Heralded Generation and Detection of Entangled Microwave--Optical Photon Pairs

TL;DR

This paper proposes a heralded scheme to generate entangled microwave-optical photon pairs using a cavity electro-optomechanical system, advancing quantum state transfer and hybrid quantum networks.

Contribution

It introduces a novel heralded entanglement generation method that bypasses efficiency thresholds, compatible with current experimental capabilities.

Findings

Scheme is feasible with recent electro-optomechanics progress

Identifies parameter regime for entanglement verification

Potential for generalization to other physical systems

Abstract

Quantum state transfer between microwave and optical frequencies is essential for connecting superconducting quantum circuits to coherent optical systems and extending microwave quantum networks over long distances. To build such a hybrid `quantum Internet,' an important experiment in the quantum regime is to entangle microwave and optical modes. Based on the model of a generic cavity electro-optomechanical system, we present a heralded scheme to generate entangled microwave--optical photon pairs, which can bypass the efficiency threshold for quantum channel capacity in direct transfer protocols. The parameter regime for entanglement verification is identified that is compatible with realistic experimental settings. Our scheme is feasible given the latest experimental progress on electro-optomechanics, and can be potentially generalized to various physical systems.

Tables

Table 1. Table 1: Experimentally feasible parameters. Unless specified otherwise, these parameters apply to all figures and evaluations in the text.

$g_{\text{em}}$/MHz	$g_{\text{om,0}}$/kHz	${\kappa }_{\text{e,i}}$/kHz	${\kappa }_{\text{o,i}}$/(GHz)	${\kappa }_{\text{e,c}}$	${\kappa }_{\text{o,c}}$	${\kappa }_{\text{m}}$/kHz	${\overline{n}}_{\text{ba}}$(T=1 K)	${\omega }_{\text{m}}$/GHz	${\omega }_{\text{e}}$/GHz	${\omega }_{\text{o}}$/THz
$2\pi \times 2.0$	$2\pi \times 5.5$	$2\pi \times 100$	$2\pi \times 0.24$	$\sim$(${\kappa }_{\text{e,i}},{10}^3{\kappa }_{\text{e,i}})$	$\sim {\kappa }_{\text{o,i}}$	$2\pi \times 20$	$\sim 1.67$	$2\pi \times 10$	$2\pi \times 10$	$2\pi \times 195$

Equations

$$\begin{array}{rl}\hat{H}/\hslash & =-{\Delta }_{\mathrm{p}}{\hat{a}}^{†}\hat{a}+{\omega }_{\mathrm{m}}{\hat{b}}^{†}\hat{b}+{\omega }_{\mathrm{e}}{\hat{c}}^{†}\hat{c}-g_{\mathrm{em}}({\hat{b}}^{†}\hat{c}+\hat{b}{\hat{c}}^{†})\\ & -g_{\mathrm{om},0}\sqrt{{\bar{n}}_{\mathrm{o}}}({\hat{a}}^{†}+\hat{a})({\hat{b}}^{†}+\hat{b}),\end{array}$$

$$\begin{array}{rl}{|{\Psi }_{\mathrm{tb}}(\lambda )⟩}_{\mathrm{oe}}& \simeq {|0,0⟩}_{\mathrm{o}}{|0,0⟩}_{\mathrm{e}}+\lambda {\hat{a}}_{\text{out,c}}^{(1)†}{\hat{c}}_{\text{out,c}}^{(1)†}{|0,0⟩}_{\mathrm{o}}{|0,0⟩}_{\mathrm{e}}\\ & +\lambda {\hat{a}}_{\text{out,c}}^{(2)†}{\hat{c}}_{\text{out,c}}^{(2)†}{|0,0⟩}_{\mathrm{o}}{|0,0⟩}_{\mathrm{e}}+O({\lambda }^2),\end{array}$$

$${\mathbf{V}}_{\mathrm{oe}}^{\mathrm{out}}=\left(\begin{array}{cccc}u(\omega )& 0& -w(\omega )& 0\\ 0& u(\omega )& 0& w(\omega )\\ -w(\omega )& 0& v(\omega )& 0\\ 0& w(\omega )& 0& v(\omega )\end{array}\right),$$

$$F_{\mathrm{lb}}=\sum _{\begin{array}{c}\nu =\pm \\ \phi =0,\frac{\pi }{2}\end{array}}\frac{p_{\phi \phi }^{\nu \nu }}{2}-\frac{p_{\frac{\pi }{2}\frac{\pi }{2}}^{+-}}{2}-\frac{p_{\frac{\pi }{2}\frac{\pi }{2}}^{-+}}{2}-\sqrt{p_{00}^{+-}{p}_{00}^{-+}},$$

$$\begin{array}{rl}S({\phi }_{\mathrm{o}},{\phi }_{\mathrm{e}};{\phi }_{\mathrm{o}}^{\prime },{\phi }_{\mathrm{e}}^{\prime })& \equiv \\ E({\phi }_{\mathrm{o}},{\phi }_{\mathrm{e}})+E& ({\phi }_{\mathrm{o}}^{\prime },{\phi }_{\mathrm{e}}^{\prime })+E({\phi }_{\mathrm{o}}^{\prime },{\phi }_{\mathrm{e}})-E({\phi }_{\mathrm{o}},{\phi }_{\mathrm{e}}^{\prime }),\end{array}$$

Full text

The modular quantum architecture—moderate-sized quantum registers and memories connected by efficient communication channels—is a competitive approach toward a scalable quantum network Cirac et al. (1997); *Kimble2008; Jiang et al. (2007); Monroe et al. (2014). Physically, it is comprised of natural or artificial “atoms”—the nodes—and flying photons—the interconnects. As engineerable mesoscopic “atoms,” superconducting qubits Esteve et al. (2004); Clarke and Wilhelm (2008) can strongly interact with microwave photons in cavities or waveguide resonators, known as the circuit quantum electrodynamics (cQED) architecture Blais et al. (2004); Girvin (2014). Lately, microwave photons have been employed to entangle remote transmon qubits Narla et al. (2016); Campagne-Ibarcq et al. (2018); Kurpiers et al. (2018) and cavity memories Axline et al. (2018); Chou et al. (2018). However, the high loss in commercial microwave cables at room temperature prevents the transmission of quantum signals over long distances Kur . In contrast, optical photons stand out as quantum information carriers at large spatial scales—entanglement and teleportation have been demonstrated over kilometers through telecommunication fibers Tittel et al. (1998); Valivarthi et al. (2016) and over one thousand kilometers in free space Yin et al. (2017); *Ren2017. Therefore, high-fidelity quantum state transfer between superconducting circuits and optical photons will greatly expand the quantum computing network, as well as bridging superconducting qubits and other quantum modules with coherent optical interfaces, including neutral atoms Matsukevich and Kuzmich (2004), trapped ions Blinov et al. (2004), defects in solids Togan et al. (2010); Koehl et al. (2011), quantum dots Gao et al. (2012); De Greve et al. (2012), etc.

However, superconducting circuits do not have an optical transition. A quantum transducer is thus needed to interface microwave and optical photons in a quantum coherent manner. So far, most investigations of quantum transducers are based on direct quantum transduction Zhang et al. (2018), which linearly converts input photons to output photons at different frequencies. Proposed direct microwave–optical (M–O) transducers involve cold alkali atoms Hafezi et al. (2012); Kiffner et al. (2016); Gard et al. (2017), rare-earth-doped crystals Williamson et al. (2014); O’Brien et al. (2014), ferromagnetic magnons Hisatomi et al. (2016), electro-optical devices Tsang (2010); *Tsang2011; Javerzac-Galy et al. (2016), and nanomechanical oscillators Andrews et al. (2014); Regal and Lehnert (2011); Bochmann et al. (2013); Taylor et al. (2011); Barzanjeh et al. (2011); *Barzanjeh2012; Wang and Clerk (2012); Tian and Wang (2010); *Tian2012; *Tian2014; Zou et al. (2016); Midolo et al. (2018); Bagci et al. (2014); Vainsencher et al. (2016); Winger et al. (2011); Pitanti et al. (2015). Recent experiments on cavity electro-optomechanics are very encouraging Andrews et al. (2014); Higginbotham et al. ; Bagci et al. (2014); Bochmann et al. (2013); Vainsencher et al. (2016); Han et al. (2014); Winger et al. (2011); Pitanti et al. (2015), but many challenges remain. In these setups, microwave and mechanical resonators are coupled by either electrostatic Bagci et al. (2014); Andrews et al. (2014); Higginbotham et al. or piezoelectric forces Bochmann et al. (2013); Vainsencher et al. (2016); Han et al. (2014). Nanomembranes for electrostatic coupling usually vibrate at megahertz frequencies Bagci et al. (2014); Andrews et al. (2014); Higginbotham et al. ; Teufel et al. (2011); *Palomaki2013; *Reed2017, resulting in a narrow conversion bandwidth and high added thermal noise. On the other hand, piezoelectric oscillators can be routinely fabricated with gigahertz frequencies O’Connell et al. (2010); Bochmann et al. (2013); Chu et al. (2017); *Chu2018; Han et al. (2014, 2015, 2016), which couple to much lower thermal noise. Nevertheless, piezo-optomechanical converters also require large matched electromechanical and optomechanical cooperativities to achieve high conversion efficiency Han et al. (2016); Vainsencher et al. (2016); Forsch et al. (2018). Although strong piezo-electromechanical coupling has been demonstrated with a cooperativity over 2000 Han et al. (2014); Zou et al. (2016), it is still challenging to achieve a matched large optomechanical cooperativity in integrated piezo-optomechanical devices.

A direct photon converter is capable of transferring quantum states only if the conversion efficiency $\eta>1/2$ eta . In principle, we can bypass this stringent requirement by introducing two-way classical communication that first heralds successful entanglement generation Duan et al. (2001); Chou et al. (2005); Moehring et al. (2007); Lee et al. (2011); Hofmann et al. (2012); Bernien et al. (2013); Narla et al. (2016); Riedinger et al. (2018) and then completes the quantum state transfer by quantum teleportation Bennett et al. (1993); *Pirandola2015. Quantum states can thus be transferred bi-directionally between microwave and optical frequencies, which connects superconducting quantum processors to an optical “quantum Internet” Kimble (2008). In this Letter, we propose such a heralded M–O entanglement generation and detection scheme, which is the first step of realizing this entanglement-based quantum transduction. We analyze its implementation in a generic cavity electro-optomechanical system with dissipation and thermal noise, and map out the preferable parameter ranges for manifesting M–O entanglement. Our scheme and analysis can be generalized to M–O transducers based on various physical platforms and therefore provide a new guideline for realizing coherent quantum state transfer between microwave and optical frequencies.

*Entanglement generation.—*Without loss of generality, our physical discussions are based on the model depicted in Fig. 1. The thickness mode of a mechanical oscillator is on one side linearly coupled to a microwave resonator by piezoelectric force Zou et al. (2016); Han et al. (2016); Chu et al. (2017), and on the other side parametrically coupled to an optical cavity by radiation pressure Aspelmeyer et al. (2014). The frequencies of the optical, the mechanical, and the microwave resonators are at $\omega_{\mathrm{o}}$, $\omega_{\mathrm{m}}$, and $\omega_{\mathrm{e}}$, respectively. A laser at frequency $\omega_{\mathrm{p}}=\omega_{\mathrm{o}}+\Delta_{\mathrm{p}}$ pumps the optical cavity and populates it by $\bar{n}_{\mathrm{o}}$ photons on average. In the rotating frame of the pump, we write the linearized Hamiltonian of the system

[TABLE]

where $\hat{a}$, $\hat{b}$, and $\hat{c}$ represent the optical, mechanical, and microwave modes; $g_{\mathrm{em}}$ and $g_{\mathrm{om,0}}$ are the piezoelectric and the single-photon optomechanical coupling rates.

To entangle microwave and optical photons, as shown in the inset (a) of Fig. 1, we generate entangled phonon–photon pairs by driving an optomechanical parametric down-conversion process with a blue-sideband pump at $\Delta_{\mathrm{p}}=\omega_{\mathrm{m}}$ Lee et al. (2011); Riedinger et al. (2018, 2016); *Hong2017. Meanwhile, mechanical excitations are swapped into the microwave resonator through the piezoelectric interaction. The M–O mode is thus approximately in a two-mode squeezed vacuum $\ket{\psi_{\mathrm{sq}}(\lambda)}_{\mathrm{oe}}\simeq\sum_{N=0}^{\infty}\frac{\lambda^{N}}{N!}(\hat{a}^{\dagger})^{N}(\hat{c}^{\dagger})^{N}\ket{0}_{\mathrm{o}}\ket{0}_{\mathrm{e}}$, in which subscripts “o” and “e” represent the optical and microwave modes. For a weak enough pump (the effective squeezing factor $\lambda\ll 1$), an output M–O photon pair can be generated through the coupling ports with probability $|\lambda|^{2}$.

To use flying photons as quantum information carriers and demonstrate non-classical correlations, we can encode qubits into multiple modes with different polarizations Freedman and Clauser (1972); Aspect et al. (1981); *Aspect1982, momenta Rarity and Tapster (1990), time bins Marcikic et al. (2002), frequency bins Ramelow et al. (2009); *Olislager2010, etc. As an example, we present the scheme of generating time-bin entanglement in our proposed setup. As shown in Fig. 1, the optical cavity is pumped by two blue-sideband laser pulses separated by time $\Delta t$, which is within the coherence time of the pump laser and superconducting qubits (alternativelty, we can pump the cavity with continuous wave and select output by pulsed gates with time separation $\Delta t$). Denoting $\hat{a}^{(1,2)}_{\text{out,c}}$ and $\hat{c}^{(1,2)}_{\text{out,c}}$ as the optical and microwave output mode operators in time-bin 1 and 2, respectively, the M–O output modes is

[TABLE]

where for either mode, $\ket{0,0}$ denotes the state with zero photon in the first and second time bin. Neglecting the $O(\lambda^{2})$ terms and discarding the zero-photon events by postselection, we obtain a time-bin Bell state $\frac{\sqrt{2}}{{2}}(\hat{a}^{(1)\dagger}_{\text{out,c}}\hat{c}^{(1)\dagger}_{\text{out,c}}+\hat{a}^{(2)\dagger}_{\text{out,c}}\hat{c}^{(2)\dagger}_{\text{out,c}})\ket{0,0}_{\mathrm{o}}\ket{0,0}_{\mathrm{e}}$ with probability $|\lambda|^{2}$.

*Flying photon measurement.—*Optical time-bin qubits can be detected with an unbalanced Mach–Zehnder interferometer Franson (1989). As depicted on the left of Fig. 1 (the gray panel), given the relative time delay between the long and short arms precisely matching $\Delta t$, a photon in the first time bin passing through the long arm and a photon in the second time bin passing through the short arm will meet and interfere at the second beam splitter. A click at $\mathrm{D}_{1,2}$ projects the optical qubit on $\ket{\varphi_{\mathrm{o}}^{\pm}}_{\mathrm{o}}=\frac{1}{\sqrt{2}}(\ket{1,0}_{\mathrm{o}}\pm e^{i\varphi_{\mathrm{o}}}\ket{0,1}_{\mathrm{o}})$, in which $\varphi_{\mathrm{o}}$ is an adjustable phase shift. Note that the unconditional maximum efficiency of this projective readout is 0.5, because half of the click events correspond to a photon in the first time bin passing through the short arm or a photon in the second time bin passing through the long arm, and thus produce no interference Tittel et al. (2000). Maximum visibility of unity can be recovered by discarding these early and late counts through postselection.

On the microwave side (the blue panel on the right of Fig. 1), flying photons in two time bins are first converted to excitations of two transmon qubits in a cQED system through stimulated two-photon Raman absorption Campagne-Ibarcq et al. (2018), after which the transmons and the optical modes are entangled as $\frac{1}{\sqrt{2}}(\ket{1,0}_{\mathrm{o}}\ket{eg}+\ket{0,1}_{\mathrm{o}}\ket{ge})$, where $g$ ($e$) denotes the ground (first excited) state of a transmon. After a joint parity measurement (to be explained in the next paragraph), a CNOT gate Paik et al. (2016) followed by a high-fidelity single-shot qubit readout Hatridge et al. (2013) projects the two transmons on $\frac{1}{\sqrt{2}}(\ket{eg}\pm e^{-i\varphi_{\mathrm{e}}}\ket{ge})$, where $\varphi_{e}$ can be continuously adjusted from 0 to $2\pi$. We thus effectively project the microwave photons on $\ket{\varphi_{\mathrm{e}}^{\pm}}_{\mathrm{e}}=\frac{1}{\sqrt{2}}(\ket{1,0}_{\mathrm{e}}\pm e^{-i\varphi_{\mathrm{e}}}\ket{0,1}_{\mathrm{e}})$.

During the experiment, heralded signals that indicate successful entanglement generation events are produced by measuring the output photons in both microwave and optical domains—an M–O Bell state contains one and only one photon pair. Specifically, on the microwave side, instead of counting photons, we can measure the parity of the two-qubit state immediately after the stimulated Raman absorption and postselect on the condition that only one transmon is excited Ristè et al. (2013); Shankar et al. (2013); *Liu2016. Excluded by heralding are null events $\ket{0,0}_{\mathrm{e}}\ket{0,0}_{\mathrm{o}}$, undecayed higher-order generation events, and the cases where the flying photons are lost in the paths. Although not increasing the generation rate, this heralded scheme significantly improves the fidelities of the obtained Bell pairs and enables the entanglement verification given non-negligible transmission loss and limited detection efficiencies.

*Dissipation and thermal noise.—*In reality, the ideal squeezed vacuum $\ket{\psi_{\mathrm{sq}}(\lambda)}_{\mathrm{oe}}$ generated in the parametric down-conversion process is degraded to a generic two-mode Gaussian state by the dissipation and thermal fluctuations of the modes. We label the optical, microwave, and mechanical dissipation rates by $\kappa_{\mathrm{o}}=\kappa_{\mathrm{o,i}}+\kappa_{\mathrm{o,c}}$, $\kappa_{\mathrm{e}}=\kappa_{\mathrm{e,i}}+\kappa_{\mathrm{e,c}}$, and $\kappa_{\mathrm{m}}$ (the subscript ‘i’ for internal loss, ‘c’ for external coupling), and assume the thermal phonon (photon) population of the mechanical (microwave) dissipation bath is $\bar{n}_{\mathrm{ba}}=(e^{\hbar\omega_{\mathrm{m(e)}}/k_{\mathrm{B}}T}-1)^{-1}$ on average, while the optical resonator and the optical and microwave coupling ports are purely subject to vacuum fluctuations. These assumptions comply with the experimental conditions: in reality, we might need to vary the temperature of the electro-optomechanical device from 10 mK to a few kelvins for higher mechanical quality factor and better power handling capability. Meanwhile, the gigahertz microwave mode can be radiatively cooled close to its ground state if it’s very overcoupled to the 10 mK bath ($\kappa_{\mathrm{e,c}}\gg\kappa_{\mathrm{e,i}}$).

The output two-mode Gaussian state can be derived analytically in the frequency domain combining the Heisenberg-Langevin equations of motion with the input-output theory. Denoting the output M–O state quadratures as $\textbf{x}_{\text{oe}}^{\text{out}}=\{\hat{q}_{\text{o}},\hat{p}_{\text{o}},\hat{q}_{\text{e}},\hat{p}_{\text{e}}\}$, we write down the corresponding covariance matrix in the standard form

[TABLE]

whose matrix elements reflect the corresponding quadrature correlations (see Ref. sup for details). To characterize the output M–O state, we show its power spectrum densities $u(\omega)=\braket{}{\hat{q}_{\text{o}}(\omega)}{{}^{2}}$ (optical mode) and $v(\omega)=\braket{}{\hat{q}_{\text{e}}(\omega)}{{}^{2}}$ (microwave mode) in Fig. 2(d). The microwave mode has higher spectrum density due to its intrinsic coupling to the thermal bath. Also, we numerically calculate the entanglement of formation $E_{\mathrm{F}}$ (see Ref. sup for the definition) of the output state as a function of the optomechanical cooperativity $C_{\mathrm{om}}=4g^{2}_{\text{om}}/\kappa_{\text{o}}\kappa_{\text{m}}$ and the microwave readout ratio $\kappa_{\text{e,c}}/\kappa_{\text{e,i}}$, which can be tuned in experiments via the optical pump power and the position of the microwave readout probe, respectively. Using feasible parameters listed in Tab. 1, $E_{\text{F}}$ with resonant frequency ($\omega=0$ in rotating frame) is plotted in Fig. 2(a), which effectively demonstrates the entanglement of the output modes. $E_{\mathrm{F}}>0$ indicates the continuous-variable entanglement under any finite two-mode squeezing and $E_{\mathrm{F}}$ reaches its maximum when the electromechanical cooperativity $C_{\mathrm{em}}=4g^{2}_{\text{em}}/\kappa_{\text{e}}\kappa_{\text{m}}\simeq C_{\mathrm{om}}-1$ where strong parametric down conversion (PDC) dominates. In these regimes, the system tends to be unstable and generates entangled state with extremely low state purity $\mu_{\text{P}}$, as shown in Fig. 2(b) (see Ref. pu for purity definition). In Fig. 2(c), the $E_{\text{F}}$ and $\mu_{\text{P}}$ are shown with $C_{\text{om}}=1$ and different thermal nosies. As expected, higher thermal noise decreases the entanglement and the state purity. It is worth pointing out that in these plots, the parameter regime $g_{\text{em}}<\kappa_{\text{e,c}}$ is chosen in order to avoid electromechanical strong coupling and mode splitting, which can potentially complicate the microwave photon detection and thus not preferred in the entanglement generation. Nevertheless, entanglement in the strong coupling regime would be interesting for further investigations.

*Entanglement verification.—*A lower bound of the entanglement fidelity of the M–O output state with respect to the ideal Bell state $\frac{\sqrt{2}}{2}(\hat{a}^{(1)\dagger}_{\text{out,c}}\hat{c}^{(1)\dagger}_{\text{out,c}}+\hat{a}^{(2)\dagger}_{\text{out,c}}\hat{c}^{(2)\dagger}_{\text{out,c}})\ket{\text{vac}}$ is given by Blinov et al. (2004); Bernien et al. (2013)

[TABLE]

in which $p^{\mu\nu}_{\varphi_{\mathrm{o}}\varphi_{\mathrm{e}}}$ is the normalized probability of detecting $\ket{\varphi_{\mathrm{o}}^{\mu}}_{\mathrm{o}}$ and $\ket{\varphi_{\mathrm{e}}^{\nu}}_{\mathrm{e}}$ on the optical and microwave sides in a Bell state measurement. $F_{\mathrm{lb}}>1/2$ strictly indicates the M–O entanglement. Such regions are delineated in Fig. 3(a), which shows that a better entanglement fidelity should avoid the strong PDC parameter regime. Although $E_{\text{F}}$ is large at these regimes, the highly-mixed-entangled state (shown in Fig. 2(b)) is not suitable for this entanglement verification. This is also consistent with the previous discussions that weakly squeezed M–O output states better approximate the two-mode squeezed vacuum and thus the ideal time-bin Bell state.

Furthermore, a stronger entanglement manifestation that excludes local hidden-variables is the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality Clauser et al. (1969), which can be tested in our proposed setup by measuring the correlation quantity

[TABLE]

in which $E(\varphi_{\mathrm{o}},\varphi_{\mathrm{e}})\equiv p^{++}_{\varphi_{\mathrm{o}}\varphi_{\mathrm{e}}}+p^{--}_{\varphi_{\mathrm{o}}\varphi_{\mathrm{e}}}-p^{+-}_{\varphi_{\mathrm{o}}\varphi_{\mathrm{e}}}-p^{-+}_{\varphi_{\mathrm{o}}\varphi_{\mathrm{e}}}$ can be acquired by detecting photonic and superconducting qubits on the $\{\ket{\varphi_{\mathrm{o}}^{\pm}}_{\mathrm{o}},\ket{\varphi_{\mathrm{e}}^{\pm}}_{\mathrm{q}}\}$ basis. Choosing $\varphi_{\mathrm{o}}^{\prime}-\varphi_{\mathrm{o}}=\varphi_{\mathrm{e}}^{\prime}-\varphi_{\mathrm{e}}=\pi/2$, we simulate the typical curves of $S$ in Fig. 3(b) for varied thermal baths. It can be seen that $S$ sinusoidally depends on the phase variable and reaches its maximum when $\varphi_{\mathrm{e}}-\varphi_{\mathrm{o}}=\pi/4+k\pi$ ($k\in\mathbb{N}$). For low bath photon numbers, clear inequality violation ($|S|_{\mathrm{max}}>2$) is observed. As the thermal noise increases, the violation is gradually destroyed. The $|S|_{\mathrm{max}}>2$ regions are delineated in Fig. 3(a), which shows that the threshold for CHSH violation is more stringent than $F_{\mathrm{lb}}>1/2$.

*Discussion.—*The coincidence detection rate can be written as $R_{\text{c}}=R_{\text{ac}}+R_{\text{cc}}$ Mandel and Wolf (1995), where the first and the second parts denote the accidental and the correlated counts, respectively. Ideally, they are approximated by $R_{\text{ac}}=R_{\text{o}}R_{\text{e}}\tau_{b}$ and $R_{\text{cc}}=\int_{0}^{\tau_{b}}\mathinner{\!\left\lvert R_{\text{oe}}\right\rvert}^{2}d\tau$, where $\tau_{b}$ is the photon collection time window, $R_{\text{o}}=\braket{\hat{a}^{\dagger}_{\text{out,c}}(t)\hat{a}_{\text{out,c}}(t)}$ and $R_{\text{e}}=\braket{\hat{c}^{\dagger}_{\text{out,c}}(t)\hat{c}_{\text{out,c}}(t)}$ are the optical and microwave photon generation rates, respectively, and $R_{\text{oe}}=\braket{\hat{a}^{\dagger}_{\text{out,c}}(t)\hat{c}^{\dagger}_{\text{out,c}}(t+\tau)}$ is the photon correlations, which reaches maximum when the detector time delay $\tau\simeq 0$. Considering a typical experiment repetition time $\sim 10\$\mathrm{\SIUnitSymbolMicro s} Campagne-Ibarcq *et al.* ([2018](#bib.bib10)) that includes $1\ $\mathrm{\SIUnitSymbolMicro s} Raman absorption (determining the detector time window) and using $C_{\text{om}}=1$ and $\kappa_{\text{e,c}}/\kappa_{\text{e,i}}=150$, the joint detection rate is found approximately on the order of $10^{4}\$\mathrm{H}\mathrm{z}$$.

In reality, the detection scheme suffers from photon losses (filter or transmission losses), detector inefficiency, and dark counts. Denoting $T_{\text{o}},D_{\text{o}},\eta_{\text{o}}$ ($T_{\text{e}},D_{\text{e}},\eta_{\text{e}}$) as the optical (microwave) photon transmissivity, dark count rate and detector efficiency, respectively, the accidental and correlated counting rates are replaced by $R_{\text{ac}}=(\eta_{\text{o}}T_{\text{o}}R_{\text{o}}+D_{\text{o}})(\eta_{\text{e}}T_{\text{e}}R_{\text{e}}+D_{\text{e}})\tau_{\text{b}}$ and $R_{\text{cc}}=\eta_{\text{o}}\eta_{\text{e}}T_{\text{o}}T_{\text{e}}\int_{0}^{\tau_{b}}\mathinner{\!\left\lvert R_{\text{oe}}\right\rvert}^{2}d\tau$, which shows that the joint detection rates become lower due to the transmission loss and detector inefficiency, while dark counts increase the accidental counts. To ensure a good entanglement fidelity, $R_{\text{cc}}\gg R_{\text{ac}}$ is required, which leads to $g^{(2)}\gg 2+\xi_{\text{o}}+\xi_{\text{e}}+\xi_{\text{o}}\xi_{\text{e}}$. $g^{(2)}=R_{\text{c}}/R_{\text{ac}}$ is the second order correlation function Guo et al. (2017), and $\xi_{\text{o}}=\frac{D_{\text{o}}}{\eta_{\text{o}}T_{\text{o}}R_{\text{o}}}$ and $\xi_{\text{e}}=\frac{D_{\text{e}}}{\eta_{\text{e}}T_{\text{e}}R_{\text{e}}}$ are the ratios of dark counts to the signal counts. As expected, suppressing detector dark counts are beneficial for identifying entanglement. Meanwhile, the detector efficiencies, the transmissivities, and the detector time window are essential for observing the M–O entanglement in a reasonable amount of time, and should be simultaneously optimized in the experimental design.

Our proposal is compatible with recent experiments on cavity electro-optomechanics. Given a relatively smaller $C_{\mathrm{om}}$ Han et al. (2014); Zou et al. (2016); Vainsencher et al. (2016), the system can be well prepared in the cooperativity mismatched regime, leading to efficient entangled photon pair generation and detection even in the presence of a few thermal noise photons. Furthermore, although our analysis in this Letter is grounded on electro-optomechanics, the entanglement generation and detection scheme can be potentially generalized to other types of M–O converters based on parametric interactions—for instance, the electro-optic converters Rueda et al. (2016); Fan et al. —and thus shed light on M–O quantum state transfer in various physical systems.