Heralding multiple photonic pulsed Bell-pairs via frequency-resolved entanglement swapping
Sofiane Merkouche, Valérian Thiel, Alex O. C. Davis, Brian J. Smith
HHeralding multiple photonic pulsed Bell-pairs via frequency-resolved entanglementswapping
Sofiane Merkouche ∗ , Val´erian Thiel , Alex O.C. Davis , and Brian J. Smith ∗ Department of Physics and Oregon Center for Optical, Molecular,and Quantum Science, University of Oregon, Eugene, Oregon 97403, USA Centre for Photonics and Photonic Materials, Department of Physics, University of Bath, Bath, BA2 7AY, UK (Dated: February 9, 2021)Entanglement is a unique property of quantum systems and an essential resource for many quan-tum technologies. The ability to transfer or swap entanglement between systems is an importantprotocol in quantum information science. Entanglement between photons forms the basis of dis-tributed quantum networks and the demonstration of photonic entanglement swapping is essentialfor their realization. Here an experiment demonstrating entanglement swapping from two inde-pendent multimode time-frequency entangled sources is presented, resulting in multiple heraldedtemporal-mode Bell states. Entanglement in the heralded states is verified by measuring condi-tional anti-correlated joint spectra as well as quantum beating in two-photon interference. Ourproof-of-concept experiment is able to distinguish up to five orthogonal Bell pairs within the samesetup, limited in principle only by the entanglement of the sources.
Introduction.—
Entanglement, the correlations dis-played between subsystems of a multipartite quantumsystem, is one of the most distinguishing properties ofquantum theory and a significant resource for quantuminformation science (QIS). Entanglement swapping [1] isa protocol that enables entanglement of quantum sys-tems that have never interacted and can be separated bylarge distances [2] or even time [3] [4]. This protocol un-derpins efforts to realize large-scale quantum networks asthe core element of quantum repeaters [5], in addition toshedding light on the fundamental nature and extent ofnon-locality of multipartite quantum systems.Entanglement swapping relies on the ability to performprojective measurements onto entangled states. For pho-tonic states, projective measurements onto two-photonentangled states can be implemented using a beam split-ter and mode-resolved measurements. Entanglementswapping has been experimentally demonstrated usingphotons entangled in their polarization [6], spatial [7],and temporal [8] degrees of freedom. Recent efforts haveshown that temporal- or pulsed-mode encoding offersunique opportunities for QIS [9]. Thus addressing pulse-mode entanglement manipulation and verification is atimely topic [10].In this Letter, we report an experiment demonstrat-ing the swapping of time-frequency two-photon entan-glement between two independent multimode photonpair sources. This is enabled by multiplexed frequency-resolved detection [11] to implement projective measure-ments of multiple temporal-mode Bell states. The entan-glement of the heralded two-photon states is verified bymeasurement of two-photon quantum beats [12], and thejoint-spectral intensity arising from four-fold frequencyresolved measurements. To the best of our knowledge,this is the first experiment to demonstrate both herald-ing and discrimination of multiple Bell pairs with a singlesource and measurement apparatus, as well as the first experiment employing simultaneous time-of-flight spec-trometry of four photons.
Theory –
The two-photon term of output state of asingle spontaneous parametric down conversion (SPDC)source can be expressed as | ψ (cid:105) = (cid:90) d ω s d ω i f ( ω s , ω i )ˆ a † ( ω s )ˆ b † ( ω i ) | vac (cid:105) (1)where ˆ a † ( ω ) (ˆ b † ( ω )) creates a photon with frequency ω inthe signal (idler) mode. The function f ( ω s , ω i ) is the nor-malized complex joint spectral amplitude (JSA), and itsmodulus squared, | f ( ω s , ω i ) | , is the joint spectral inten-sity (JSI). The JSI is the two-photon probability densityfunction in frequency space. The state contains spectralentanglement when the JSA is not factorable; that is,when f ( ω s , ω i ) (cid:54) = f s ( ω s ) f i ( ω i ). We assume that the pro-cess is single-mode in the polarization and transverse spa-tial degrees of freedom, so that only the time-frequencydegrees of freedom are relevant.Our experiment makes use of two independent, identi-cal SPDC sources, producing the state | ψ (cid:105) = (cid:90) d ω s d ω i d ω (cid:48) s d ω (cid:48) i f ( ω s , ω i )ˆ a † ( ω s )ˆ b † ( ω i ) × f ( ω (cid:48) s , ω (cid:48) i )ˆ a † ( ω (cid:48) s )ˆ b † ( ω (cid:48) i ) | vac (cid:105) (2)where the 1 and 2 subscripts denote the first and secondsources, respectively. Entanglement swapping requiresperforming a partial Bell-state measurement (BSM) onthe idler fields b and b , which is achieved by inter-fering the fields at a 50:50 beamsplitter and performinga frequency-resolved coincident detection at the output,at frequencies Ω j and Ω k . This measurement projectsthe input idler fields onto the two-color singlet Bell state | ψ − jk (cid:105) = √ ( | Ω j (cid:105) b | Ω k (cid:105) b − | Ω k (cid:105) b | Ω j (cid:105) b ), and the state a r X i v : . [ qu a n t - ph ] F e b FBS
Entanglement verificationState characterization
FBS
TOFSbTOFSb
BSM
S2S1
415 nm 100 fsTOFSa
In OutSMF 500mTOFSb:
Delay
CFBGIn OutTOFSa: a)b) c) d) e)
TOFSa
FIG. 1. a) Experimental setup - see main text for description. BSM: frequency-resolved Bell-state measurement implementedon the idler photons. FBS: fiber beamsplitter. SMF: single mode fiber. CFBG: chirped fiber Bragg grating. b) Statecharacterization - joint spectral measurement of the signal photons, conditioned on the BSM. c) Entanglement verification -two-photon interference as a function of relative delay τ , conditioned on the BSM. TOFSa and TOFSb are disperion-basedtime-of-flight spectrometers. d) Labeling convention for Ω j ( k ) measurements, (e.g. Ω corresponds to a bin centered at 830nm). e) Measured JSI for each of the sources, where λ s ( λ i ) is the signal (idler) wavelength. heralded in the signal fields is well-approximated by (seeSupplemental) | Ψ − jk (cid:105) = 1 (cid:112) C jk (cid:0) | φ j (cid:105) | φ k (cid:105) − | φ k (cid:105) | φ j (cid:105) (cid:1) . (3)Here we have defined | φ j ( k ) (cid:105) = (cid:90) d ωφ j ( k ) ( ω )ˆ a † ( ω ) | vac (cid:105) , (4)where φ j ( k ) ( ω ) ∝ f ( ω, Ω j ( k ) ) are normalized Gaus-sian amplitude functions given by φ j ( k ) ( ω ) ∝ exp [ − ( ω − ω j ( k ) ) / σ ], with ω j ( k ) and σ determined bythe JSA, and C jk = 1 − | (cid:104) φ j | φ k (cid:105) | (see Supplemental).The heralded state | Ψ − jk (cid:105) can be characterized by mea-suring its JSI, which is given by F jk ( ω , ω ) = 12 C jk | φ j ( ω ) φ k ( ω ) − φ k ( ω ) φ j ( ω ) | . (5)In order to verify entanglement in the state | Ψ − jk (cid:105) , beyondclassical correlations, two-photon interference is used ina manner similar to the method employed in reference[10]. Here the heralded signal photons are detected incoincidence at the output of a 50:50 beamsplitter, as afunction of relative arrival time delay τ . For the inputstate | Ψ − jk (cid:105) , the coincidence probability is given by (seeSupplemental) P jk ( τ ) = 12 + 12 e − τ σ cos [( ω j − ω k ) τ ] , (6) which oscillates at the difference frequency ω j − ω k . Theseoscillations, obtained without filtering of the interferingfields, are a hallmark of two-color entanglement of theinput state (see, for instance, reference [13]).If the heralding idler photons are not resolved in the(Ω j , Ω k ) space, the signal photons are heralded in thestate ˆ ρ = (cid:88) j,k p jk | Ψ − jk (cid:105) (cid:104) Ψ − jk | , (7)where p jk , normalized as (cid:80) j,k p jk = 1, is the joint spec-tral distribution of the idler photons at the output of thebeamsplitter, and gives the probability of heralding thestate | Ψ − jk (cid:105) . Likewise, the JSI of this state will be givenby F ( ω , ω ) = (cid:88) j,k p jk F jk ( ω , ω ) . (8)The state ˆ ρ is a mixed state which retains the antisym-metry of its constituent states | Ψ − jk (cid:105) . This is evidencedby its two-photon interference pattern, given by P ( τ ) = (cid:88) j,k p jk P jk ( τ ) . (9)where, notably, we still expect a coincidence peak to sur-vive at τ = 0. That is, the antisymmetry of the state ˆ ρ is the basis for this predicted antibunching [14]. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C oun t s x10 ov e r s FIG. 2. Left, state characterization: array of the measured JSI’s F jk of the signal photons, conditioned on the (Ω j , Ω k ) outcomesof the BSM. Right, entanglement verification: array of interference fringes P jk ( τ ), conditioned on the same (Ω j , Ω k ) outcomes,verifying entanglement of the states. The background color indicates the total number of counts for each plot, and thus theentire array maps out the p jk matrix. Each array was obtained in a single measurement run. Experiment.—
Our experimental setup is shownschematically in Fig. 1. For the light source we use ul-trashort (100 fs) pulses from a titanium-doped sapphire(Ti:Sapph) laser oscillator at a central wavelength of 830nm and a repetition rate of 80 MHz. These pulses arefrequency-doubled in a 1 mm-long birefringent BiB O crystal (BiBO) to generate the blue (415 nm) pump forthe SPDC sources. SPDC occurs at a second, 2.5 mm-long BiBO, which is double-passed to generate a pairof frequency-entangled photons on the first pass (source1), and on the second pass (source 2). This double-passconfiguration ensures that the two sources are identical.Type II phase matching permits the deterministic sep-aration of the signal and idler photons using polarizingbeamsplitters, after the blue pump has been filtered outusing dichroic mirrors. Signal and idler photons fromboth sources are collected into polarization-maintainingsingle-mode fibers (PM fibers) and directed to the re-mainder of the set-up for analysis and entanglementswapping. We measure a pair detection rate of up to 300kHz from each source using superconducting nanowiresingle-photon detectors (SNSPDs) from IDQuantique.The joint spectral intensity of each source is measuredefficiently using a time-of-flight spectrometer consistingof a pair of 500 m-long fibers (TOFSb). Each photonfrom the signal-idler pair is passed through the dispersivefiber, imparting a wavelength-dependent delay relative tothe Ti:Sapph reference pulse train. Time-resolved coin-cidence detections at the output, using a time-to-digitalconverter (ID900) with a resolution of about 30 ps, pro- vide a direct measure of the joint spectral intensity with aresolution of about 0.5 nm (see Fig. 1e). Assuming neg-ligible phase correlations [15], we estimate the amountof entanglement in the state by taking the square rootof the JSI and calculating the Schmidt number [16], forwhich we obtain a value of K ∼ j , Ω k ) heraldsa distinct state | Ψ − jk (cid:105) , as defined in Eq. 3, in the signalphotons.We characterize the heralded state | Ψ − jk (cid:105) by measuringits JSI using a similar set-up to TOFSb. Here, a chirpedfiber Bragg gratings (TOFSa) are used instead of thelong fibers, imparting a large dispersion and giving a highspectral resolution (0.1 nm, vs 1.5 nm for TOFSb), butwith higher losses (10 dB, vs 1 dB for TOFSb) [11]. TheBSM on the idler photons associates with each (Ω j , Ω k )pair a distinct signal-pair JSI F jk ( ω , ω ), correspondingto the state | Ψ − jk (cid:105) . In Fig. 2 a) we display an array ofthe measured JSI’s, with j, k ∈ [ − ,
4] according to the - - τ [ ps ] C oun t s ( × ) ov e r s
825 830 835825830835
825 830 835825830 λ [ nm ] λ [ n m ] ( ) - -
825 830 835825830 λ [ nm ] λ [ n m ] ( ) - -
825 830 835825830 λ [ nm ] λ [ n m ] ( ) -
825 830 835825830 λ [ nm ] λ [ n m ] ( )
825 830 835825830 λ [ nm ] λ [ n m ] ( ) - - - τ [ ps ] - - - - τ [ ps ] - - - - τ [ ps ] - - - τ [ ps ] - - τ [ ps ] - FIG. 3. Left: The integrated interference fringes P ( τ ) as given by Eq. 9. The red plot is not a fit, but the sum of the fits fromFig. 2b. The inset is the integrated JSI F ( ω , ω ) as given by Eq. 8. On the right is a set of JSI’s and interference fringescorresponding to 5 quasi-orthogonal modes, satisfying an overlap of ≤ .
15 as described in the text, chosen from the full setfrom Fig. 2. convention in Fig. 1. This data was all obtained in asingle measurement run.To perform the entanglement verification, the signalphotons are routed through another 50:50 FBS, and de-tected in coincidence at the output while scanning a free-space time delay τ in the arm of the signal from source1. As for the idler case, delay matching was obtainedby measuring the unheralded signal coincidences and ob-serving a Hong-Ou-Mandel dip. For each of the heraldedstates | Ψ − jk (cid:105) , we observe coincidence fringes that oscillateat the angular frequency difference ( ω j − ω k ), as pre-dicted in equation 6. We plot the corresponding arrayof these results in Fig. 2 b. Because we use probabilis-tic sources, it should be recalled that the probability ofone of the sources firing two pairs of photons is on thesame order as the probability of each source firing a sin-gle pair. This contributes to additional terms in the her-alded state which are inherent to all similar entanglementswapping experiments [17], and which appear as a con-stant background that we measure and subtract in theentanglement verification measurements [10]. After thisbackground subtraction, the measured visibility of theinterference fringes is roughly 75%, consistent with themaximum visibility expected from source matching mea-surements (see Supplemental). Finally, it is notable thatthe total number of counts measured in each ( j, k ) binof the arrays in figure 2 is in fact a measure of p jk . Wehighlight this by coloring the background of the interfer-ence plots as a function of the total number of counts,such that the entire array can be seen as a plot of theJSI of the idler photons after the beamsplitter. Note theridge along the diagonal ( j = k ) due to Hong-Ou-Mandelinterference.As pointed out in the previous section, performing anunresolved BSM on the idler photons, where the measure-ment is integrated over all (Ω j , Ω k ), projects the signalphotons onto the mixed state ρ denoted in Eq.7. Al-though mixed, this state is a convex combination of an- tisymmetric Bell pairs over frequency space. Two no-table features arise from this fact. First, there are nocoincidences where ω j = ω k , due to the aforementionedHong-Ou-Mandel interference in the idler BSM, and thiscontributes to a ridge along the diagonal of the integratedJSI F ( ω , ω ), displayed as an inset in Fig. 3a. Second,the antisymmetry of the state is preserved, and this isevidenced by the fully visible peak in the integrated two-photon interference scan of P ( τ ), which is displayed inthe main plot of Fig. 3a. Note also that the red curvein that plot is not a new fit to the data, but rather justthe sum of the individual fits to the P jk ( τ ). Taken alltogether, our data nicely highlights the quantum natureof measurement, whereby different quantum states ariseas a consequence of different measurement results.As a final point, we note that although the number ofstates | Ψ − jk (cid:105) that we can resolve is essentially limited bythe resolution of the spectrometers on the heralding side,not all these states will be orthogonal, because the num-ber of available orthogonal modes in the sources is finiteto begin with. This is indeed the property that is quan-tified by the Schmidt number. A combinatorics argu-ment shows that, for two identical sources with Schmidtnumber K , one can herald at most K ( K − / (cid:82) d ω d ω F jk ( ω , ω ) F j (cid:48) k (cid:48) ( ω , ω ) ≤ (cid:15) , ∀ ( j, k ) (cid:54) = ( j (cid:48) , k (cid:48) ),where (cid:15) can be chosen arbitrarily small. In an appli-cation setting such as multiplexed entanglement distri-bution, one could in principle restrict consideration tothis smaller set of j, k heralding events which correspond Alice
BobCharlie
FIG. 4. Frequency-multiplexed entanglement swappingscheme. Charlie performs a frequency-resolved BSM on pho-tons from sources with high-dimensional entanglement. Eachindependent result C jk enables distribution of a photonic Bellpair to a separate pair of parties A j and B k on the Alice andBob sides, even if using a single fiber transmission line. to orthogonal states, in order to avoid any unwantedcrosstalk between frequency channels. In Fig. 3b, weshow a representative set of quasi-orthogonal modes se-lected with (cid:15) = 0 .
15. In this case there are 5 orthogonalmodes, while our estimated K of 4 predicts a maximumof 6 orthogonal modes. Discussion.—
One conceivable application of this workis a multi-party quantum key distribution network, asdepicted schematically in Fig. 4. Several parties on the“Alice” side, denoted by A j , are to share entanglementwith several parties on the “Bob” side, denoted by B k ,such that each A j is connected to each B k by an indepen-dent channel. Such a network is enabled by “Charlie”,who possesses two identical sources of photon pairs withhigh-dimensional frequency entanglement. By perform-ing a frequency-resolved entanglement swapping protocolas we describe in this work, Charlie is able to convert themultimode entanglement of the sources into one of manydistinct Bell pairs, dependent on his outcome C jk , eachof which can be routed to a distinct user pair A j and B k .In this way a single quantum repeater can serve multiplechannels, multiplexed in the frequency domain.In conclusion, we have demonstrated a multimode fre-quency entanglement swapping scheme that is easily im-plemented with generic SPDC sources and readily avail-able measurement apparatus. Our design provides a sim-ple way of heralding a high number of orthogonal fre-quency Bell pairs that is completely measurement-basedand requires no source engineering. Alternatively, ourprotocol could be combined with frequency translators inthe signal beams [19] to generate multiple copies of thesame Bell state using broadband sources in a versatilemanner, again without the requirement of source engi-neering. Finally, with the advent of push-button sourcesof entangled photon pairs [20], multiplexed quantum re-peaters of the kind that our protocol allows could proveto be a scalable solution for quantum communication net-works. Acknowledgements.—
This project has received fund-ing from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No.665148, the United Kingdom Defense Science and Tech-nology Laboratory (DSTL) under contract No. DSTLX-100092545, and the National Science Foundation underGrant No. 1620822. ∗ Corresponding author: sofi[email protected][1] M. ˙Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ek-ert, Phys. Rev. Lett. , 4287 (1993).[2] H. de Riedmatten, I. Marcikic, J. A. W. van Houwelin-gen, W. Tittel, H. Zbinden, and N. Gisin, Phys. Rev. A , 050302 (2005).[3] A. Peres, Journal of Modern Optics , 479(2012).[5] H.-J. Briegel, W. D¨ur, J. I. Cirac, and P. Zoller, PhysicalReview Letters , 5932 (1998).[6] J.-W. Pan, D. Bouwmeester, H. Weinfurter, andA. Zeilinger, Physical Review Letters , 3891 (1998).[7] Y. Zhang, M. Agnew, T. Roger, F. S. Roux, T. Konrad,D. Faccio, J. Leach, and A. Forbes, Nature Communi-cations , 632 (2017), 1609.06094.[8] M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon,and H. Zbinden, Nature Physics , 692 (2007).[9] B. Brecht, D. V. Reddy, C. Silberhorn, and M. G.Raymer, Physical Review X , 041017 (2015).[10] F. Graffitti, P. Barrow, A. Pickston, A. M. Bra´nczyk,and A. Fedrizzi, Phys. Rev. Lett. , 053603 (2020).[11] A. O. C. Davis, P. M. Saulnier, M. Karpi´nski, and B. J.Smith, Opt. Express , 12804 (2017).[12] Z. Y. Ou and L. Mandel, Phys. Rev. Lett. , 54 (1988).[13] S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Lang-ford, and A. Zeilinger, Physical Review Letters ,253601 (2009).[14] A. Fedrizzi, T. Herbst, M. Aspelmeyer, M. Barbieri,T. Jennewein, and A. Zeilinger, New Journal of Physics , 103052 (2009).[15] A. O. C. Davis, V. Thiel, and B. J. Smith, Optica ,1317 (2020).[16] C. K. Law, I. A. Walmsley, and J. H. Eberly, PhysicalReview Letters , 5304 (2000).[17] C. Wagenknecht, C.-M. Li, A. Reingruber, X.-H. Bao,A. Goebel, Y.-A. Chen, Q. Zhang, K. Chen, and J.-W.Pan, Nature Photonics , 549 (2010).[18] D. V. Reddy and M. G. Raymer, Optica , 423 (2018).[19] L. J. Wright, M. Karpi´nski, C. S¨oller, and B. J. Smith,Phys. Rev. Lett. , 023601 (2017).[20] F. Basso Basset, M. B. Rota, C. Schimpf, D. Tedeschi,K. D. Zeuner, S. F. Covre da Silva, M. Reindl, V. Zwiller,K. D. J¨ons, A. Rastelli, and R. Trotta, Phys. Rev. Lett. , 160501 (2019).[21] V. Torres-Company, J. Lancis, and P. Andres, Progressin Optics , 1 (2011).[22] K. Goda and B. Jalali, Nature Photonics , 102 (2013).[23] A. O. C. Davis, P. M. Saulnier, M. Karpi´nski, and B. J.Smith, Opt. Express , 12804 (2017).[24] V. Ansari, J. M. Donohue, B. Brecht, and C. Silberhorn,Optica , 534 (2018), 1803.04316. [25] P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk,A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, Phys.Rev. Lett. , 133601 (2008). SUPPLEMENTAL MATERIALSFour photon state
To correctly describe the four photon state in our entanglement swapping setup, we begin by labeling the pathswith bosinic operators according to Fig5. Each source generates two pairs of photons by spontaneous parametricdown conversion whose signal and idler paths are respectively labelled ˆ a n and ˆ b n , where n ∈ { , } denotes the sourcenumber. These follow the standard bosonic commutation rules. The Hamiltonian describing source n is generallywritten as: ˆ H n = √ η n (cid:90) d ω s d ω i u n ( ω s + ω i ) sinc (cid:20) ∆ k n ( ω s , ω i ) L (cid:21) ˆ a † n ( ω s )ˆ b † n ( ω i ) + h.c. (Supp.1)where u n represents the spectral mode of the pump, ∆ k is the wave-vector mismatch between the pump, signal andidler waves and η n is the gain of the parametric process, which depends on the crystal length L , the non-linear strengthof the material and the number of photon in the pump beam.In the low gain regime, it is straightforward to compute the state at the output of the n th source: | ψ n (cid:105) = ∞ (cid:88) k =0 √ η nk k ! (cid:18)(cid:90) d ω s d ω i f n ( ω s , ω i )ˆ a † n ( ω s )ˆ b † n ( ω i ) (cid:19) k | vac (cid:105) (Supp.2)The function f is the joint spectral amplitude (JSA) which defines the energy conservation between the daughterphotons. In general, the two sources can be different, but for the sake of generality, we’ll assume that they areequivalent, thus having an equal JSA and effective non-linearity.To derive the state ˆ ρ RB heralded by the BSM, we begin by writing the SPDC state due to two independent andidentical sources as a tensor product | ψ SPDC (cid:105) = | ψ (cid:105) ⊗ | ψ (cid:105) = (cid:40) ˆ1 + √ η (cid:90) d ω s d ω i f ( ω s , ω i )ˆ a † ( ω s )ˆ b † ( ω i ) + η (cid:18)(cid:90) d ω s d ω i f ( ω s , ω i )ˆ a † ( ω s )ˆ b † ( ω i ) (cid:19) + . . . (cid:41) ⊗ (cid:40) ˆ1 + √ η (cid:90) d ω s d ω i f ( ω s , ω i )ˆ a † ( ω s )ˆ b † ( ω i ) + η (cid:18)(cid:90) d ω s d ω i f ( ω s , ω i )ˆ a † ( ω s )ˆ b † ( ω i ) (cid:19) + . . . (cid:41) | vac (cid:105) . (Supp.3)We expand this and keep only terms of order η , which are responsible for the four-photon contribution: | ψ η (cid:105) (cid:39) (cid:90) d ω s d ω i d ω (cid:48) s d ω (cid:48) i f ( ω s , ω i ) f ( ω (cid:48) s , ω (cid:48) i )ˆ a † ( ω s )ˆ b † ( ω i )ˆ a † ( ω (cid:48) s )ˆ b † ( ω (cid:48) i ) | vac (cid:105) + 12 (cid:90) d ω s d ω i d ω (cid:48) s d ω (cid:48) i f ( ω s , ω i ) f ( ω (cid:48) s , ω (cid:48) i )ˆ a † ( ω s )ˆ b † ( ω i )ˆ a † ( ω (cid:48) s )ˆ b † ( ω (cid:48) i ) | vac (cid:105) + 12 (cid:90) d ω s d ω i d ω (cid:48) s d ω (cid:48) i f ( ω s , ω i ) f ( ω (cid:48) s , ω (cid:48) i )ˆ a † ( ω s )ˆ b † ( ω i )ˆ a † ( ω (cid:48) s )ˆ b † ( ω (cid:48) i ) | vac (cid:105) . (Supp.4)For convenience, we will denote these three terms | Ψ (cid:105) , | Ψ (cid:105) , and | Ψ (cid:105) , so that we have, with the proper renormal-ization | ψ η (cid:105) = (cid:114) (cid:18) | ψ (cid:105) + 12 | ψ (cid:105) + 12 | ψ (cid:105) (cid:19) , (Supp.5)and the density matrix for this state isˆ ρ η = 23 (cid:18) | ψ (cid:105) (cid:104) ψ | + 14 | ψ (cid:105) (cid:104) ψ | + 14 | ψ (cid:105) (cid:104) ψ | (cid:19) + (cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40) cross terms . (Supp.6)The cross terms correspond to coherence between the terms in | ψ η (cid:105) , which is ultimately due to the optical phase ofthe pump. Because our sources are pumped by the same laser, we do indeed expect them to be mutually coherent.However, over the course of a measurement run (several hours), the phase drifts significantly, so it is reasonable to Source 1Source 2
FIG. 5. General scheme. average over it, and thus these cross terms vanish. For the rest of the manuscript, we will neglect the terms | φ (cid:105) and | φ (cid:105) from our computations and focus solely on | φ (cid:105) .Most of the experiments that we performed rely on performing a spectrally-resolved Bell state measurement betweenthe idler photon. There are two cases that we should consider, whether a spectral coincidence between the idlersprojects the signal into a pure state or into a mixed state. Pure state approximation
Heralded state and JSI
A Bell state measurement is performed on the idler photons by interfering them at a beamsplitter and detectingcoincidences at the output while monitoring the frequency of the interfering idler photons. The beamsplitter operationis defined by the following operators:ˆ c † ( ω ) = ˆ b † ( ω ) + ˆ b † ( ω ) √ , ˆ d † ( ω (cid:48) ) = ˆ b † ( ω (cid:48) ) − ˆ b † ( ω (cid:48) ) √ , (Supp.7)and coincidences are detected between ˆ c and ˆ d . We consider that we have a perfect resolution in that spectralmeasurement, such that the POVM element for this detection is simply given byˆΠ BSM jk = ˆ c † (Ω j ) ˆ d † (Ω k ) | vac (cid:105) (cid:104) vac | ˆ c (Ω j ) ˆ d (Ω k ) , (Supp.8)which is a projector onto the monochromatic frequencies Ω j and Ω k .We then proceed to compute the heralded signal state. It is defined by: | Ψ − jk (cid:105) = ˆΠ BSM jk | ψ (cid:105)√ p jk , (Supp.9)where the norm p jk is given by p jk = (cid:104) ψ | ˆΠ BSM jk | ψ (cid:105) . (Supp.10)This the probability density of a coincidence between the idler photons at (Ω j , Ω k ), or equivalently, the JSA of theidlers after the beamsplitter. Upon computing p jk , we obtain p jk = 12 (cid:104) ρ (Ω j , Ω j ) ρ (Ω k , Ω k ) − ρ (Ω j , Ω k ) ρ (Ω k , Ω j ) (cid:105) , (Supp.11)where ρ (Ω , Ω (cid:48) ) = (cid:90) d ω f ( ω, Ω) f ∗ ( ω, Ω (cid:48) ) (Supp.12)is the idlers’ density matrix. From Eq.(Supp.9), we obtain the following expression for the heralded state: | Ψ − jk (cid:105) = 1 (cid:112) C jk | φ j (cid:105) | φ k (cid:105) − | φ k (cid:105) | φ j (cid:105)√ , (Supp.13)where we defined the the functions φ j ( k ) ( ω ) = f ( ω, Ω j ( k ) ) /ρ (Ω j ( k ) , Ω j ( k ) ) and the states | φ j ( k ) (cid:105) as | φ j ( k ) (cid:105) = (cid:90) d ω φ j ( k ) ( ω )ˆ a † ( ω ) | vac (cid:105) . (Supp.14)The normalization of Eq. (Supp.13) is given by C jk = 1 − |(cid:104) φ j | φ k (cid:105)| for any heralded frequencies Ω j and Ω k . The φ j ( k ) functions are normalized but not orthogonal. Their definition follows from the fact that when the idlers frequency isinfinitely resolved, the signals are heralded is a pure state. It is therefore convenient to approximate the JSA f as aGaussian distribution (for instance by approximating the Sinc function by a Gaussian of the same width), such that f ( ω s , ω i ) = C exp (cid:34) − (cid:18) ω s − ω σ s (cid:19) − (cid:18) ω i − ω σ i (cid:19) − α ( ω s − ω )( ω i − ω ) , (cid:35) (Supp.15)where σ s ( σ i ) is the spectral width projected on the ω s ( ω i ) axis, ω is the center frequency, α quantifies the amountof spectral entanglement and C = (cid:16)(cid:82) d ω | f ( ω, ω (cid:48) ) | (cid:17) − / is a normalization constant. From there, the expression of φ j ( k ) is given by φ j ( k ) ( ω ) = 1 (cid:112) σ s √ π exp[ − ( ω − ω j ( k ) ) / σ s ] , (Supp.16)where the center frequency of this heralded marginal dependent on a coincidence with an idler at frequency Ω j ( k ) is ω j ( k ) = 2 α σ s Ω j ( k ) . (Supp.17)Under this prescription, any energy variation due to the idler’s frequency is absorbed in the normalization, hence toretrieve any quantity that depends on the JSA, a proper weight has to be applied.The heralded joint spectrum is then easily obtained by computing F jk ( ω , ω ) = (cid:12)(cid:12)(cid:12)(cid:68) ω , ω (cid:12)(cid:12)(cid:12) Ψ − jk (cid:69)(cid:12)(cid:12)(cid:12) = 12 C jk (cid:12)(cid:12)(cid:12) φ j ( ω ) φ k ( ω ) − φ j ( ω ) φ k ( ω ) (cid:12)(cid:12)(cid:12) . (Supp.18)The heralded JSI is zero when j = k , i.e. when the heralding idler photons are indistinguishable. The photons bunchat either output of the idler beamsplitter, and therefore the probability to measure four-fold coincidences is null.Each heralded color of the JSI consists of two identical separable Gaussian joint spectra centered at ω j and ω k whichseparation depends on the heralding frequency Ω j , Ω k . A representation of these JSI is given in Fig6(b), using theGaussian model which parameters are fitted to the experimental distribution Eq. (Supp.15) of our source (see Fig8).In the absence of spectral resolution in the BSM, the measurement operator becomesˆΠ BSM = (cid:88) j,k ˆΠ BSM jk , (Supp.19)and the heralded state is mixed, given by ˆ ρ = (cid:88) jk p jk | Ψ − jk (cid:105) (cid:104) Ψ − jk | . (Supp.20)The JSI due to this mixed state is then given by F ( ω , ω ) = Tr (ˆ ρ | ω , ω (cid:105) (cid:104) ω , ω | ) = (cid:88) jk p jk (cid:12)(cid:12)(cid:12)(cid:68) ω , ω (cid:12)(cid:12)(cid:12) Ψ − jk (cid:69)(cid:12)(cid:12)(cid:12) = (cid:88) jk p jk F jk ( ω , ω ) , (Supp.21)0 ω [ fs - ] ω [ f s - ] Heralded state JSI - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - FIG. 6. Left: full heralded JSI defined by the sum of Eq.(Supp.21) over all j, k . Right: F jk for different values labeled as λ j and λ k . All the plots on this figures have the same axes. which we represented in Fig6(a). Upon performing the calculation, we find that the expression of the JSI is given by: F ( ω , ω ) = 12 (cid:104) ρ s ( ω , ω ) ρ s ( ω , ω ) − | ρ s ( ω , ω ) | (cid:105) , (Supp.22)where ρ s ( ω, ω (cid:48) ) = (cid:82) dΩ f ( ω, Ω) f ∗ ( ω (cid:48) , Ω) is the signals’ density matrix. The expression is similar to (Supp.11). Indeed,equation (Supp.22) is exactly what would be obtained were the beamsplitter placed in the signal paths, rather thanthe idler paths, and this equivalence is ultimately due to the non-separability of the JSA f ( ω, Ω).
Entanglement verification
To verify that the heralded state is indeed entangled, we combine both signal photon with another beamsplitter,and label its output according to Fig5:ˆ x † ( ω ) = ˆ a † ( ω ) e iωτ + ˆ a † ( ω ) √ , ˆ y † ( ω (cid:48) ) = ˆ a † ( ω (cid:48) ) e iω (cid:48) τ − ˆ a † ( ω (cid:48) ) √ τ between the two inputs. We apply this transformation to the heralded state(Supp.13) and define our verification POVM as a coincidence between the output of the signal beamsplitter:ˆΠ verif = (cid:90) d ω d ω (cid:48) ˆ x † ( ω )ˆ y † ( ω (cid:48) ) | vac (cid:105) (cid:104) vac | ˆ x ( ω )ˆ y ( ω (cid:48) ) . (Supp.24)The probability of getting a coincidence heralded by a BSM at frequencies Ω j ,Ω k then given by: P jk ( τ ) = (cid:68) Ψ − jk (cid:12)(cid:12)(cid:12) ˆΠ verif (cid:12)(cid:12)(cid:12) Ψ − jk (cid:69) (Supp.25)= (cid:90) d ω d ω (cid:48) (cid:12)(cid:12)(cid:12) (cid:104) ω, ω (cid:48) | Ψ − jk (cid:105) (cid:12)(cid:12)(cid:12) where, | ω, ω (cid:48) (cid:105) = ˆ x † ( ω )ˆ y † ( ω (cid:48) ) | vac (cid:105) . As for the heralded JSI, the full probability summed over all possible heraldingfrequency bins is P ( τ ) = (cid:88) jk p jk P jk ( τ ) . (Supp.26)1 - - τ [ ps ] P ( τ ) FIG. 7. Left: probability P ( τ ) of coincidence heralded by a BSM without spectral resolution from Eq.(Supp.29). Right: P jk for different values labeled as λ j and λ k , Eq.(Supp.25). The color gradient indicates the probability of getting a coincidenceon the herald photons. All the plots on this figures have the same axes. Upon evaluating Eq.(Supp.25), we obtain P jk ( τ ) = 12 (cid:32) e − σ s τ cos (cid:2) ( ω j − ω k ) τ (cid:3) − | (cid:104) φ j | φ k (cid:105) | (1 + e − σ s τ )1 − | (cid:104) φ j | φ k (cid:105) | (cid:33) , (Supp.27)which oscillate at the difference frequency ω j − ω k between the heralded states. These fringes are a witness ofentanglement swapping, which can be simply demonstrated by setting α → ω j ( k ) (see Eq.(Supp.17)), thusremoving the oscillating term in Eq.(Supp.27). Therefore, non entangled state will only manifest as a constant termas a function of τ .This probability depends on the overlap integral (cid:104) φ j | φ k (cid:105) which quantifies the overlap between the marginals of theheralded state. This quantity goes to zero while the heralding bins Ω j , Ω k are further apart while it goes to unitwhen they become degenerate. However, asymptotic analysis of Eq.(Supp.26) shows that the weight factor p jk asdefined in Eq.(Supp.11) constrains the total probability to be zero. Therefore, it is reasonable to approximate thespectrally-resolved probability as P jk ( τ ) ≈ (cid:16) e − σ s τ cos (cid:2) ( ω j − ω k ) τ (cid:3)(cid:17) , (Supp.28)to which we fit our experimental results. Finally, we can evaluate (Supp.26) without any approximations to obtain P ( τ ) = 14 (cid:32) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d ω dΩ f ( ω, Ω) e iωτ (cid:12)(cid:12)(cid:12)(cid:12) − (cid:90) d Ω ρ (Ω , Ω (cid:48) ) ρ (Ω (cid:48) , Ω) − (cid:90) d ω ρ s ( ω, ω (cid:48) ) ρ s ( ω (cid:48) , ω ) e i ( ω − ω (cid:48) ) τ (cid:33) , (Supp.29)which also depends on the signals’ density matrix as in (Supp.22). The entanglement verification signal thereforecontains four terms. The first one is simply background, while the second one is the overlap integral between thetwo sources. Evaluating this term while scanning the delay τ reveals a peak (which is Gaussian in our approximatedmodel) which width depends on the joint temporal distribution of the sources. The last two terms consist respectivelyon the overlap integrals between the idlers’ and the signals’ density matrices of each source. The latter consists of anunheralded HOM dip between the signals photons. Hence, the full verification signal corresponds to peak centered ina HOM dip. In Fig7), we plotted a simulation of the full signal P ( τ ) as well as the spectrally resolved probabilities P jk ( τ ) for different heralding frequencies Ω j , Ω k . Mixed state model
Heralded state and JSI
In the realistic case, the idlers’ BSM is not performed with perfect resolution, but rather over a certain spectralwindow. In our case, this is due to the resolution of the time-of-flight spectrometer, which is a convolution of2multiple response function in the frequency-to-time conversion. It is dominated by the timing jitter ( (cid:39)
30 ps) of thesuperconducting nanowires.When this resolution is not perfect, then we can show that the signal photons are heralded into a mixed state. Webegin by rewritting the idlers’ POVM as:ˆΠ
BSM jk = (cid:90) dΩdΩ (cid:48) | t j (Ω) | | t k (Ω (cid:48) ) | ˆ c † (Ω) ˆ d † (Ω (cid:48) ) | vac (cid:105) (cid:104) vac | ˆ c (Ω) ˆ d (Ω (cid:48) ) (Supp.30)where t j ( k ) (Ω) is a filter transmission amplitude centered at Ω j ( k ) . It is straightforward to show that the POVM(Supp.8) is obtained by setting t j ( k ) (Ω) → δ (Ω − Ω j ( k ) ).It can be seen from the Fig8 that filtering the idler with a finite filter function results in a joint spectrum that is notnecessarily separable, and therefore the signal photons are heralded in a mixed state defined by the density matrix:ˆ ρ jk = Tr ˆ b (cid:104) ˆΠ BSM jk | ψ (cid:105) (cid:104) ψ | (cid:105) Tr (cid:104) ˆΠ BSM jk | ψ (cid:105) (cid:104) ψ | (cid:105) (Supp.31)where Tr ˆ b is the partial trace over the subspace defined by operators ˆ b and ˆ b .Similar to the pure state case, the p jk are defined as p jk = Tr (cid:104) ˆΠ BSM jk | ψ (cid:105) (cid:104) ψ | (cid:105) = 12 (cid:90) dΩdΩ (cid:48) | t j (Ω) | | t k (Ω (cid:48) ) | (cid:16) ρ (Ω , Ω) ρ (Ω (cid:48) , Ω (cid:48) ) − ρ (Ω , Ω (cid:48) ) ρ (Ω (cid:48) , Ω) (cid:17) (Supp.32)where the idlers’ density matrix is defined as Eq.(Supp.12), and we again obtain Eq.(Supp.11) by setting the filters t j ( k ) as δ functions.We may now compute the heralded state density matrix:ˆ ρ jk = (cid:90) d ω d ω (cid:48) d˜ ω d˜ ω (cid:48) ρ jk ( ω, ω (cid:48) ; ˜ ω, ˜ ω (cid:48) ) | ω (cid:105) | ˜ ω (cid:105) (cid:104) ω (cid:48) | (cid:104) ˜ ω (cid:48) | (Supp.33)where ρ jk ( ω, ω (cid:48) ; ˜ ω, ˜ ω (cid:48) ) =12 p jk (cid:90) dΩdΩ (cid:48) | t j (Ω) | | t k (Ω (cid:48) ) | (cid:0) f ( ω, Ω) f (˜ ω, Ω (cid:48) ) − f ( ω, Ω (cid:48) ) f (˜ ω, Ω) (cid:1)(cid:0) f ∗ ( ω (cid:48) , Ω) f ∗ (˜ ω (cid:48) , Ω (cid:48) ) − f ∗ ( ω (cid:48) , Ω (cid:48) ) f ∗ (˜ ω (cid:48) , Ω) (cid:1) (Supp.34)The heralded JSI is then given by by: F jk ( ω , ω ) = (cid:104) ω , ω | ˆ ρ jk | ω , ω (cid:105) (Supp.35)where | ω , ω (cid:105) = ˆ a † ( ω )ˆ a † ( ω ) | vac (cid:105) . Finally, as in the pure state case, in the absence of frequency resolution at theBSM, the heralded mixed state is ˆ ρ = (cid:88) jk p jk ˆ ρ jk , (Supp.36)and the JSI for this state is again given by F ( ω , ω ) = (cid:88) jk p jk F jk ( ω , ω ) . (Supp.37) Entanglement verification
When the signal photons in the state ˆ ρ jk are incident on a 50:50 beamsplitter, the coincidence fringes at the outputare given by3 P jk ( τ ) = Tr (cid:16) ˆΠ verif ˆ ρ jk (cid:17) = (cid:90) d ω (cid:104) ω, ω (cid:48) | ˆ ρ jk | ω, ω (cid:48) (cid:105) , (Supp.38)where | ω, ω (cid:48) (cid:105) = ˆ x † ( ω )ˆ y † ( ω (cid:48) ) | vac (cid:105) as before. When evaluated, this gives P jk ( τ ) = 12 p jk (cid:90) dΩdΩ (cid:48) | t j (Ω) | | t k (Ω (cid:48) ) | (cid:16) ρ (Ω , Ω) ρ (Ω (cid:48) , Ω (cid:48) ) − ρ (Ω , Ω (cid:48) ) ρ (Ω (cid:48) , Ω) (cid:17) P (Ω , Ω (cid:48) , τ ) , (Supp.39)where P (Ω , Ω (cid:48) , τ ) = 12 (cid:32) e − σ s τ cos [2 ασ s (Ω , Ω (cid:48) ) τ ] − O (Ω , Ω (cid:48) )(1 − e − σ s τ )1 − O (Ω , Ω (cid:48) ) (cid:33) , O (Ω , Ω (cid:48) ) = ρ (Ω , Ω (cid:48) ) ρ (Ω (cid:48) , Ω) ρ (Ω , Ω) ρ (Ω (cid:48) , Ω (cid:48) ) , (Supp.40)is the pure state interference expression from (Supp.27).Since, by construction, (cid:80) j,k | t j (Ω) | | t k (Ω (cid:48) ) | = 1, the integrated coincidence probability is, as before, P ( τ ) = (cid:88) j,k p jk P jk ( τ ) = 14 (cid:90) dΩdΩ (cid:48) (cid:16) ρ (Ω , Ω) ρ (Ω (cid:48) , Ω (cid:48) ) − ρ (Ω , Ω (cid:48) ) ρ (Ω (cid:48) , Ω) (cid:17) P (Ω , Ω (cid:48) , τ ) , (Supp.41)which is equivalent to (Supp.29). Detection
The single photon are detected utilizing superconducting nanowire single photon detector (SNSPD) from IDQuan-tique (ID281) which can detect the arrival time of photons with a resolution of 20 ps. This temporal resolution istranslated into spectral resolution using time-of-flight spectrometers (TOFS), thanks to frequency-to-time conversion[21, 22]. For coarse spectral resolution, we used two spools of 500 meters-long HP780 fiber. These imprint a dispersionof about 50 ps/nm, hence the spectrometers have a resolution of 0.4 nm. The losses per spool at 830 nm are about33%. For fine resolution, we used two chirped fiber Bragg gratings (CFBG from Teraxion) with a dispersion of 1000ns/nm [23], or a spectral resolution of less than 0.02 nm. This extra resolution comes with a heavy loss of over80% and also with a finite spectral window of 10 nm. The signals coming out of the detectors are registered witha time-to-digital converter (TDC, ID900 from IDQuantique). The time reference is provided by the clock generatedby the laser source, thus ensuring that each time tag is taken with respect to a stable signal for each pulse. Withthis setup, it is possible to register coincidences between any combination of the four photons with the advantage ofmeasuring their wavelength. This allows for “pixelization” of any event into spectral bins.To ensure that our TOFS are accurate, it is necessary to calibrate them. This procedure is usually realized usingsingle frequency emission from known white light sources, but this isn’t possible with TOFS since they require a pulsedsignal to extract their time tags. While it is usually sufficient to use rough estimate of the dispersion imprinted bythe fiber spool or by the CFBG, this doesn’t take into account any other source of dispersion in the setup. Therefore,we opted for an in-situ calibration utilizing the single photons from the SPDC.We utilized a pulse shaper based on putting a spatial light modulator (SLM) in a 4-f line, enabling to address boththe amplitude and phase over a 30 nm range with a resolution of 0.02 nm. By scanning a narrow interference filterof 1 nm FWHM over the SLM mask while recording the resulting time tags, we obtain a linear dependency betweenthe recorded time tags and the wavelength of the filter set on the pulse shaper. The slope of that function is then thedispersion parameter of the TOFS. With the CFBG-based spectrometers, we obtained a dispersion of 944 ± ± − ±
820 825 830 835 840810820830840850 [ nm ] [ n m ] Source 1
820 825 830 835 840810820830840850 [ nm ] [ n m ] Source 2
FIG. 8. Joint spectral intensity of both sources. Insets: marginal spectra.
Entangled photon source
Source distinguishability
The entanglement verification protocol we use, that is, the two-photon interference of the state | Ψ − jk (cid:105) , ultimatelyrelies on the indistinguishability of the two source states. To see this, we relabel the source JSA’s as f ( ω, Ω) and f ( ω, Ω), and for simplicity, we assume that they are identical up to a translation in frequency space. Note now thatthis leads to a heralded state | Ψ − jk (cid:105) ∝ | φ j (cid:105) | φ k (cid:105) − | φ k (cid:105) | φ j (cid:105) , (Supp.42)where | φ j ( k ) (cid:105) = (cid:90) d ωφ j ( k ) ( ω )ˆ a † ( ω ) | vac (cid:105) , (Supp.43)and φ j ( k ) = f ( ω, Ω j ( k ) ) ρ (Ω j ( k ) , Ω j ( k ) ) . (Supp.44)Although this state is still entangled, this entanglement cannot in general be assessed through measuring coincidencefringes in P jk ( τ ), because the distinguishability of f and f will reduce the visibility of these fringes. To see this, werecalculate P j k ( τ ) in its approximate form (Supp.28), and find P jk ( τ ) ≈ (cid:32) − V jk e σ s τ cos (cid:34)(cid:32) ω j + ω j − ω k + ω k (cid:33) τ (cid:35)(cid:33) (Supp.45)where the visibility V jk is given by V jk = (cid:18)(cid:90) d ωφ ∗ j ( ω ) φ j ( ω ) (cid:19) (cid:18)(cid:90) d ωφ ∗ k ( ω ) φ k ( ω ) (cid:19) . (Supp.46)5 [ a.u. ] C o i n c i d e n ce s r a t e [ H z ] FIG. 9. Measured coincidence fringes P cc ( τ ) with a contrast of 80%. We can maximize this visibility by maximizing the overlap f and f . We see that this latter provides a lowerbound on V jk by writing (cid:90) d ωφ ∗ j ( ω ) φ j ( ω ) = (cid:82) d ωf ∗ ( ω, Ω j ) f ( ω, Ω j ) (cid:112) ρ (Ω j , Ω j ) ρ (Ω j , Ω j ) ≥ (cid:90) d ω dΩ f ∗ ( ω, Ω) f ( ω, Ω) , (Supp.47)and likewise for k .It is relatively straightforward to maximize the quantity on the left by tuning experimental parameters, namely pumpwavelength, phasematching angle, and transverse optical fiber position (due to residual spatial chirp), and observingtwo-fold coincidences resulting from first order interference of the sources. Because both sources are pumped with thesame pulse, the two-photon term of the state is given by | ψ (cid:105) ∝ (cid:90) d ω dΩ (cid:16) f ( ω, Ω)ˆ a † ( ω )ˆ b † (Ω) + f ( ω, Ω)ˆ a † ( ω )ˆ b † (Ω) (cid:17) | vac (cid:105) . (Supp.48)A straightforward calculation shows that the probability of a two-fold coincidence between ports ˆ c (or ˆ d ) and ˆ x (orˆ y ) is given by P cc = 14 (cid:90) d ω (cid:12)(cid:12)(cid:12) f ( ω, Ω) ± f ( ω, Ω) (cid:12)(cid:12)(cid:12) = 12 (cid:16) ± Re (cid:90) d ωf ∗ ( ω, Ω) f ( ω, Ω) (cid:17) (Supp.49)In the following, we will outline additional measurements to quantify the source indistinguishability. In our case,our dual-pass geometry implies that we need to match the JSD of both sources, which is achieved when both signalsand idlers from both sources have maximum overlap. We opted for a bulk crystal source in Type II to enable pumpingin both directions while being able to separate our four photons into different paths. We used a BiBO crystal due toits relatively high non linearity.First, we measured the JSI by directing the two daughter photons from either source into the fiber spools, sincetheir large spectral bandwidth would be cropped with the CFBG. The JSI from each source is depicted in Fig8.They show that both sources are nearly indistinguishable; a singular value decomposition yields a Schmidt numberof K = 2 . ± . K = 2 . ± .
1. These values are lower than the theoretical expectation ( K ∼
5) because ofthe timing jitter of our detectors that result in a wider distribution. This was confirmed by measuring the JSI withthe CFBG that have a better resolution but are limited in range. The correlation width was found to be lower andtherefore the Schmidt number can be expected to be at least K = 4.Note that this method is insensitive to any spectral phase difference, such as dispersion from the pump, sincethe second pump is slightly more dispersed than the first due to propagation. This has been shown to increasethe entanglement and the Schmidt number [15, 24]. However, this difference should be negligable, and the methodpresented latter that relies on Eq.(Supp.49) allows for a more accurate estimation of the overlap. Nevertheless, theJSI measurement showed perfect correspondance between the intensity of the two sources which is a critical step toensure indistinguishably between the uncorrelated photon pairs.6 - - τ [ ps ] C oun t s ( × ) ov e r s - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - FIG. 10. Left: HOM dip between the signal photons heralded by a coincidence between the idler photons. Right: samemeasurement but with spectral resolution of the heralding photons, labelled j, k for Ω k , Ω k , where index j, k = 0 correspondsto the center frequency ω . To further characterize the indistinguishability of the sources, we measure their heralded g (2) by splitting theirsignal photon into a beamsplitter. This yields a value of g (2)1 = 0 . ± .
003 and g (2)2 = 0 . ± . g (2) for source 2 is consistent with the fact that it also has ahigher heralding efficiency than source 1. The reason is not entirely clear, but it is likely that the previous interactionwith the PDC crystal on the first pass results in an additional filtering on the pump as well as a slight reduction inoptical power because of absorption.Finally, in Fig9 we measured the coincidences between ports ˆ c and ˆ x (see Fig5) while scanning the relative phasebetween the two pump fields with a piezoelectric stack, which is related to the probability from Eq.(Supp.49). Wescanned using a slow voltage ramp resulting in a few micrometers of displacement over a few seconds. The visibilityof those fringes is of 80%, which is a direct measurement of the overlap between the two sources, and therefore aquantification of distinguishability.Note that we also performed this measurement with spectral resolution, essentially measuring those interferencesin narrower spectral bins using our time-of-flight spectrometer. This results in interferences over a much narrowerbandwidth, therefore restricting the degrees of freedom of the single photon mode-function. Notably this method hasthe advantage of being less sensitive to higher order phase mismatch between both sources, such as dispersion. Themeasured contrast across all spectral bins was found to be 90%. Purity of the heralded states
Since the state | ψ (cid:105) from the sources is assumed to be a pure state, the purity of the heralded states | Ψ − jk (cid:105) isultimately dependent on the amount of spectral filtering in the heralding BSM. To assess this purity, we measureHOM interference between the heralded signal photons when there is no beamsplitter in the idler arms. In this case,upon a coincidence detection of the idler photons at (Ω j , Ω k ), the reduced state of the signal photons is separable,and given byˆ ρ j ⊗ ˆ ρ k = (cid:18)(cid:90) d ωρ j ( ω, ω (cid:48) ) (cid:19) (cid:18)(cid:90) d ˜ ωρ k (˜ ω, ˜ ω (cid:48) ) (cid:19) ˆ a † ( ω )ˆ a † (˜ ω ) | vac (cid:105) (cid:104) vac | ˆ a ( ω (cid:48) )ˆ a (˜ ω (cid:48) ) , (Supp.50)where7 Source 1Source 2 - - τ [ ps ] C oun t s ( × ) ov e r s - - - - - - - - - - Source 1 - - - - - - - - - - Source 2
FIG. 11. Left: P ( τ ) without removing the constant two photon contribution from source 1 (dot) and source 2 (square). P ( τ )ressembles Eq.(Supp.29) and the fit is obtained by summing the individual fits of P jk as given by (Supp.28). Right: distributionof these background terms as a function of the heralding frequencies Ω j , Ω k ρ j ( k ) ( ω, ω (cid:48) ) = (cid:90) dΩ | t j ( k ) (Ω) | f ( ω, Ω) f ∗ ( ω (cid:48) , Ω) . (Supp.51)When the signal photons in this state are incident on a 50:50 beamsplitter, the expected visibility of the HOMinterference is given by [25] V = Tr(ˆ ρ j ˆ ρ k ) , (Supp.52)and when the idlers are detected in identical frequency bins ( j = k ), this becomes V = Tr(ˆ ρ j ( k ) ) = P (ˆ ρ j ( k ) ) , (Supp.53)where P ( · ) denotes the purity of a state. Thus, for ( j = k ) the visibility of the HOM dip gives a lower bound on thepurity of the state ˆ ρ j ( k ) , and by extension, the state ˆ ρ jk . Our measurements, shown in Fig. 10, indicate that purity ofthe heralded states is at least j = k line. By comparison, a directcalculation of the expected purity using our experimental parameters gives ∼ Background signal
As shown by Eq.(Supp.6), the full four photon state in the interferometer (see Fig. 5) contains a contribution fromphoton pairs emitted by individual sources due to the stochastic nature of parametric down conversion. These termscontribute to P ( τ ) in the form of interferences that get averaged over the course of a measurement. It is thereforepossible to remove that contribution from the signal subsequently to the measurement by blocking a source andrecording the rate of four-fold coincidences.We therefore repeated the measurement of P jk ( τ ) with either source blocked to obtain the constant backgroundsignal for each j, k frequencies, as shown in Fig. 11. This shown that the background terms are similar betweenboth sources, therefore the two sources are similar. Summing over all the bins, we can plot on the same scale thecontribution of all term in Fig. 11. The peak corresponds to interferences from | ψ (cid:105) while the flat terms represent | ψ (cid:105) and | ψ (cid:105) . As expected from the theory, both source contribute to 1 / j , Ω k from P jk , we obtain the fringes from the main paper with optimal visibility.8
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825 830 835825830 λ [ nm ] λ [ n m ] ( ) FIG. 12. Sets of orthogonal modes F jk that have less than 15 % of mutual overlap within each part of the grid. The insetslabel j and k . Orthogonal modes
From Eq.(Supp.13), we see that the heralded state | Ψ jk (cid:105) is dependent on the modes | φ j (cid:105) and | φ k (cid:105) , which, in thepure state case, results in a heralded joint spectrum (Eq.(Supp.18)) dependent on the outer products φ j ( ω ) φ k ( ω ).For each heralding bin j and k , we label the heralded JSI from (Supp.13) as F n ( ω , ω ), where n indexes a pair( j, k ). These are normalized as (cid:82) d ωF n ( ω , ω ) = 1 ∀ n but are not orthogonal, even in the pure state case, i.e (cid:82) d ωF n ( ω , ω ) F m ( ω , ω ) (cid:54) = δ nm . Orthogonality is usually a corner stone in any quantum protocol, and it istherefore necessary to select the heralded states from our measurement that are orthogonal. To do so, we utilize ourmeasurement of F jk , obtained by measuring the spectral coincidences between the signal’s photon heralded by a BSMon the idlers. We then obtain a figure similar to Fig. 6 albeit without perfect spectral resolution, putting us in themixed state configuration, but the strategy to select orthogonal modes within this set is similar to the pure statemodel.First, it is important to notice the symmetry in (Supp.13), where F jk = F kj . Since our TOFS are well-calibrated,it is reasonnable to symmetrize our measured heralded JSI by averaging the experimentally obtained F jk and F kj (for j (cid:54) = k ) thus defining the F n functions. Then we compute the mutual overlaps (cid:82) d ωF n ( ω , ω ) F m ( ω , ω ) and use analgorithm to select a set of modes { F n } which all have an overlap below a certain threshold of 15%. We representeda few of these JSI in Fig. 12. Since the spectral range of our high resolution TOFS is limited, so is the range overwhich we can compute overlap, as can be seen from the modes that are labelled with a large j, k . Nevertheless, thereis a sufficient amount of spectral coincidence in those cases to infer orthogonality with the other JSI.Note that while this overlap is computed between the joint spectral intensities and not between the states, it canbe shown that if the overlap in intensity is zero, then the states are necessarily orthogonal, hence the strategy is validto select which | Ψ jk (cid:105) are mutually orthogonal. Therefore, it is fair to say that the JSI F jk from Fig. 12 correspond tothe heralded states | Ψ jk (cid:105)(cid:105)