We propose an optical scheme to generate an entangled state between a trapped ion and a solid state donor qubit through which-path erasure of identical photons emitted from the two systems. The proposed scheme leverages the similar transition frequencies between In donor bound excitons in ZnO and the $P21/2$ to $S21/2$ transition in Yb^{+}. The lifetime of the relevant ionic state is longer than that of the ZnO system by a factor of 6, leading to a mismatch in the temporal profiles of emitted photons. A detuned cavity-assisted Raman scheme weakly excites the donor with a shaped laser pulse to generate photons with a 0.99 temporal overlap to the Yb^{+} emission and partially shift the emission of the defect toward the Yb^{+} transition. The remaining photon shift is accomplished via the dc Stark effect. We show that an entanglement rate of $2.1\xd7104$ s^{−1} and an entanglement fidelity of 94% can be attained using a weak excitation scheme with reasonable parameters.

Hybrid quantum systems offer the opportunity to combine the benefits of different qubit types while avoiding some of their pitfalls. Task-dependent qubit selection allows the usage of long-lived qubits for memory and qubits with rapid gate speeds for operations. For optical systems, a photon bus can be used to remotely link these systems via photon-heralded entanglement. To successfully generate entanglement, the two different qubit systems must emit identical photons, requiring spectro-temporal engineering of at least one qubit's photon wavepacket. While significant progress has been made toward efficient quantum-frequency conversion,^{1–4} post-emission temporal photon pulse-shaping^{5–7} techniques for the narrow-band photons from both trapped ions and solid-state defects are an outstanding challenge.

We have identified two disparate, complementary qubit systems in which high-fidelity photon-mediated entanglement should be possible by direct control over the photon emission process. Trapped ions are a well-studied qubit system with high operational fidelities^{8} and long coherence times,^{9} but relatively slow initialization and gate speeds.^{10} Electron spins in semiconductors have rapid initialization and gate speeds,^{11–13} but have shorter coherence times. A hybrid system consisting of ions and electrons bound to donor defects would have the ability to use ions for quantum memory and defects for gate operations, producing a system more rapid and reliable than either qubit alone.

Yb^{+} and the ZnO donor were chosen as the target systems for their shared transition near 369 nm: the $P21/2$ to $S21/2$ transition in ^{171}Yb^{+} and the In neutral donor bound exciton (D^{0}X) to neutral donor (D^{0}) transition in ZnO (Fig. 1). In:ZnO is analogous in structure to the better-known P:Si qubit system;^{14} however, ZnO is a direct bandgap semiconductor enabling efficient donor coupling to photons. While the two transition frequencies are quite close (*δ* = 0.36 THz), the excited state lifetimes differ by a factor of 6, resulting in a large temporal mismatch. Previous semiconductor spin—trapped ion entanglement schemes addressed a similar temporal mismatch by using coherent scattering^{15} or sacrificing fidelity.^{16} Here, we demonstrate that pulse shaping can be a powerful tool to attain high-fidelity entanglement and show that an entanglement rate of $2.1\xd7104$ s^{−1} and a fidelity of 94% are feasible.

A heralded entanglement scheme based on weak excitation, single-photon detection, and which-path erasure can be used to entangle the two systems, similar to the proposal by Cabrillo *et al.*^{17} Figure 1 depicts the relevant energy levels and excitation/decay pathways for the donor and ion. Here, the D^{0} system is in the Voigt (B $\u22a5k\u0302$) geometry, but the Faraday geometry could also be utilized. The donor is coupled to an optical cavity detuned by Δ from the D^{0}X-D^{0} transition.

The diagram of the experiment is shown in Fig. 2. The Yb^{+} and In donor are first initialized using optical pumping to $|F=0,mF=0\u27e9$ and $|ms=\u22121/2\u27e9$, respectively, producing the initial state $|\Psi \u27e9i=|0\u27e9Yb\u2297|0\u27e9In\u2297|vac\u27e9\u2261|0;0;vac\u27e9$. Next, each system is excited to $|e\u27e9In$ or $|e\u27e9Yb$, using resonant or near-resonant pulsed excitation. Here, we assume the weak excitation limit (excitation probability $p1,x<10%,\u2009x={Yb,In}$).

The state of the ZnO donor and ion is now given by

where the emitted photons on paths A and B of Fig. 2 $|\zeta Yb\u27e9=\u2211\omega \xi Yb,\omega a\omega \u2020|vac\u27e9$ and $|\zeta In\u27e9=\u2211\omega \u2032\xi In,\omega \u2032b\omega \u2032\u2020|vac\u27e9$ are given by a sum over all modes *ω* ($\omega \u2032$) with coefficients $\xi Yb,\omega $ ($\xi In,\omega \u2032$) and creation operators $a\omega \u2020$ ($b\omega \u2032\u2020$). The coefficients *β* emerge from the excitation ($p1,x$) probabilities of the two systems, the phase gained from excitation laser phases ($\varphi x,L$), and the distance traveled by the collected photon ($\varphi x,d$),

By phase locking the laser pulses, we can ignore $\varphi x,L$.

Collected photons from both systems interfere on the beam splitter, which erases which-path information.

Entanglement is heralded by the detection of a single photon at one of the two photodetectors. With the appropriate choice for $p1,Yb,\u2009p1,In$, and the collection efficiency from each system (supplementary material), photon detection in path D projects the ion-donor qubits onto the renormalized entangled state,

where $\Delta \varphi $ is determined by the optical path length difference. A similar expression can be derived for detector C. Tracing over all photon modes, we get the reduced $Yb+\u2014$ In density matrix,

where $\u27e8\zeta Yb|\zeta In\u27e9=\u2211\omega \u0303\xi Yb,\omega \u0303*\xi In,\omega \u0303$ is the overlap of the photons from the Yb^{+} and ZnO systems.

Factors that affect the entanglement fidelity are the photon overlap, false identification of both-system excitation as a single-system excitation, and atomic recoil from the ion interacting with the excitation laser. Accounting for these sources of error, the final fidelity is

where $c1$ depends on the excitation probabilities and detection efficiencies of both systems (supplementary material) and *F _{dyn}* is related to the photon recoil.

^{17}Motion of the trapped ion due to photon recoil during the absorption/emission process can shift the frequency of the photon and reduce fidelity of the entangled state. Note that for the ZnO donor, absorption/emission is recoilless due to the Mössbauer effect. For a Doppler-cooled

^{171}Yb

^{+}in a 1 MHz trap in geometry where the ion is excited by a laser pulse parallel to the light collection direction, the expected $Fdyn$ is 96%.

^{17}In addition, uncertainty in both $\varphi x,L$ and $\varphi x,d$ leading to an undesired phase factor $ei\epsilon $ between the terms in Eq. (3) can further degrade the fidelity according to $Re(\u27e8\zeta Yb|\zeta In\u27e9)\u2192Re(ei\epsilon \u27e8\zeta Yb|\zeta In\u27e9)$. Other factors that may further decrease the fidelity include photodetector dark counts, background luminescence from ZnO, and D

^{0}X spectral diffusion.

^{18,19}

Photon collection efficiency primarily affects the protocol's probability of success. For trapped ions, light collection is challenging due to the high-vacuum environment and the need to isolate ions from decoherence-inducing surfaces. The typical light collection efficiency is 2%–4% utilizing off-the-shelf long working distance microscope objectives,^{20} while optics based on in-vacuum lenses^{21} and custom high-NA objectives^{22} are capable of collecting up to 10% of the emitted photons. Further enhancement is possible by integrating a metallic parabolic mirror as an RF electrode of the ion trap.^{23} Ions are trapped at the focus of the mirror, so that the emitted photons are collimated upon reflection from the mirror with an expected 32% overall coupling efficiency into a single-mode optical fiber. As we show below, the parabolic mirror trap also provides a mechanism for polarization filtering. Longer-term, integrated-photonics platforms may provide a path toward high-NA collection from scalable arrays of ions.^{24}

For the donor, a photonic cavity can be fabricated in ZnO to enhance collection efficiency. As shown in Fig. 3, cavities that satisfy high cooperativity $C=g2/\kappa \Gamma In$ (here, *g* is the donor-cavity coupling strength, *κ* is the cavity decay rate, and $\Gamma In$ is the spontaneous decay rate) in the “bad cavity” limit necessity for the pulse-shaping procedure described below, which lie in a band of readily achievable *Q*/*V* ratios with today's nanophotonic fabrication techniques (here, *Q* is the quality factor and *V* is the mode volume of the cavity). Due to intrinsic band edge absorption, the high quality factor region in Fig. 3 may not be achievable at D^{0}X-D^{0} transition,^{25} and thus, low mode volume cavities with moderate quality factors should be targeted. While nanophotonic fabrication in ZnO is relatively immature compared to other quantum defect host crystals, small mode volume ZnO nanowire cavities have enabled UV lasers^{26} and ZnO cavities fabricated by focused ion beam milling,^{27} a method that has been used to achieve high cooperativity in rare-earth doped systems,^{28} and exhibit quality factors up to 1000. In the limit that the cavity photon loss rate *κ* is dominated by coupling to the output mode, over 50% collection efficiency into a waveguide for planar geometry cavities^{29} or into an objective lens for nanowire cavities^{30} is possible.

As shown in Eq. (5), for high fidelity entanglement, the frequency, polarization, and temporal shape of the photons emitted by the two systems must be matched to maximize $Re(\u27e8\zeta Yb|\zeta In\u27e9)$. The type of donor used affects the amount of frequency shift required to match the emission frequency of Yb^{+}. Of the three primary donor candidates, Al, Ga, and In, the In D^{0}X transition is closest to the Yb^{+} transition, $vIn=vYb+0.36$ THz,^{31} where $vIn$ and $vYb$ are the values of the $|0\u27e9\u2192|e\u27e9$ transitions with zero magnetic field, and in the absence of a dc Stark shift. By employing the dc Stark effect, the frequency gap can be reduced. In the quantum dot trion system, Stark tuning of up to several THz has been demonstrated.^{32} If one cannot achieve the full 360 GHz via Stark tuning, the remainder can be bridged by detuning the cavity resonance. For concreteness, here, we assume a dc Stark shift of 160 GHz and a cavity detuning of 200 GHz.

Decay from $|e\u27e9Yb$ ($P21/2|F=1,mF=0\u27e9$) can occur along three different channels, producing either a $\sigma \xb1$ Raman photon or a *π* Rayleigh photon (see Fig. 1). A pure polarization state is required for polarization matching with the photon emitted by the ZnO donor. While the use of a high-NA collection optic increases the photon collection efficiency, it can pose problems for polarization purity. However, the parabolic mirror can be utilized to filter out the undesired *π* polarized photons when the optical axis is oriented along the quantization axis defined by the applied magnetic field.^{33} In this geometry, the *π*-polarized photons reflected off the mirror have a radial polarization pattern, which completely destructively interferes when focused into a single-mode optical fiber. The *σ*-polarized photons, on the other hand, have an elliptical polarization upon reflection from the mirror. The eccentricity increases with the radial distance from the center, with perfectly circular polarization at the center of the reflected beam and linear polarization at the edge. The linear component is filtered out by destructive interference in the optical fiber.

In the Voigt geometry, with the applied magnetic field perpendicular to the crystal axis, the branching ratio between the ZnO donor Raman transitions $|e\u27e9In\u2192|0\u27e9In$ and $|e\u27e9In\u2192|1\u27e9In$ is approximately 1:1.^{34,35} For a cavity with large *V* and high *Q* (e.g., ring resonator^{36}), the cavity resonance will be narrower than the Zeeman splitting of $D0X$, allowing for frequency selective coupling of the desired Raman transition for small detunings. For high *V*, the size of the cavity is large compared to the excitation beam diameter, and so polarization selection can be attained by selectively exciting a small area of the cavity, where only one dipole moment is coupled to the cavity mode. For cavities with low *Q* and *V*, polarization and frequency selection can be achieved via cross polarization,^{16} waveguide excitation,^{37} and spectral filtering.

Matching the temporal profiles of the emitted photons poses a greater challenge. The $P21/2$ Yb^{+} state lifetime is 8.1 ns,^{38} while that of the D^{0}X state in ZnO is only 1.4 ns.^{39} Post-emission pulse shaping^{40,41} is not feasible because the ZnO and Yb photons are too narrow bands for these dispersive methods. Instead, the photons emitted by the ZnO donor can be pulse-shaped at their creation^{42} by modulating the intensity of the excitation pulse. The ZnO cavity is constructed with parameters within the “bad cavity” regime ($\kappa \u226bg2/\kappa \u226b\Gamma In$).^{42} The large cavity decay rate ensures that we are not in the strong coupling regime, and so the donor excitation follows the optical pulse, while the high cooperativity ensures that the donor decays via Raman emission into the cavity.

While it is possible to obtain an analytical expression for the ideal excitation pulse shape for the maximum photon overlap,^{43} in this work, we limit ourselves to experimentally attractive Gaussian pulses and perform numerical simulations to determine the photon temporal overlap, given the practical cavity considerations discussed above and previously demonstrated progress in engineering temporal wavepackets.^{44} The donor defect is modeled as a three level system with initial state $|0\u27e9In$ (Fig. 4) connected to the excited state $|e\u27e9In$ by an excitation pulse of Rabi frequency $\Omega In(t)$ and detuning Δ. We neglect the effect of the other excited state level. The cavity is coupled to the $|e;0\u27e9\u2194|1;1\u27e9$ transition with detuning Δ and coupling strength *g*. Photons from this transition have a spontaneous radiative decay rate of $\Gamma In$. Photons escape the cavity at the cavity decay rate *κ*. The equations of motion for the population amplitudes are^{42,43}

where $aIn(t)=[a0,In(t),ae,In(t),a1,In(t)]T$.

Yb^{+} is modeled in a similar manner but without a cavity. The ground state $|0\u27e9Yb$ is coupled to the excited state $|e\u27e9Yb$ by the Rabi pulse $\Omega Yb(t)$. Decay from the excited state occurs with the rate $\Gamma Yb$. The equations of motion are

The emission rates of the photons from the ZnO and Yb^{+} systems are $\kappa |a1,In(t)|2$ and $\Gamma Yb|ae,Yb(t)|2$, respectively,^{42} with temporal wavefunctions given by normalizing the population amplitudes $a1,In(t)\u2192A1,In(t)$ and $ae,Yb(t)\u2192Ae,Yb(t)$. By controlling the Rabi frequencies $\Omega In(t)$ and $\Omega Yb(t)$, it is possible to engineer the real component of the photon overlap $\u222b\u2212\u221e\u221eAe,Yb*(t)A1,In(t)dt=\u27e8\zeta Yb|\zeta In\u27e9$ to ∼0.99 for practical experimental parameters using the control pulses shown in Fig. 5. The optimized pulse is restricted to a Gaussian pulse shape with adjustable rise time *σ*_{1}, fall time *σ*_{2}, time to pulse max *τ*, hold time *t _{h}*, maximum pulse height Ω

_{max}, and phase factor $ei\alpha (t)$, where $\alpha (t)=\theta 0+\theta 1t$ describes a linear time-dependent phase. Setting either pulse to achieve a desired excitation probability $p1,x$, we iteratively sweep the pulse parameters for the other system to obtain local maxima in the overlap.

The probability of successful entanglement is

where $p2,x$ is the collection efficiency from each system and *η* is the quantum efficiency of the detector, which can be as high as ∼80% using superconducting nanowire single photon detectors (SNSPDs)^{45} for photons at 369 nm. With a parabolic mirror ion trap, the collection efficiency for Yb^{+} systems is 32%; the ZnO system is set to 34% collection efficiency to match the coefficients in Eq. (2) to create a maximally entangled state. Excitation probabilities depend on the pulse shaping requirements and need to be kept low (<10%) to minimize error. For good fidelity while still maintaining a reasonably high success probability, we use excitation probabilities around 5%.

Each experimental run begins with ∼1 *μ*s of optical pumping, followed by the ∼10 ns excitation pulse. If a single photon is detected, then the state readout is performed, taking ∼10 *μ*s and limited by the ion.^{45} We find a success probability of ∼2.7%, leading to an entanglement generation rate of $2.1\xd7104$ s^{−1}. Practically, this rate will be further decreased by the interferometer phase stabilization and defect frequency stabilization steps.^{18,46}

With all experiments using this type of protocol, there is a trade-off between success probability and fidelity.^{18,19,46} One can always increase the success probability by increasing the excitation probability, but this degrades the fidelity according to Eq. (5). Further, in order to be useful, the entanglement rate needs to be comparable to the rate of decoherence. While the demonstrated coherence time for trapped ytterbium ions is long^{9} (10 min), the spin echo time $T2$ of ensemble donor-bound excitons in ZnO is only 50 *μ*s. However, the fundamental limit of $T2$ is the longitudinal spin relaxation time $T1$, which exceeds 100 ms,^{34} and may allow for improvement through chemical and isotope purification.^{47}

In summary, a ZnO donor defect qubit and a single trapped Yb^{+} ion can be remotely entangled via a photonic link at 369 nm. Pulse shaping techniques can be used to alter the temporal profile of the photon emitted by the donor to attain the temporal wavefunction overlap of 0.99 with the photon emitted by the trapped ion, leading to an entangled state fidelity of 94% with realistic parameters.

See the supplementary material for a derivation of the fidelity expression (Eq. 5).

We thank Xiayu Linpeng for assistance in creating Fig. 2. This material is based upon the work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0020378.

## DATA AVAILABILITY

The data that support the findings of this study are available within this article.