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Efficient photon sorter in a high-dimensional Hilbert space Warner A. Miller1,2, and Mark T. Gruneisen2, ∗ ∗ 1 Department of Physics, Florida Atlantic University, Boca Raton, Florida 33431 2 Air Force Research Laboratory, Directed Energy Directorate, Kirtland AFB, New Mexico 87117 (Dated: September30 ,2008) An increase in the dimension of Hilbert space for quantum key distribution (QKD)can decrease 8 its fidelity requirements while also increasing its bandwidth. A significant obstacle for QKD with 0 qudits (d 3) has been an efficient and practical quantum state sorter for photons with complex 0 ≥ wavefronts. We propose such a sorter based on a multiplexed thick hologram constructed from 2 photo-thermalrefractive glass. Wevalidatethisapproach usingcoupled-modetheory tosimulate a t holographic sorter for states spanned by three planewaves. The utility of such a sorter for broader c O quantuminformation processing applications can be substantial. 3 PACSnumbers: 42.50.Tx,03.67.Hk,42.40.Pa,42.50.Ex ] h Weconcernourselvesherewiththesecuredistribution Molina-Terriza et. al. [13] introduced a scheme to pre- p of a one time only key from a sender (Alice) to a re- pare photons in multidimensional vector states of OAM - t ceiver (Bob). The three elements of any quantum key commencing OAM QKD. Recently a practical method n distribution (QKD) system are: (1) Alice must be able hasbeendemonstratedto producearbitraryOAMMUB a u to prepare at will a single photon state chosen from a statesusingcomputer-generatedholographywithasingle q set of mutually-unbiased bases (MUB) [1, 2], (2) each of spatial light modulator (SLM) [14]. [ these quantum amplitudes must be propagatedfrom Al- While the advantage of OAM QKD lies in its ability 2 ice to Bob with reasonable fidelity, and finally (3) Bob to increase bandwidth while simultaneously tolerating a v must have the ability to choose between one or another higher bit error rate (BER) [6], two potential problems 6 of the MUBs and, if he is lucky, be able to efficiently confront this approach. First, such transverse photon 3 sort and detect each of these photon states. This QKD 3 wave functions are more fragile in propagation than the scenario has been exhaustively studied in the literature 0 photon’s spin [15, 16], and the divergence of the states . and is replete with security proofs for numerous proto- ( √l) mayrequirelargerapertures. Despite this, multi- 0 cols [3, 4, 5]. These security proofs have been extended ∝ 1 conjugate adaptive optical communicationchannels may in many cases to higher-dimensional Hilbert spaces [6], 8 be able to ameliorate these problems [17, 18]. A second 0 and all of the protocols have been or are currently being obstacleinvolvestheefficientsortingofOAMMUB-state : implemented successfully [7]. v photons with small Fock-state quantum numbers, and i ConventionalrealizationsofQKDtodayinvolvetrans- thepaperwilladdressthisparticularproblem. Currently, X mittingheavily-attenuatedlaserpulsesfromAlicetoBob the only solutionto this problemis the use if a cascaded r a and encoding qubit information in each packet by utiliz- Mach-Zehnder interferometric system [19, 20, 21]. Pro- ing the spin of the photon. This allows Alice and Bob, posed systems of this kind have been demonstrated only who are suitably authenticated, the possibility to estab- for4-dimensionsandaresimplynotpractical. Otherap- lishandshareanarbitrarily-secureone-timeonlykeybe- proachesthatusecrossedthindiffractiongratingsarenot tweenthem. Here they have accessto a two-dimensional efficient enough to establish a secure key. Hilbert space and can therefore form three MUBs each We focus here on the efficient sorting of single pho- withtwoorthogonalpolarizationstates. Suchasix-state tons with arbitrary complex wavefronts. Ideally, what is QKD scheme [8] has limited bandwidth and optical fi- neededisanOAMversionofapolarizer,i.e. asingleop- delity constraints. These constraints can be ameliorated ticalelement,oneperMUBbasis,thatcanefficientlysort by extending the QKD to higher-dimensional Hilbert each of the qudit states in that basis while equally dis- space [6]. tributingeveryotherquditstate. Thickholographicgrat- The potential of extending photon-based QKD to ingsfortunatelyproducehighdiffractionefficiencyinthe higherdimensionswasmadepossiblein1992whenAllen first order [22, 23, 24]. If several predominant diffracted et.al.[9]showedthatLaguerre-Gaussianlightbeamspos- ordersarerequired,asisthecaseforsorting,severalinde- sesseda quantizedorbitalangularmomentum (OAM)of pendentfringestructurescanexistintheemulsion. Such l~ per photon. This opened up an arbitrarily high di- multiplexedhologramshavebeenusedformultiple-beam mensional quantum space to a single photon[10]. Fol- splittersandrecombiners[24]andmorerecentlyforwide- lowing this discovery Mair et. al. [11, 12] unequivocally angle beam steering[25]. In this paper we propose such demonstrated the quantum nature of photon OAM by a MUB-state sorter based on a multiplexed thick holo- showing that pairs of OAM photons can be entangled graphicelementconstructedfromcommerciallyavailable using parametric down conversion. Shortly thereafter, photo thermal refractive (PTR) glass [26]. Due to the 2 unique properties of PTR glass the grating’s thickness L can approach several mm and be highly Bragg selective. There is evidence that such sorters can be highly effi- r1 Λ cient, > 95% [25]. Our simulations presented here and r 2 empiricaldataonthickBragggratingsindicatethatthey mayprovideanadequatesolutiontothiscriticalandlong standing problem. Before we describe our proposed thick holographic a aτ MUBsorterwewillbrieflyfocusourattentionontwelve- b cτ D state QKD and work in a 3-dimensional Hilbert space. c This is one more dimension than that available to pho- rτ 3 ton polarization states and serves to illustrate our approach. Nevertheless, our work is equally applicable to higher dimensional Hilbert spaces. Its limitations will r3 require further investigation. In 3-dimensions there are a maximum of four MUBs which we refer to here as MUB , MUB , MUB and 1 2 3 MUB . Each of these orthonormal bases contain three 4 FIG. 1: An illustration of our proposed thick holographic state vectors. If we identify a , b and c as the or- | i | i | i MUB4 sorter. The three signal waves are the appropriate thonormal ket vectors of MUB1, then the other nine linearsuperpositionsofthethreequditstatesofMUB4shown qudit states fromthe other three MUB basesarespecific in Table I. linear combinations of these (Table I). ForourapplicationwecanfreelychooseasourMUB 1 our 3-dimensional Hilbert space. The other nine MUB any three pure OAMstates (a , b and c ) corresponding to an angular momentum| i, l| i= a~|,il = b~, and states (Table I) can be obtained from these by linear su- a b l = c~ with integers a, b and c being the azimuthal perposition and will represent wavefronts with both am- c plitude and phase variations. quantum numbers. For the purpose of our calculations For each MUB we consider a multiplexed thick holo- we cansimplify our analysisandretainthe physicalcon- graphic sorter, i. e. a triple-exposed grating structure tent by quantizing in the space of linear momentum (k- formed by the incoherentsuperposition of three gratings QKD) rather than in angular momentum (OAM-QKD). within a single emulsion. Here, each grating is formed As a result we can freely choose as MUB any three 1 bythesuperpositionofthe respectiveMUB stateandits non co-linear planewaves. In this case, our three integer own unique plane reference wave. In this paper we con- quantum numbers will be the number of waves of tilt of centrateontheconstructionoftheMUB sorter(Fig.1), these planewaveswith respectto the normalofthe holo- 4 as the other three MUB sorterswill be of similar design. graphicemulsionofapertureD. Thesewavescorrespond to a transverse linear momentum px = a~kx, px = b~kx a b and px = c~kx; respectively. Here, kx = kλ/D is the We construct the MUB4 sorting in three steps. First, c werecordtheinterferencepatternofourfirstsignalwave x-componentofaplanewaveoffrequencywithonewave a with a corresponding reference planewave, r hav- of tilt (τ λ/D). In the frame of the hologram and in | 4i | 1i ∼ ing r 1 waves of tilt. After this initial recording is units where the speed of light is unity, the components 1 ≫ of the 4-momentum (p = pt,px,py,pz ) of each of our complete, we then record the interference pattern of our { } second signal state from MUB , namely b with a sec- three photons can be expressed in terms of their trans- 4 | 4i verse momentum kx and wavenumber (k). ond reference planewaveh|r2i. Finally, we recorda third independentsetoffringepatternsbyinterferingthequdit p =~k =a~kx 1,1,0, (k/akx)2 1 , signal state c with a third reference planewave r . a a 4 3 { − } | i | i p =~k =b~kx 1,1,0,p(k/bkx)2 1 , This produces a triple-multiplexed hologram. b b { − } We show that the hologramdescribed above faithfully pc =~kc =c~kx 1,1,0,p(k/ckx)2 1 . represents a quantum projection operator for MUB . { − } 4 p In the remainder of this paper the transverse linear mo- = r a + r b + r c (1) 4 1 4 2 4 3 4 mentumwavenumbersrepresentourthreequantumnum- P | ih | | ih | | ih | bers for k-QKD. These three planewaves define our first Its operation on any one of the 12 MUB states should MUB. produce the desired result. In other words, if the hologram is illuminated by MUB state a , b or c it MUB1 = a , b , c . | 4i | 4i | 4i {| i | i | i} shouldproduce a planewavein state r , r or r , re- 1 2 3 | i | i | i EachofthesestatesrepresentsatransverseFouriermode spectively. If it is illuminated by any of the other nine of a photon; they areorthogonal( ij =δ ) and define MUB states it should then produce an equally weighted i,j h | i 3 MUB1 MUB2 MUB3 MUB4 a a2 a + b + c a3 a + b +z c a4 a + b +z2 c |bi |b2i ∝ |ai+z |bi+z2 |ci |b3i ∝ |ai+z |bi+ |ci |b4i ∝ |ai+z |bi+z |ci |ci |c2i ∝ |ai+z2 |bi+z |ci |c3i ∝ |ai+z2 |bi+z2 |ci |c4i ∝ |ai+z2 |bi+ |ci | i | i ∝ | i | i | i | i ∝ | i | i | i | i ∝ | i | i | i TABLEI:ThefourMUBs inour3-dimensional Hilbertspace. Notethat phasez=exp(i2π/3) isacuberoot of unity,and we havesuppressed thenormalizing factor of 1/√3 in each of thenine MUB states in thelast threecolumns. response into all three reference states, e.g. Assuming thatthe interactionbetween the diffractedor- dersisslow,wecanneglectsecondordertermsandarrive 1 r b = , i 1,2,3 . at the CM equations for the mode amplitudes, i 4 3 h |P | i √3 ∀ ∈{ } The unique property of PTR glass with its bulk in- 0 0 0 1 1 z d∆enx/onf0re=fra3c3ti6opnp,mn0, =pla1c.4e86it5,saqnudardeleyptihnotfhmeordeuallmatsionof, RRR21′′   00 00 00 ρρ1112 zρρz∗12 zρρ1∗12 RRR12  sFwBcuriatarlhtgahgrBedrrsmiaefflgoregarc-ectp,itvliaaoitnnPyeTthpReeohrΛrioyo/ldLoasgn,rcdaaΛmcno∼cuaappλnlp.ebrdoe-Camtochohnidcskeethq(LueCe∼MndDti)fflyrt∼,ahcte1thoiceromyinr.  SSSacb′3′′′ =iκ2 σzσσ11∗abc σσσ1zzabc σzσσ11∗abc ρ0001 ρ0001 ρ0001  SSSabc3 , ∼ limit of one wave of tilt across its aperture [25]. For λ=1085nm and D L 1cm this yields a minimal where κ2 β2 ∆n , and ρ = kz and σ = kz divergence of our thre∼e sig∼nal waves of a few arcsec. are the z-≡com(cid:16)p6(o3n+e√n3t)s(cid:17)o(cid:0)f nth(cid:1)e wave viectorsi for thej referj- ToexaminetheMUB sorterwewillfollowcloselythe 4 ence and signal states, respectively. The solution of this CM approach of Kogelnik [23] and the notation used by equationforeachofthetwelveinitialsignalstates(MUB Case [24] to solve the scalar wave equation for polariza- states) are shown in Fig. 2 and faithfully reproduce the tion perpendicular to the plane of incidence. desired projection operator of Eq. 1. We independently 2E +k2E =0 (2) examined the far-field pattern for such gratings using a y y ∇ finite difference time domainsolutionofMaxwell’sequa- Here, the linearly-polarized electric field Ey(x,z) of fre- tions andobservedthat the CW assumptionswerevalid, quency ν is assumed to be independent of y. Following i.e. only the primary modes were dominant, and there CM theory we keep only primary modes for the electric were no relevant polarization changes in the field. field. These are the transverse harmonic modes given While the analysispresentedhere suggeststhat ahigh by the k-vectors ~k , ~k , ~k , ~k , ~k and ~k associated to 1 2 3 a b c efficiency single optical element sorter is feasible with planewavereferencestates r , r , r andMUB sig- | 1i | 2i | 3i 1 commerciallyavailablematerialsandholographicrecord- nal states a , b , c , respectively. | i | i | i ing techniques, further work is needed. First, we need E (x,z) =R (z)exp~k1~r+R (z)exp~k2~r+R (z)exp~k3~r to understand the quantum nature of such an element. y 1 · 2 · 3 · Second, we need to examine its sensitivity to alignment +Sa(z)exp~ka·~r+Sb(z)exp~kb·~r+Sc(z)exp~kc·~r. (linearandrotational)andexaminethe wavelengthscal- ing issues involved in recording OAM photons. Our first Here, the six mode amplitudes, R and S are only { i} { i} step will be to produce a 3-state k-QKD MUB1 sorter. functions of z. They are set initially to R (z = 0) = { i } Ourgoalistotestitsperformanceusingstatesgenerated 1,1,1 and to the corresponding amplitude and phase { } by a single phase SLM [14]. If the MUB-state sorters factors of one of the twelve corresponding signal states described here can be produced, they should have far shown in Table I. For example, signal state c > | 4 moreutilityinquantuminformationprocessingthanjust one would set S = 1/√3 1,z2,1 . The wavenumber { i} { } QKD,e.g. asanessentialelementinlinearquantumcom- k(x,z)ofEq.2representsthethreeincoherentlyrecorded puting [27]. gratings mentioned above, We wish to acknowledge important discussions with ∆n(I +I +I ) Glenn Tyler, Robert Boyd, Leonid Glebov, Geoff An- R1 R2 R3 k =n(x,z)k =n k 1+ , 0 0 0(cid:18) n0 6(1+√3) (cid:19) derson, David Reicher and Raymond Dymale. We are β alsogratefulforthe adviceprovidedby AngelaGuzman, |{z} Grigoriy Kreymerman, William Rhodes, Chris Beetle, where I is the intensity modulation of the ith grating, Ri and Ayman Sweiti of the FAU Quantum Optics group. e.g. We thank Anna Miller for reviewing this letter. This 1 work was supported by the Air Force Office of Scientific I =2 eik3r+ eikar+zeikbr+eikc r 2. R3 −| · √3 · · · | Research,and by AFRL/RDSE under the IPA program. (cid:0) (cid:1) 4 y 1 Signal a y 1 Signal a2 y 1 Signal a3 y 1 Signal a4 abilit2/3 abilit2/3 abilit2/3 abilit2/3 b b b b o o o o Pr1/3 Pr1/3 Pr1/3 Pr1/3 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 z (mm) z (mm) z (mm) z (mm) y 1 Signal b y 1 Signal b2 y 1 Signal b3 y 1 Signal b4 abilit2/3 abilit2/3 abilit2/3 abilit2/3 b b b b o o o o Pr1/3 Pr1/3 Pr1/3 Pr1/3 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 z (mm) z (mm) z (mm) z (mm) y 1 Signal c y 1 Signal c2 y 1 Signal c3 y 1 Signal c4 abilit2/3 abilit2/3 abilit2/3 abilit2/3 b b b b o o o o Pr1/3 Pr1/3 Pr1/3 Pr1/3 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 z (mm) z (mm) z (mm) z (mm) FIG.2: HereweshowthepredictedperformanceusingCMtheoryforthemultiplexedMUB4sorter. Thissingleopticalelement is a triplemultiplexed volumehologram constructed usingPTR glass (n0 =1.4865, ∆n=0.0005 at λ=1085nm). This small ratioofdepthofmodulationtobulkindex(∆n/n0 336ppm)placesitsquarelyinthelinearregimeofCMtheory. Thereisone ∼ graphforeachofthetwelveMUBstateswhichshowtheprobabilitythatthesignalphotonwillbeobservedtobediffractedinto oneoranotherofthethreeoriginalreferencedirections( ~rr1 =R1eik1·r, ~rr2 =R2eik2·r and ~rr3 =R3eik3·r)asafunction h | i h | i h | i of the depth (z)of the emulsion. The curves have been terminated near their maximum efficiency depth (zmax 8.5mm). ≈ Each of the three curves in each graph has its own unique value of zmax. They can be made roughly equivalent by proper choice of wavevectors ρi’s and the σj’s. The solid curve corresponds to r1 , the dashed curve to r2 and the dotted curve to | i | i r3 . A large probability amplitude in the r1 corresponds to a signal wave with a large projection along the qudit state a4 . | i | i | i The same is true for the other two MUB4 states. The twelve graphs are portrayed in a matrix. Each column represents each of thefour allowable MUBbases, and each row corresponds toone of thethreeMUB states within each MUB basis (Table I). 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