Teleportation Systems Towards a Quantum Internet

Quantum teleportation is essential for many quantum information technologies including long-distance quantum networks. Using fiber-coupled devices, including state-of-the-art low-noise superconducting nanowire single photon detectors and off-the-shelf optics, we achieve quantum teleportation of time-bin qubits at the telecommunication wavelength of 1536.5 nm. We measure teleportation fidelities of>=90% that are consistent with an analytical model of our system, which includes realistic imperfections. To demonstrate the compatibility of our setup with deployed quantum networks, we teleport qubits over 22 km of single-mode fiber while transmitting qubits over an additional 22 km of fiber. Our systems, which are compatible with emerging solid-state quantum devices, provide a realistic foundation for a high-fidelity quantum internet with practical devices.

Quantum teleportation is essential for many quantum information technologies including longdistance quantum networks. Using fiber-coupled devices, including state-of-the-art low-noise superconducting nanowire single photon detectors and off-the-shelf optics, we achieve quantum teleportation of time-bin qubits at the telecommunication wavelength of 1536.5 nm. We measure teleportation fidelities of ≥ 90% that are consistent with an analytical model of our system, which includes realistic imperfections. To demonstrate the compatibility of our setup with deployed quantum networks, we teleport qubits over 22 km of single-mode fiber while transmitting qubits over an additional 22 km of fiber. Our systems, which are compatible with emerging solid-state quantum devices, provide a realistic foundation for a high-fidelity quantum internet with practical devices.

I. INTRODUCTION
Quantum teleportation [1], one of the most captivating predictions of quantum theory, has been widely investigated since its seminal demonstrations over 20 years ago [2][3][4]. This is due to its connections to fundamental physics [5][6][7][8][9][10][11][12][13][14], and its central role in the realization of quantum information technology such as quantum computers and networks [15][16][17][18][19]. The goal of a quantum network is to distribute qubits between different locations, a key task for quantum cryptography, distributed quantum computing and sensing. A quantum network is expected to form part of a future quantum internet [20][21][22]: a globally distributed set of quantum processors, sensors, or users there-of that are mutually connected over a network capable of allocating quantum resources (e.g. qubits and entangled states) between locations. Many architectures for quantum networks require quantum teleportation, such as star-type networks that distribute entanglement from a central location or quantum repeaters that overcome the rate-loss trade-off of direct transmission of qubits [19,[23][24][25][26].
Quantum teleportation of a qubit can be achieved by performing a Bell-state measurement (BSM) between the qubit and another that forms one member of an entangled Bell state [1,18,27]. The quality of the teleportation is often characterized by the fidelity F = ψ| ρ |ψ of the teleported state ρ with respect to the state |ψ accomplished by ideal generation and teleportation [15]. This metric is becoming increasingly important as quantum networks move beyond specific applications, such as quantum key distribution, and towards the quantum internet.
Qubits encoded by the time-of-arrival of individual photons, i.e. time-bin qubits [28], are useful for networks due to their simplicity of generation, interfacing with quantum devices, as well as independence of dynamic transformations of real-world fibers. Individual telecom-band photons (around 1.5 µm wavelength) are ideal carriers of qubits in networks due to their ability to rapidly travel over long distances in deployed optical fibers [17,[29][30][31] or atmospheric channels [32], among other properties. Moreover, the improvement and growing availability of sources and detectors of individual telecom-band photons has accelerated progress towards workable quantum networks and associated technologies, such as quantum memories [33], transducers [34,35], or quantum non-destructive measurement devices [36].
Teleportation of telecom-band photonic time-bin qubits has been performed inside and outside the laboratory with impressive results [29][30][31][37][38][39][40][41][42]. Despite this, there has been little work to increase F beyond ∼ 90% for these qubits, in particular using practical devices that allow straightforward replication and deployment of quantum networks (e.g. using fiber-coupled and commercially available devices). Moreover, it is desirable to develop teleportation systems that are forward-compatible with emerging quantum devices for the quantum internet.
In the context of Caltech's multi-disciplinary multiinstitutional collaborative public-private research program on Intelligent Quantum Networks and Technologies (IN-Q-NET) founded with AT&T as well as Fermi Na-tional Accelerator Laboratory and Jet Propulsion Laboratory in 2017, we designed, built, commissioned and deployed two quantum teleportation systems: one at Fermilab, the Fermilab Quantum Network (FQNET), and one at Caltech's Lauritsen Laboratory for High Energy Physics, the Caltech Quantum Network (CQNET). The CQNET system serves as an R&D, prototyping, and commissioning system, while FQNET serves as an expandable system, for scaling up to long distances and is used in multiple projects funded currently by DOE's Office of High Energy Physics (HEP) and Advanced Scientific Research Computing (ASCR). Material and devices level R&D in both systems is facilitated and funded by the Office of Basic Energy Sciences (BES). Both systems are accessible to quantum researchers for R&D purposes as well as testing and integration of various novel devices, such as for example on-chip integrated nanophotonic devices and quantum memories, needed to upgrade such systems towards a realistic quantum internet.Importantly both systems are also used for improvements of the entanglement quality and distribution with emphasis on implementation of protocols with complex entangled states towards advanced and complex quantum communications channels. These will assist in studies of systems that implement new teleportation protocols whose gravitational duals correspond to wormholes [43], error correlation properties of wormhole teleportation, on-chip codes as well as possible implementation of protocols on quantum optics communication platforms. Hence the systems serve both fundamental quantum information science as well as quantum technologies.
Here we perform quantum teleportation of time-bin qubits at a wavelength of 1536.5 nm with an average F ≥ 90%. This is accomplished using a compact setup of fiber-coupled devices, including low-dark-count single photon detectors and off-the-shelf optics, allowing straight-forward reproduction for multi-node networks. To illustrate network compatibility, teleportation is performed with up to 44 km of single-mode fiber between the qubit generation and the measurement of the teleported qubit, and is facilitated using semi-autonomous control, monitoring, and synchronization systems, with results collected using scalable acquisition hardware. Our systems, which operates at a clock rate of 90 MHz, can be run remotely for several days without interruption and yield teleportation rates of a few Hz using the full length of fiber. Our qubits are also compatible with erbiumdoped crystals, e.g. Er:Y 2 SiO 5 , that are used to develop quantum network devices like memories and transducers [44][45][46]. Finally, we develop an analytical model of our system, which includes experimental imperfections, predicting that the fidelity can be improved further towards unity by well-understood methods (such as improvement in photon indistinguishability). Our demonstrations provide a step towards a workable quantum network with practical and replicable nodes, such as the ambitious U.S. Department of Energy quantum research network envisioned to link the U.S. National Laborato-ries.
In the following we describe the components of our systems as well as characterization measurements that support our teleportation results, including the fidelity of our entangled Bell state and Hong-Ou-Mandel (HOM) interference [47] that underpins the success of the BSM. We then present our teleportation results using both quantum state tomography (QST) [48] and projection measurements based on a decoy state method [49], followed by a discussion of our model. We conclude by considering improvements towards near-unit fidelity and GHz level teleportation rates.

II. SETUP
Our fiber-based experimental system is summarized in the diagram of Fig. 1. It allow us to demonstrate a quantum teleportation protocol in which a photonic qubit (provided by Alice) is interfered with one member of an entangled photon-pair (from Bob) and projected (by Charlie) onto a Bell-state whereby the state of Alice's qubit can be transferred to the remaining member of Bob's entangled photon pair. Up to 22 (11) km of single mode fiber is introduced between Alice and Charlie (Bob and Charlie), as well as up to another 11 km at Bob, depending on the experiment (see Sec. III). All qubits are generated at the clock rate, with all of their measurements collected using a data acquisition (DAQ) system. Each of the Alice, Bob, Charlie subsystems are further detailed in the following subsections, with the DAQ subsystem described in Appendix A 1.

A. Alice: single-qubit generation
To generate the time-bin qubit that Alice will teleport to Bob, light from a fiber-coupled 1536.5 nm continuous wave (CW) laser is input into a lithium niobate intensity modulator (IM). We drive the IM with one pulse, or two pulses separated by 2 ns. Each pulse is of ∼65 ps full width at half maximum (FWHM) duration. The pulses are produced by an arbitrary waveform generator (AWG) and amplified by a 27 dB-gain high-bandwidth amplifier to generate optical pulses that have an extinction ratio of up to 22 dB. We note that this method of creating time-bin qubits offers us flexibility not only in terms of choosing a suitable time-bin separation, but also for synchronizing qubits originating from different nodes in a network. A 90/10 polarization-maintaining fiber beam splitter combined with a power monitor (PWM) is used to apply feedback to the DC-bias port of the IM so as to maintain a constant 22 dB extinction ratio [50]. In order to successfully execute the quantum teleportation protocol, photons from Alice and Bob must be indistinguishable in all degrees of freedom (see Sec. III B). Hence, the optical pulses at the output of the IM are band-pass filtered using a 2 GHz-bandwidth (FWHM) fiber Bragg  grating (FBG) centered at 1536.5 nm to match the spectrum of the photons from the entangled pair-source (described in Sec. II B). Furthermore, the polarization of Alice's photons is determined by a manual polarization controller (POC) in conjunction with a polarizing beam splitter (PBS) at Charlie. Finally, the optical pulses from Alice are attenuated to the single photon level by a variable optical attenuator (VOA), to approximate photonic time-bin qubits of the form |A = γ |e A + 1 − γ 2 |l A , where the late state |l A arrives 2 ns after the early state |e A , γ is real and set to be either 1, 0, or 1/ √ 2 to generate |e A , |l A , or |+ A = (|e A + |l A )/ √ 2, respectively, depending on the experiment. The complex relative phase is absorbed into the definition of |l A . The duration of each time bin is 800 ps. B. Bob: entangled qubit generation and teleported-qubit measurement Similar to Alice, one (two) optical pulse(s) with a FWHM of ∼ 65 ps is (and separated by 2 ns are) created using a 1536.5 nm CW laser in conjunction with a lithium niobate IM driven by an AWG, while the 90/10 beam splitter and PWM are used to maintain an extinction ratio of at least 20 dB. An Erbium-Doped Fiber Amplifier (EDFA) is used after the IM to boost the pulse power and thus maintain a high output rate of photon pairs.
The output of the EDFA is sent to a Type-0 periodically poled lithium niobate (PPLN) waveguide for second harmonic generation (SHG), upconverting the pulses to 768.25 nm. The residual light at 1536.5 nm is removed by a 768 nm band-pass filter with an extinction ratio ≥ 80 dB. These pulses undergo spontaneous para-metric down-conversion (SPDC) using a Type-II PPLN waveguide coupled to a polarization-maintaining fiber (PMF), approximately producing either a photon pair |pair B = |ee B , or the time-bin entangled state |φ + B = (|ee B + |ll B )/ √ 2, if one or two pulses, respectively, are used to drive the IM.
The ordering of the states refers to so-called signal and idler modes of the pair of which the former has parallel, and the latter orthogonal, polarization with respect to the axis of the PMF. As before, the relative phase is absorbed into the definition of |ll B . Each photon is separated into different fibers using a PBS and spectrally filtered with FBGs akin to that at Alice. Note the bandwidth of the FBG is chosen as a trade-off between spectral purity and generation rate of Bob's photons [51].
The photon in the idler mode is sent to Charlie for teleportation or HOM measurements (see Sec. III B), or to the MZI (see below) for characterizations of the entangled state (see Sec. III A), with its polarization determined using a POC.The photon in the signal mode is sent to a Mach Zehnder interferometer (MZI) by way of a POC (and an additional 11 km of single-mode fiber for some measurements), and is detected by superconducting nanowire single photon detectors (SNSPDs) [52] after high-pass filtering (HPF) to reject any remaining 768.25 nm light. The MZI and detectors are used for projection measurements of the teleported state, characterization of the time-bin entangled state, or measuring HOM interference at Charlie. The time-of-arrival of the photons is recorded by the DAQ subsystem using a timeto-digital converter (TDC) referenced to the clock signal from the AWG.
All SNSPDs are installed in a compact sorption fridge cryostat [53], which operates at a temperature of 0.8 K for typically 24 h before a required 2 h downtime. Our SNSPDs are developed at the Jet Propulsion Laboratory and have detection efficiencies between 76 and 85%, with low dark count rates of 2-3 Hz. The FWHM temporal resolution of all detectors is between 60 and 90 ps while their recovery time is ∼50 ns. A detailed description of the SNSPDs and associated setup is provided in Appendix A 2.
The MZI has a path length difference of 2 ns and is used to perform projection measurements of |e B , |l B , and (|e B + e iϕ |l B )/ √ 2, by detecting photons at three distinct arrival times in one of the outputs, and varying the relative phase ϕ [28]. Detection at the other output yields the same measurements except with a relative phase of ϕ + π. Using a custom temperature-feedback system, we slowly vary ϕ for up to 15 hour time intervals to collect all measurements, which is within the cryostat hold time. Further details of the MZI setup is described in Appendix A 3.

C. Charlie: Bell-state measurement
Charlie consists of a 50/50 polarization-maintaining fiber beam splitter (BS), with relevant photons from the Alice and Bob subsystems directed to each of its inputs via a PBSs and optical fiber. The photons are detected at each output with an SNSPD after HPFs, with their arrival times recorded using the DAQ as was done at Bob. Teleportation is facilitated by measurement of the |Ψ − AB = (|el AB − |le AB )/ √ 2 Bell state, which corresponds to the detection of a photon in |e at one detector followed by the detection of a photon in |l at the other detector after Alice and Bob's (indistinguishable) qubits arrive at the BS [54]. Projection on the |Ψ − AB state corresponds to teleportation of |A up to a known local unitary transformation, i.e. our system produces −iσ y |A , with σ y being the Pauli y-matrix.

III. EXPERIMENTAL RESULTS
Prior to performing quantum teleportation, we measure some key parameters of our system that underpin the teleportation fidelity. Specifically, we determine the fidelity of the entangled state produced by Bob by measuring the entanglement visibility V ent [55], and also determine to what extent Alice and Bob's photons are indistinguishable at Charlie's BS using the HOM effect [47].

A. Entanglement visibility
The state |pair B (and hence the entangled state |φ + B ) described in Sec. II B is idealized. In reality, the state produced by Bob is better approximated by a two-mode squeezed vacuum state √ p n |nn B after the FBG filter and neglecting loss [56]. Here, n is the number of photons per temporal mode (or qubit), p is the emission probability of a single pair per mode (or qubit), with state ordering referring to signal and idler modes. However, |TMSV B approximates a photon pair for p << 1, with p ≈ µ B mean number of pairs per mode (or qubit), conditioned on measurement of a pair such that the n = 0 term is eliminated. As a compromise between the pair-creation rate ∝ p and the quality of entanglement, here and henceforth we set the mean photon number of our pair source to be µ B = (8.0 ± 0.4) × 10 −3 per time bin, which is feasible because of the exceptionally low dark counts of our SNSPDs. Measurement of µ B is outlined in Appendix B. We generate |φ + B and measure V ent by directing the idler photon to the second input port of the MZI, slightly modifying the setup of Fig. 1. The idler photon is delayed compared to the signal, allowing unambiguous measurement of each qubit. We vary ϕ and project each qubit of the entangled state onto phase-varied superpositions of |e and |l by accumulating coincidence events of photons at both the outputs of the interferometer [55].
The results shown in Fig. 2 are fit proportional to with R x(n) denoting the maximum (minimum) rate of coincidence events [55], ω and Φ are unconstrained constants, and T is the temperature of the MZI, finding V ent = 96.4 ± 0.3%.
The deviation from unit visibility is mainly due to nonzero multi photon emissions [57], which is supported by an analytical model that includes experimental imperfections [58]. Nonetheless, this visibility is far beyond the 1/3 required for non-separability of a Werner state [59] and the locality bound of 1/ √ 2 [55,60]. Furthermore, it predicts a fidelity F ent = (3V ent + 1)/4 = 97.3 ± .2% with respect to |φ + [59], and hence is sufficient for quantum teleportation.

B. HOM interference visibility
The BSM relies on quantum interference of photons from Alice and Bob. This is ensured by the BS at Charlie, precise control of the arrival time of photons with IMs, identical FBG filters, and POCs (with PBSs) to provide the required indistinguishabiliy. The degree of interference is quantified by way of the HOM interference visibil- denoting the rate of coincident detections of photons after the BS when the photons are rendered as distinguishable (indistinguishable) as possible [47]. Completely indistinguishable single photons from Alice and Bob may yield V HOM = 1. However in our system, Alice's qubit is approximated from a coherent state |α A = e −|α| 2 /2 ∞ n=0 α n √ n! |n A with α << 1, akin to how Bob's pair is approximated from |TMSV B (see Sec. III A), with µ A = |α| 2 being Alice's mean photon number per mode (or qubit) [56]. Therefore, the contribution of undesired photons from Alice and Bob lowers the maximum achievable V HOM below unity, with a further reduction if the interfering photons are not completely indistinguishable. The dependence of V HOM with varied µ A and µ B , including effects of losses or distinguishable photons in our system is analytically modelled in Ref. [58], and briefly discussed in Sec. IV.
We measure V HOM by slightly modifying the setup of Fig. 1: We prepare |A = |e A with µ A = 2.6 × 10 −3 and Bob as |pair B and direct Alice's photon and Bob's idler to Charlie, with Bob's signal bypassing the MZI to be directly measured by an SNSPD. Alice's IM is used to introduce distinguishability by way of a relative difference in arrival time ∆t AB of Alice and Bob's photons at Charlie's BS. Using Charlie's SNSPDs and the third detector at Bob, a three-fold coincidence detection rate is measured for varying ∆t AB , with results shown in Fig. 3a. Since the temporal profiles of our photons are approximately Gaussian, we fit our results to where A is the maximum coincidence rate when the photons are completely distinguishable and σ = 300 ps is the 1/e temporal duration of the optical pulses [47,61], finding V HOM = 70.9 ± 1.9%. The maximum V HOM for this experiment is 83.5% if the photons were completely indistinguishable [58], with the difference ascribed to slight distinguishability between our photons as supported by the further measurements and analytical modelling in Sec. IV. Improvements to our system to remove this distinguishability is discussed in Sec. V. To test our system for quantum teleportation over long distances, we introduce the aforementioned 22, 11, and 11 km lengths of single-mode fiber between Alice and Charlie, Bob and Charlie, and in the path of Bob's signal photon, respectively, repeat our measurement of V HOM and fit the results as before (see Fig. 3b). We find V HOM = 63.4 ± 5.9%, which is consistent with the maximum V HOM we expect when including the impact of the additional 5.92 (2.56) dB loss between Charlie and Alice (Bob) as well as the effect of photon distinguishability (analyzed in Sec. IV). This suggests that the additional fiber importantly does not introduce any further distinguishability (that we cannot account for), thereby supporting our system's use in quantum networking. Overall, the presence of clear HOM interference suggests our system (with or without the additional fiber) introduces relatively little imperfections that can negatively impact the BSM and hence the fidelity of quantum teleportation.

C. Quantum teleportation
We now perform quantum teleportation of the timebin qubit basis states |e , |l and |+ , so as to measure the teleportation fidelities, F e , F l , an F + , respectively, of the teleported states with respect to their ideal counterparts, up to the local unitary introduced by the BSM (see Sec. II C). Since measurement of |+ in our setup by symmetry is equivalent to any state of the form (|e + e iϕ |l )/ √ 2 (and in particular the remaining three basis states (|e − |l )/ √ 2 and (|e ± i |l )/ √ 2), we may determine the average teleportation fidelity F avg = (F e + F l + 4F + )/6 of any time-bin qubit.
First, we prepare |e A and |l A with µ A = 3.53 × 10 −2 , with Bob's idler bypassing the MZI to be detected by a single SNSPD. We measure F e = 95 ± 1% and F l = 96 ± 1%, conditioned on a successful measurement of |Ψ − AB at Charlie, with fidelity limited by multiphoton events in Alice and Bob's qubits and dark counts of the SNSPDs [58]. We then repeat the measurement with µ A = 9.5 × 10 −3 after inserting the aforementioned 44 km length of fiber as before to emulate Alice, Charlie and parts of Bob being separated by long distances. This gives F e = 98 ± 1% and F l = 98 ± 2%, with no reduc-tion from the additional fiber loss owing to our low noise SNSPDs.
Next, we prepare |+ A with µ A = 9.38 × 10 −3 , insert the MZI and, conditioned on the BSM, we measure F + = (1 + V + )/2 = 84.9 ± 0.5% by varying ϕ. Here, V + = 69.7 ± 0.9% is the average visibility obtained by fits to the resultant interference measured at each output of the MZI, as shown in Fig. 4a. The reduction in fidelity from unity is due to multiphoton events and distinguishability, consistent with that inferred from HOM interference, as supported by further measurements and analytical modelling in Sec. IV.
The measurement is repeated with the additional long fiber, giving V + = 58.6±5.7% and F + = 79.3±2.9% with results and corresponding fit shown in Fig. 4b. The reduced fidelity is likely due to aforementioned polarization variations over the long fibers, consistent with the reduction in HOM interference visibility, and exacerbated here owing to the less than ideal visibility of the MZI over long measurement times (see Sec. A 3).
To glean more information about our teleportation system beyond the fidelity, we reconstruct the density matrices of the teleported states using a maximum-likelihood QST [48] described in Appendix C. The results of the QST with and without the additional fiber lengths are summarized in Figs. 8 and 9, respectively. As can be seen, the diagonal elements for |+ are very close to the expected value indicating the preservation of probabilities for the basis states of |e and |l after teleportation, while the deviation of the off-diagonal elements indicate the deterioration of coherence between the basis states. The decoherence is attributed to multiphoton emissions from our entangled pair source and distinguishability, consistent with the aforementioned teleportation fidelities of |+ A , and further discussed in Sec. IV. Finally, we do also extract the teleportation fidelity from these density matrices, finding the results shown in Fig. 5, and F avg = 89 ± 1% (88 ± 3%) without (with) the fiber spools, which are consistent with previous measurements given the similar µ A used for QST.
We point out that the 2/3 classical bound may only be applied if Alice prepares her qubits using genuine single photons, i.e. |n = 1 , rather than using |α << 1 as we do in this work [63]. As a way to account for the photon statistics of Alice's qubits we turn to an analysis using decoy states.

Teleportation fidelity using decoy states
To determine the minimum teleportation fidelity of qubits prepared using single photons, we use a decoy state method [49] and follow the approach of Refs. [29,64]. Decoy states, which are traditionally used in quantum key distribution to defend against photonnumber splitting attacks, are qubits encoded into coherent states |α with varying mean photon number µ A = |α| 2 . Measuring fidelities of the teleported qubits for different µ A , the decoy-state method allows us to calculate a lower bound F d A on the teleportation fidelity if Alice had encoded her qubits using |n = 1 .
We prepare decoy states |e A , |l A , and |+ A with varying µ A , as listed in Table I, and perform quantum teleportation both with and without the added fiber, with teleportation fidelities shown in Table I. From these results we calculate F d A as shown in Fig. 5, with F d avg ≥ 93 ± 4% (F d avg ≥ 89 ± 2%) without (with) the added fiber, which significantly violate the classical bound and the bound of 5/6 given by an optimal symmetric universal cloner [65,66] tem for high-fidelity teleportation. As depicted in Fig.  5 these fidelities nearly match the results we obtained without decoy states within statistical uncertainty. This is due to the suitable µ A , as well as low µ B and SNSPD dark counts in our previous measurements [58].

IV. ANALYTICAL MODEL AND SIMULATION
As our measurements have suggested, multi-photon components in, and distinguishability between, Alice and Bob's qubits reduce the values of key metrics including HOM interference visibility and, consequently, quantum teleportation fidelity. To capture these effects in our model, we employ a Gaussian-state characteristicfunction method developed in Ref. [58], which was en-qubit without long fiber with long fiber abled by work in Ref. [67]. This approach is well-suited to analyze our system because the quantum states, operations, and imperfections (including losses, dark counts, etc.) of the experiment can be fully described using Gaussian operators, see e.g. Ref. [68]. We now briefly outline the model of Ref.
[58], and employ it to estimate the amount of indistinguishability ζ between Alice and Bob's qubits in our measurements of HOM interference and quantum teleportation.
Distinguishability in any degree-of-freedom may be modelled by introducing a virtual beam splitter of transmittance ζ into the paths of Alice and Bob's relevant photons. As shown in Fig. 6, indistinguishable components of incoming photon modes are directed towards Charlie's BS where they interfere, whereas distinguishable components are mixed with vacuum at the BS and do not contribute to interference. Here ζ = 1 (ζ = 0) corresponds to the case when both incoming photons are perfectly indistinguishable (distinguishable). Now we may calculate the probability of a three-fold coincidence detection event P 3f between D 1 , D 2 (Charlies' detectors), and D 3 (detects Bob's signal photon) for a given qubit state ρ AB from Alice and Bob: (1) where theâ andb operators refer to modes, which originate from Alice and Bob's virtual beam splitters and are directed to D 1 and D 2 , respectively, andĉ corresponds to Bob's idler mode, which is directed to D 3 , see Fig. 6. This allows the derivation of an expression for the HOM interference visibility consistent with that introduced in Sec. III B. Since Alice and Bob ideally produce ρ AB = (|α α|) ⊗ (|TMSV TMSV|), and recognizing that all operators in P 3f are Gaussian, we analytically derive for varied ζ, where η i and η s are the transmission efficiencies of the signal and idler photons, including detector efficiencies. We similarly calculate the impact of distinguishability on the teleportation fidelity of |+ : where ϕ max (ϕ min ) is the phase of the MZI added into the path of the signal photon, corresponding to maximum (minimum) three-fold detection rates. To compare the model to our measurements, we use the experimental mean photon numbers for the photon-pair source η i = 1.2 × 10 −2 and η s = 4.5 × 10 −3 as determined by the method described in Appendix B. We then measure the teleportation fidelity of |+ and HOM interference visibility (keeping the MZI in the system to ensure η s remains unchanged) for different values µ A . The results are plotted in Fig. 7. The data is then fitted to the expressions V HOM (ζ) and F (ζ) derived in our model and graphed in Fig. 7. The fitted curves are in very good agreement with our experimental values and consistently yield a value of ζ = 90% for both measurements types. This implies that we have only a small amount of residual distinguishability between Alice and Bob's photons. Potential effects leading to this distinguishability are discussed in Sec. V.
Overall, our analytic model is consistent with our experimental data [58] in the regime of µ A << 1, which is the parameter space most often used in quantum networking protocols (e.g. key distribution). Our model, thus, offers a practical way to determine any underlying distinguishability in a deployed network where a full characterization of the properties of Alice and Bob's photons may not be possible.

V. DISCUSSION AND OUTLOOK
We have demonstrated quantum teleportation systems for photonic time-bin qubits at a quantum channel-and device-compatible wavelength of 1536.5 nm using a fiberbased setup comprising state-of-the-art SNSPDs and offthe-shelf components. We measure an average fidelity above 90% using QST and a decoy state analysis with up to 44 km of single mode fiber in our setup. Our results are further supported by an analytical model in conjunction with measurements of the entanglement and HOM interference visibilities.
The decoy state analysis indicates that the maximum teleportation fidelity is currently restricted by that of the teleported qubits prepared in superposition states, for which a 10% distinguishability between the qubits undergoing BSM and the contribution of multiple photon pairs play the largest role. Our model predicts that the average fidelity will increase to 95% with completely indistinguishable photons, while fidelities close to unity can be achieved with lowered mean number of photon pairs. Alternatively, we may replace our SNSPDs with efficient photon-number resolving (PNR) SNSPDs [69] to allow postselection of multiphoton events at the MZI or BSM [70]. Both approaches must be accompanied by increased coupling efficiencies of the pair source beyond the current ∼ 1% either to maintain realistic teleportation rates (above the system noise and current rate of phase drift of the MZI), or to derive any advantage from PNR capability.
As suggested by the width of our HOM interference fringe -which predicts an average photon bandwidth of 0.44/σ ∼1.5 GHz (see Sec. III B), i.e. less than the 2 GHz bandwidth of our FBGs -the indistinguishability in our system could be limited by the large difference in the bandwidth between the photons originating from the SPDC (>100 GHz) and those generated at Alice by the IM (15 GHz), leading to nonidentical filtering by the FBG. This can be improved by narrower FBGs or by using a more broadband pump at Alice (e.g. using a mode locked laser or a higher bandwidth IM, e.g > 50 GHz, which is commercially available). Alternatively, pure photon pairs may be generated by engineered phase matching, see e.g. Ref. [71]. Distinguishability owing to nonlinear modulation during the SHG process could also play a role [72]. The origin of distinguishability in our system, whether due to imperfect filtering or other device imperfections (e.g. PBS or BS) will be studied in future work. Coupling loss can be minimized to less than a few dB overall by improved fiber-to-chip coupling, lowerloss components of the FBGs (e.g. the required isolator), spliced fiber connections, and reduced losses within our MZI. Note that our current coupling efficiency is equivalent to ∼50 km of single mode fiber, suggesting that our system is well-suited for quantum networks provided loss is reduced. While the fidelities we demonstrate are sufficient for several applications, the current ∼Hz teleportation rates with the 44 km length of fiber are still low. Higher repetition rates (e.g. using high-bandwidth modulators with wide-band wavelength division multiplexed filters and low-jitter SNSPDs [73]), improvements to coupling and detector efficiencies, enhanced BSM efficiency with fastrecovery SNSPDs [74], or multiplexing in frequency [64] will all yield substantial increases in teleportation rate. Note that increased repetition rates permits a reduction in time bin separation which will allow constructing the MZI on chip, providing exceptional phase stability and hence, achievable fidelity. Importantly, the aforementioned increases in repetition rate and efficiency are afforded by improvements in SNSPD technology that are currently being pursued with our JPL, NIST and other academic partners.
Upcoming system-level improvements we plan to investigate and implement include further automation by the implementation of free-running temporal and polarization feedback schemes to render the photons indistinguishable at the BSM [29,30]. Furthermore, several electrical components can be miniaturized, scaled, and made more cost effective (e.g. field-programmable gate arrays can replace the AWG).We note that our setup prototype will be easily extended to independent lasers at different locations, also with appropriate feedback mechanisms for spectral overlap [75,76]. These planned improvements are compatible with the data acquisition and control systems that were built for the systems and experiments at FQNET and CQNET presented in this work.
Overall, our high-fidelity teleportation systems achieving state-of-the-art teleporation fidelities of time-bin qubits serve as a blueprint for the construction of quantum network test-beds and eventually global quantum networks towards the quantum internet. In this work, we present a complete analytical model of the teleporation system that includes imperfections, and compare it with our measurements. Our implementation, using approaches from High Energy Physics experimental systems and real-world quantum networking, features near fullyautomated data acquisition, monitoring, and real-time data analysis. In this regard our Fermilab and Caltech Quantum Networks serve as R& D laboratories and prototypes towards real-world quantum networks. The high fidelities achieved in our experiments using practical and replicable devices are essential when expanding a quantum network to many nodes, and enable the realization of more advanced protocols, e.g. [18,77,78].
is equipped with monitoring and active feedback stabilization systems (e.g. for IM extinction ratio), or has capability for remote control of critical network parameters (e.g. varying the qubit generation time). Each subsystem has a central classical processing unit with the following functions: oversight of automated functions and workflows within the subsystem, data acquisition and management, and handling of input and output synchronization streams. As the quantum information is encoded in the time domain the correct operation of the classical processing unit depends critically on the recorded timeof-arrival of the photons at the SNSPDs. Thus significant effort was dedicated to build a robust DAQ subsystem capable of recording and processing large volumes of time-tagged signals from the SNSPDs and recorded by our TDCs at a high rate. The DAQ is designed to enable both real-time data analysis for prompt data quality monitoring as well as post-processing data analysis that allows to achieve the best understanding of the data.
The DAQ system is built on top of the standalone Linux library of our commercial TDC. It records time tags whenever a signal is detected in any channel in coincidence with the reference 90 MHz clock. Time tags are streamed to a PC where they are processed in realtime and stored to disk for future analysis. A graphical user interface has been developed, capable of real-time visualization and monitoring of photons detected while executing teleportation. It also allows for easy control of the time-intervals used for each channel and to configure relevant coincidences between different photon detection events across all TDC channels. We expect our DAQ subsystem to serve as the foundation for future real-world time-bin quantum networking experiments (see Sec. V).

Superconducting nanowire single photon detectors
We employ amorphous tungsten silicide SNSPDs manufactured in the JPL Microdevices Laboratory for all measurements at the single photon level (see Sec. II B) [52]. The entire detection system is customized for optimum autonomous operation in a quantum network. The SNSPDs are operated at 0.8 K in a closed-cycle sorption fridge [53]. The detectors have nanowire widths between 140 to 160 nm and are biased at a current current of 8 to 9 µA. The full-width at half maximum (FWHM) timing jitter (i.e. temporal resolution) for all detectors is between 60 and 90 ps (measured using a Becker & Hickl SPC-150NXX time-tagging module). The system detection efficiencies (as measured from the fiber bulkhead of the cryostat) are between 76 and 85 %. The SNSPDs feature low dark count rates between 2 and 3 Hz, achieved by short-pass filtering of background black-body radiation through coiling of optical fiber to a 3 cm diameter within the 40 K cryogenic environment, and an additional bandpass filter coating deposited on the detector fiber pigtails (by Andover Corporation). Biasing of the SNSPDs is fa-cilitated by cryogenic bias-Ts with inductive shunts to prevent latching, thus enabling uninterrupted operation. The detection signals are amplified using Mini-Circuits ZX60-P103LN+ and ZFL-1000LN+ amplifiers at room temperature, achieving a total noise figure of 0.61 dB and gain of 39 dB at 1 GHz, which enables the low system jitter. Note that FWHM jitter as low as 45 ps is achievable with the system, by biasing the detectors at approximately 10 µA, at the cost of an elevated DCR on the order of 30 cps. Using commercially available components, the system is readily scalable to as many as 64 channels per cryostat, ideal for star-type quantum networks, with uninterrupted 24/7 operation. The bulkiest component of the current system is an external helium compressor, however, compact rack-mountable versions are readily available [53].

Interferometer and phase stabilization
We use a commercial Kylia 04906-MINT MZI, which is constructed of free-space devices (e.g mirrors, beam spliters) with small form-factor that fits into a hand-held box. Light is coupled into and out of the MZI using polarization maintaining fiber with loss of ∼2.5 dB. The interferometer features an average visibility of 98.5% that was determined by directing |+ with µ A = 0.07 into one of the input ports, measuring the fringe visibility on each of the outputs using an SNSPD. The relative phase ϕ is controlled by a voltage-driven heater that introduces a small change in refractive index in one arm of the MZI. However, this built-in heater did not permit phase stability sufficient to measure high-fidelity teleportation, with the relative phase following the slowly-varying ambient temperature of the room. To mitigate this instability, we built another casing, thermally isolating the MZI enclosure from the laboratory environment and controlled the temperature via a closed-loop feedback control system based on a commercial thermoelectric cooler and a LTC1923 PID-controller. The temperature feedback is provided by a 10 kΩ NTC thermistor while the set-point is applied with a programmable power supply. This control system permits us to measure visbilities by slowly varying ϕ over up to 15 hour timescales. We remark that no additional methods of phase control were used beyond that of temperature.
Appendix B: Estimation of mean number of photon pairs and transmission efficiencies of signal and idler photons Using a method described in Ref. [55], we measure the mean number of photon pairs produced by Bob µ B as a function of laser excitation power before the PPLN waveguide used for SHG. To this end, we modify the setup of Fig. 1 and direct each of Bob's signal and idler photons to a SNSPD. We then measure detection events while varying the amplification of our EDFA by way of an applied current. We extract events when photon pairs which originated from the same clock cycle are measured in coincidence, and when one photon originating from a cycle is measured in coincidence with a photons originated from a preceding or following clock cycle, in other words we measure the so-called coincidence and accidental rates. The ratio of accidentals to coincidences approximates µ B << 1, with all results shown in Table II. For all measurements we use µ B = (8.0 ± 0.4) × 10 −3 per time bin, which corresponds to an EDFA current of 600 mA.
The transmission efficiencies of the signal and idler photons, η s and η i , respectively, mentioned in Sec. IV are determined by calculating the ratio of the independent rates of detection of the idler and signal photons, respectively, with the coincidence rate of the photons pairs (in the same clock cycle) [55]. We repeat the measurements using the setup shown in Fig. 1, which is that used to generate the results of Fig. 7 (i.e. we direct the signal and idler photons through the setup as if we are to perform teleportation). We find η s = 4.5 × 10 −3 and η i = 1.2 × 10 −2 , which take into account losses between when the photons are produced to when they are detected by their respective SNSPDs.