1. Introduction

Quantum computing represents an upcoming threat to public-key cryptography [1,2]. Quantum key distribution (QKD) is a method for overcoming this threat by sharing secret cryptographic keys between parties in a manner that is sufficiently secure against potential eavesdroppers and the decryption capabilities of quantum computers. Point-to-point QKD networks are a precursor to more advanced quantum networks which enable the transfer of quantum states for multiple applications including distributed quantum computing, sensing, or secure communication. We characterize any quantum network as “advanced” if it enables protocols and capabilities that go beyond point-to-point QKD [3]. These include teleportation [4,5], entanglement swapping [6], memory-assisted networks [7], and others. Entangled photons are a fundamental resource for such demonstrations, and entanglement distribution is therefore a key component of premier quantum network initiatives including the European Quantum Communication Infrastructure (EuroQCI) project, the Illinois Express Quantum Network (IEQNET), the Chinese Quantum Experiments at Space Scale (QUESS) initiative, the United Kingdom UKQNTel network, and the Washington DC-QNet Research Consortium. Future quantum networks should enable high-fidelity and high-rate transfer of individual quantum states across multiple quantum nodes, mediated by distribution of entangled photons, quantum memories, and entanglement swapping measurements.

High-rate entanglement distribution enables high-rate entanglement-based QKD, as well as more general operations that characterize advanced quantum networks. Entanglement distribution and entanglement-based QKD have been demonstrated with impressive performance across a number of metrics. These include 40-kbps data rates in a QKD system deployed over 50 km of fiber [8] as well as multiple polarization entangled sources that leverage spectral multiplexing. These polarization sources include a demonstration of 181 kebits/s across 150 ITU channel pairs and a high-throughput source potentially capable of gigabit rates with many added channels and detectors [9,10]. Multiple works have highlighted the need to leverage high total brightness, spectral brightness, collection efficiency, and visibility from pair-generating nonlinear crystals to realize practical high-rate entanglement distribution [10–16].

A time-bin entangled photon source has certain advantages over a polarization-based system [17]. Time-bin entanglement can be measured with no moving hardware and does not require precise polarization tracking to maximize visibility [18,19]. Also, with suitable equipment, robust time-bin modulation is possible over free space links with turbulence [20]. Therefore, the possibility of simplified fiber-to-free-space interconnects and larger quantum networks based on a shared time-bin protocol motivates development of improved time-bin sources. Furthermore, time-bin encoding is suited for single-polarization light–matter interfaces [21].

We direct 4.09-GHz mode-locked laser light into a nonlinear crystal via 80-ps delay-line interferometers (12.5-GHz free-spectral range) to realize a high-rate entanglement source. The ability to resolve time-bin qubits into 80-ps-wide bins is enabled by newly developed low-jitter differential superconducting nanowire single-photon detectors (SNSPDs) [22]. Wavelength multiplexing is used to realize multiple high-visibility channel pairings which together sum to a high coincidence rate. Each of the pairings can be considered an independent carrier of photonic entanglement [23,24] and therefore the system as a whole is applicable to flex-grid architectures through the use of wavelength selective switching [25,26]. However, we focus on maximizing the rate between two receiving stations, Alice and Bob [Fig. 1(a)]. Each station is equipped with a DWDM that separates the frequency multiplexed channel into multiple fibers for detection. The SNSPDs are used with a real-time pulse pileup and time-walk correction technique [27] to keep jitter low even at high count rates.

 

Fig. 1. (a) Pulses from a 1539.47-nm mode-locked laser (Pritel UOC) are split into two by an 80-ps delay-line interferometer before up-conversion and amplification in a second harmonic generation + erbium doped fiber amplifier (SHG + EDFA) module (Pritel). A short PM fiber from the SHG module connects to a nonlinear crystal generating photon pairs by spontaneous parametric downconversion (SPDC). The coarse wavelength division multiplexing (CWDM) module separates the photon pair spectrum into eight 13-nm-wide bands around 1530 and 1550 nm, for the signal and idler photon, respectively. The signal and idler are directed to the Bob and Alice stations, respectively. The readout interferometers introduce the same time delay as the source interferometer. Polarization controllers are used to maximize the coincidence rates, as the detection efficiencies of each SNSPD is polarization sensitive ($\pm 10{\% }$). Entanglement visibility is unaffected by readout polarization. The polarization controllers could be removed if future systems adopt polarization insensitive SNSPDs [28]. 100-GHz spacing dense wavelength division multiplexer (DWDM) modules are used to direct each frequency channel into a distinct fiber. Two superconducting nanowire single-photon detectors (SNSPDs) are used to measure a specific frequency multiplexed channel pair. Measurements for different multiplexed channels are performed in succession to resolve full system performance. (b) ITU channels used in the experiment. Pairs of channels highlighted with the same color obey the phase and pump-energy matching condition for SPDC. To asses the full 16 channels (27–42) of Alice’s DWDM multiplexer, Bob’s 8-channel DWDM is replaced with a narrowband filter with tunable resonance frequency (not shown in the figure).

Download Full Size | PDF

We quantify per-channel brightness and visibility as a function of pump power, as well as collection efficiencies, coincidence rates across eight channel pairs. Performance of a 16-channel pair configuration is discussed in Supplement 1 note 4. We show that the 8-channel system achieves visibilities that average to 99.3% at low mean photon number $\mu _{L} = 5.6{\times } 10^{- 5}\,\pm \,9{\times } 10^{-6}$. At a higher power ($\mu _{H} = 5.0{\times } 10^{-3}\,\pm \,3{\times } 10^{-4}$), we demonstrate a total coincidence rate of 3.55 MHz with visibilities that average to 96.6%. Through quantum state tomography, we bound the distillable entanglement rate of the system to between 69% and 91% of the $\mu _{H}$ coincidence rate (2.46–3.25 Mebits/s).

Quantifying a source’s spectral mode purity is important for gauging its utility in advanced quantum networks that rely on interferometric measurements like two-photon interference which enables Bell-state measurements (BSMs) [5]. With Schmidt decomposition, we quantify the modal purity of single DWDM channel pairs and derive the inverse Schmidt number that serves as an estimate for two-photon interference visibility between two such sources. Ultimately, we demonstrate that an entanglement generation source of this design makes for a robust and powerful building block for future high-rate quantum networks.

2. System

Figure 1 shows the experimental setup. Pulses from the 4.09-GHz mode-locked laser, with a center wavelength at 1539.47 nm, are sent through an 80-ps delay-line interferometer (Optoplex DPSK Phase Demodulator). All interferometers used are the same type; they have insertion loss of $1.37\,\pm \,0.29\,\mathrm {dB}$, are polarization independent, and have extinction ratios greater than 18 dB. The source interferometer produces two pulses each clock cycle used to encode early/late basis states ($|e\rangle,|l\rangle$), which are subsequently upconverted by a second harmonic generation (SHG) module (Pritel) and downconverted into entangled photon pairs by a type-0 spontaneous parametric downconversion (SPDC) crystal (Covesion) [17]. The SPDC module uses a 1-cm-long waveguide-coupled MgO-doped lithium niobate crystal with an 18.3 $\mathrm{\mu} \textrm{m}$ polling period. The upconverted pulses at 769 nm have a FWHM bandwidth of 243 GHz (0.48 nm), which, along with the phase matching condition of the SPDC waveguide, defines a wide joint spectral intensity (JSI) function [29].

The photon pairs are separated by a coarse wavelength division multiplexer (CWDM) which serves to split the SPDC spectrum into two wide-bandwidth halves. For a system using more than 16 DWDM channels at Alice and Bob, the CWDM would be replaced with a splitter that efficiently sends the full SPDC spectrum shorter than 1540 nm to Bob, and the spectrum longer than 1540 nm to Alice. A dichroic splitter with a sharp transition at 1540 nm would also enable the use of DWDM channels 43–46 and 48–51. The pairs are of the form $|\psi \rangle =\frac {1}{\sqrt {2}}\left (|e\rangle _{s}|e\rangle _{i}+e^{i \phi }|l\rangle _{s}|l\rangle _{i}\right )$. Entangled idler and signal photons are sent to the receiving stations labeled Alice and Bob, respectively. One readout interferometer at each station projects all spectral bands into a composite time-phase basis. From here, DWDMs divide up the energy-time entangled photon pairs into spectral channels.

DWDM outputs are sent to differential niobium nitride (NbN) single-pixel SNSPDs [22] with 22 $\times$ 15 $\mathrm{\mu}$m active areas formed by meanders of 100-nm-wide and 5-nm-thick NbN nanowires on a 500-nm pitch. These measure the arrival time of photons with respect to a clock signal derived from the mode-locked laser. Use of the high system repetition rate and compact 80-ps delay interferometers is only possible due to the high timing resolution of these detectors. Low jitter performance is achieved by incorporating impedance matching tapers for efficient RF coupling, resulting in higher slew rate pulses, and by enabling RF pulse readout from both ends of the nanowire. The dual-ended readout allows for the cancellation of jitter caused by the variable location of photon arrival along the meander when the differential signals are recombined with a balun. SNSPDs of this type reach system jitters down to 13.0 ps FWHM, and 47.6 ps FW(1/100)M [22]. We use two SNSPDs for this demonstration with efficiencies at 1550 nm of 66% and 74%. They exhibit 3-dB maximum count rates of 15.1 and 16.0 MHz. A full 8-channel implementation of this system would require 16 detectors operating in parallel at both Alice and Bob. To read out both outputs of both interferometers, four detectors per channel are required, resulting in 32 detectors total.

A novel time-walk or pulse-pileup correction technique is used to extract accurate measurements of SNSPD pulses that arrive between 23 and 200 ns after a previous detection on the same RF channel. Without special handling of these events, timing jitter will suffer due to RF pulse amplitude variations and pileup effects. As detailed in Supplement 1 note 3, the correction method works by subtracting off predicable timing distortions based on the inter-arrival time that precedes them [27,30]. An in situ calibration process is used to build a lookup table that relates corrections and inter-arrival time. At the highest achievable pump power, this correction method leads to 320% higher coincidence rates compared to a data filtering method that rejects all distorted events arriving within $\simeq$200 ns of a previous pulse.

3. Results

By pairing up particular 100-GHz DWDM channels and recording coincidence rates, a discretized form of the JSI of our pair source is measured [Fig. 2(a)]. Due to the wide pump bandwidth, the spectrum of signal photons spans several ITU channels for a given idler photon wavelength. Pairs along the main diagonal are optimized for maximum coincidence rates by tuning the pump laser frequency, and are therefore used for all remaining measurements. In Fig. 1(b), these pairs are highlighted with matching colors. Coupling or heralding efficiencies $\eta$ shown in Fig. 2(a) are derived from a JSI fitting analysis (see Supplement 1 note 6) and include all losses between the generation of entangled pairs in the SPDC and final photo-detection.

 

Fig. 2. (a) Singles rates at Alice $S_A$ and Bob $S_B$ (gray bars), path coupling efficiencies (purple bars), and coincidence rates $C_{AB}$ (black text in colored boxes) for different DWDM channel pairings. All measurements are recorded at a SHG pump power of 14.6 mW, for which $\mu$ of the channel pairs along the main diagonal are shown in red text. The joint spectral intensity envelope spans several 100-GHz channels. As detailed in Supplement 1 note 2, the coincidence rates (kHz, black) are scaled to represent two branches of the total wavefunction, so that they are consistent with the coupling efficiencies $\eta$ (purple bars) and the singles rates (gray bars). There are four branches for each pairing of the four interferometer output ports. In practice, one output each of Alice and Bob’s interferometers is measured, thereby capturing one branch. See Supplement 1 note 6 for details of the fitting method used to solve for the coupling efficiencies $\eta$. (b) Visualization used to motivate the geometric factor $\delta$. The portion of the JSI that is captured by one 100-GHz filter (singles) is on the left, and that captured by two filters (coincidences) is on the right. Assuming 100% coupling efficiency inside the filter passbands, the ratio of captured coincidences on the right over singles on the left is $\delta$. For context, the portion of the JSI that is filtered away is faintly visible.

Download Full Size | PDF

The mean pair rate is commonly derived as $\mu = S_A S_B / (R C_{AB})$, where $S_A, S_B$ are the detector count rates at Alice and Bob, $C_{AB}$ is the coincidence rate, and $R$ is the system repetition rate. This is appropriate when losses on the signal and idler arm do not significantly depend on wavelength. However, the formula gives misleading $\mu$ values when DWDM channel filtering is narrowband relative to the width of the JSI. In this experiment, only a fraction of idler (signal) photons that exit one DWDM channel will be detected with their corresponding signal (idler) photon, even for ideal DWDM channels with 100% transmission within their passbands. This property is illustrated in Fig. 2(d). This implies a geometric limit on the ratio of coincidence to singles rates in this narrowband multiplexing regime. We account for this in calculations of $\mu$ by adding a geometric compensation factor $\delta$ to the commonly used equation:

In the following, rigorous tests of entanglement are primarily done with the eight ITU 100-GHz channel pairings: Ch. 35–42 at Alice and Ch. 52–59 at Bob. However, as shown in Supplement 1 note 4, source brightness measurements were conducted on a partially realized 16-channel configuration which makes use of all 16 channels available on Alice’s DWDM.

Signals from the SNSPDs are directed to a free-running time tagger (Swabian) and processed with custom software. The resulting histograms, referenced from a shared clock [Fig. 3(a)], depict three peaks, which are caused by the sequential delays of the source and readout interferometers. Some intensity imbalance between long and short paths is present in these interferometers, which explains the asymmetry between early and late peaks in Fig. 3(a). Such imbalances are present in both the source and readout interferometers to varying degrees. The interferometer used for the source exhibits an early/late intensity balance ratio of 1.13. Alice and Bob’s interferometers exhibit early/late imbalances of 1.24 and 1.15, respectively. These induce imperfect overlap of certain time-bin modes of differing amplitudes. This mismatch lowers interference visibilities, as detailed in Supplement 1 note 10.

 

Fig. 3. (a) Histogram of photon arrival events with respect to the 4.09-GHz clock. Dashed black and gray lines show the response functions for coincidence events. Events within 10-ps guard regions centered at 80 and 160 ps (shaded red) are discarded for analysis of coincidences between individual bins. This is done to maximize visibility in the presence of some minor overlap of the pulses (see Supplement 1 note 5 for discussion). The coincidence histograms include pairings from any combination of early, middle, and late time-bins. Therefore, the height of the center peak in the phase-min state is not near zero, as non-phase-varying terms contribute. (b) Coincidence rate interference fringes for the center time bin in isolation. Based on the good agreement between the fringe data and a cosine fit, we make subsequent tomographic measurements assuming that phase is linear with the electrical power applied to the interferometer phase shifter.

Download Full Size | PDF

The coincidence rate across Alice and Bob’s middle bins varies sinusoidally with respect to the combined phase relationship of the source and readout interferometers (see Supplement 1 note 1) [17,31]. In Fig. 3(a), the coincidences shown are for any combination of early, middle, or late bins. For tomography and visibility measurements, coincidence detections across specific bin pairings are considered.

Due to the small size ($3\,\times \,3\,\mathrm {cm}$) and temperature insensitivity of the interferometers, minimal temporal phase drift is observed. Without active temperature control or phase feedback, we observe minimized coincidence rates of the center time bin stay within 6% of their original values after 50 minutes. Nevertheless, software is used to lock the voltage-controlled phase at a minimum or maximum with a simple hill-climbing algorithm. This varies the phase by small amounts over several minutes to search for or maintain an extremum. This is simpler to implement than the techniques needed to stabilize interferometers of longer path length difference, including the use of precise temperature control [5] or co-propagating stabilization lasers [32].

Channels 35 and 59 are chosen for an analysis of entanglement visibility and rates versus pump power. Visibility with respect to pump power or mean entangled pair rate is shown in Fig. 4(a). We define the entanglement visibility as $V = 100{\% }*(C_{max} - C_{min})/(C_{max} + C_{min})$, where $C_{min}$ and $C_{max}$ are the minimum and maximum coincidence rates in the middle bin for varied phase. As this coincidence rate depends on the total phase across the source and readout interferometers, only Bob’s interferometer is actively controlled to scan the full state space.

 

Fig. 4. (a) Visibility versus pump power. Error bars are calculated by taking multiple measurements of the center bin coincidence rate over some integration time. These measurements span small ranges of interferometer phase, as the extremum-finding algorithm jitters the interferometer voltage. $V_C$ (gray data, red line) is a construction that models how visibility would be affected if accidental coincidences from mutually incompatible spectral modes could be mitigated in future systems. (b) Bounded distillable entanglement rate versus pump power. Multiple such measurements are made for all the tomographic measurements. These are used to calculate standard deviations for visibility, log negativity, and coherent information. Error bars for the log negativity and coherent information are smaller than the line width shown. Rates shown assume readout of all four available interferometer ports, based on data measured using one port each at Alice and Bob.

Download Full Size | PDF

The raw visibility versus $\mu$ is shown in blue in Fig. 4(a). Relative to similar measurements [33], this drops quickly with increasing $\mu$, and one reason is the presence of accidental coincidences across mutually incompatible spectral modes. The presence of these unwanted coincidences is a consequence of the narrowband filtering regime, and depends on factors included the singles rates $S_A$ and $S_B$, and the geometric compensation factor $\delta$ (see Supplement 1 note 9 for derivation). We model this type of accidental coincidence rate $C_{Acc}$ versus $\mu$, and subtract it off from coincidence measurements to produce the gray data in Fig. 4(a). This simulated visibility’s more gradual drop with increasing $\mu$ highlights the detrimental effect of our high single-to-coincidence rates $S_A/C_{AB}$, $S_B/C_{AB}$. As detailed in the discussion section below, this motivates special source engineering techniques for future systems. Supplement 1 note 12 infers the auto-correlation function $g^2(0)$ versus $\mu$ for this source in the small $\mu$ limit.

We quantify the rate of useful entanglement by supplying bounds for the distillable entanglement rate $C_D$. Measured in ebits/s, $C_D$ is the maximal asymptotic rate of Bell-pair production per coincidence using only local operations and classical communications [9,34]. It is bounded above by log-negativity $C_N = C_{AB} E_N$ and below by coherent information $C_I = C_{AB} E_I$ [9]. For each pump power setting in Fig. 4, a series of tomographic measurements is performed and density matrices are calculated. The values of $E_I$ and $E_N$ are calculated from the density matrices as detailed in Supplement 1 note 8.

Figure 5 shows visibilities, raw coincidence rates, and bounded distillable entanglement rates for two pump powers and all eight channel pairings. The highest pump power is currently limited by our EDFA-amplified SHG module. The pump power in principle could be increased until the SNSPD efficiency drops due to saturation, and the net coincidence rate plateaus. Without the time-walk correction, high-rate jitter becomes an issue well before the gradual drop of SNSPD efficiency. At the $\mu _H$ (22.9 mW) power, the singles rates $S_A, S_B$ average to 3.84 MHz, for which SNSPD efficiencies are approximately 78% of nominal.

 

Fig. 5. (a) Visibility for the main eight channel pairs, measured at a high (22.9 mW) and a low (0.21 mW) SHG pump power setting. Each power setting results in similar $\mu$ for all channels: $\mu _L = 5.6{\times }10^{-5} \pm 9{\times }10^{-6}$ and $\mu _H = 5.0{\times }10^{-3} \pm 3{\times }10^{-4}$. (b) Rate metrics for the eight channel pairs at the same high and low power settings. The range of possible values for distillable entanglement rate is spanned by the yellow regions, bounded above by log-negativity and below by coherent information. Rates shown assume readout of all four available interferometer ports, based on data measured using one port each at Alice and Bob.

Download Full Size | PDF

Using the data in Fig. 2(a), we model the JSI of our pair source as a product of pump envelope and phase matching condition functions

We calculate the Schmidt decomposition of the pair source JSI, taking into account the DWDM filters at Alice and Bob, and derive an average inverse Schmidt number $1/K$ of $0.87$. This value quantifies the spectral purity of the entangled photon source, and is theoretically equivalent to the visibility of a two-source HOM (Hong–Ou–Mandel) interferogram [35]. If 50-GHz ITU channels are used instead, the resulting filtered JSI better approximates a single mode, and the model predicts $1/K = 0.96$. The increase of $1/K$ with tighter filtering, as predicted by our model, is supported by simpler analytical expressions [36]. Relative to the 100-GHz channels used here, we expect the use of 50-GHz channels to decrease singles rate by roughly a factor of two, and coincidence rate by a factor of four for equivalent pump power. Our definition of $\mu$ would require a larger geometric factor $\delta$, and visibility would therefore scale less favorably with $\mu$ due to the greater impact of accidental coincidences. Ultimately, due to this narrowband filtering regime and geometric filter considerations, there are trade-offs to using narrower filters which may be acceptable for some applications, but not others.

4. Discussion

We have demonstrated that a time-bin entanglement source based on a mode-locked laser, spectral multiplexing, and low-jitter detectors produces high entangled photon rates suitable for QKD or advanced quantum networks. The distillable entanglement rate, achievable secret key rate, and visibilities of this source are highly competitive relative to other multiplexed entanglement distribution systems (Table 1). Still, there is potential to increase rates beyond those measured here with some straightforward changes to the setup. First, a higher power EDFA-amplified SHG module or tapered amplifier may be used. With this, we predict a single channel pair could sustain rates up to those specified in the first column of Table 2. These metrics all depend on both entanglement quality and coincidence rate $C_{AB}$. Due to the trade-off between $C_{AB}$ and entanglement quality or visibility, they all reach maximum values for particular pump powers extrapolated in Supplement 1 note 11. Our measurements of 8-channel and 16-channel configurations imply the approximately multiplicative scalings in columns 2 and 3 of Table 2, as coincidence rates of these channels pairs are all within 27% of each other. From measurements of the SPDC spectrum, it is also possible to extrapolate rates to a 60-channel 100-GHz DWDM configuration that includes channels spanning the L, C, and S ITU bands. This configuration could sustain a 34.9-MHz total coincidence rate, and a distillable entanglement rate between 27.7 ($C_N$) and 15.9 Mebits/s ($C_I$). These rates are impressive considering they are achievable with existing SNSPDs and other technology. A measurement of the full SPDC spectrum and extrapolation details are found in Supplement 1 note 11.

Table 1. Architecture and Rates Comparison^a

View Table | View all tables in this article

Table 2. Extrapolated Rates (MHz)^a

View Table | View all tables in this article

The ratio of singles rates $S_A, S_B$ to coincidence rates $C_{AB}$ are high in this system due to the relatively wide-band JSI and narrow filters. Each DWDM channel at Alice picks up a large fraction of photons that cannot be matched with pairs passing though the corresponding channel passband at Bob, a feature quantified by the $\delta$ factor. The high singles rates lead to accidental coincidences from mutually incompatible spectral modes that lower visibility and load the detectors with useless counts. However, there is potential to mitigate these extra counts by embedding the nonlinear crystal undergoing SPDC in a cavity that enhances emission at the center frequencies of multiple DWDM channels [37–39]. Also, there are other approaches to achieving such intensity islands that require dispersion engineering [40,41]. With such periodically enhanced emission, the resulting JSI would exhibit a series of intensity islands lying along the energy-matching anti-diagonal, easily separable with DWDMs at Alice and Bob. The photon flux for each channel would originate primarily from these islands covered by both signal and idler DWDM passbands, resulting in a higher ratio of coincidences to singles. The probability of accidental coincidences $C_{Acc}$ would be lower, and therefore bring the decrease of visibility with $\mu$ more in line with the modeled $V_C$ data in Fig. 4. We intend for the $V_C$ construction to represent how visibility would degrade primarily due to multi-pair effects, assuming accidental coincidences from incompatible spectral modes could be mitigated. The more gradual decrease in visibility with $\mu$ would enable substantially higher maximum rate metrics than those in Table 2.

This source is a fundamental building block for future space-to-ground and ground-based quantum networks. It leverages the strengths of the latest SNSPD developments – namely simultaneous high count rates, low jitter, and high efficiency – and in doing so adopts interferometers and DWDM systems that are compact, stable, and accessible. By elevating the system clock rate to 4.09 GHz and shrinking the time bin size to 80 ps, we have demonstrated a new state of the art in quantum communication that enables adoption of mature and extensively developed technologies from classical optical networks. Also, the spectral multiplexing methods used here are potentially compatible with those demonstrated in broadband quantum memories [42] and optical quantum computing [43].