Compact source of narrow-band counterpropagating polarization-entangled photon pairs using a single dual-periodically poled crystal
Yan-Xiao Gong, Zhen-Da Xie, Ping Xu, Xiao-Qiang Yu, Peng Xue, Shi-Ning Zhu
aa r X i v : . [ qu a n t - ph ] D ec Compact source of narrow-band counterpropagating polarization-entangled photonpairs using a single dual-periodically poled crystal
Yan-Xiao Gong,
1, 2
Zhen-Da Xie, Ping Xu, ∗ Xiao-Qiang Yu, Peng Xue, and Shi-Ning Zhu Department of Physics, Southeast University, Nanjing, 211189, People’s Republic of China National Laboratory of Solid State Microstructures and School of Physics,Nanjing University, Nanjing, 210093, People’s Republic of China
We propose a scheme for the generation of counterpropagating polarization-entangled photon pairsfrom a dual-periodically poled crystal. Compared with the usual forward-wave type source, thissource, in the backward-wave way, has a much narrower bandwidth. With a 2-cm-long bulk crystal,the bandwidths of the example sources are estimated to be 3 . / (s GHz mW). Two concurrent quasi-phase-matched spontaneous parametricdown-conversion processes in a single crystal enable our source to be compact and stable. Thisscheme does not rely on any state projection and applies to both degenerate and non-degeneratecases, facilitating applications of the entangled photons. PACS numbers: 42.65.Lm, 42.50.Dv,03.67.Bg
I. INTRODUCTION
Polarization-entangled photons play a key role notonly in testing the foundations of quantum mechan-ics [1] but also in various photonic quantum technolo-gies [2]. A compact, robust, and high-brightness source ofpolarization-entangled photons is therefore desirable forpractical implementation of a variety of entanglement-based applications.Spontaneous parametric down-conversion (SPDC) innonlinear crystals is a successful technique to generatepolarization-entangled photon pairs. A typical methodinvolves using the type-II birefringence phase-matching(BPM) in a nonlinear crystal [3], such as beta barium bo-rate (BBO). However, only a small fraction of the totalemitted photons, the intersecting locations of two non-overlapping cones, are polarization-entangled, and there-fore, such a source is inefficient. A more efficient sourceconsists of two type-I nonlinear crystals via BPM [4, 5],from which polarization-entangled photons are emittedin a cone. However, generally only a small fraction of thecone is collected for use, and thus, such a source is againless efficient.One way to solve the inefficiency problem in theconelike sources is by means of quasi-phase-matching(QPM) [6, 7] in periodically poled (PP) crystals [8], suchas periodically poled lithium niobate (PPLN) and pe-riodically poled potassium titanyl phosphate (PPKTP).QPM has advantages over BPM due to its higher effi-ciency and the fact that it enables flexible frequency-tunable processes. In particular, QPM enables the pho-ton pairs in a collinear and beam-like configuration. Con-sequently, it is possible to make a much bigger fraction ofthe created photons polarization-entangled than conelikesources, thus leading to more efficient sources. However ∗ [email protected] a new problem arises, namely the need to spatially sepa-rate the collinear photon pairs.A simple method to solve this problem is by usingdichroic mirrors when the photon pairs are generatedat substantially different frequencies. In this way, sev-eral non-degenerate polarization entanglement sourceshave been designed by coherently combining two SPDCsources at a polarizing beam splitter [9–15], by manip-ulating polarization ququarts [16], by overlapping twocascaded PP crystals [17, 18] , or by two cascaded[19] or concurrent [20, 21] SPDC processes in a singlePP crystal. Theses non-degenerate sources have vari-ous applications, for instance, in quantum communica-tion [22]. However, in many entanglement-based ap-plications, for example, in quantum computation [23],frequency-degenerate polarization-entangled photons arerequired. A straightforward way to build degenerateentangled sources based on PP crystals is by separat-ing collinear orthogonally polarized photon pairs with abeam splitter followed by twofold coincidence measure-ment as a postselection [24]. However, this method suf-fers a 50% loss. A postselection-free method employsinterferometers to combine two pairs of orthogonally po-larized photons [25, 26], but such interferometric sources(also the non-degenerate sources in Refs. [9–15]) requirestringent phase control and stabilization.Another problem of SPDC sources lies in the broadbandwidth determined by the phase-matching condition,which is usually on the order of several THz or hundredsof GHz. The broadband SPDC source becomes very dimin many applications requiring narrow-band photons,such as long-distance fiber optical quantum communica-tion ( ∼ GHz [27]), strong interaction of the photons withatoms and molecules( ∼ MHz [28], and recently relaxed toseveral GHz [29, 30]), and interference of independentsources without time synchronization ( ∼ GHz [31]). Pas-sive filtering is a straightforward way to obtain narrow-band sources [27, 31], but it will greatly reduce the gen-eration rate. Cavity-enhanced SPDC can provide high-brightness narrow-band photon paris [32–34]. However,additional spectral filtering is required to obtain single-mode output due to the broad gain bandwidth.In this paper, we succeed in solving all the above prob-lems by building a compact and narrow-band polariza-tion entanglement source based on the backward-wavetype SPDC in a dual-periodically poled crystal. Thebackward-wave type SPDC [35–37], has a much narrowerbandwidth than the forward-wave interaction. The coun-terpropagating photon pair generation has also been ex-tensively studied in waveguide structures [38–41]. More-over, it not only has the same advantage as the usualcollinear, beam-like output SPDC on photon collectionand overlapping for possible polarization entanglement,but it also does not suffer from the problem of spatial sep-aration. Our scheme relies on the coherence of two con-current backward-wave type SPDC processes in a singlePP crystal, rather than any interferometer and postse-lection. Furthermore, this scheme can work in frequencydegenerate and non-degenerate cases, for which we de-sign two experimentally feasible structures, respectively.With a 2-cm-long bulk crystal, the bandwidths of thetwo sources are estimated to be 3 . / (s GHz mW), respec-tively.The rest of this paper is organized as follows. Inthe next section, we give a description of the dual-periodically poled crystal and design the structures re-quired in our scheme. In Sec. III, we introduce ourscheme and make detailed calculations on the sources wepropose. Sec. IV are conclusions. II. DESCRIPTION OF ADUAL-PERIODICALLY POLED CRYSTAL
QPM originates from modulation of the second-ordernonlinear susceptibility χ (2) . It has been advanced to avariety of domain structures which allow multiple andflexible nonlinear processes in a single crystal, leading tocompact and integrated devices. A dual-periodic struc-ture is one of the QPM structures, which permits twocoupled optical parametric interactions [42, 43]. Here,taking the potassium titanyl phosphate (KTP) crystalfor example, we design a dual-periodic structure to sat-isfy two concurrent SPDC processes, H p → H s + V i , and H p → V s + H i , where p , s , i represent the pump, signal,idler fields, respectively, with H ( V ) denoting the hori-zontal (vertical) polarization.The schematic of a dual-periodically poled KTP (DPP-KTP) crystal is shown in Fig. 1, in which inverted do-mains (with − χ (2) ) distribute on a + χ (2) background asa dual-periodic structure. It is formed by twice-periodicmodulation of χ (2) . Suppose g ( x ) and g ( x ) are two pe-riodic functions as the sign of nonlinearity χ (2) . Then FIG. 1. Schematic of a dual-periodically poled potassiumtitanyl phosphate crystal. Gray and blank areas are inverted( − χ (2) ) and background positive ( χ (2) ) domains, respectively. their Fourier expansions can be written as g ( x ) = X m G m e − iG m x , (1) g ( x ) = X n G n e − iG n x , (2)respectively, where the reciprocals are G m = 2 mπ Λ , G n = 2 nπ Λ , (3)and the Fourier coefficients G m = 2 mπ sin( mD π ) , G n = 2 nπ sin( nD π ) . (4)with Λ and Λ (Λ < Λ ) denoting the two modulationperiods, D and D representing the duty cycles, andnonzero integers m and n indicating the orders of recip-rocals. Then we can write the dual-periodic structureas g ( x ) = g ( x ) g ( x ) = X m,n G m,n e − iG m,n x , (5)where G m,n = G m G n = 4 mnπ sin( mD π ) sin( nD π ) , (6) G m,n = G m + G n = 2 mπ Λ + 2 nπ Λ . (7)Then the modulation of the second-order nonlinear sus-ceptibility χ (2) can be described as χ (2) ( x ) = dg ( x ) = d X m,n G m,n e − iG m,n x , (8)where d is the effective nonlinear coefficient.An arbitrary twice-periodic modulation could result insmaller domains which may make fabrication more diffi-cult. A straightforward way to avoid the unwanted smalldomains, is by designing the structure such that Λ / Λ = l/ D = 1 / D = ⌊ l/ ⌋ /l , l is an integer bigger than 2 , (9)where ⌊·⌋ is the floor function to get the integer part ofa number. In practice, this condition can be satisfied bytuning the temperature and wavelengths.We consider the pump wave vector along the x direc-tion and H ( V ) in the y ( z ) directions. By choosingthe right wavelengths and temperature we are able toobtain the following QPM conditions for two backward-wave type SPDC processes∆ k = k p,H − k s,H + k i,V − G m ,n = 0 , (10)∆ k = k p,H − k s,V + k i,H − G m ,n = 0 , (11)where G m ,n and G m ,n are given by Eq. (7) in thecase of { m, n } = { m , n } and { m, n } = { m , n } , re-spectively. Here we require the two SPDC processes tohave the same signal frequency ω s and the same idler fre-quency ω i , with the energy conservation condition ω p = ω s + ω i , where ω p is the pump frequency. In addition, aswe shall see in Sec. III, we require m n = ± m n . Inorder to show the experimental feasibility of such a struc-ture, in the following we design two possible structuresbased on the temperature-dependent Sellmeier equationgiven by Emanueli and Arie [44].We first design a structure for a degenerate source of λ p = 655 nm, λ s = λ i = 1310 nm. Such a source couldfind applications in long-distance fiber-based quantuminformation processing, as the wavelength of the photonsis in the the second telecom window. At a working tem-perature of 75 ◦ C, we get the two reciprocals for QPMas G , = 17 . µ m − and G , − = 18 . µ m − , corre-sponding to the two modulation periods Λ = 1 . µ mand Λ = 16 . µ m, respectively. The ratio of the twoperiods is Λ / Λ = 15 .
5, and thus the duty cycle D should be 15 / λ p = 532 nm, λ s = 807 . λ i = 1560nm. This choice is motivated by the photon source re-quirements in real-world quantum networks, for examplephotonic memories in quantum repeaters. The shorter-wavelength photon of this source can be used for cou-pling and entangling atomic systems, and the other pho-ton at 1560 nm can be transmitted over a long dis-tance in fiber because its wavelength lies in the low-loss transmission window of optical fibers. By choos-ing the working temperature as 75 . ◦ C, we obtain thetwo reciprocals for QPM as G , = 14 . µ m − and G , − = 15 . µ m − , with the two corresponding mod-ulation periods as Λ = 1 . µ m and Λ = 12 . µ m,respectively, the ratio of which is Λ / Λ = 10 .
5, andtherefore the duty cycle D = 10 / III. GENERATION OFPOLARIZATION-ENTANGLED PHOTONS
We consider a classical pump wave illuminating theDPPKTP crystal with a length of L in the x directionand the interaction volume denoted by V . The inducedsecond-order nonlinear polarization is given by [48] P (2) i ( ~r, t ) = ε χ (2) ijk E j ( ~r, t ) E k ( ~r, t ) , (12)where ε is the vacuum dielectric constant and χ (2) ijk isthe second-order nonlinear susceptibility tensor, where i , j , k refer to the cartesian components of the fields.Here, we use the Einstein notation of repeated indices fortensor products. The Hamiltonian of the electromagneticsystem can be expressed as H = 12 Z V d ~r (cid:18) ~D · ~E + 1 µ ~B · ~B (cid:19) , (13)where µ is the vacuum permeability constant. Since ~D = ε ~E + ~P , we obtain the interaction Hamiltonian inthe parametric down-conversion process H I ( t ) = 12 Z V d ~r ~P · ~E = ε Z V d ~rχ (2) E p ( ~r, t ) E s ( ~r, t ) E i ( ~r, t ) , (14)where we replace χ (2) ijk / χ (2) [48], which has the form of Eq. (8)for an ideal structure. After quantization of the electro-magnetic fields, E ( ~r, t ) becomes a Hilbert space operatorˆ E ( ~r, t ), which can be decomposed into its positive andnegative parts ˆ E ( ~r, t ) = ˆ E (+) ( ~r, t ) + ˆ E ( − ) ( ~r, t ). Then wecan rewrite the interaction Hamiltonianasˆ H I ( t ) = ε Z V d ~rχ (2) ( x ) ˆ E (+) p ( ~r, t ) ˆ E ( − ) s ( ~r, t ) ˆ E ( − ) i ( ~r, t )+ H.c. , (15)where H.c. denotes the Hermitian conjugate part. Here,we only write the two terms that lead to energy conserv-ing processes, and we neglect the other six terms thatdo not satisfy energy conservation and are therefore ofno importance in the steady state. Note that neglectingthese contributions is equivalent to making the rotating-wave approximation.Since the transverse structure of DPPKTP is homo-geneous, we ignore the transverse vectors of interactingwaves and only consider the interaction along the propa-gating direction. We consider the case of signal and idlerphotons in forward and backward directions, respectively.Then the negative parts of the field operators of the sig-nal and idler ˆ E s , ˆ E i are represented by Fourier integralsas ˆ E ( − ) s ( x, t ) = X q = H,V Z dω s E ∗ s,q e − i ( k s,q x − ω s t ) ˆ a † s,q ( ω s ) , (16)ˆ E ( − ) i ( x, t ) = X q = H,V Z dω i E ∗ i,q e i ( k i,q x + ω i t ) ˆ a † i,q ( ω i ) , (17)where E j,q = i p ~ ω j / (4 πε cn q ( ω j )), j = s, i . For sim-plicity, here we consider a continuous-wave (cw) plane-wave pump with horizontal polarization. In addition, thepump field is treated as an undepleted classical wave, andthus the positive part of its field operator is replaced withits complex amplitude E (+) p ( x, t ) = E p e i ( k p,H x − ω p t ) . (18)Then by substituting Eqs. (8), (16), (17), and (18) intoEq. (15), we obtainˆ H I ( t ) = − ~ E P πc X q = H,V X q ′ = H,V X m,n d G m,n Z − L dx Z dω s × Z dω i r ω s ω i n q ( ω s ) n q ′ ( ω i ) ˆ a † s,q ( ω s )ˆ a † i,q ′ ( ω i ) × e i ( ω s + ω i − ω p ) t e − i ( k s,q − k i,q ′ − k p,H + G m,n ) x + H.c. , (19)For the SPDC process, the interaction is weak, so un-der first-order perturbation theory the state evolutionfrom time t ′ to t can be written as | Ψ i = | vac i + 1 i ~ Z tt ′ ˆ H I ( τ ) dτ | vac i . (20)Considering steady state output we may set t ′ = −∞ and t = ∞ . Then we have Z ∞−∞ dτ e i ( ω s + ω i − ω p ) τ = 2 πδ ( ω s + ω i − ω p ) , (21)which gives the energy conservation relation ω s + ω i − ω p = 0 . (22)The integral over crystal length can be calculated as Z − L dxe − i ( k s,q − k i,q ′ − k p,H + G m,n ) x = Lh ( L ∆ k qq ′ ) , (23)where ∆ k qq ′ = k p,H − k s,q + k i,q ′ − G m,n and the h -function has the following form h ( x ) = 1 − e − ix ix = e − i x sinc x . (24) h ( L ∆ k qq ′ ) determines the natural bandwidth of the two-photon state, as we shall see. In the case of infinite crystallength, Eq. (23) becomes a δ -function, thus leading to the momentum conservation, i.e., the perfect phase-matchingcondition, ∆ k qq ′ = 0.Suppose that perfect phase matching conditions givenby Eqs. (10) and (11) can be satisfied at frequencies Ω s and Ω i , with corresponding wave vectors K s,H , K s,V , K i,H , and K i,V , such thatΩ s + Ω i = ω p , K j,q = n q (Ω j )Ω j c , (25)with j = s, i and q = H, V . Due to the existence ofbandwidth, and constrained by Eq. (22), we let ω s = Ω s + ν, ω i = Ω i − ν, (26)where | ν | ≪ Ω j , j = s, i . Then in the case of the QPMconditions given by Eqs. (10) and (11), we can write thestate of SPDC as | Ψ i = | vac i + A HV d HV L Z dνh ( L ∆ k HV )ˆ a † s,H (Ω s + ν ) × ˆ a † i,V (Ω i − ν ) | vac i + A V H d V H L Z dνh ( L ∆ k V H ) × ˆ a † s,V (Ω s + ν )ˆ a † i,H (Ω i − ν ) | vac i , (27)where d HV = d G m ,n = 4 dπ m n sin m π n D π ) , (28) d V H = d G m ,n = 4 dπ m n sin m π n D π ) , (29) A HV = iE p c s Ω s Ω i n s,H n i,V , (30) A V H = iE p c s Ω s Ω i n s,V n i,H , (31)with n j,q denoting the refraction index of a photon withpolarization q at frequency Ω j . Here A HV d HV and A V H d V H are slowly varying functions of frequency, whichhave been taken outside the integral.We can see that the maximally polarization-entangled state can be obtained under the conditionof | A HV d HV h ( L ∆ k HV ) | = | A V H d V H h ( L ∆ k V H ) | . Thecondition of d HV = d V H = d ′ can be satisfied straight-forwardly by choosing m n = ± m n . In the following,we make calculations on h ( L ∆ k HV ) and h ( L ∆ k V H ), i.e.,the spectrum of the photon pairs. In other words, thetwo-photon correlation time is on the order of severalhundred picoseconds.
A. Characterizing the spectrum of photon pairsgenerated from our source
We first expand the magnitudes of the wave vectors forsignal and idler photons around the central frequenciesΩ s and Ω i respectively, up to first order in ν , k s,q = n q ( ω s ) ω s c ≈ K s,q + νu q (Ω s ) , (32) k i,q = n q ( ω i ) ω i c ≈ K i,q − νu q (Ω i ) , (33)where u q (Ω j ) = d Ω j /dK j,q are the group velocities ofsignal and idler photons at central frequencies, with j = s, i and q = H, V . Therefore we obtain∆ k HV = − νS HV , S HV = (cid:20) u H (Ω s ) + 1 u V (Ω i ) (cid:21) , (34)∆ k V H = − νS V H , S
V H = (cid:20) u V (Ω s ) + 1 u H (Ω i ) (cid:21) . (35)We thus obtain the joint spectral densities for the com-ponents | H, V i and | V, H i , | h ( L ∆ k HV ) | =sinc νLS HV , (36) | h ( L ∆ k V H ) | =sinc νLS V H , (37)and the corresponding bandwidths are ∆ ω HV ≈ . π/ ( LS HV ) and ∆ ω V H ≈ . π/ ( LS V H ), respec-tively. Compared with the usual forward-wave type-II SPDC under the same conditions on crystal lengthand frequencies [49], the backward-wave source has amuch narrower bandwidth, with a reducing factor of( u − H + u − V ) / | u − H − u − V | .More explicitly, we consider the two example struc-tures given in Sec. II, and the crystal length is set to 2cm. For the degenerate source, we get the bandwidth∆ ω HV = ∆ ω V H ≈ π × .
66 GHz and the reducingfactor is 41. For the non-degenerate source, we obtainthe two bandwidths as ∆ ω HV ≈ π × .
61 GHz and∆ ω V H ≈ π × .
63 GHz, corresponding to reducing fac-tors of 25 . .
2, respectively. Note that, comparedwith the asymmetric spectrum in the forward-wave case,our backward-wave source has an almost symmetric spec-trum.
B. Quantifying the polarization entanglementproduced by our source
To quantify the polarization entanglement produced byour source, we employ a commonly used entanglementmeasure, namely, concurrence [50], whose value rangesfrom zero, for a non-entangled state, to one, for a max-imally entangled state. For a pure two-qubit state | ψ i ,expressed in a fixed basis such as {| i , | i , | i , | i} ,the concurrence C = |h ψ | σ y ⊗ σ y | ψ i| , where σ y is thesecond Pauli matrix (cid:0) i − i (cid:1) in the same basis. For oursource, we need to treat the two-photon term of the stategiven by Eq. (27), denoted as | Ψ i . Note that the state | Ψ i is unnormalized and the reciprocal of the square of its normalization constant is the two-photon generationrate, given by R = h Ψ | Ψ i = d ′ L h | A HV | Z dν | h ( L ∆ k HV ( ν )) | + | A V H | Z dν | h ( L ∆ k V H ( ν )) | i . (38)Substituting Eqs. (36) and (37) into the above equation,we obtain R = 2 πd ′ L (cid:18) | A HV | S HV + | A V H | S V H (cid:19) . (39)Then we can calculate the concurrence C = |h Ψ | σ y ⊗ σ y | Ψ i|h Ψ | Ψ i = d ′ L | A HV A V H | R (cid:12)(cid:12)(cid:12) Z dνh ∗ ( L ∆ k HV ( ν )) h ( L ∆ k V H ( ν ))+ Z dνh ∗ ( L ∆ k V H ( ν )) h ( L ∆ k HV ( ν )) (cid:12)(cid:12)(cid:12) . (40)By substituting Eqs. (24), (34) and (35) into the aboveequation, we arrive at C = 2 S min δ n S HV + S V H /δ n , (41)where S min = min { S HV , S V H } and δ n = p n s,H n i,V / ( n s,V n i,H ).For degenerate case, i.e., Ω s = Ω i , S HV = S V H , δ n = 1, and thus C = 1, so our source can generate de-generate maximal polarization entanglement. This fea-ture can also be seen directly from the two-photon termof the state given by Eq. (27), which shows a maximallyentangled state in the form of ( | HV i + | V H i ) / √ s = Ω i , S HV = S V H , δ n = 1, and therefore C <
1, so the entanglementis nonmaximal. However, actually there is not a big dif-ference between δ n S HV and S V H /δ n , so the concurrenceis very near to 1. Explicitly, let us consider the examplestructure given in Sec. II, the concurrence of the entan-glement generated from which is found to be as high as0 . C. Generation rate of the entangled photon pairs
The photon pair generation rate can be estimated fromEq. (39), by substituting Eqs. (30), (31), and | E p | =2 P/ ( ε n p cS ) into it, where P denotes the pump powerand S represents the transverse area of the pump beam.Therefore we obtain R = πd ′ LP Ω s Ω i ε n p c S (cid:18) n s,H n i,V S HV + 1 n s,V n i,H S V H (cid:19) . (42)Let us consider the two specific example sources, forwhich we set P = 1 mW, S = 0 .
01 mm , and L = 2 cm.The nonlinear coefficient d ′ is given by Eq. (28), where m = 3, n = 1, and the effective nonlinear coefficient d ,stemming from d , is found to be 3 . / V. Then wefind the generation rate of the degenerate source to be421 pairs / s, and thus we get the spectral brightness as2 πR/ ∆ ω ≈
115 pairs / (s GHz mW). The generation rateof the non-degenerate source is found to be 554 pairs / s,corresponding to the spectral brightness of 154 pairs / (sGHz mW). We have to emphasize that the experimentalvalue of the photon pair rate and the two-photon spec-trum will definitely be affected by the poling quality, suchas the deviations and fluctuation of the poling period andduty cycle [51]. However, the state of the art of the pol-ing technique can enable us to engineer a nearly idealizedpoled structure. IV. CONCLUSIONS
In conclusion, we have presented a scheme for build-ing polarization-entangled photon pair sources utiliz-ing backward-wave type SPDC processes in a dual-periodically poled crystal. Our scheme does not rely onany state projection and can work in degenerate and non-degenerate cases. The backward-wave type SPDC en-ables the entangled photon pairs from our source to trans-mit in a beam-like way, exhibiting more efficient pho-ton collection and mode overlapping. Furthermore, thebackward-wave type SPDC has a much narrower band-width than the usual forward-wave one. In addition, ourscheme employs two concurrent SPDC processes in a sin-gle crystal rather than any interferometer, and therefore our source is compact and stable. By proper engineer-ing on the domain structure a complete set of Bell statescan be achieved directly from this DPPKTP crystal [52].This implies further applications in integrated photonicquantum technologies.We have designed two possible DPPKTP structures fordegenerate and non-degenerate sources, respectively. Us-ing a 2-cm-long bulk crystal, the bandwidths of the twosources were found to be ∼ . / (s GHz mW), respectively.Our high-spectral-brightness narrow-band sources shouldfind applications in large-scale quantum networks andother fields requiring narrow-band entangled photons.Furthermore, we have also quantified the polarization en-tanglement via concurrence and found that the degener-ate source can provide maximally polarization-entangledphoton pairs while the concurrence of the polarizationentanglement generated from the non-degenerate sourceis as high as 0 . ACKNOWLEDGMENTS
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