Flexible entanglement-distribution network with an AlGaAs chip for secure communications
F. Appas, F. Baboux, M. I. Amanti, A. Lemaître, F. Boitier, E. Diamanti, S. Ducci
FFlexible entanglement-based secure communication with an AlGaAs chipfor quantum networks
F´elicien Appas, Florent Baboux, Maria I. Amanti, AristideLemaˆıtre, Fabien Boitier, Eleni Diamanti, and Sara Ducci Laboratoire Mat´eriaux et Ph´enom`enes Quantiques,Universit´e de Paris, CNRS-UMR 7162, Paris 75013, France Universit´e Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France Nokia Bell Labs, Nozay, France Sorbonne Universit´e, CNRS, LIP6, 4 place Jussieu, F-75005 Paris, France (Dated: February 10, 2021)Quantum communication networks enable applications ranging from highly secure communica-tion to clock synchronization and distributed quantum computing. Miniaturized, flexible, and cost-efficient resources will be key elements for ensuring the scalability of such networks as they progresstowards large-scale deployed infrastructures. Here, we bring these elements together by combining anon-chip, telecom-wavelength, broadband entangled photon source with industry-grade flexible-gridwavelength division multiplexing techniques, to demonstrate reconfigurable entanglement distribu-tion between up to 8 users in a resource-optimized quantum network topology. As a benchmarkapplication we use quantum key distribution, and show low error and high secret key generationrates across several frequency channels, over both symmetric and asymmetric metropolitan-distanceoptical fibered links and including finite-size effects. By adapting the bandwidth allocation to spe-cific network constraints, we also illustrate the flexible networking capability of our configuration.Together with the potential of our semiconductor source for distributing secret keys over a 60 nmbandwidth with commercial multiplexing technology, these results offer a promising route to thedeployment of scalable quantum network architectures.
I. INTRODUCTION
Quantum technologies have the potential to enhancein an unprecedented way the security of communica-tions across network infrastructures. Services and ap-plications that large-scale quantum communication net-works can provide span secure communication with secu-rity guarantees impossible to achieve only with classicalsystems [1], delegated and blind quantum computing [2],or distributed quantum computing and sensing [3], even-tually leading to the full capabilities of a quantum inter-net [4]. The choice of network topology and the optimiza-tion of the corresponding required resources are crucialwhen designing the architecture of such infrastructures.This is particularly true for the entangled photonic re-sources that need to be deployed to enable some of themost advanced applications, but also to alleviate the needfor trusted nodes in cryptographic scenarios [5, 6].In this context, a topology that has naturally at-tracted much attention exploits the presence of polariza-tion and time-energy photonic entanglement in state-of-the-art sources to distribute polarization-entangled pho-tons from a single source to multiple receivers (users)that hold energy-matched channel pairs [7–14]. This con-cept was in particular ingeniously pushed in [11], whichproposed and demonstrated a scheme allowing each userof the network to share quantum correlations with ev-ery other user, hence forming a fully connected entangle-ment network. Several challenges come, however, withsuch an architecture. First, the number of required fre-quency channels increases quadratically with the numberof users, making the bandwidth of the source a critical feature for ensuring scalability to large multi-user net-works. Although the scaling was improved in [13], therelatively narrowband polarization entangled photon pairsource based on periodically poled crystals that was usedwould not permit to further increase the number of users.Furthermore, scalability can also be hindered by the useof cascaded passive elements such as dense wavelengthdivision multiplexers (DWDM) and beamsplitters, whichalso come with high losses [11, 13]. Employing insteadadvanced multiplexing techniques based on wavelengthselective switch (WSS) technology offers a number of at-tractive features for flexible entanglement distribution, aswas recently shown in [14], albeit again with a relativelynarrowband entangled photon source.In this work, we demonstrate a scalable approach tothe fully connected architecture of [11]. To this end,we use a broadband source, emitting photon pairs attelecom wavelength that is based on an AlGaAs semi-conductor chip. AlGaAs exhibits direct bandgap, strongelectro-optical effect, and small birefringence, which en-ables an effective miniaturization and the generation ofpolarization-entangled states without any off-chip com-pensation [15–17]. We show that thanks to the featuresof our source, we can maintain an entanglement fidelityabove 85% over a 60 nm-wide spectral region. To bench-mark the performance of our system, we then run theflagship quantum communication application, namelyQKD, and in particular the BBM92 protocol [18], bothin the asymptotic and finite-size regime, over symmetricand asymmetric fibre-optic links, separating the users byup to 50 km in the symmetric case. We distribute en-tanglement using Flexgrid multiplexing based on WSS a r X i v : . [ qu a n t - ph ] F e b technology, which is the current industry standard inclassical communications, and offers straightforward re-configurability in terms of central frequency and channelbandwidth adjustment, wavelength-insensitive insertionloss and polarization diversity. This allows us to showthat each of the two-user links in a fully connected en-tangled network of 4, 5 or 8 users, can support high-performance QKD at metropolitan scale distances. Wefurther showcase the flexibility of our scheme in an unbal-anced network scenario, featuring several local users andone distant user. We demonstrate that the bandwidthreallocation enabled by the WSS can be used to equili-brate the resulting rates, hence materializing an elasticnetwork configuration and highlighting its potential forthe deployment of flexible, scalable quantum network ar-chitectures. II. ALGAAS PHOTONIC CHIP ANDEXPERIMENTAL SETUP
The schematic of our experimental setup is presentedin Fig. 1 (c) and consists of three stages: entanglementgeneration with an AlGaAs chip, frequency demultiplex-ing/multiplexing (demux/mux) of the generated signaland a distribution stage. Note that two different fre-quency demultiplexing/multiplexing approaches can beimplemented, depending on the spectral region of inter-est: in the telecom C-Band (1530 nm to 1565 nm) weuse a WSS and in the L-Band (1565 nm to 1625 nm) acoarse wavelength division multiplexing unit (CWDM)with 13 nm wide channels followed by a tunable filter.The WSS enables the implementation of flexible reconfig-urable fully-connected multi-user entanglement networksas shown in Fig. 1 (b).The AlGaAs chip, depicted in Fig. 1 (a), consists ofa Bragg reflection ridge waveguide emitting photon pairsthrough type II spontaneous parametric down conversion(SPDC) in the C+L-Band [10]. The sample is pumpedwith a tunable CW diode laser leading to a pair produc-tion rate of the order of 10 pairs/s at the chip output,corresponding to a brightness of 3 . × pairs/s/mW(see Appendix A for details).Thanks to the dispersion and nonlinear properties ofthe AlGaAs platform, the quantum state produced bythe chip presents several features making it appealing forthe implementation of quantum networks. Our chip gen-erates time-energy and polarization entangled photonsemitted in a | Ψ + (cid:105) polarization Bell state [10, 15]. Thepairs being generated in a broad spectral band, we canmultiplex the generated photons in a large number ofwavelength channels using standard telecom componentsand share polarization entanglement between users re-ceiving energy-matched channels [10]. Besides, the groupvelocity mismatch between orthogonally polarized pho-tons in the AlGaAs chip is so small that no off-chipwalk-off compensation is required to obtain polarizationentanglement [15, 16]. This is a key feature enabling the direct use of the emitted pairs at the output of the chip,opening the way to its easy integration into simple androbust architectures.The biphoton bandwidth has been measured by inject-ing the generated photon pairs into a fibered 50/50 beamsplitter (BS) with a fibered C+L-Bands tunable filter in-serted in one of its output ports and detecting the co-incidences between the two output ports via supercon-ducting nanowire single photon detectors (SNSPD) anda time-to-digital converter (TDC). The result is shownin the inset of Fig. 2: the full-width at half-maximumof the output signal is 60 nm, corresponding to approxi-mately 72 ITU 100 GHz channels, 36 on each side of thequantum state degeneracy frequency.In order to assess the effective biphoton bandwidthfor entanglement distribution we have selected symmet-ric output channels with respect to the biphoton degen-eracy frequency and we have estimated the fidelity ofthe transmitted photon pairs to the | Ψ + (cid:105) Bell state asa function of the channel number. In the spectral re-gion around degeneracy (C-Band) the measurement hasbeen performed by defining channels using the WSS,while in the spectral region far from degeneracy (L-Band)we have used the CWDM followed by a fibered tun-able filter. As shown in Fig. 1 (c), after demultiplex-ing, the photons are sent through a polarization anal-ysis stage consisting of a set of λ/ , λ/ , λ/ X = {| H (cid:105) , | V (cid:105)} and Z = {| D (cid:105) = √ ( | H (cid:105) + | V (cid:105) ) , | A (cid:105) = √ ( | H (cid:105) − | V (cid:105) ) } . A lower bound to the fidelity F to the | Ψ + (cid:105) Bell state is estimated by coincidence measurementsin those two bases by recording the number of coincidencecounts between the two detectors in 8 different config-urations : C HH , C HV , C V H , C
V V , C DD , C DA , C AD , C AA (see Appendix C for details). From these quantities, wecan access the diagonal elements of the density matrixin both MUBs. For ( α, β ) ∈ B × B , with B ∈ { X , Z } ,the matrix element ρ αβ corresponding to the first (sec-ond) photon in α ( β ) polarization state is calculated as ρ αβ = C αβ / ( (cid:80) B×B C α (cid:48) β (cid:48) ). Note that the normalizationfactor has to be calculated independently for the twobases. The lower bound to F is then obtained throughthe relation [19, 20]: F ≥
12 ( ρ HV + ρ V H − ρ HH − ρ V V (1)+ ρ DD + ρ AA − ρ DA − ρ AD ) . The experimental results, along with the fidelity of thenumerically simulated quantum state emitted by the Al-GaAs chip, are shown in Fig. 2. The observed lowerbound of the fidelity is above 95 % over a 26 nm spec-tral range corresponding to the first 13 pairs of 100 GHzITU channels around degeneracy, and stays above 85 %for up to 38 channels pairs around degeneracy spanning a λ /4 λ /2 λ /4 SNSPDSNSPD 25 km SMF2825 km SMF28 λ /4 λ /2 λ /4 Demux / Mux Demux / MuxMO MO HPF TFCWDMWSSLaser 780 nm input port 1port 2port 1port 2inputinput port 1port 2
AlGaAs chip
A BCDESource AliceBobCharlieDaveEmilyDemux MuxPhysical architecture Quantum correlationsA BCD A B CDEFGHSM fi berPM fi ber10 μ m FPBSFPBS FIG. 1. (a) Artist view of an AlGaAs chip under operation. An incoming pump photon is depicted on the top left corner,at the input of the waveguide. The generated entangled two-photon state is represented on the bottom right corner. (b)Schematic view of the entanglement network architecture. The physical structure of the network consists of an entangledphoton source, a demux/mux stage and a distribution stage. The resulting quantum correlation topology takes the form of afully-connected graph of variable size (here N = 4 , , WSS
CWDM +Tunable Filter
Detuning to biphoton degeneracy (nm) B e ll s t a t e f i de li t y ( ne t ) Simulation (no cavity effect)Simulation (with cavity effect)Experiment
100 GHz channel number offset C o i n c . C oun t s ( a . u . ) FIG. 2. Main graph: Lower bound to the fidelity to a (cid:12)(cid:12) Ψ + (cid:11) Bell state as a function of the detuning to biphoton degen-eracy wavelength (1556 .
55 nm) in nm (lower x-axis) and inunits of 100 GHz channels (upper x-axis). Error bars takeinto account the Poissonian statistics of both signal countsand subtracted background counts. The different color ar-eas correspond to the two different wavelength demultiplex-ing schemes (see main text for details). Inset: Coincidencecounts as a function of filter wavelength. total 60 nm wavelength range. Since the channels of theCWDM are not centered around the biphoton degener-acy wavelength, measurements in the L-Band are limitedto the conjugate channels that fall within the transmis-sion window of the CWDM ports. For this reason only3 data points have been acquired in the orange region ofFig. 2.Close to degeneracy, the experimental data are ingood agreement with theoretical predictions describingthe SPDC process in our sample in a simple pass con-figuration (solid lines). However, moving away from de-generacy, cavity effects arising from facets reflectivity ofthe AlGaAs waveguide have to be taken into account.The inclusion of these effects in our theoretical modelleads to the calculated fidelity represented with a dashedline, reproducing the experimental data with a very goodquantitative agreement. Details on the calculation of thefidelity can be found in Appendix B.The ability to maintain high quality polarization en-tanglement over a broad spectral region establishes thehigh potential of AlGaAs-based devices as ready-to-useminiature sources of broadband polarization entangledbiphoton states. A sy m p t. K e y R a t e ( b i t s / s )
100 GHz channel number offset
Detuning to biphoton degeneracy (nm) Q BE R ( % ) Positive key rate threshold
FIG. 3. Asymptotic key rate (upper panel) and QBER (lowerpanel) as a function of the detuning to biphoton degeneracyin nm (lower x-axis) and in units of 100 GHz channels (upperx-axis). The 11 % error threshold for a positive key rate isgiven for an error correcting code operating at the Shannonlimit.
III. SECRET KEY SHARING BETWEEN USERPAIRSA. BBM92 protocol with polarization-entangledphotons
Next, we exploit the broadband polarization entangledbiphoton state generated with the AlGaAs chip to imple-ment the BBM92 QKD protocol [18]. In this scenario,two users share a maximally entangled Bell state, in ourcase: | Ψ + (cid:105) = √ ( | H (cid:105) s | V (cid:105) i + | V (cid:105) s | H (cid:105) i ) with signal ( s )and idler ( i ) denoting the high and low energy photonrespectively. We deterministically separate the photonsof each pair into two fiber paths according to their fre-quency ( s or i ) using the WSS. As explained previously,each photon is then sent to a measurement station, con-sisting of a polarization analysis stage and a SNSPD asshown in Fig. 1 (c). In the full BBM92 protocol, the ba-sis choice is random, each user having 4 detectors, onefor each measurement outcome ( H, V, D, A ). The ran-dom basis choice can be implemented by using, for in-stance, passive 50/50 BS [21] or polarization-to-time-binconversion [12]. However, in our proof-of-principle mea-surement, we choose for simplicity to manually configurethe waveplates in each path ( s and i ) and record the coin-cidence counts in the same 8 configurations as presentedin section II. This allows us to assess the performance ofour QKD scheme. B. Performance after the WSS stage
To evaluate the intrinsic QKD performance of ourscheme, we measure the figures of merit of the BBM92protocol right after the demultiplexing stage: a) theasymptotic secret key rate R key , which corresponds tothe number of secret bits per second established by thetwo users following basis reconciliation (sifting), errorcorrection and privacy amplification, and b) the quan-tum bit error rate e (QBER), which is the fraction oferroneous bits in the raw key. We repeat the measure-ment for 13 different choices of signal and idler 100 GHzchannel pairs. Note that this spectral range is limited bythe upper cutoff wavelength (1565 nm) of the WSS cor-responding to the boundary of the C-Band, and not bythe spectral bandwidth of the generated biphoton state.We first express R key and e in terms of the recordedcoincidence counts [12]. The raw coincidence counts ineach basis ( X and Z ) read: C (raw) X = C HH + C HV + C V H + C V V (2) C (raw) Z = C DD + C DA + C AD + C AA (3)The QBER can be readily obtained by taking the ratioof accidental counts against the total number of recordedraw coincidence counts: e = C HH + C V V + C DA + C AD C (raw) X + C (raw) Z . (4)Next, we compute the raw key rate as the arithmeticmean between the raw count rate in the X and in the Z basis: R raw = 12 C (raw) X + C (raw) Z τ , (5)with τ the integration time. Finally, the asymptotic keyrate is obtained as [22, 23]: R key ≥ R raw
12 (1 − f ( e ) H ( e ) − H ( e )) . (6)where H ( e ) = − e log ( e ) − (1 − e ) log (1 − e ) is thebinary entropy function. This corresponds to the max-imum number of key bits per unit time that can be se-curely extracted by the two parties in the limit of infi-nite key length. The factor 1 / f ( e ) H ( e ) is the portion of the key that has to be usedfor error correction, with f ( e ) ≥ f ( e )[22] assuming the standard bidirectionnal code presentedin ref. [24]. Finally, the last term in H ( e ) is the lowerbound of the fraction of the raw key that is lost afterprivacy amplification [25].The calculated R key and QBER are plotted in Fig. 3.We see that both quantities are stable over the 13 ITU100 GHz channels spanning a 10 . − f ( e ) H ( e ) − H ( e ) =0, assuming an error-correcting code operating at theShannon limit ( f ( e ) = 1). This stability can be at-tributed to the flatness of the source spectrum in thisspectral region (inset of Fig. 2) and to the wavelength-independent insertion losses of the WSS. The entangle-ment quality of the emitted quantum state, combinedwith the high conversion efficiency of AlGaAs, yields highasymptotic key rates of 28 to 39 bits/s, over a very broadspectral range. The ability to support high key rateswith low error over many frequency channels makes ourAlGaAs entangled-photon source an ideal technology fora chip-based QKD network, as will be illustrated in Sec-tion IV. C. Performance across long-distance fiber links
We assess the long-distance performance of our ar-chitecture by estimating R key and e after adding 25 kmSMF28 fiber spools between the demux/mux stage andthe user’s measurement station. This measurement isperformed in two configurations: symmetric (both usersseparated from the source by 25 km of fiber) and asym-metric (one user at 0 km and the other at 25 km) in or-der to compare the performances and provide the op-timized solution for secure communication. All mea-surement runs are performed by selecting entangled pho-tons from 100 GHz ITU channels 23 (1558 .
98 nm) and29 (1554 .
13 nm) which are considered representative ofall the available channels. The results are shown asblack symbols on Fig. 4 (a) and (b) for the symmetricand asymmetric configurations respectively.To estimate the maximum distance of a repeaterlesslink we also perform this measurement by replacing thefiber spools by variable channel attenuation programmedon the WSS. The attenuation can then be converted intoequivalent length of SMF28 fibers assuming a standardvalue of 0 .
22 dB km − for the SMF28 optical losses. Re-sults for a symmetric and asymmetric link are shown inFig. 4 (a) and (b) respectively as color symbols. We com-pare the experimental value of the asymptotic key rateto theoretical predictions (continuous lines) for a BBM92scheme with passive basis selection following the modelof refs. [23, 26].In order to highlight the relevance of these results forreal-world implementations, we perform a finite key se-curity analysis of our scheme. As mentioned in the pre-vious section, the secret key rate in Eq. (6) is only validin the limit of infinite key length. However, in practice,raw key bits are exchanged in blocks of finite size. Toguarantee a certain security threshold for a given blocksize, we resort to the framework developed in refs. [5, 27].We set correctness and security bounds to (cid:15) corr = 10 and (cid:15) sec = 10 respectively for a 10 min block size. Theresulting key rates are shown in Fig. 4 (a) and (b) as tri- angles for experimental values and dashed lines for theo-retical predictions.We see that the measurements performed with vari-able attenuation closely follow our theoretical predictions(solid and dashed lines) while measurements with fiberspools (black symbols) deviate slightly from our model.Since the QBER does not increase in fiber-based exper-iments (black filled circles), this small discrepancy canbe attributed solely to the insertion losses induced bythe presence of extra fiber connectors between the WSSand the fiber spools and not to the potential thermal andmechanical instability of our fiber link.Based on this data, we can compare experimentallythe long-distance performance of the BBM92 protocol insymmetric and asymmetric configurations. We observethat, in both scenarios, the key rate including finite sizeeffects stays positive for distances of up to 75 km. How-ever, we also observe that, at very long distances, theasymptotic key rate for a symmetric link stays positiveup to 250 km, while in an asymmetric link, it drops ataround 215 km. Indeed, in the latter configuration, astrong attenuation in the link between the source andthe distant user induces a strong increase of the QBER asthe signal approaches the noise background. On the con-trary, in the symmetric case, the losses are distributed be-tween the two users and this critical situation is reachedat higher levels of attenuation. In the finite key securityregime in which we are operating, the finite key rates be-come negative before the difference between symmetricand asymmetric links comes into play. This result haspractical implications in the context of deployed QKDschemes; indeed this proves that a scenario involving anentangled pair source connected to one local user andone distant user will not be detrimental to the BBM92protocol efficiency.Furthermore, the performance of our device beingpractically insensitive to wavelength in the C-band, asshown in Fig. 3, the measured performances for the twoparticular channels (23 and 29) can be extrapolatedto the 12 other available 100 GHz channel pairs. Thismeans that each channel pair can support a long-distancefiber link, making our scheme compatible with large-scalefibered QKD networks. These results establish a solidground for a future deployment of our scheme in fiberedmetropolitan networks. IV. MULTI-USER ENTANGLEMENTNETWORKS WITH FLEXIBLE BANDWIDTHALLOCATIONA. Reconfigurable multi-user entanglementnetworks
Building on the previously presented results, we nowdemonstrate the operation of a scalable multi-user net-work architecture by taking advantage of the broad en-tanglement bandwidth of our source and of the flexible
Equivalent SMF28 fiber link distance (km) -8 -6 -4 -2 S e c r e t k e y r a t e ( b i t s / s ) Q BE R ( % ) Asymptotic rateFinite key rateQBER
Total attenuation (dB)
Equivalent SMF28 fiber link distance (km) -8 -6 -4 -2 S e c r e t k e y r a t e ( b i t s / s ) Q BE R ( % ) Asymptotic rateFinite key rateQBER
Total attenuation (dB)
50 km fi ber 25 km fi ber Symmetriclink Asymmetriclink
FIG. 4. Asymptotic and finite size key rates and QBER for a (a) symmetric and an (b) asymmetric two-user link supportedby 100 GHz ITU channels 23 (1558 .
98 nm) and 29 (1554 .
13 nm) as a function of SMF28 fiber distance (lower x-axis) andattenuation (upper x-axis). Symbols represent experimental data and continuous lines theoretical predictions for a BBM92scheme. Finite key rates are estimated assuming 10 min block size. frequency management enabled by the WSS.We consider a simple physical network consisting ofone central server node which hosts the entangled pho-ton source, and N user nodes. Each user has a polar-ization analyzing device and single photon detectors tocarry out the BBM92 protocol. We focus on a particularclass of such networks, where each of the N users sharesan entangled state with the remaining N − N = 4 , , N is N ( N − /
2. Hence to get a completeentanglement network, one needs to establish N ( N − / N ( N − / N ( N −
1) frequency channels. Thenwe recombine those channels into single optical fibers,one for each user. This is done via the WSS which im-plements both operations. Each user then receives pho-tons from N − N − N . In this context, we see thatthe broadband entanglement of the biphoton state emit-ted by the AlGaAs chip is a crucial asset. In addition, theuse of a WSS at the demultiplexing/multiplexing stageenables a reconfigurable multi-user network where thecentral frequency and the bandwidth of each channel can be adjusted. In our setup we can increase the number ofusers in the network by simply reducing the bandwidthallocated to each channel. Using all the available wave-length overlap between the biphoton state and the WSSoperating range, our reconfigurable complete-graph net-work can accommodate from 4 users, with 200 GHz chan-nels, to 8 users, where each link is supported by a 50 GHzconjugate channel pair.We sequentially characterize the N ( N − / N = 4 , N = 5 and N = 8. Due to the lack of additional single photon detec-tors, we did not run real-time QKD sessions for N usersin parallel. Instead, we measure the time-correlation his-tograms for all N ( N − / N = 4 , , e and R key using the protocol de-
50 100 150
Temporal bin (81 ps) C o i n c i den c e c oun t s i n s ABACBCBDCDDA T o t a l c oun t s
400 500 600 700 800
Temporal bin (81 ps) C o i n c i den c e c oun t s i n s ABACADAECBCDDBEBECED T o t a l c oun t s A sy m p t. k e y r a t e ( b i t s / s ) Channel width (GHz) Q BE R ( % )
200 300 400 500 600 700 800 900 1000
Temporal bin (81 ps) C o i n c i d e n c e c o un t s i n s ABACADAEAFAGAHBFBGBHCBCDCGDB EBECEDEGEHFCFDFEFGFHGDGHHCHD T o t a l c o un t s A BCD A BCDEA B CDEFGH
FIG. 5. Reconfigurable multi-user entanglement network. (a-c): Time-correlation histograms for all two-user links in a fullyconnected network of size (a) N = 4 with 200 GHz channels, (b) N = 5 with 100 GHz channels and (c) N = 8 with 50 GHzchannels. (d): Measured asymptotic key rate and QBER for ITU channels 23 (1558 .
98 nm) and 29 (1554 .
13 nm) as a functionof channel width. scribed in section III B for ITU channels 23 and 29 as afunction of channel bandwidth. The result is shown inFig. 5 (d). We see that the QBER is essentially insen-sitive to the channel bandwidth and the correspondingasymptotic key rates show a linear dependence with thebandwidth, as expected. We conclude that these valuesfor R key and QBER, alongside with those of Fig. 3, pro-vide an estimate for the achievable QKD performancesin the various N -user graphs presented in Fig. 5 (a-c).Our scheme presents several advantages in terms ofscalability over passive WDM QKD schemes. Indeed, theflexibility offered by the WSS makes it possible to fullyreconfigure the network without modifying the hardware.On the contrary, using passive WDM demux/mux the ad-dition of more users to the network requires either chang-ing the complex combination of cascaded WDM filters ornon-deterministically splitting some channels by combin-ing WDM filters and multiport fiber splitters [13, 28],which, either way, comes at the cost of extra opticallosses. In contrast, the fixed insertion losses of the WSSmakes it possible to extend the network without degrad-ing the signal, a major asset in a fully deployed networkscenario. B. Towards a smart quantum network with flexiblebandwidth allocation
Finally, we exploit the flexibility of our scheme todemonstrate a quantum network where the signal of eachlink is optimized following a structural constraint. Thisexperiment represents a first step towards a fully au-tonomous smart network, where communication ratesare automatically optimized based on bidirectional in-formation exchange between the users and the networkprovider. We consider a fully-connected entanglementnetwork where one of the user nodes is located far apartfrom the entangled photon source, while all the othernodes are located close to the source as schematicallydepicted in Fig. 6 (a). We implement this situation for N = 5 by separating node B’s detector from the sourceby a 25 km-long SMF28 fiber spool. If all two-user linkshave the same fixed bandwidth, the key rate of the fourlinks connecting the distant user to the rest of the net-work, i.e. , AB CB DB EB, will be lower than the others.To avoid this problem, we reallocate the state bandwidthby changing the channel widths, assigning more band-width to long links, and less bandwidth to short links,as schematically depicted in Fig. 6 (b). We use a simplealgorithm and distribute the available bandwidth with a12 . AB AC CB CD CE DA DB DE EA EB010002000300040005000 C o i n c i den c e c oun t s i n s Before reallocationAfter reallocation
25 km SMF28 source
A BCDE
Signal/idler wavelength B e f o r e r e a ll o c a t i o n A f t e r r e a ll o c a t i o n Degeneracy(CH26, 1556.55 nm)
FIG. 6. Illustration of flexible bandwidth allocation. (a)Sketch of the experiment. In a fully connected network with N = 5 users, user B is separated from the source by a 25 kmlong SMF28 fiber spool. (b) Schematic of the bandwidth dis-tribution between all two-user links before and after applyingthe bandwidth reallocation algorithm. After reallocation, thefour links (red, dark green, light blue and dark blue) that con-nect user B to the rest of the network are allowed a broaderbandwidth. (c) Coincidence counts for each two-user link be-fore (squares) and after (triangles) bandwidth reallocation. the measured signal across the whole network, we mea-sure one by one the 10 two-user coincidence counts ratesfor each link. The results are shown in Fig. 6 (c). Redsquares represent the raw coincidence counts of the 10two-user links for fixed 100 GHz channels and blue trian-gles are the raw coincidence counts recorded after band-width reallocation. We observe that, starting from avery unbalanced initial configuration, we can bring allusers to a similar level of signal. The same techniquecould be applied to other ends, such as boosting the sig-nal across specific links according to user needs. Thisproof-of-principle experiment shows that QKD is fullycompatible with state-of-the-art telecom network man-agement, opening the way for flexible metropolitan-scaleentanglement distribution. V. DISCUSSION AND OUTLOOK
We have demonstrated a scalable approach to a fully-connected entanglement distribution network using anAlGaAs chip emitting broadband polarization-entangled photon pairs in the telecom band. The lower bound onentanglement fidelity of the quantum state generated byour chip stays above 95 % over a 26 nm wide spectralrange around biphoton degeneracy and above 85 % overa 60 nm range. We deterministically separate the pho-tons of each pair into energy-matched frequency channelsand distribute them to the network users using a wave-length selective switch. We benchmark the performanceof our quantum network by running the entanglement-based BBM92 QKD protocol. We perform key distri-bution between two users with a QBER below 2% acrossfibered optical links of up to 50 km including finite-key ef-fects and we extrapolate a positive key rate for distancesof up to 75 km in both symmetric and asymmetric con-figurations. We have extended our study to the multi-user case, taking advantage of the flexibility offered byour setup. By reconfiguring the frequency grid, we showthat our network can accommodate up to 8 users over 50GHz ITU channels. We further demonstrate the insensi-tivity of the QBER with respect to channel width, indi-cating that every two-user link in the network can sup-port high-performance QKD at metropolitan-scale dis-tances. Finally, we demonstrate that the bandwidth re-allocation enabled by the WSS can be used to equilibrateunbalanced network scenarii, which is compatible withan elastic network configuration. Future work includesthe implementation of our architecture in a metropolitanquantum communication network to test its performanceand robustness in a real-world situation.Further progress is possible in different ways. On onehand, an extreme miniaturization of the AlGaAs sourcecan be achieved thanks to its compliance with electri-cal injection as already shown in [29], giving a clear ad-vantage to our approach in terms of portability, energyconsumption and integration with future quantum tech-nologies. On the other hand, the cavity effects presentlylimiting the fidelity to a polarization Bell state far fromdegeneracy can be avoided by applying an anti-reflectioncoating to the waveguide facets. An optimization of thedesign can also be implemented to reduce the modalbirefringence in order to further broaden the entangledphoton pair bandwidth [30] and to place the degeneracywavelength at the center of the telecom C-band to fullyexploit the WSS bandwidth. Using WSS technology cov-ering both the C and L bands would further allow toexploit the full bandwidth of our source. Besides, thebiphoton bandwidth and brightness could be optimizedby choosing the optimal sample length taking into ac-count SPDC efficiency and propagation losses.In addition to the improvement of the photon-pairsource, future investigations of different entanglement-distribution topologies can benefit from the flexibility ofour setup. In particular, using the full potential of our60 nm entanglement bandwidth, non-fully connected net-works which can host up to 28 users over 50 GHz ITUchannels could readily be implemented. In this topol-ogy, the network is broken up into independent fully-connected 4-user subnets which are all interconnectedvia extra links. Owing to the broad bandwidth of oursource, this scheme can be demonstrated without theneed to resort to passive BS multiplexing, as suggestedin refs. [13, 28] thus avoiding the reduction of the signalcaused by extra optical losses.Finally, in another approach we could take advantageof the cavity effects present in our sample to generateand coherently control biphoton frequency combs [31],opening the way to the utilization of frequency for quan-tum information processing [32, 33] and to the combinedexploitation of both polarization and frequency entangle-ment for future quantum networks.
ACKNOWLEDGEMENTS
This work has been supported by ANR (AgenceNationale de la Recherche) through the QUANTIFYProject (Project No. ANR-19-ASTR-0018-01) and byParis ˆIle-de-France R´egion in the framework of DIMSIRTEQ through the LION Project. The authors ac-knowledge Perola Milman for fruitful discussions andYves Jaouen for the C+L band tunable filter loan.
APPENDIX A: SAMPLE AND EXPERIMENTALSETUP
The sample consists of a 6-periodAl . Ga . As/Al . Ga . As Bragg reflector (lowermirror), a 366 nm Al . Ga . As core and a 2-periodAl . Ga . As/Al . Ga . Bragg reflector (uppermirror). Waveguides are fabricated using wet chemicaletching to define 5 µ m wide and 790 nm deep ridges ; thewaveguide length is 4 mm. The sample is pumped with atunable CW diode laser (TOPTICA TM Photonics DLpro 780) which is coupled into the waveguide through amicroscope objective (NA = 0.95, 63 × ); light emergingfrom the opposite end is collected with a second identicalmicroscope objective and sent to a fibre coupler, afterfiltering out the pump wavelength with a high pass filter.A thermocouple and a Peltier cooler, connected to a PIDcontroller, monitor and keep the waveguide temperatureconstant at 19 . ° C, fixing the wavelength degeneracy ofthe photon pairs to 1556 .
55 nm, which corresponds tothe center of the ITU 100 GHz channel number 26. Thedemux/mux stage is realized either with a WavelengthSelective Switch (model Finisar 4000s) or with a CoarseWavelength Division Multiplexing unit (model FS 78163)followed by a tunable C+L band filter (model AlnairLabs CVF-220-CL) depending on the spectral regionof interest. After the analysis and distribution stagephotons are detected with superconducting nanowiresingle photon detectors (SNSPD, Quantum Opus) andtemporal correlation measurement are performed with atime-to-digital converter (TDC, quTools).
APPENDIX B: THEORY
In this section, we give the expression of the quantumstate of the source and derive the theoretical fidelity F as a function of channel frequency. A. Quantum state emitted by the AlGaAs source
We start from the generic form of the two-photon stategenerated by collinear Type-II parametric downconver-sion: | ψ (cid:105) = (cid:90) (cid:90) + ∞−∞ d ω d ω C ( ω , ω ) | ω , H (cid:105) | ω , V (cid:105) (7)where | ω, α (cid:105) = a † α ( ω ) | vac (cid:105) denotes the state of the elec-tromagnetic field with one photon in a mode of po-larization α = H, V and angular frequency ω . Thecomplex function C ( ω , ω ) is called the joint spec-tral amplitude (JSA) and it is normalized to unity (cid:82)(cid:82) d ω d ω |C ( ω , ω ) | = 1. By rewriting the quan-tum state in the rotated basis: ω + = ω + ω and ω − = ω − ω , the JSA takes the form [34]: C ( ω , ω ) = f + ( ω + ) f − ( ω − ). The first term of the product corre-sponds to energy conservation during the downconver-sion process and the second term is related to momen-tum conservation (phase matching). The latter can benumerically calculated using the dispersion properties ofthe guided modes involved in the downconversion pro-cess. Cavity effects arising from the nonzero reflectiv-ity of the AlGaAs waveguide facets can be included inthe numerically-calculated JSA using a two-mode Airydistribution [31]. Since the waveguide is pumped by anarrow-linewidth CW laser, one can approximate f + toa Dirac delta function centered on the pump laser angularfrequency f + ( ω + ) = δ ( ω + − ω p ) and the resulting gener-ated quantum state is strongly frequency-anticorrelated.In this case, the state can be rewritten: | ψ (cid:105) = (cid:90) + ∞−∞ dΩΦ(Ω) | ω d + Ω , H (cid:105) | ω d − Ω , V (cid:105) , (8)where we defined for convenience ω d = ω p / ω − / f − ( ω − ).We then rewrite the state as a continuous superposi-tion of bipartite polarization-entangled states. To do so,we split the summation in Eq. (8) into two parts usingthe identity: (cid:82) ∞∞ = (cid:82) ∞ − (cid:82) −∞ then make the changeof variable Ω → − Ω into the second term and finallyrecombine the two integrals to obtain: | ψ (cid:105) = (cid:90) ∞ dΩ [Φ(Ω) | ω d + Ω , H (cid:105) | ω d − Ω , V (cid:105) + Φ( − Ω) | ω d − Ω , H (cid:105) | ω d + Ω , V (cid:105) ] . (9)0 B. Bell state fidelity
To calculate the fidelity to a | Ψ + (cid:105) Bell state as a func-tion of channel frequency, we derive the reduced densitymatrix in polarization space after spectral filtering. Atthe demultiplexing stage, signal and idler photons lyingin energy matched frequency channels are sent onto sep-arate fiber paths A and B . Namely, all signal photonswithin the frequency window [ ω d +(Ω − ∆ / , ω d +(Ω +∆ / A and all idler photons within thefrequency window [ ω d − (Ω + ∆ / , ω d − (Ω − ∆ / B , where ∆ is the channel width and Ω the de-tuning of the channel central frequency with respect todegeneracy. The resulting post-selected quantum statetakes the form: | ψ (cid:48) (cid:105) = (cid:90) ∞ dΩ f (Ω) [Φ(Ω) | ω d + Ω , H (cid:105) A | ω d − Ω , V (cid:105) B + Φ( − Ω) | ω d − Ω , H (cid:105) B | ω d + Ω , V (cid:105) A ] (10)with f (Ω) the filter lineshape, which in our case is as-sumed to be rectangular: f (Ω) = (cid:26) , for Ω ∈ [Ω − ∆ / , Ω + ∆ / , elsewhere . (11)The corresponding density operator is ˜ ρ = | ψ (cid:48) (cid:105) (cid:104) ψ (cid:48) | . Byfollowing the approach of ref. [35], we compute the re-duced polarization density matrix ρ by tracing out thefrequency part of the density operator: ρ = 1 N (cid:90) (cid:90) d ω (cid:48) d ω (cid:48)(cid:48) A (cid:104) ω (cid:48) | B (cid:104) ω (cid:48)(cid:48) | ˜ ρ | ω (cid:48)(cid:48) (cid:105) B | ω (cid:48) (cid:105) A . (12)with N a normalization constant. After some straight-forward algebra, one obtains: ρ = α | HV (cid:105) ABAB (cid:104) HV | + D | HV (cid:105) ABAB (cid:104)
V H | + D ∗ | V H (cid:105)
ABAB (cid:104) HV | + β | V H (cid:105)
ABAB (cid:104)
V H | , (13)where the 4 non-zero matrix elements are: α = 1 N (cid:90) ∞ dΩ f (Ω) | Φ(Ω) | (14) β = 1 N (cid:90) ∞ dΩ f (Ω) | Φ( − Ω) | (15) D = 1 N (cid:90) ∞ dΩ f (Ω)Φ(Ω)Φ ∗ ( − Ω) (16) and the normalization constant is set to N = (cid:82) ∞ dΩ f (Ω) (cid:2) | Φ(Ω) | + | Φ( − Ω) | (cid:3) such that Tr ρ = 1. Fi-nally, the fidelity to a | Ψ + (cid:105) Bell state can be evaluatedfrom the definition: F = (cid:0) Tr (cid:112) √ ρ | Ψ + (cid:105) (cid:104) Ψ + | √ ρ (cid:1) . APPENDIX C: COINCIDENCE HISTOGRAMSAND BELL CORRELATION CURVES
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