A multinode quantum network over a metropolitan area

We generate remote entanglement between a pair of distant quantum nodes via a single photon scheme [34]. We start by preparing the light-matter entanglement in each quantum node (Fig. 1B) using the DLCZ protocol [34]. A weak write pulse induces a spontaneous Raman-scattered photon (write-out) in a small probability $\chi\approx 1\%$, whose presence and absence is associated to that of collective excitation of the atomic ensemble. This forms a Fock-basis light-matter entanglement $\ket{\Phi_{\textrm{ap}}}=\ket{00}+\sqrt{\chi}\ket{11}$, where $0$ and $1$ refer to the number of photon or atomic excitation, indicated by the subscript a and p, respectively, from hereafter. The atomic excitation can be stored up to 560 $\upmu$s until being retrieved for following operations [30]. The write-out photon is down-converted from 795 nm (3.5 dB/km) at Rubidium resonance to 1342 nm (0.3 dB/km) in the telecom O band in a periodically poled lithium niobate (PPLN) waveguide chip with the help of a strong 1950 nm Pump laser. The telecom photon then is sent to the server node for interference. The QFC module, including the down-conversion, noise filtering and fiber coupling, constitute an end-to-end efficiency of 46% and noise of $\sim 100$ Hz. In the server node (Fig. 1C), a beamsplitter combines the write-out photon from two quantum nodes and erases their which-path information. A click from a superconducting nanowire single photon detector (SNSPD) heralds the maximally entangled states between two ensembles

where the $\pm$ sign depends on the heralding detector. $\Delta\varphi=\Delta\varphi_{\text{w}}+\Delta\varphi_{\text{p}}+\Delta\varphi_{\text{f}}$ accounts for the optical phase difference accumulated during the entangling process, including $\Delta\varphi_{\text{w}}$ and $\Delta\varphi_{\text{p}}$, the phase difference of Write lasers and of Pump lasers in two nodes, respectively, as well as $\Delta\varphi_{\text{f}}$, the phase difference of two fiber channels.

The key to the success of the single-photon scheme lies in accurately setting $\Delta\varphi$ to a known value. To this end, previous laboratory experiments have typically used an ad hoc laser shared by separated nodes, and locked the closed-loop interferometer composed of the laser splitting paths and the quantum channels [9, 20, 32, 22, 17]. It adds complexity to the networking by requiring additional fiber and hardware resources and exposes the quantum nodes to the risk of potential attacks during quantum communication [35]. Crucially, additional unknown fiber phases are introduced to the generated state, which erases the state’s coherence.

To address these challenges, we developed a network architecture based on a weak-field remote phase stabilization method. The method relies on detecting the phase difference $\Delta\varphi$ in situ at the entanglement generation facilities using weak-field phase probe (PP) pulses and photon counting, and compensating for it immediately, similar to that being realized recently in quantum key distribution experiments [36]. To obtain an accurate measurement of $\Delta\varphi$, we prepare a PP pulse with a length of $\tau_{\textrm{pp}}=4~{}\upmu$s at the write-out photon’s frequency from the Read laser (due to its frequency likeness to the write-out photon) in each node, and use it to simulate the write-out photon’s transmission $\tau_{\Delta}=5~{}\upmu$s earlier than the entanglement trial. Analogous to the entangling process, two PP pulses interfere in the server node, with the results reflected in the counts of SNSPDs. We extract the phase difference $\Delta\varphi_{\text{pp}}$ from the interference and compensate for it in an electro-optical modulator (EOM) before the interference beamsplitter. This method has three critical requirements for the PP pulses: their frequency difference is much smaller than $1/\tau_{\textrm{pp}}$, their linewidths are much narrower than $1/\tau_{\textrm{pp}}$ and $1/\tau_{\Delta}$, and they inherit the same laser phase as the write-out photons. To meet these requirements, we lock all lasers in each node onto a reference cavity and get a linewidth of sub-kilohertz and a drift of a few kilohertz per day for all lasers. We perform a one-off frequency calibration of the corresponding lasers in different nodes. Additionally, we synchronize the Write and Read lasers using an optical phase-locked loop (OPLL). As a result, each PP pulse differs from the corresponding write-out photon with a phase $\theta$, leading to a residual phase $\Delta\theta$ (the difference of $\theta$ in two nodes) in Eq. 1 after compensation. One can either remove $\Delta\theta$ by synchronizing the OPLLs in two nodes or choose $\Delta\theta$ as the phase reference to observe the entanglement. We chose the latter method in the following experiments.

In practice, we conduct twice the phase stabilization in each trial for a better performance (Fig. 2A). To characterize the stabilization performance, we send an additional PP pulse from each quantum node 5 $\upmu$s after the feedback and find a Gaussian phase distribution with a standard deviation of 17^∘, as shown in Fig. 2B. By changing the relative phase between the additional pair of PP pulses, we observe a sinusoidal oscillation of their interference visibility (Fig. 2B), which confirms a good phase correlation between two nodes.

An additional challenge in operations arises from the slow frequency drift of the reference cavities, which can transfer to the lasers and eventually to write-out photons. To avoid this, we employ a continuous frequency calibration using the PP interference data. We determine the minor laser frequency offset $\Delta\omega$ by analyzing the phase evolution between successive entanglement trials over a duration of $t$ and compensate it to the Pump lasers. Considering the long coherence time of the lasers ($\approx 1$ ms) and the slow fluctuations in the fiber, the phase evolves solely due to $\Delta\omega t$. We verify this by observing the interference visibility as a function of $\Delta\omega$ shown in Fig. 2C. Through the implementation of the calibration, we are able to maintain $\Delta\omega\leq 0.1$ kHz over a span of 76 hours, despite an observed drift of 5 kHz (Fig. 2D).

We first generate remote entanglement between Alice and Bob. In addition to laser frequency and phase stabilization, we apply polarization filtering using a fiber polarizing beamsplitter (PBS) for the photons when they arrive the server node and continuously adjust an electrical polarization controller before the PBS to maximize the filtering efficiency (not shown in the setup schematic). To measure the distant entanglement between the two atomic ensembles, we retrieve each atomic state to a photonic mode via a read pulse generated from the Read laser, and obtain $\ket{\Psi_{\text{pp}}^{\pm}}=(\ket{01}\pm e^{i\Delta\varphi_{r}}\ket{10})/\sqrt{2}$. To evaluate the entanglement, one needs the knowledge of coherence $d$ between $\ket{01}$ and $\ket{10}$. As it is experimentally hard to perform qubit operations in the Fock basis, prior experiments mostly inferred $d$ by interfering two read-out modes using a beamsplitter [9, 32, 17], which is not favorable for spatially separate cases. To overcome this challenge, we develop a method based on the weak-field homodyne measurement schemes [37, 38] that allows us to witness remote Fock-basis entanglement through local operations and measurements. Additionally, our method effectively converts the Fock-basis entanglement into a polarization maximally entangled (PME) state [34], making it accessible for entanglement-based communication schemes.

Our method starts by generating an entanglement probe (EP) pulse from the Write laser in each node. It is a weak coherent light field that has the same frequency and profile as the read-out field, but orthogonally polarized. When choosing the phase reference of $e^{i\Delta\theta}$, one can write the joint state of two EP pulses as $\ket{\textrm{EP}}_{\textrm{A}}\ket{\textrm{EP}}_{\textrm{B}}=|\alpha|_{A}|\alpha|_{B}e^{i\Delta\varphi_{r}}$, which is in phase with $\ket{\Psi_{\text{pp}}^{\pm}}$. Next, we combine the read-out field with the EP pulse in every node using a PBS and perform projection measurements under different polarization bases (Fig. 3A). Initially, we measure the mixed field in both nodes in the $\ket{H}/\ket{V}$ basis. On the detector corresponding to horizontal polarization, the EP is removed, thus we find $p_{00}$, $p_{01}$, $p_{10}$, and $p_{11}$ through statistical analysis, where $p_{ij}$ represents the probability of having $i$ excitations in Alice and $j$ excitations in Bob. When measuring in the superposition basis, the read-out field interfere with the EP pulses, resulting in remote correlation. Specifically, when $\ket{H}\pm\ket{V}$ basis is chosen, we have the correlation function

where $N_{mn}$ represent the coincidence events between the $m$ mode in Alice and $n$ mode in Bob ($m,n$ takes $+$ or $-$, referring to $\ket{H}+\ket{V}$ and $\ket{H}-\ket{V}$, respectively). In this case, we find $|d|\geq(p_{01}+p_{10}+2\sqrt{p_{00}p_{11}})|E_{\pm}|/2$. Now we can estimate the degree of entanglement with concurrence $\mathcal{C}=\textrm{max}(0,2|d|-2\sqrt{p_{00}p_{11}})$. This measure ranges from 0 for a separable state to 1 for a maximally entangled state.

Although being a good measure of entanglement, $\mathcal{C}$ does not indicate how useful the entangled state is in entanglement-based communication. To answer this question, we consider the mixed field right after the combining PBS at each node, which can be approximated (with arbitrary phase factors and no normalization)

where $\mathds{1}$ is the identity operator and $a^{\dagger}$ is the photon creation operator with the subscript indicating its node (Alice or Bob) and polarization ($H$ or $V$). Selecting at least one photon at each node, Eq. 3 becomes

Neglecting the higher-order term $o(\alpha^{2})$, Eq. 4 is a PME state between two photonic modes. This allows local Pauli operations ($X$, $Y$, and $Z$) and enables entanglement based communication protocols. More details about the weak-field entanglement verification method are discussed in Ref. [39].

The weak-field method to verify remote entanglement relies on the EP pulse’s likeness to the read-out field. We examine this by performing a Hong-Ou-Mandel (HOM) experiment between these two fields. By scanning the detuning of the EP pulse, we observe a HOM dip in the correlation function $g^{(2)}$ shown in Fig. 3B, and get a minimum value of $0.716(19)$, close to the thermal-coherent interference limit of $2/3$ (the EP pulse is a coherent state and the read-out field without conditioning by the write-out field is a thermal state). This will lead to a 10% underestimation of $d$ in measurements [39].

Next, we verify the remote entanglement generated between Alice and Bob using the weak-field method. In the first experiment, we verify the entanglement in a delayed-choice fashion [40]. The atomic states are retrieved 5 $\upmu$s after the atom-photon entanglement is created, earlier than the entanglement is heralded. Fig. 3C and D summarize the reconstructed density matrices between the read-out fields and the corresponding $\mathcal{C}$. $\mathcal{C}>0$ for both $\ket{\Psi^{+}_{\textrm{pp}}}$ and $\ket{\Psi^{-}_{\textrm{pp}}}$ witness the existence of remote entanglement. By deducting the loss during retrieval and detection, we infer the density matrices between two atomic ensembles and their corresponding concurrence $\tilde{\mathcal{C}}$ to two atomic ensembles and show them in the same figures. Meanwhile, we find the fidelity $\mathcal{F}=0.672(32)$ for $\ket{\Psi_{\textrm{PME}}^{+}}$ and 0.666(31) for $\ket{\Psi_{\textrm{PME}}^{-}}$ to corresponding maximally entangled state, both above 0.5 to certify the entanglement (Fig. 3E and H). In the second experiment, we create the remote entanglement in the same fashion but store it for $107~{}\upmu$s, exceeding the round-trip transmission time between the quantum and the server nodes. We get a slightly lower concurrence result for each case, as summarized in Fig. 3F and G. However, the fidelities $\mathcal{F}$ of $\ket{\Psi_{\textrm{PME}}^{\pm}}$ are on par with that of the former experiment (Fig. 3E and H). Comparing the variation of $\mathcal{C}$ and $\mathcal{F}$ between two experiments, we confirm that a large portion of noise in the generated entangled states is the vacuum component and can be effectively filtered out using our weak-field method and other protocols [34]. The measured heralding probability is $6.9\times 10^{-4}$ and $8.0\times 10^{-4}$ for the first and second experiments, respectively, which corresponds to an entangling rate of 1.93 Hz and 0.83 Hz, considering the experiment repetition rate of 2.76 kHz and 1.03 kHz.

Now we complete the network architecture by including the third quantum node, Charlie, and concurrently generate entanglement $\ket{\Psi^{\pm}}_{\textrm{AB}}$, $\ket{\Psi^{\pm}}_{\text{BC}}$ and $\ket{\Psi^{\pm}}_{\text{AC}}$ on three links, Alice-Bob, Bob-Charlie and Alice-Charlie, respectively. To this end, the server node is equipped with a multi-input-two-output optical switch and assigns two of the three users to the detection upon request. In our demonstration, we assign a 5.8 s time slot for each pair of users and dynamically switch user pairs according to a quantum random number generator (QRNG), simulating unknown requests in real-world applications. Since different links are frequently switched, a ‘hot swapping’ feature is crucial, i.e. the entangling conditions are meet immediately after switching. In our architecture, the phase stabilization is executed for every entanglement trial and is unaffected by switching. The good laser stability guarantees ignorable relative frequency drift although a 1/3 effective calibration time is assigned for each pair of nodes. Furthermore, we store the optimal EPC parameters for the polarization filtering to each specific fiber and load them during switching.

We first observe the local manipulation of non-local states. For instance, when applying a unitary operation locally at one of the quantum nodes (e.g., Charlie), its relevant non-local states ($\ket{\Psi^{\pm}}_{\text{AC}}$ and $\ket{\Psi^{\pm}}_{\text{BC}}$) change in form while the entanglement is preserved. The irrelevant non-local states $\ket{\Psi^{\pm}}_{\textrm{AB}}$ remain unchanged. In practice, we vary the phase $\theta$ at Charlie and measure the $\braket{XX}$ correlation of each pair of entanglement. As shown in Fig. 4A and B, we observe sinusoidal oscillations in the expected pairs $\ket{\Psi^{\pm}}_{\text{BC}}$ and $\ket{\Psi^{\pm}}_{\text{AC}}$, while $\ket{\Psi^{\pm}}_{\text{AB}}$ contributes a constant in this measurement. A $\pi/2$ delay between the oscillations of $\ket{\Psi^{\pm}}_{\text{BC}}$ and $\ket{\Psi^{\pm}}_{\text{AC}}$ arises from the phase stabilization mechanism [39].

In the end, we characterize the generated remote entangled states on three links in the delayed-choice fashion. Tab. 1 summarizes the concurrence results $\mathcal{C}$ and $\mathcal{\tilde{C}}$ and Fig. 4C, D show the correlation measurement results of the PME states. For all three links, we find $\mathcal{C}>0$ to confirm the presence of atomic entanglement and $\mathcal{F}>0.5$ to witness the entanglement in corresponding PME state. The entangled states between Alice and Bob are comparable to that generated in the fixed connection case, indicating no extra decoherence being introduced in the switching process. The lower entanglement quality in other links arises from experimental imperfections [39]. The entangling rates of all three links are roughly 1/3 of that measured in the fixed connection case as expected. Some simple upgrades to the server node will immediately improve the networking performance, such as parallel (rather than concurrent) creation of multiple entanglement pairs with the help of wavelength division multiplexing technique [41] and genuine three-body entanglement creation using a modified interferometer [42, 21, 43].

In this work, we present the debut implementation of a metropolitan area entanglement-based quantum network. With this facility, we have successfully demonstrated the heralded generation of remote entanglement between two quantum nodes separated by 12.5 km and store it longer than the two-way communication time. We also showed the concurrent generation of multiple entanglement pairs within the network. The high entangling rate in our network is enabled by a series of methods, two of which are crucial. Firstly, the remote phase stabilization guarantees the basis to implement the single photon scheme. In addition, it provides a flexible and scalable network infrastructure that can accommodate more quantum nodes within the metropolitan area. Secondly, the weak-field method simplifies the verification of remote Fock-basis entanglement. It removes the necessity to interfere two remote Fock-basis photons, instead only requires local linear optics operations and measurements. Moreover, using this method, one can effectively convert Fock-basis entanglement to polarization maximally entanglement, making entanglement-based communication within the reach. Adapting these methods to other physical platforms will accelerate their networking process.

Further advances of the network performance will happen in four main aspects, increasing the entangling rate by shifting the wavelength to telecom C band [27, 31] and by adapting multiplexed atomic quantum memories [44, 45, 46]; decreasing the entanglement infidelity by suppressing unwanted higher order atomic excitations with the help of Rydberg blockade mechanism [38, 47]; realizing a better entanglement verification by filtering out the vacuum components in the remote entangled state [34, 10] or by using nonlinear atomic state detection techniques [48, 49]; and prolonging the entanglement storage time to the second scale by further limiting atomic thermal motion with optical lattice technique [50, 51]. With these improvements, we expect remote entanglement can be generated much faster than it is lost, laying the basis for mediating serval metropolitan area quantum networks via quantum repeater protocols [7, 8].

We acknowledge QuantumCTek for the allocation of node Bob, and Hefei Institutes of Physical Science for the allocation of node Charlie. Funding: This research was supported by the Innovation Program for Quantum Science and Technology (No. 2021ZD0301104), National Key R&D Program of China (No. 2020YFA0309804), Anhui Initiative in Quantum Information Technologies, National Natural Science Foundation of China, and the Chinese Academy of Sciences. Competing interests: The authors declare no competing interests. Data and materials: The data are archived at Zenodo [52].

Here we give a complete description of the phase involved in the remote entanglement generation and verification. Fig. S3 depicts a detailed schematic of a quantum node. The notations used in the following derivation and their corresponding description are listed in Tab. S1.

We start from the local atom-photon entanglement pair generation. The interference and the detection of the write-out photons herald the entanglement of two atomic ensembles. Then the atomic excitations are retried as photons for entanglement verification. The retrieval and write-out photon detection processes are communitive, so we consider a delayed-choice experiment where the atomic state is retrieved immediately after its creation. In this case, the local entanglement between the write-out and the read-out photon mode reads

where

$\varphi_{f}=\phi_{w}+\phi_{wo}$ correspond to the phase of the fiber channel in the main text. With a click event of an SNSPD, the remote entangled state reads

Apart from the phase of photon pairs $\phi_{wr}^{\textrm{A/B}}$, the phase $\phi_{\textrm{EOM}}$ is involved because the phase feedback is applied before the remote entanglement generation.

During the entanglement verification, we combine an EP pulse with the read-out photons at each node. We generate the EP pulse from the Write laser (because of its frequency likeliness with the read-out photon). Hence the EP pulse inherits the initial phase of the Write laser. In the limit of weak power, we can write the joint state of two EP pulses in two nodes as follows,

We encode the read-out photon modes in Eq. S3 in the horizontal polarization ($\ket{H}$), the EP pulse mode in Eq. S4 in the vertical polarization mode ($\ket{V}$) and combine them using PBSs (shown in the Fig. 3A of the main text). By selecting one photon at each node, we obtain a polarization maximally entangled (PME) state

We define $\phi_{\textrm{PME}}$ as the phase of the PME state,

where we divide the total phase into the local phase at nodes A and B ($\phi^{\textrm{A/B}}_{lo}$) and the remote phase between two nodes ($\phi_{rm}$). They are given by:

In the expression of $\phi^{\textrm{A/B}}_{lo}$ (Eq. S7), the first term $\phi_{a}^{\textrm{A/B}}$ stands for the phase during the storage in the atomic ensembles. The first bracket represents the phase difference of some optical paths, constituting an interferometer highlighted in Fig. S3. The second bracket indicates the phase inherited from the lasers at different times.

To keep $\phi_{a}^{\textrm{A/B}}$ stable in the 100 $\upmu$s time scale, we reduce the Zeeman energy level fluctuation of atoms by securing a constant magnetic environment (see Ref. [1]) and transferring the spin wave to clock transition to reduce the sensitivity to the magnetic field. The stable laser phase is guaranteed by locking lasers to a stable and narrow linewidth reference cavity. To stabilize the local interferometer, we inject a locking beam during the MOT loading phase and apply the feedback onto a mirror glued with a piezo-electric transducer. The acousto-optic modulators (AOMs) on the optical paths function as frequency shifters. One shifts the PP pulse to the write-out photon’s frequency during the phase probing phase, and the other shifts the EP pulse to the read-out photon’s frequency during the entanglement verification phase.

We lock the remote phase $\phi_{rm}$ with the weak-field remote phase stabilization methods. For practical reasons, we add an addition $\pi/2$ in the feedback (i.e. $\phi_{rm}=\pi/2$), which implies,

By adding three equations in Eq. S9, we get:

which explains the $\pi/2$ phase delay between $\ket{\Psi^{\pm}}_{\textrm{BC}}$ and $\ket{\Psi^{\pm}}_{\textrm{AC}}$ in Fig. 4A and 4B in the main text.

Using the EP pulses, we can transfer the Fock state entanglement to PME states, which makes entanglement verification much simpler. Thus, we can use entanglement fidelity to quantify the quality of entanglement. Here, we give the form of entanglement fidelity and show the limitation under our experimental imperfections.