1. Introduction

Quantum entanglement is a unique phenomenon in quantum physics and enables many quantum technologies that surpass their classical counterparts [1–8]. It is worth noticing that large-scale quantum communication and distributed quantum computation require quantum entanglement between distant nodes in quantum networks [9–12]. However, quantum entanglement could only be created locally [13,14]. The propagation of photonic pulses becomes a prerequisite for creating nonlocal quantum entanglement between remote quantum nodes, owing to the stable and robust property of photons to transmit quantum information across long distances [15,16]. In practice, the transmission loss in optical fibers or free-space channels sets a limit on the efficiency and range of entanglement generation between distant nodes [17]. This bottleneck can be overcome by dividing the long entangled channel into multiple intermediate segments and then creating entanglement within each segment, a crucial element of quantum repeaters [18–20].

In general, there are three methods for generating entanglement between spatially separated stationary quantum memories (QMs) [21–23]. The first method entangles two remote QMs by creating a hybrid entangled state between a photonic pulse and a QM at each node, followed by the interference of the two photonic pulses at an intermediate node [24–28]. The second approach generates hybrid entanglement between a QM and a photonic pulse, which is then sent to interact with another QM to entangle the two QMs [29–31]. The third method utilizes an entangled photon source and sends two photons from each entangled pair to two separate nodes to entangle two remote QMs [32–37]. In these approaches, a photonic pulse (single photon) or an entangled photon pair is transmitted over a distance equal to the separation between two nodes to entangle a pair of QMs. They require the transmission of many single photons or photon pairs, as most applications require multiple pairs of entangled QMs, such as quantum error correction [38,39] and multiqubit state teleportation [40].

Quantum multiplexing encodes a single photon with more information than a single qubit [41], and it has been introduced to entangle two pairs of QMs using only a single photonic pulse, encoding in both the polarization and the time-bin degrees of freedom (DOFs) [42,43]. This multiplexing approach takes advantage of the fact that a photonic pulse possesses multiple DOFs, enlarging the corresponding Hilbert space. It can also be extended to entangle additional pairs of QMs by encoding a photonic pulse with more DOFs [42]. Apart from qubits hyperencoded in multiple DOFs [41], qudits (i.e. high-dimensional quantum systems) can also form high-dimensional Hilbert spaces and are implementable in practical experiments. By introducing enhanced-capacity quantum multiplexing, which encodes a single photon in both polarization and high-dimensional time-bin DOFs, Xie et al.,[44] presented a general protocol for the parallel entanglement of multiple pairs of QMs using a single photon. Zhou et al., [45] showed that the efficiency of nonlocal multiqubit entanglement generation can be significantly improved. Meanwhile, parallel entanglement can be achieved using time-bin photonic qudits without the need for polarization encoding [46]. These protocols in principle entangle several QM qubits in one node with a high-dimensionally encoded single photon and then evolve the hybrid entanglement between QM qubits and the photonic qudit into entanglement between multiple pairs of QMs, after the photon reaches the other node and interacts sequentially with QMs in that location. Therefore, these protocols impose a constraint on the coherence time of QMs in the initial node and also constrain their ability to achieve spatial-temporal multiplexing [18]. Meanwhile, the polarization DOF of the photon is susceptible to polarization channel noise, leading to a reduction of hybrid entanglement. This, in turn, decreases the entanglement between QMs. A potential approach to address this issue is to employ advanced entanglement purification protocols [47,48].

Alternatively, it is worthwhile to generalize the original quantum multiplexing entangling protocol [42] to include optical entanglement, i.e., a quantum multiplexing version of the entangling method that makes use of entangled photon sources. This approach has the potential to efficiently create multiple entangled pairs of QMs in combination with the higher ability to spatial-temporal multiplexing with lower QM coherence times. In comparison to two-dimensional optical entanglement, high-dimensional entanglement offers more pronounced benefits [49,50], including robustness against noise [51–55], heightened communication capacity [56–58], and enhanced security in quantum cryptography [59,60]. It has been demonstrated that high-dimensional entanglement can be used to efficiently transmit information over long distances [61]. Moreover, there are multiple DOFs of a single photon that can be harnessed for qudit encoding [62–64], such as spatial modes, frequency modes, and time bins. The time-bin photonic qudit is particularly suitable for long-distance transmission, owing to its resilience against decoherence and transmission noise, and it is easy to obtain high-dimensional time-bin entanglement [65,66].

In this article, we present a heralded scheme for the parallel creation of entanglement between multiple pairs of QMs by utilizing a pair of high-dimensional time-bin entangled photons. By distributing two photons from each entangled pair to two remote nodes and interacting each photon with the QMs in its respective node, the parallel entanglement between multiple pairs of nonlocal QMs can be created in a heralded way, signaled by the detection of one photon in each node. Therefore, our protocol has the potential to be expanded to simultaneously accommodate both spatial-temporal multiplexing and quantum multiplexing. In our protocol, each photon interacts with $n$ QMs to entangle $n$ pairs of QMs and travels half the distance between the nodes. This reduces the time required, when compared to protocols where photons traverse the entire channel and interact with all QMs. Furthermore, our protocol can potentially be extended to entangle additional pairs of QMs by increasing time-bin dimensions. These distinguishing features make our protocol efficient to entangle multiple pairs of nonlocal QMs and thus provide a viable route toward the construction of large-scale quantum networks.

2. Controlled-polarization flip unit based on the spin-photon interface

An interface between single photons and individual spins [67] is an essential platform for constructing quantum networks and distributed quantum computation [68–70]. Such an interface can be implemented by coupling a four-level atom to a single-sided optical cavity, shown in Fig. 1, where the frequency of the cavity mode $\omega _c$ is nearly resonant with the transition from coupled ground state $\vert g\rangle$ to excited state $\vert \tilde {g}\rangle$ at a frequency $\omega _0$. Meanwhile, the transition $\vert e\rangle \leftrightarrow \vert \tilde {e}\rangle$ at a frequency $\omega _1$ decouples from the cavity mode, either due to a large detuning or polarization mismatch. A polarized photon with frequency $\omega \simeq \omega _c$ will be reflected by the atom-cavity system with a state-dependent reflection coefficient [71–73]

A controlled-polarization flip unit can be constructed by using such an atom-cavity system and two circularly polarizing beam splitters (CPBSs) [41–44,72], shown in Fig. 1(b). The CPBS reflects the right-circularly polarized photon $\vert R\rangle$ that matches the transition $\vert g\rangle \leftrightarrow \vert \tilde {g}\rangle$, and transmits left-circularly polarized photon $\vert L\rangle$. For a polarized photon in state $\vert D\rangle =\frac {1}{\sqrt {2}}(\vert L\rangle +\vert R\rangle )$ or $\vert A\rangle =\frac {1}{\sqrt {2}}(\vert L\rangle -\vert R\rangle )$ impinging into the input port of the unit, the output state of the combined system consisting of the photon and the atom can be described as follows:

 

Fig. 1. (a) Level structure and transitions of a four-level atom. (b) Schematic of a controlled-polarization flip unit. CPBS represents a circular-polarization beam splitter that reflects the right-circularly polarized photon $\vert R\rangle$ and transmits left-circularly polarized photon $\vert L\rangle$.

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3. Heralded parallel protocol for entangling two pairs of QMs

In this section, we show that parallel entanglement between distant stationary qubits can be achieved by distributing high-dimensional optical entanglement, and present a specific scheme for entangling two pairs of QMs placed in two remote nodes, as shown in Fig. 2. The TBES represents a time-bin entangled source that prepares pairs of polarized photons entangled in four-dimensional time-bin DOF. OS$_i$ ($i=1, 2$) are optical switches that combine photon components in different spatial modes with different time bins into one spatial mode, and vice versa [74,75]. The H is a half-wave plate that flips the polarization of photons passing through it. The PBS is a polarizing beam splitter that reflects $A$-polarized photons and transmits $D$-polarized ones, while the PBS$'$ is a polarizing beam splitter that reflects $D$-polarized photons and transmits $A$-polarized ones. The two time-bin entangled photons prepared by the TBES are sent to nodes A and B, respectively. They experience parallel operation procedures in the two nodes, which can lead to entanglement between QM1 (QM3) and QM2 (QM4). The principle of this process can be detailed as follows.

 

Fig. 2. Schematics of generating two pairs of QMs via distribution of optical entanglement. (a) Step-by-step protocol with (b) circuit of PSE for photon state exchanges. Here QM$k$($k$=1,…,4) denotes a quantum memory. PC is a Pockel cell that flips the polarization of photons in the last two time-bin modes. OS$_1$ is an optical switch that directs photons in the time-bin states $\vert 0\rangle$ ($\vert 1\rangle$) through the upper (lower) path. OS$_2$ directs two spatial modes of a photon with different time bins into one spatial mode. $\mathcal {T}{(i)}$ introduces an optical delay of $it_\Delta$. H is a half-wave plate and flips the polarization. PBS is polarization beam splitter that transmits the $D$-polarized photon and reflects the $A$-polarized photon, while PBS$'$ transmits the $A$-polarized photon and reflects the $D$-polarized photon.

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Suppose QM$k$ ($k=1,\ldots,4$) is initialized to superposition state $\vert \Psi _{k}\rangle =(\vert g_k\rangle +\vert e_k\rangle )/\sqrt {2}$, and one pair of $D$-polarized photons p$1$p$2$ are initialized to the time-bin-entangled state $\vert \Psi _{P_0} \rangle =\frac {1}{2}\sum _{i=0}^{3}\mathcal {T}_1{(i)}\mathcal {T}_2{(i)}\vert DD\rangle$, where time operator $\mathcal {T}_{1(2)}{(i)}$ introduces a time delay of $it_{\Delta }$ for photons p$1$ and p$2$, which satisfies the relationship $\mathcal {T}_{1(2)}{(l+p)}=\mathcal {T}_{1(2)}{(l)}\mathcal {T}_{1(2)}{(p)}$ [65,66]. The two photons p$1$ and p$2$ are sent to node A and node B, respectively. The time-bin entangled state $\vert \Psi _{P_0} \rangle$ is converted into a hyper-entangled state $\vert \Psi _{P_1} \rangle =\frac {1}{\sqrt {2}}\sum _{i=0}^{1}\mathcal {T}_1{(i)}\mathcal {T}_2{(i)}\vert \phi _P\rangle$ with $\vert \phi _P\rangle =(|DD\rangle +|AA\rangle )/\sqrt {2}$ by respectively passing p$1$ and p$2$ through a converting unit consisting of a Pockel cell (PC) and an unbalanced polarizing Mach–Zehnder interferometer, shown in Fig. 2(a). Here PC flips the polarization of photons in the last two time-bin modes, and the two time delays of photons in state $\vert AA\rangle$ are compensated after they pass through the short arm of the interferometer.

The entangled photons p$1$ and p$2$ then interact with QM1 and QM2, respectively. Their polarization states are entangled with QM1 and QM2, since photon p$1$ (p$2$) flips its polarization when QM1 (QM2) is in state $|e\rangle$. Subsequently, the photon states are exchanged as $\mathcal {T}{(0)} |A\rangle \leftrightarrow \mathcal {T}{(1)}|D\rangle$ by passing the photons through the PSE circuit, consisting of OS$_{1,2}$, PBSs, PBS$'$s, H, and time delayers $\mathcal {T}{(1)}$. The combined state of QM1,2 and photons p$1$p$2$ then evolves into

The two spatial modes of photons p$1$ and p$2$ are combined into one spatial mode by OS$_{2}$ at the output port of each PSE, which is then directed to interact with QM3 and QM4, respectively. The combined state of the two photons and four QMs evolves into a hybrid entangled state

4. Heralded parallel protocol for entangling multiple pairs of QMs

Our protocol can be generalized to entangle more pairs of QMs placed in two remote nodes in parallel using one pair of photons entangled in the high-dimensional time-bin DOF. A parallel schematic for entangling three pairs of QMs, i.e., (QM1, QM2), (QM3, QM4), and (QM5, QM6) is shown in Fig. 3, where all QMs are all initialized to superposition states $\vert \Psi _{k}\rangle =(\vert g_k\rangle +\vert e_k\rangle )/\sqrt {2}$ ($k=1,\ldots,6$). The TBES produces a pair of $D$-polarized photons p$1$p$2$ that are entangled in the time-bin mode with $\vert \Psi _{P_2} \rangle =\frac {1}{2\sqrt {2}}\sum _{i=0}^{7}\mathcal {T}_1{(i)}\mathcal {T}_2{(i)}\vert DD\rangle$. Photons p$1$p$2$ are separated and distributed to nodes A and B, respectively. They pass through the same photonic circuit placed in the corresponding node and experience parallel operation procedures as detailed below.

 

Fig. 3. Schematics of generating three pairs of QMs via distribution of optical entanglement. (a) Step-by-step protocol with (b) and (c) circuits of PSE$_1$ and PSE$_2$ for photon state exchanges. Here QM$k$($k$=1,…,6) denotes a quantum memory. PC$'$ is a Pockel cell that flips the polarization of photons in the last four time-bin modes. OS$_1$ (OS$_3$) is an optical switch that directs photons in the time-bin states $\vert 0\rangle$ and $\vert 2\rangle$ ($\vert 0\rangle$ and $\vert 1\rangle$) to the upper path and photons in states $\vert 1\rangle$ and $\vert 3\rangle$ ($\vert 2\rangle$ and $\vert 3\rangle$) to the lower path. OS$_2$ (OS$_4$) directs two spatial modes of a photon with different time bins into one spatial mode. H is a half-wave plate and flips the polarization. $\mathcal {T}{(i)}$ describes a time delayer, introducing an optical delay of $it_\Delta$. PBS is polarization beam splitter that transmits the D-polarized photon and reflects the A-polarized photon. PBS$'$ transmits the A-polarized photon and reflects the D-polarized photon.

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Upon arriving at nodes A and B, the state of photons p$1$p$2$ is transformed into a hyper-entangled state $\vert \Psi _{P_3} \rangle =\frac {1}{2}\sum _{i=0}^{3}\mathcal {T}_1{(i)}\mathcal {T}_2{(i)}\vert \phi _P\rangle$ utilizing a converting unit similar to the one used for entangling two pairs of QMs, while the PC′ now flips the polarization of photons in the last four time-bin modes and four time delays are compensated by passing them through the short arm of the interferometer. Photons p$1$ and p$2$ are then directed to interact with QM1 and QM2, resulting in their entanglement. Subsequently, PSE$_1$, shown in Fig. 3(b), engineers photon states and completes the exchange $\mathcal {T}{(0)} |A\rangle \leftrightarrow \mathcal {T}{(1)} |D\rangle$ and $\mathcal {T}{(2)} |A\rangle \leftrightarrow \mathcal {T}{(3)} |D\rangle$, by directing photons in time-bin states ($\vert 0\rangle$, $\vert 2\rangle$) through the upper path and those in states ($\vert 1\rangle$, $\vert 3\rangle$) through the lower path with OS$_1$. After two spatial modes with different time bins are combined into one spatial mode by OS$_2$ for each photon, the combined state of p$1$p$2$ and QM1,2 can be described as

The photons then interact with QM3 and QM4, and pass through PSE$_2$ that engineers photon states as $\mathcal {T}{(0)} |A\rangle \leftrightarrow \mathcal {T}{(2)} |D\rangle$ and $\mathcal {T}{(1)} |A\rangle \leftrightarrow \mathcal {T}{(3)} |D\rangle$, by directing photons in time-bin states ($\vert 0\rangle$, $\vert 1\rangle$) to the upper path and photons in states ($\vert 2\rangle$, $\vert 3\rangle$) to the lower path of PSE$_2$ with OS$_3$. The combined state of two photons and four QMs evolves into

To entangle the third pair of QMs, photons p$1$ and p$2$ then interact with QM5 and QM6, respectively. The combined state of three pairs of QMs and two photons evolves into

5. Physical implementation with practical atoms

Nitrogen-vacancy (NV) color center, a kind of isolated point defect in diamond, is a promising candidate serving as a quantum stationary system for implementing quantum networks [76–78]. The electron spins of NV centers can be optically manipulated and read out with high fidelity [79,80], and can be used to initialize and manipulate nearby nuclear spins with long coherence time for quantum storage [81,82]. So far, NV centers have been used to perform various quantum information processing tasks, such as quantum teleportation over remote distance [83,84], quantum gates on electron spins and photons [85,86], and multinode quantum networks [87].

An NV$^-$ center can be treated as an approximate four-level system [71,72]. The electronic energy level structure of an NV$^-$ center consists of three subspaces: ground state manifold (GSM), excited state manifold (ESM), and meta-stable state manifold (MSM). A static magnetic field of approximately 20 mT along the NV$^-$ axis at low temperatures causes the level $|-1\rangle$ of the GSM to be far detuned by about 2.3 GHz above $\vert g\rangle =\vert 0\rangle$ and about 1.1 GHz below $\vert e\rangle =\vert +1\rangle$ [88]. The ESM comprises six basis states $\{M_{1,\ldots,6}\}$ of the excited-state Hamiltonian. The optical transitions between GSM and ESM, coupled to the cavity mode, can be described by an effective Hamiltonian [71,73,88,89]

When only a $\sigma _-$ polarized photon is involved, we assume that $g_{-1,1}$, $g_{0,3}$, $g_{+1,5}$, and $g_{+1,6}$ are equal and labeled as $g$ for simplicity, while all other $g_{s,i}$ vanish due to the electric-dipole selection rules [88]. A $\sigma _-$ polarized photon impinging onto the NV$^-$-cavity system will be reflected with NV$^{-}$ state-dependent reflection coefficients. These coefficients can be obtained by solving the dynamic equations of the motion for the cavity field operator $\hat {a}$ and the NV$^{-}$ operators $\hat {\sigma }_{s,i}$ in combination with the input-output relation,

For an NV$^-$-cavity system with resonant frequency $\omega _{c}\approx \omega _{0,3}$ and the NV$^-$ center in the GSM state $|0\rangle$ or $|+1\rangle$, a $\sigma _-$ polarized photon with frequency $\omega \approx \omega _{0,3}$ can be scattered with state-dependent reflection coefficients [71–73]

6. Performance of the heralded parallel protocol

In our heralded parallel protocol, one pair of high-dimensional time-bin entangled photons are utilized and their states are engineered by linear-optical elements and QMs. We have assumed an ideal case where the photon scattering process from the four-level atom-cavity system is deterministic with reflection coefficients $r_g=-r_e=1$. However, the finite cooperativity $C_i$ and normalized detunings including $\Delta _{i}$ and $\Delta _{c}$ in combination with additional excited states deviate the scattering from an ideal one, which decreases the efficiency and fidelity of the entanglement creation. Through proper parameter tuning, a practical NV$^-$-cavity system can actively suppress the detrimental effects of non-ideal scattering conditions on fidelity, although this may come at the cost of reduced efficiency. For instance, when parameters $C_3=110$, $\Delta _{c}=0.5$, $\Delta _{3}=0$, $\Delta _{5}=400$, $\Delta _{6}=550$, $(\gamma _{3}, \gamma _{5}, \gamma _{6})/2\pi =(6, 11, 13)$ MHz [71] are used, the reflection coefficients $r_g=-r_e\simeq 0.999$ can be achieved with a phase difference of $\pi$, leading to a unity fidelity but a decreased efficiency with $\eta _0=|r_{g}|^2$ for each single photon scattering.

The efficiency of our protocol for entangling $n$ pairs of QMs can be described as

The efficiencies $\eta _2$ and $\eta _3$ of our protocol for entangling $n=2$ and $n=3$ pairs of QMs as a function of the distance $L$ are shown in Fig. 4(a). Here, we assume that the attenuation rate of optical fiber channels is $0.2$ dB/km $(\alpha \simeq$1/22 km$^{-1})$ and single-photon detector efficiency is $\eta _d=0.96$.The corresponding efficiencies $\eta '_2$ and $\eta '_3$ of conventional schemes with the same parameters are also shown in Fig. 4(a). Meanwhile, the efficiency $\eta '_1$ for generating a single pair of entangled QMs is shown using the same parameters as a reference. When the QMs are located in the same node, that is $L\simeq 0$, we can entangle $n=2$ and $n=3$ pairs of QMs with the efficiencies of $\eta _2=0.768$ and $\eta _3=0.709$, which exceed the corresponding efficiencies $\eta '_2=0.723$ and $\eta '_3=0.614$ of the previous schemes. When QM pairs are separated by a medium distance $L=25$ km, we can entangle $n=3$ pairs of QMs with the efficiency $\eta _{3}=0.228$ which is approximately three times higher than that for preparing two pairs of QM entanglement with two photon pairs with $\eta '_2=0.074$, while approximately $83{\%}$ of that for generating a single pair of entangled QMs. Theoretically, our protocol demonstrates a further advantage in efficiency as either the distance between two nodes or the size of the time-bin dimension increases.

 

Fig. 4. (a) Efficiency of entanglement creation for $n$ pairs of QMs versus distance $L$. (b) Efficiency enhancement versus distance $L$ for spatial multiplexing with $x=5$ entangled photon pairs. Here we set $\eta _0=0.96$, $\eta _d=0.96$, and $\alpha \simeq 1/22$ km$^{-1}$.

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Spatial multiplexing can further increase the efficiency of entanglement creation by transmitting several photon pairs in parallel [38]. Suppose there are $x$ high-dimensional entangled photon pairs in each transmission round and there are $y$ receiver units in each quantum node [36]. Each receiver unit is composed of photon-state engineering blocks and $m\geq 2$ QMs, shown in Figs. 2 and 3. When a unit $y_i$ receives a photon $x_j$, heralded by a successful measurement of the photon, the photon $x_{j+1}$ is directed to the next unit $y_{i+1}$, otherwise $x_{j+1}$ is directed to the unit $y_{i}$ after it has been re-initialized. Finally, two nodes exchange information about the received photons and project two units receiving photons from the same photon pair into the target entangled states [36].

For simplicity, we consider the case where entanglement has been established between all units of two quantum nodes, corresponding to the situation that $y$ pairs of entangled photons have arrived at their respective nodes, interacted with the receiver units, and clicked $2y$ photon detectors. The efficiencies of our scheme ($P_1$) and conventional schemes ($P_2$) are [36]

Here $C^a_b=\frac {b!}{a!(b-a)!}$ is the binomial coefficient. $p_1=\eta _2$ ($p_2=\eta '_1$) is the efficiency of the parallel (conventional) scheme without spatial multiplexing, in which a pair of four-dimensional (two-dimensional) entangled photons have been successfully detected by two single-photon detectors. $p'_l=(1-\sqrt {p_l})^2$ for $l=1$ and $2$ is the probability that both photons are lost. Note that the efficiency $P_2$ corresponds to the scenario in which $2y$ pairs of QMs are entangled with $2x$ two-dimensional entangled photon pairs using conventional schemes.

For a specific example of spatial multiplexing, with $x=5$ and $y=2$, the efficiency of entangling at least one pair of receiver units is

To show the enhanced efficiency achieved by spatial multiplexing, we introduce the parameters

In our protocol, optical switches are used to manipulate the time-bin states of photons. So far, we assume that the OS is ideal with the unity efficiency. In practice, the efficiency $\eta _s$ of the OS is always less than unity due to the intersection loss, and our efficiency should be multiplied by a parameter $\eta ^m_s$, where we assume the efficiency of all OSs are identical and $m$ is the number of OSs used in our parallel protocol. To show the potential enhancement of our parallel protocol, we introduce an efficiency enhancement $\eta ^{\rm enh}_n$ with ideal OSs and a threshold OS efficiency $\eta _n^{\rm thr}$, shown in Fig. 5, as follows:

 

Fig. 5. Efficiency enhancement of entanglement creation and threshold efficiency of optical switches for the parameters assumed in Fig. 4(a).

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For the specific case with $n=2$ and $n=3$, we have $m=4$ and $m=8$, shown in Fig. 2 and Fig. 3. Interestingly, we have the coincidence that $\eta _2^{\rm thr}\simeq \eta _3^{\rm thr}=\sqrt {\eta _d/\eta _{p}}\exp ({-0.25\alpha L})$, which is independent of the single-photon scattering probability $\eta _0$. Our protocol can tolerate a lower OS efficiency threshold for a longer distance $L$. For instance, we have $\eta _{3}^{\rm thr}=0.739$ for $L=25$ km and $\eta _3^{\rm thr}=0.557$ for $L=50$ km.

To show the robustness against variations in detunings and cooperativities, we simulate the average fidelity of the parallel protocol for entangling two pairs of QMs, which is defined as

The average fidelity $F$ as a function of $C_3$ is shown in Fig. 6 for five different detunings $\Delta _3$, conditioned on $\Delta _c=0.5$, $\Delta _{5}=400$ and $\Delta _{6}=550$. As anticipated, when $\Delta _3=0$ and $C_3=110$, the average fidelity approaches unity. In addition, for $C_3\geq 80$ and $\Delta _3\in [-10,10]$, the average fidelity retains a large value with $F\geq 0.95$ and the average fidelity of each QM pair is $F_{1}=\sqrt {F}\geq 0.97$. The fidelity $F_{1}$ corresponds to the moment when QMs become entangled after single-photon measurements, and it is independent of the distance $L$ between the two nodes. In contrast, in multiplexing protocols using single photons [42,44], the QMs in the sender node must store the quantum states during photon transmission. The corresponding fidelity $F'_{1}$ of each entangled QM pair, at the moment when the QMs become entangled using ideal spin-photon interfaces, can be given by $F'_{1}=[1+\exp (-L/cT_2)]/2$ [42], where $c$ represents the speed of light in fiber channels and $T_2$ denotes the QM coherence time. Therefore, our protocol can achieve a higher fidelity in entangling QMs for distances $L>10.5$ km, assuming $T_2=1$ ms [42]. Subsequently, a confirmation signal is transmitted across the entire channel for both protocols before further operations can proceed.

 

Fig. 6. Average fidelity of the protocol for entangling two pairs of QMs in parallel. Here $\Delta _c=0.5$, $\Delta _{5}=400$, $\Delta _{6}=550$, $(\gamma _{3}, \gamma _{5}, \gamma _{6})/2\pi =(6, 11, 13)$ MHz.

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7. Discussion and summary

Our protocol for entanglement creation between multiple pairs of QMs is achieved by the distribution of a single pair of high-dimensional entangled photons. Two photons experience identical operational procedures in parallel, including the interaction with the QMs, which convert high-dimensional time-bin entanglement into hybrid entanglement between QMs and photons, and then into parallel entanglement between QM pairs. Our protocol thus presents a specific connection between high-dimensional optical time-bin entanglement and two-dimensional stationary entanglement [52]. The QMs are projected into an anticipated product state, comprising one Bell state for each QM pair, which is heralded by the detection of two photons in a proper basis. By the distribution of high-dimensional optical entanglement, the limitation of exponential scaling channel loss on entangling distant QMs is significantly suppressed. This leads to a noteworthy improvement in efficiency when compared to that achieved by entangling each distant QM pair with a single pair of entangled photons [32–35]. In fact, the efficiency of our protocol for generating multiple pairs of entangled QMs becomes comparable to that of generating one pair of QMs, when utilizing nearly ideal photon-spin interfaces and optical switches.

The attenuation rate of $0.2$ dB/km in Sec. 6 is suitable for telecom wavelengths. To convert the telecom wavelength to the wavelengths where the NV center operates, a quantum frequency converter should be inserted in each node before interacting the photons with the NV centers. This will decrease the efficiency of our protocol by $\eta _{\rm con}^2$, while the efficiency of a conventional scheme, which involves entangling $n$ QM pairs with $n$ entangled photon pairs, will decrease by $\eta _{\rm con}^{2n}$. Here, $\eta _{\rm con}$ represents the converter efficiency, and a quantum frequency converter with $\eta _{\rm con}=57{\%}$ has been recently used to demonstrate atom-telecom-photon entanglement [91].

The previous parallel protocols [42–46] utilize a high-dimensional single photon, sequentially interacting with each QM, whereas our protocol uses an entangled high-dimensional photon pair, interacts each photon with half of all QMs, and works in a parallel style rather than a cascaded one. Considering spatial multiplexing, the previous parallel protocols make efficient use of QMs and promptly announce QM entanglement after photon measurements [38], while our parallel protocol optimizes the utilization of the entangled photon source and QMs experience reduced dephasing during a given number of attempts [37]. Moreover, with a limited number of QMs, the capability of our protocol for spatial multiplexing surpasses the number of QMs within each node [36]. The optimization of entanglement generation rates, achieved by combining high-dimensional entangled photon pairs or high-dimensional single photons with spatial multiplexing, requires further investigation for the construction of large-scale quantum networks in the future. Additionally, our protocol can be extended to simultaneously entangle multiple qubits in parallel across several quantum nodes [92–96].

In summary, we have proposed a heralded protocol for entangling multiple pairs of QMs located in remote nodes with one pair of high-dimensional entangled photons. This protocol exploits quantum multiplexing of the time-bin entanglement and can entangle additional pairs of QMs by increasing the dimensions of the time-bin DOF, leading to a substantial enhancement of entanglement efficiency. Our protocol is heralded by the detection of one photon in each node, and thus can be extended to simultaneously support spatial-temporal multiplexing. These distinct features make our protocol particularly useful for long-distance quantum communication and large-scale quantum networks.