High-fidelity, quasi-deterministic entanglement generation using phase-matched spectral islands in a zero-added-loss multiplexing architecture

Qubit teleportation [1, 2, 3] is a key primitive for quantum communication because it affords error-free, repeaterless transmission of quantum states over pure-loss bosonic channels with $\eta<1/2$ transmissivities, viz., links with zero quantum capacity [4]. At power levels below the onset of nonlinear effects, and employing post-propagation dispersion compensation, $L$-km-long transmission over standard single-mode fiber at 1.55 $\upmu$m wavelength may be regarded as a pure-loss channel [5] with transmissivity $\eta=10^{-0.02L}$. Hence its quantum capacity vanishes for $L>15\,$km. Because the fiber channel’s classical capacity is not subject to this catastrophic collapse, the internet’s fiber backbone uses erbium-doped fiber amplifiers (EDFAs)—spaced 80–100 km apart—in lieu of classical repeaters. Thus, because co-locating quantum repeaters with those amplifiers is highly desirable for efficient build-out [7] of a quantum internet [8, 9, 10], transmission via qubit teleportation becomes necessary, and that requires creating reservoirs of shared qubit entanglement between these EDFA sites.

Spontaneous parametric down-converters (SPDCs) have long been workhorses for generating polarization-entangled [11, 12, 13, 14] and time-bin entangled [15, 16, 17] photon pairs. Polarization entanglement is preferable for satellite-to-ground entanglement distribution, because atmospheric turbulence is not depolarizing [18]. For terrestrial entanglement distribution over optical fiber, however, time-bin entanglement is the right choice, because standard fiber does not preserve polarization and polarization-maintaining fiber is currently too lossy for long-distance operation. Sadly, in both cases, record-setting demonstrations leave much to be desired with respect to entanglement distribution over long distances. For example, Neumann et al. [19] realized only a 9 s^-1 distribution rate over 248 km of deployed fiber, while Yin et al. [20] achieved only a 1.1 s^-1 distribution rate from the Micius satellite to a pair of ground stations separated by 1200 km.

The preceding demonstrations used bidirectionally-pumped Sagnac SPDCs [21, 22], which are renowned for their excellent stability and high efficiency. Nevertheless, a Sagnac SPDC—like SPDCs in general—is intrinsically probabilistic, a characteristic that adversely impacts its entanglement-distribution rate. In particular, to suppress unwanted multipair events, SPDCs are typically operated at low brightness, e.g., an average of 0.01 entangled pairs per pump pulse. Multiplexing an array of SPDCs could push up the entangled-pair generation rate to near-deterministic, or at least quasi-deterministic, performance. But multiplexing schemes that rely on path switching at the source, such as Mower et al. [23] and Dhara et al. [24], suffer significant performance penalties from their switch losses. Chen et al.’s zero-added-loss multiplexing (ZALM) [25] is different. ZALM uses the equivalent of two Sagnac sources [26] and partial Bell-state measurements (BSMs) [27] to herald the generation of polarization-entangled biphotons across a large number of frequency-multiplexed signal channels that collectively span the down-converters’ multi-THz phase-matching bandwidth [28]. According to Chen et al., this arrangement has the potential to keep multipair emissions negligible on a per-channel basis while approaching quasi-deterministic performance over the SPDCs’ full phase-matching bandwidth.

ZALM uses a source-in-the-middle configuration, which has higher efficiency than its meet-in-the-middle and sender-receiver alternatives [29]. For the quantum internet’s fiber backbone, every classical-internet EDFA site would contain a ZALM quantum transmitter (QTX) to distribute entangled photon pairs to ZALM quantum receivers (QRXs) at that site’s nearest neighbors, Alice and Bob. As in Thomas et al. [6], classical traffic—including the identities of the frequency channels containing the heralded biphotons—could be confined to the C-band, while optical transmission of the heralded biphotons could occur in the O-band. Alice and Bob’s QRXs would then employ quantum mode conversion—frequency conversion [30, 31] and bandwidth compression [32]—to efficiently couple the biphoton received in each heralded channel to intracavity color-center quantum memories.

Chen et al. [25] presented a broad treatment of ZALM, including its source, mode-conversion, and memory-loading characteristics, as well as its overall performance scaling. Their ZALM relied on a burst of pump pulses to generate a striped frequency-domain biphoton wave function over a very broad (10 THz) bandwidth that, together with narrowband (1 GHz) filtering in the partial BSM and suitable (12.5 GHz) channelization at the receivers, led to a large number (800) of channels whose frequency-domain biphoton wave functions each approximated that of a spectrally-factorable, i.e., single-temporal-mode, state. Such near single-temporal-mode biphoton generation ensured the production of high-fidelity ($\geq$ 99%) biphotons, and the 800 channels provided quasi-deterministic ($\geq$ 25% generation probability per pump burst) operation. Unfortunately, Chen et al.’s analysis did not fully account for the combined effects of multipair events and losses.

To overcome these limitations, we take the essence of Chen et al.’s approximately-factorable channelized biphoton wave function to its ideal limit. Motivated by Morrison et al. [33], who domain engineered a $\chi^{(2)}$ crystal, pumped by a transform-limited Gaussian pulse with an appropriately chosen bandwidth, to realize a biphoton wave function with 8 discrete and spectrally factorable frequency bins, we assume the ZALM source’s SPDCs each have a modest number ($\ll$ 800) of phase-matched spectral islands that each generate two-mode squeezed-vacuum states. As a result, our analysis is able to go well beyond Chen et al.’s, in that ours includes multipair events of all orders, losses in the partial BSM and in propagation to Alice and Bob’s QRXs, and the use of both same-island and cross-island heralding. More importantly, it shows that entangled photon pairs can be generated and delivered with high ($\geq$ 99%) Bell-state fidelity and very high ($\geq$ 99.9%) Bell-state fraction despite losses in the partial BSM and in propagation. Furthermore, the generation rate can be quasi-deterministic ($\geq$ 25% generation probability per pump pulse) with a modest number ($\ll$ 800) of islands.

The remainder of the paper is organized as follows. Section II introduces the islands-based ZALM source and shows that in ideal, i.e., lossless, operation it can easily generate entanglement that has high fidelity and is near-deterministic. Section III extends our treatment of islands-based ZALM to include losses in the partial BSM and in the signal photons’ propagation to Alice and Bob’s QRXs. Here, same-island plus cross-island heralding permits the QTX’s entanglement generation to be quasi-deterministic, while delivering entangled pairs to Alice and Bob’s QRXs with high Bell-state fidelity and very high Bell-state fraction even with 1% QTX-to-QRX transmissivity. Concluding remarks are presented in Sec. IV, and derivations underlying the performance metrics reported in Secs. III.1 and III.2 appear in Appendices A and B, respectively.

To set the stage, we first introduce and analyze islands-based ZALM in the ideal case of lossless operation. Modifying figures from our previous treatment of dense wavelength-division multiplexed (DWDM) ZALM using type-0 phase matching [34], islands-based ZALM’s QTX is shown schematically in Figs. 1 and 2 and explained below.

Islands-based ZALM’s QTX uses a pair of Sagnac sources that employ identical periodically-poled lithium niobate (PPLN) crystals each with $N_{I}$ phase-matched spectral islands. The SPDCs’ islands are type-0 phase matched and pulse-pumped in phase to produce signal-idler pairs from each island that, except for their island-dependent signal and idler center frequencies, are in identical temporal modes. To achieve such a state, the spectral islands must be identical, nonoverlapping, and spectrally factorable, i.e., expressible as a product of functions of the signal frequency and the idler frequency [35] as sketched in Fig. 3 for 6 islands.

Restricting our attention to the foregoing temporal modes, and assuming operation in the SPDCs’ usual no-pump-depletion regime, we have that the output states from Sagnacs $k=1,2$ resulting from a single pump pulse are the following tensor products of two-mode squeezed-vacuum (TMSV) states,

Here: $|m\rangle_{K_{n_{P}}}$ is the $m$-photon Fock state of the $P$-polarized ($P=H,V$) signal ($K=S$) and idler ($K=I$) modes emitted by the $n$th phase-matching island; and

is the Bose-Einstein probability distribution with $G$ being the squeezing gain, so that $G-1>0$ is the average number of signal-idler pairs emitted per SPDC-island per pump pulse. The joint signal-idler state of the two Sagnacs is then $|\psi\rangle_{{\bf SI}}=|\psi\rangle_{S_{1}I_{1}}|\psi\rangle_{S_{2}I_{2}}.$

Figure 2’s partial-BSM setup first combines the idlers from the two Sagnacs on a 50–50 beam splitter, after which it separates their horizontal ($H$) and vertical ($V$) polarization components with polarizing beam splitters (PBSs). Coarse wavelength-division multiplexing (CWDM) filters then perfectly separate the modes from the different phase-matching islands, and the CWDM filters’ outputs illuminate single-photon detectors (SPDs) that have partial number-resolution (PNR), i.e., 0, 1, or $>1$, capability and no dark counts. A same-island herald from a particular pump pulse is declared for the $n$th island when exactly two idler photons are detected from the $n$th island, with one being $H$ polarized and the other $V$ polarized. If those detections both occur in Fig. 2’s $I_{+}$ branch or in its $I_{-}$ branch, the signal modes’ heralded state is the polarization Bell-state triplet [34]

Alternatively, if one of those detections occurs in Fig. 2’s $I_{+}$ branch, while the other occurs in its $I_{-}$ branch, the signal modes’ heralded state is the polarization Bell-state singlet

Because here we are assuming lossless optics and unit-efficiency, PNR-capable detectors with no dark counts in our partial-BSM apparatus, it is clear that we will get an $n$th-island herald from a particular pump pulse if and only if two $n$th-island idler photons enter the partial-BSM apparatus with one being $H$-polarized and the other being $V$-polarized. There are four ways—all equally likely—in which this $n$th-island heralding event, $\mathcal{H}_{n}$, can occur: Sagnacs 1 and 2 emit, respectively, an $n$th-island $H$-polarized idler photon and an $n$th-island $V$-polarized idler photon, or vice versa; and Sagnac 1 emits an $n$th-island $H$-polarized idler photon and an $n$th-island $V$-polarized idler photon with no $n$th-island emission from Sagnac 2, or vice versa. The probability of an $n$th-island herald from a particular pump pulse is therefore

and independent of $n$ because we have assumed all modes have the same squeezing gain, cf. Eq. (1). Note that only the first two $n$th-island heralding possibilities are true heralds, i.e., heralds for which the heralded $|\psi^{\pm}\rangle_{S_{1_{n}}S_{2_{n}}}$ state is indeed the one that is transmitted to Alice and Bob. The second two $n$th-island heralding possibilities are called false heralds, viz., they result in two $n$th-island signal photons being sent to Alice and none to Bob, or vice versa. Both the true and false $n$th-island heralds arise because the partial-BSM’s 50–50 beam splitter erases which-path information from the ZALM QTX’s two Sagnacs.

At this point we introduce the three metrics by which we will measure the ZALM QTX’s performance. The first is the per-pump-pulse probability that Alice and Bob are sent a true $n$th-island herald, $\Pr(\mathcal{H}_{\rm true})$, which, like $\Pr(\mathcal{H}_{n})$, is independent of $n$. The second is the Bell-state fraction, $\mathcal{B}$, of the state transmitted after a true $n$th-island herald, i.e., the probability that this state lies in the $S_{1_{n}}S_{2_{n}}$ Hilbert space spanned by those modes’ polarization Bell states. The third is the Bell-state fidelity, $\mathcal{F}$, viz., the conditional probability that the state sent to Alice and Bob is the state that was heralded, given a true $n$th-island herald has occurred and the state sent to Alice and Bob lies in the $S_{1_{n}}S_{2_{n}}$ polarization Bell-state Hilbert space.

To evaluate these metrics for same-island heralding we will assume the ZALM QTX sends at most one herald per pump pulse, in keeping with a ZALM QRX architecture that allocates only one qubit of quantum memory to each pump pulse. The probability that Alice and Bob are sent a true $n$th-island herald on a particular pump pulse in this case is therefore

i.e., half the probability that at least one island heralds. Our assumption of lossless QTX optics and unit-efficiency, PNR-capable SPDs with no dark counts immediately implies, from Eq. (1), that a $|\psi^{\pm}\rangle_{S_{1_{n}}S_{2_{n}}}$ true herald results in a $|\psi^{\pm}\rangle_{S_{1_{n}}S_{2_{n}}}$ biphoton being transmitted. So, the ideal islands-based ZALM QTX with same-island heralding realizes unit Bell-state fraction ($\mathcal{B}=1$) and unit Bell-state fidelity ($\mathcal{F}=1$). Figure 4 plots the per-pump-pulse probability of a true herald versus the average number of signal-idler pairs per SPDC island per pump pulse for $N_{I}=2,4,6,\ldots,12$ islands.

The preceding results tell us that the ideal ZALM QTX with same-island heralding and at most one herald per pump pulse is near perfect, i.e., it has unit Bell-state fraction and unit Bell-state fidelity with a true-herald per-pump-pulse probability that exceeds 0.25 for 8 islands at $G-1=0.5$, and is more than 80% of that metric’s ultimate (0.5) limit for 16 islands at $G-1=0.5$. In contrast, the lossless ZALM QTX from Chen et al. [25] needed 800 channels to achieve high-fidelity operation with 0.25 per-pump-pulse probability of a true herald.

Having painted such a rosy picture for islands-based ZALM QTX with ideal equipment, same-island heralding, and at most one herald per pump pulse, it behooves us to explore how rapidly that QTX’s performance degrades in non-ideal operation. We should also quantify the quality of the state delivered to Alice and Bob after loss in propagation from the ZALM QTX to their QRXs. Thus, in this section, we first consider a lossy QTX whose losses are symmetrically distributed on the idler paths through the partial-BSM setup and lumped together into sub-unit quantum efficiencies, $\eta_{T}$, at each SPD [37], but with no losses in the Sagnac sources’ signal paths. The second case we treat augments the preceding partial-BSM losses with symmetrically-distributed sub-unit signal-path transmissivities, $\eta_{R}$, from the ZALM QTX to Alice and Bob’s QRXs.

The per-pump-pulse $n$th-island heralding probabilities for both cases described above coincide, because they depend only on the reduced density operator for the four idler modes entering the partial BSM and the SPDs’ quantum efficiency. From Eq. (1), the $4N_{I}$ signal-idler mode pairs, $\{(S_{k_{n_{P}}},I_{k_{n_{P}})}\}$ are in independent, identically-distributed (iid) two-mode squeezed-vacuum states. Tracing out the signal states leaves the $\{I_{k_{n_{P}}}\}$ modes in iid thermal states with average photon number $G-1$. Passing those idler modes through Fig. 2’s 50–50 beam splitter produces idler modes, $\{I_{\pm P_{n}}\}$, that are also in iid thermal states with average photon number $G-1$. The well-known result for thermal-state photodetection with sub-unit quantum efficiency then gives

for the per-pump-pulse probability that an $n$th-island $H$-polarized idler photon and an $n$th-island $V$-polarized photon are detected. Moreover, as before, half of these detections are false heralds. Furthermore, continuing to assume that the ZALM QTX transmits at most one herald per pump pulse, we have that the per-pump-pulse probability of an $n$th-island true herald is given by Eq. (6) with $\Pr(\mathcal{H}_{n})$ now from Eq. (7) instead of Eq. (5).

There is no need to replot $\Pr(\mathcal{H}_{\rm true})$ for a lossy partial BSM; we need only change $G-1$ to $\eta_{T}(G-1)$ on Fig. 4’s abscissa. That said, before using $G-1=1/(2\eta_{T})$ to get $\Pr(\mathcal{H}_{\rm true})>0.25$ for an 8-island QTX with a sub-unit $\eta_{T}$, we must evaluate that transmissivity’s impact on the Bell-state fraction and Bell-state fidelity. This need arises because multipair events are increasingly important as $G$ increases, and $\eta_{T}<1$ implies that the SPDs do not completely ward them off. Finding $\mathcal{B}$ and $\mathcal{F}$ for a sub-unit $\eta_{T}$ is the task of Sec. III.1.

Chen et al. [25] proposed ZALM as a means to avoid the ill-effects of switch losses at the source in multiplexing SPDCs, and thus achieve high-fidelity ($>$ 99%), quasi-deterministic ($\geq$ 25% per-pump-pulse probability) entanglement generation. Their proposal relies on a burst of pump pulses to generate a striped frequency-domain biphoton wave function over a very broad (10 THz) bandwidth that, together with narrowband (1 GHz) filtering in the partial BSM and suitable (12.5 GHz) channelization at the receivers, leads to the need for a large number (800) of channels whose frequency-domain biphoton wave functions each approximate that of a spectrally factorable, i.e., single-temporal-mode, state. Chen et al.’s ZALM comes, however, with a heavy technological burden in that an 800-channel system requires a partial-BSM apparatus containing 3200 high quantum-efficiency SPDs, which would typically be cryo-cooled superconducting-nanowire single-photon detectors. Temporal multiplexing of the detection process could reduce the number of required SPDs, but that comes at the expense of reducing the maximum pump-pulse rate that can be employed. (In that regard, note that Chen et al. had assumed a $3\times 10^{10}\,{\rm s}^{-1}$ pump-pulse rate.) Furthermore, Chen et al.’s analysis did not fully account for multipair events and losses in the partial BSM and in propagation from the ZALM QTX to Alice and Bob’s QRXs.

In seeking a better alternative for the ZALM QTX, as well as providing a more comprehensive analysis, we drew inspiration from Morrison et al.’s demonstration of an SPDC producing a biphoton wave function comprised of 8 near-ideal single-temporal-mode spectral islands [33]. Thus, we assumed SPDCs whose nonlinear crystals had $N_{I}$ ideal spectral islands, and then derived the anti-normally-ordered characteristic functions for the output signals’ state from a lossy partial BSM, and for the joint state delivered, after lossy propagation, to Alice and Bob’s QRXs. From these characteristic functions we obtained analytic results for the Bell-state fidelities and Bell-state fractions of the output signals’ state and the joint state arriving at Alice and Bob’s QRXs. Then, assuming at most one herald is sent to Alice and Bob per pump pulse and SPCI heralding, we found that 99% Bell-state fidelity, $>$ 99.98% Bell-state fraction, and $\geq$ 25% per-pump-pulse probability of a true herald can be obtained with 28 islands if the partial BSM is 90% efficient, and with 62 islands if the partial BSM is 80% efficient. If the number of islands is limited by practical considerations to, for example, 14, then a 95% efficient partial BSM will give 99% Bell-state fidelity, $>$ 99.98% Bell-state fraction, and $\geq$ 25% per-pump-pulse probability of a true herald.

The rate, $\mathcal{R}_{E}$, at which islands-based ZALM delivers heralded polarization-entangled photon pairs to Alice and Bob is this paper’s final takeaway. It is given by [39]

where $\mathcal{R}_{P}$ is the pump’s pulse rate, and $\Pr(\mathcal{H})=2\Pr(\mathcal{H}_{\rm true})$, the per-pump-pulse herald probability, includes both true and false heralds. Assuming SPCI heralding with at most one herald sent to Alice and Bob per pump pulse, and $R_{P}=10^{10}\,{\rm s}^{-1}$, we get $R_{E}=2.52\times 10^{7}\,{\rm s}^{-1}$ when $\eta_{T}=0.9$, $G-1=0.0173$, $\eta_{R}=0.01$, and $N_{I}=28$. Reducing $\eta_{T}$ to 0.8 and $G-1$ to $8.59\times 10^{-3}$ while increasing $N_{I}$ to 62, the delivery rate drops to $\mathcal{R}_{E}=2.53\times 10^{5}\,{\rm s}^{-1}$. Both of these examples provide 99% Bell-state fidelity and $>$ 99.98% Bell-state fraction in the presence of propagation loss commensurate with: (a) 100 km fiber connections between the ZALM QTX and Alice and Bob’s QRXs; or (b) 1500-km-long satellite-to-ground free-space (no absorption, scattering, or turbulence) links at 1.55 $\upmu$m wavelength between a ZALM QTX with 10-cm-diameter optics and Alice and Bob’s QRXs with 1-m-diameter optics.

These preceding rates are admittedly optimistic, but, because they are orders-of-magnitude higher than current state of the art and with lower equipment burden than DWDM-channelized ZALM, they strongly suggest continued study of islands-based ZALM. There are a variety of theoretical directions in which such follow-on work could be pursued. One is a more comprehensive exploration of the Bell-state fidelity and Bell-state fraction as functions of $G-1$, $\eta_{T}$, $\eta_{R}$, and the available $N_{I}$. Such an exploration, in conjunction with a capabilities assessment of domain engineering PPLN, could set realistic goals for an initial islands-based ZALM source.

Another theory area to study is the Duan-Kimble loading [40] of Alice and Bob’s pair of intracavity color-center quantum memories. Islands-based ZALM affords an extremely high Bell-state fraction, permitting us to assume that Bell states are the only loadable states arriving at Alice and Bob’s QRXs. For Bell-state illumination we can use results from Raymer et al. [41] and Shapiro et al. [34] to obtain the density operator for the loaded memories. In particular, Shapiro et al. already showed that narrowband operation—in which the ZALM QRXs’ bandwidth-compression module is such that the quantum memories’ state-dependent reflectivities are constant across the photon bandwidth—leads to a perfect transfer of the incoming biphoton state from DWDM-channelized ZALM to the memories. It turns out that such is not the case for narrowband loading of the Bell state received from islands-based ZALM unless the memories are completely lossless [42]. Nevertheless, Shapiro et al.’s loading analysis establishes a route to quantifying islands-based ZALM’s Duan-Kimble loading both inside and outside the narrowband regime.

The just-mentioned follow-on studies are natural progressions from Chen et al., Raymer et al., Shapiro et al., and the present paper. A somewhat different tack, but nonetheless an important one, is to consider replacing the Sagnac sources and partial BSM with a new alternative, proposed by Chahine et al. [43], for heralded production of polarization-entangled Bell states. Chahine et al.’s proposal uses only two SPDCs and has a simpler heralding procedure than the dual-Sagnac arrangement considered here. For islands-based ZALM it then offers the possibility of lower loss in the heralding, which could increase the $G-1$ value at which high Bell-state fidelity and high Bell-state fraction are obtained from which an increased delivery rate would follow. That said, the Chahine proposal’s joint density operator for the signal states sent to Alice and Bob does not match what we have derived in this paper unless the ZALM QTX is lossless, so an analysis of its performance metrics is needed to see whether it truly offers an improvement over what we have found in this paper for the dual-Sagnac source.