. The spin-phonon swap is performed on all the brown trajectories, but ideally only the green ones are the good trajectories, which means that the red ones, lead to the infidelity of our heralding plus swap protocol. Using the trajectory approach, we can easily estimate, Pherald = \(\frac{{x}_{1}}{N}\) and pgood−traj = \(\frac{{x}_{5}}{{x}_{1}}\).

We begin by red-sideband optomechanical cooling of the phonon mode with the spin-mechanical coupling on for time tB − tA. This cools the phonon and coupled spin modes to their ground state. Then we utilize the blue-detuned Hamiltonian to perform the heralding protocol for a time tC − tB shorter than the time for the spin or optomechanical cavities to thermally repopulate. After the heralding, the controlled swap operation is implemented depending on if we get a photon click. Then the full system Hamiltonian is

$${\hat{H}}_{cool}={\hat{H}}_{0}+{\hat{H}}_{{\rm{om-rd}}}+{\hat{H}}_{{\rm{sm}}}.$$
(5)
$${\hat{H}}_{herald}={\hat{H}}_{0}+{\hat{H}}_{{\rm{om-bl}}}+{\hat{H}}_{{\rm{sm}}}.$$
(6)
$${\hat{H}}_{swap}={\hat{H}}_{0}+{\hat{H}}_{{\rm{sm}}}.$$
(7)

Below we study the fidelity and success probability of the single-phonon heralding protocol, as its performance directly affects the performance of the overall entanglement protocol. In this process, when writing down kets, we will use the Fock basis of the optical and mechanical modes, written down in that order, e.g., \(| 01\left.\right\rangle\) is zero photons and one phonon. Two processes are involved in the single-phonon heralding: the two-mode squeezing in Eq. (6) which leads to the mapping \(| 00\left.\right\rangle \to | 00\left.\right\rangle +\varepsilon | 11\left.\right\rangle +{\mathcal{O}}({\varepsilon }^{2})\); and the leakage into a waveguide and subsequent detection of the photon, which projects on the \(\varepsilon | 11\left.\right\rangle +{\mathcal{O}}({\varepsilon }^{2})\) branches. To properly derive the dynamics, we will use a stochastic master equation and we will track the most-probable quantum trajectories manually. The dynamics is governed by the equation

$${\hat{H}}_{{\rm{stoch}}}={\hat{H}}_{0}+{\hat{H}}_{{\rm{om-bl}}}+{\hat{H}}_{{\rm{sm}}}-\frac{i}{2}\sum _{c\in \{a,b,{b}^{\dagger }\}}{\gamma }_{c}{\hat{c}}^{\dagger }\hat{c},$$
(8)

where the the sum over jump operators \(\hat{c}\) provides a way to track the chance for discontinuous jumps. If \(\left\vert \right.\psi (t)\left.\right\rangle\) is the state obtained after evolving \(\left\vert \right.00\left.\right\rangle\) under \({\hat{H}}_{{\rm{stoch}}}\), then the probability density for a jump \(\hat{c}\) is \({{\rm{pdf}}}_{c}(t)={\gamma }_{c}\frac{\langle \psi (t)| {\hat{c}}^{\dagger }\hat{c}| \psi (t)\rangle }{\langle \psi (t)| \psi (t)\rangle }\). The operator \(\hat{a}\) represents the chance of photon leakage at rate \({\gamma }_{a}=\frac{{\omega }_{a}}{{Q}_{{\rm{opt}}}}\) with Qopt the optical quality factor; \(\hat{b}\) corresponds to a phonon leaking to the heat bath at rate \({\gamma }_{b}=\frac{{\gamma }_{m}({n}_{{\rm{th}}}+1)}{2}\), where \({\gamma }_{m}=\frac{2\Omega }{{Q}_{{\rm{mech}}}}\) with Qmech the quality factor of the mechanical resonator (notice the different convention leading to a factor of 2 difference); lastly \({\hat{b}}^{\dagger }\) corresponds to receiving a phonon from the bath at rate \({\gamma }_{{b}^{\dagger }}=\frac{{\gamma }_{m}{n}_{{\rm{th}}}}{2}\), where \({n}_{{\rm{th}}}=\frac{{k}_{b}\tau }{\Omega }\) is the average number of phonons in the bath, kb is the Boltzman constant, and τ is the temperature of the bath. Solving for the dynamics and the probability densities of various jumps, as done in details in the interactive supplementary materials34 leads to:

  1. 1.

    To zeroth order, no jump occurs.

  2. 2.

    To first order, a photon–phonon pair is heralded. The probability of that event is \({P}_{a}=\mathop{\int}\nolimits_{0}^{T}{\rm{d}}t{\mathrm{pdf}}_{a}(t)\).

  3. 3.

    To second order, a photon–phonon pair is heralded and then followed by any other event, for an overall of probability Pa* = 1 − 〈ψ(T)∣ψ(T)〉.

  4. 4.

    Also to second order, a b event at time τ is followed by an a event, happening with \({P}_{ba}=\mathop{\int}\nolimits_{0}^{T}{\rm{d}}\tau \,{{\rm{pdf}}}_{b}(\tau )\mathop{\int}\nolimits_{\tau }^{T}{\rm{d}}t\,{{\rm{pdf}}}_{ba}(t)\).

  5. 5.

    Similarly for b† followed by a we have probability \({P}_{{b}^{\dagger }a}\).

Above, T is the duration of the pump pulse. These are all the branches of the dynamics that have a chance of triggering a heralding event (to leading order). The total chance for heralding is \(P={P}_{a}+{P}_{ba}+{P}_{{b}^{\dagger }a}\), while the fidelity of the heralded single phonon is \(F=\frac{{P}_{a}-{P}_{a* }}{{P}_{a}+{P}_{ba}+{P}_{{b}^{\dagger }a}}{f}_{0}\), where f0 = 〈1∣ρa∣1〉 is the fidelity of “good heralding” branch of the dynamics. ρa is the density matrix for the state conditioned on only one a event having happened during the pump pulse of duration T. The above trajectories can be seen pictorially in Fig. 4c.

After simplifying and taking into account that the decay of the optical cavity is much faster than the optomechanical interaction (\({T}_{a}={\gamma }_{a}^{-1}\ll {(\overline{a}{g}_{OM})}^{-1}\)), we obtain:

$$P=4{\overline{a}}^{2}{g}_{{\rm{om}}}^{2}{T}_{a}T,$$
(9)
$$1-F=8{\overline{a}}^{2}{g}_{{\rm{om}}}^{2}{T}_{a}T+\frac{3}{4}{\gamma }_{m}{T}_{a}T\left(3{n}_{{\rm{th}}}+1\right).$$
(10)

Notice the term in the infidelity that scales exactly as the heralding probability: This is due to the (\({\mathcal{O}}\))(ε2) next-to-leading-order effect in the two-mode squeezing, leading to a proportionally larger chance of more-than-one excitations being heralded. There is also a second term, purely related to the detrimental effects of the thermal bath on the mechanical resonator. As long as \({k}_{b}\tau\, \ll\, {Q}_{{\rm{mech}}}{\alpha }^{2}{g}_{{\rm{om}}}^{2}\) we can neglect the bath heating term, however this can be difficult to quantify as Qmech strongly depends on τ. This transition between leading sources of infidelity can be seen in Fig. 5.

Fig. 5: Heralding probability and single-phonon infidelities as a function of temperature (facet) and pump power (color), parameterized by pump pulse duration (each line spans T = Ta to T = 103Ta).
Fig. 5: Heralding probability and single-phonon infidelities as a function of temperature (facet) and pump power (color), parameterized by pump pulse duration (each line spans T = Ta to T = 103Ta).
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Shorter pulses have lower probability and infidelity. However, the rate of heralding is independent of T as the shorter the pulse (the higher the repetition rate), the lower the heralding probability for that attempt is. Therefore short pump pulses are preferable as that leads to lower infidelity. In this setup, at τ = 40K, \(\alpha =\sqrt{1000}\), and T = Ta, we can theoretically achieve rates of successful single-phonon heralding in the tens of kHz at infidelity lower than 10%. The performance is even better at lower temperatures. At around 4K we see that the detrimental effects from the bath of the mechanical resonator become negligible compared to the infidelity due to multi-phonon excitations.

These are the heralding probability and fidelity of a single phononic excitation. The heralding probability and fidelity for the complete entangling protocol, in which two nodes are pumped in parallel and the photon is looked for only after “path-information erasure" differ. To leading order, the probability Pe = 2P is twice as high as either node can produce a photon, and the infidelity scales the same.

For long term storage, we coherently swap the phononic excitation into the spin. The swap gate contributes an additional infidelity of nthγm/gsm which is much lower than other sources of infidelity.

These results, given the design parameters of the previous section, are detailed in Fig. 5. Of note is that Qmech is very strongly dependent on the bath temperature due to scattering processes among the thermal phonons. At low temperatures, only clamping losses due to the design of the resonator are of importance, but as the temperature increases, Akhieser and then Landau-Rumer processes become important35,36,37,38. The typical dependence for our design and material parameters can be seen in Fig. 6. The Akhieser limited quality factor is \({Q}_{A}=\frac{1}{\Omega \tau }\frac{\rho {c}^{4}}{2\pi {\gamma }^{2}\kappa }\), where ρ is density, c is speed of light, γ is the Grüneisen coefficient, and κ is the thermal conductivity. Only κ depends strongly on temperature, and we use the values reported in39,40,41,42,43,44. At even higher temperature the Landau-Rumer processes dominate with \({Q}_{LR}=\frac{2\rho {c}^{2}}{\pi {\gamma }^{2}{C}_{v}\tau }\), where Cv is the diamond heat capacity as reported in45,46.

Fig. 6: The processes limiting the quality factor of a mechanical resonator.
Fig. 6: The processes limiting the quality factor of a mechanical resonator.
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At low temperature only clamping losses matter (green), but past a certain temperature Akhieser (blue) and Landau-Rumer (orange) processes dominate. These estimates depend on thermal properties of bulk diamond as reported in the literature. Thin-sheet diamond, as used in our devices, can have slightly differing properties.

Thus, with our design we can theoretically achieve single-phonon generation at tens of kHz and infidelity lower than 10% at temperature τ = 40 K, number of photons in the pump mode α2 = 1000, pump pulse duration T = Ta. At lower temperatures the performance significantly improves, giving limiting infidelities far below 1%.

Quantum Monte Carlo verification and Design Feedback

Finally, we use the protocol above to feedback on the optomechanical crystal design parameters (namely, ad, \({h}_{{x}_{d}}\), and \({h}_{{y}_{d}}\)) (Fig. 7a, b) to minimize the cost-function which is a function of fidelity and rate of heralding (see Supplementary). We verify the theoretical equations Eq. (9) and Eq. (10) using a quantum monte-carlo approach (see Supplementary) and proceed to use a COMSOL-to-Python (theory + quantum master equation) feedback loop to optimize the crystal design.

Fig. 7: Design feedback using FEM-QuTiP optimization.
Fig. 7: Design feedback using FEM-QuTiP optimization.
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(a) Shows the change in ad, \({h}_{{x}_{d}}\), and \({h}_{{y}_{d}}\) at each Bayesian optimizer iteration, where the resulting performance parameter variation is shown in (b). For each iteration of {\({a}_{d},{h}_{{x}_{d}},{h}_{{y}_{d}}\)}, optimizer individually selects an optimum pump time Tpump which minimizes the cost function for that iteration and (c) shows the change in Tpump by iteration. (d) Shows the rate-fidelity tradeoff of the cost function at each iteration (red), with the optimized point highlighted (green). The red (green) lines represent curves in the rate-fidelity space with equal cost function to the iteration points.

We select a bridge width b of 60 nm, taking into account the nanofabrication considerations, and defect unit cell number nd as 6. Starting with the free variables \({h}_{{y}_{d}}\), \({h}_{{x}_{d}}\), and ad, we run our Bayesian optimization routine (described in Supplementary Section 5.4). We run a COMSOL finite element method (FEM) simulation to extract device parameters {ωa, Ω, Qopt, Qmech, gom, gsm} as described in Section 2.1 and 2.2. The COMSOL simulation is interfaced with python API using the MPh module, which for each iteration transfers the COMSOL results to QuTiP. This then evaluates the heralding success rate and fidelity for the protocol described in Section 2.3 (see Supplement for more details). After evaluating the cost-function (CF) for the protocol (\(C(F,R,{T}_{2})=1-F{e}^{-1/R{T}_{2}}\)), we proceed to next iteration. Fig. 7a shows how the optimizer updates the variables \({h}_{{y}_{d}}\), \({h}_{{x}_{d}},{a}_{d}\) through each iteration. Fig. 7b shows the variation of {gom, gsm, Qopt, Qmech} with progressive iterations. Tpump acts like a hyperparameter such that for each iteration, the optimizer selects the value of Tpump which minimizes the CF for that iteration (shown in Fig. 7c). Figure 7d shows the optimization trajectory in Fidelity-Rate tradespace with the optimum fidelity ~ 0.96 and rate ~ 20 kHz. Through this optimization, we arrive at a final design parameter set in Table 1.

Table 1 Spin-optomechanical cavities optimized by the FEM-QuTiP optimization in Quantum Monte Carlo verification and Design Feedback
Full size table

Possible improvements to the protocol include (1) spectral and spatial multiplexing (2) use of a Dicke state of multiple nearby color centers to enhance gsm (3) use the nuclear registers for even longer storage times (4) entanglement purification with the nuclear registers which greatly increase the entanglement fidelity while only marginally decreasing the entanglement rate.

Discussion

In this paper, we bring the idea of a self-similar concentrator from photonic crystal devices26 to a 1D optomechanical crystal and explore the usage of the resulting cavity in spin-optomechanical interfacing. This system poses the advantages afforded by highly concentrated optical and mechanical modes: high strain in a central region while retaining optomechanical coupling in diamond relative to previously proposed and demonstrated devices47, and thus strong spin–phonon interactions. From FEM simulations, we demonstrate that this spin-optomechanical interface can achieve 820 kHz single photon–phonon coupling alongside 32 MHz spin–phonon coupling to a Group IV spin. The strength of this spin–phonon interaction is such that we can effectively ignore losses incurred when swapping a quantum between a cavity phonon and the spin state.

We explore implementation of our interface in an optically heralded entanglement protocol31,33. In this scheme, identical cavities are entangled via heralding, and the resulting entangled phonons are swapped into their respective coupled spins. This entanglement procedure completely circumvents standard issues related to spin-addressing, including the need to operate at the emitter’s optical transition wavelength (we define the optical wavelength with a telecom photonic mode) and concerns related to spectral diffusion of emitters (we never optically excite the emitter). Additionally, this scheme places no strong requirements on the optical quality factors required by other works to accomplish spin-mechanical addressing48,49–instead operating with low optical Qs to increase the rate of heralding–and requires on-chip devices that are well within fabricable parameters.

Our spin-optomechanical architecture applies to other material platforms besides diamond. For example, silicon (Si) and silicon carbide (SiC) have been used for optomechanics5,7,50,51,52,53 and have quantum emitters including carbon-based T-centers, phosphorus vacancies, and boron impurities19,54. In particular, Si with B:Si acceptor impurities has been considered for operating spin-phonon coupled systems as an acoustic alternative to circuit-cavity QED19. Here, we have shown that with an intentionally designed optomechanical cavity, one can achieve gsm much larger than previously proposed–which should be the case irrespective of the material, whether diamond, silicon, or another alternative–alongside respectable gom, such that the full spin-optomechanical interface’s performance can be evaluated (see Table 2). We have analyzed this interface assuming a SiV− spin, which has well-documented spin-strain parameters17,23; however the spin-dephasing time is highly limited above single-Kelvin temperatures due to electron-phonon dephasing55. As such, future works may use this spin-optomechanical framework while selecting a suitable combination of material platform and temperature-robust, highly strain-tunable spin defect. The beauty of this platform is that, given sideband-resolved cooling of nanomechanical oscillators at a few Kelvin56 or ~20 K57, quantum operation of a solid-state spin would not be limited by optical lifetimes and instead enabled by state-of-the-art optomechanics. The ability to separately engineer quantum memories and spin-photon interfaces, while retaining efficient interfacing between them even at moderate temperatures up to 40 K, will provide much-needed design freedom in applications from quantum networks to computing to sensing. Our COMSOL file, code for COMSOL-python optimizer, QuTiP simulation can be found in our GitHub repository58.

Table 2 Spin defect candidates for optomechanical interfacing
Full size table

Methods

We conduct FEM-and-QuTiP optimization using COMSOL Multiphysics 5.4 and the QuTiP package in Python. We run electromagnetic (EM) FEM simulations using a COMSOL-Python API to determine the optical mode properties of the optomechanical crystal. We perform structural mechanics (SM) FEM simulation through the COMSOL-Python API to determine the mechanical modes of the system. We feed the FEM simulation results into custom QuTiP code for estimating entanglement rate and fidelity. Finally, we feed the FEM and QuTiP steps into a built-in SciPy Bayesian optimizer for up to 100 iterations to optimize the optomechanical crystal parameters.