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Two-photon speckle as a probe of multi-dimensional entanglement C. W. J. Beenakker,1 J. W. F. Venderbos,1 and M. P. van Exter2 1Instituut-Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands 2Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands (Dated: January 2009) We calculate thestatistical distribution P2(I2) of thespeckle pattern produced by a photon pair current I2 transmitted through a random medium, and compare with the single-photon speckle distributionP1(I1). WeshowthatthepurityTrρ2 ofatwo-photondensitymatrixρcanbedirectly extracted from the first two moments of P1 and P2. A one-to-one relationship is derived between P1 and P2 if the photon pair is in an M-dimensional entangled pure state. For M ≫1 the single- photon speckle disappears, while the two-photon speckle acquires an exponential distribution. The 9 exponential distribution transforms into a Gaussian if the quantum entanglement is degraded to a 0 classical correlation of M ≫ 1 two-photon states. Two-photon speckle can therefore discriminate 0 between multi-dimensional quantumand classical correlations. 2 n PACSnumbers: 42.30.Ms,42.25.Dd,42.50.Dv,42.65.Lm a J Opticalspeckleistherandominterferencepatternthat 5 1 is observed when coherent radiation is passed through a diffusor or reflected from a rough surface. It has been ] much studied since the discovery of the laser, because h the speckle pattern carries information both on the co- p - herence properties of the radiation and on microscopic t details of the scattering object [1–3]. The superposition n a of partial waves with randomly varying phase and am- u plitudeproducesawidedistributionP(I)ofintensitiesI FIG. 1: Schematic layout (not to scale) of a setup to detect q around the average I . For full coherence and complete two-photon speckle. [ h i randomizationthe distribution has the exponential form 1 P(I) exp( I/ I ). The speckle contrast or visibility, ∝ − h i v alsodiscriminatebetweenquantummechanicalandclas- 2 V ≡hI2i/hIi2−1, (1) sical correlations of M modes. For classicalcorrelations, 3 on the one hand, the distributions P (I ) and P (I ) of equals to unity for the exponential distribution. 1 1 2 2 2 2 These textbook results [4] refer to single-photon prop- single-photon and two-photon speckle both tend to narrow Gaussians upon increasing M (with visibilities that 1. iesrtqieusaodfrathtiecriandtiahteiofine,ldexapmrepslsietdudbeys.anBoipbhsoertovnabolepIt1ictsh[a5t] vanish as 1/M). For quantum correlations, on the other 0 hand, P tends to the same narrow Gaussian while P 9 is concerned with observables I2 that are of fourth order 1 2 becomes an exponential distribution. 0 inthefieldamplitudes,containinginformationontheen- : tanglement of pairs of photons produced by a nonlinear Weconsideramonochromatictwo-photonstateofelec- v i optical medium. A variety of biphoton interferometers tromagnetic radiation (density operator ρîn), scattered X have been studied [6–9], but the statistical properties of by a random medium (scattering matrix S). A pair of r the biphotoninterference patternproducedby arandom photodetectors in a coincidence circuit is located in the a medium remain unknown. It is the purpose of this work far field behind the random medium (see Fig. 1). The to provide a theory for such “two-photon speckle”. coincidence detection projects the scattered two-photon There is a need for a such a theory, because of recent state (density operator ρôut) onto quantum numbers k developments in the capabilities to produce entangled and k′, which label the transverse wave vectors of an or- two-photonstatesofhighdimensionality. Thefamiliar[4] thonormal basis of modes. For a random medium of N polarization entangled two-photon state has dimension- cross-sectionalarea andforradiationofwavelengthλ, ality two and encodes a qubit [10]. Multi-dimensionally one has π /λ2Aper polarization degree of freedom. N ≃ A entangled two-photon states include spatial degrees of Thespatialstructureofthemodes(andtheprecisevalue freedom [11–16] and encode a “qudit”. The dimension- of ) depends on the experimental geometry [17, 18], N ality of the entanglement is quantified by the Schmidt but all our statistical results are independent of it (for rankM,whichcountsthenumberofpairwisecorrelated, 1)soweneednotspecifythemodesfurtherforour N ≫ orthogonal modes that have appreciable weight in the purpose. two-photon wave function [17] and is an experimentally In the far field (at a distance √ from the ran- D ≫ A adjustable parameter [18]. dommedium),thetransmittedphotoncurrentI (k)ata 1 As we will show in this paper, two-photon speckle not givenkisdetectedasabrightspotofareaδ 2/ only provides information on the value of M, but it can λ2. The random arrangement of bright anAd≃daDrk sNpo≫ts 2 (the speckle pattern) depends sensitively on the realiza- then b = S a a = S† b. Substitution into Eq. (2) · ⇔ · tion of the randomness (for example, on the precise con- gives the density operator of the outgoing state, figuration of the scattering centra), and by varying the random medium [19] one samples a statistical distribu- ρôut = 12 ρq1q2,q1′q2′(ST ·b†)q1(ST ·b†)q2 tion P1(I1). qX1,q2qX1′,q2′ speTchkeleq.uTahnetitbiiepshIo1t(okn)caunrdrePn1t(II1)(kr,ekfe′)rctoousnintsglteh-pehnoutmon- ×|0ih0|(S†·b)q1′(S†·b)q2′. (6) 2 ber of coincidence detection events per unit time, with From ρˆ we obtain the biphoton current I (k,k′) by a out 2 onephotonatkandtheotheratk′. (Weassumek =k′.) projection, 6 The distribution of I in an ensemble of random realiza- 2 tions of the disorder is denoted by P2(I2) and describes I2(k,k′)= 21α2Trρôutb†kb†k′bkbk′, (7) two-photon speckle. Our goal is to find out what new where the coefficient α accounts for a nonideal detec- information on the quantum state of the radiation can 2 tion efficiency and also contains the repetition rate of be extracted from P , over and above what is available 2 the photon pair production. from P . 1 Wenowsubstitute Eq.(6)intoEq.(7)toarriveatthe The most general two-photon density operator at the required relation between the biphoton current and the input has the form scattering matrix, ρîn = 21qX1,q2qX1′,q2′ ρq1q2,q1′q2′a†q1a†q2|0ih0|aq1′aq2′, (2) I2(k,k′)=α2qX1,q2qX1′,q2′ ρq1q2,q1′q2′Skq1Sk′q2Sk∗q1′Sk∗′q2′. (8) with a† the photon creation operator in state q and 0 q | i Here we have assumed that ρ is symmetric in both the the vacuum state. The coefficients in this expansion are first and second set of indices, collected in the 2 2 Hermitian density matrix ρ. N × N Normalization requires that Trρ = 1. If the two-photon ρq1q2,q1′q2′ =ρq2q1,q1′q2′ =ρq1q2,q2′q1′. (9) state is a pure state, then also Trρ2 = 1, while more [We canassume this without lossof generality,since any generally the purity antisymmetric contribution to ρ would drop out of Eq. =Trρ2 [0,1] (3) (2).] P ∈ In order to compare with the single-photon current quantifies how close the state is to a pure state [10]. I (k), we give the corresponding expressions, 1 We will present an exact and general theory of the speckle statistics for arbitrary ρîn, and also consider two I1(k)= 21α1Trρôutb†kbk =α1 SkqSk∗q′ρ(q1q)′, (10) specific simple examples: A maximally entangled pure q,q′ X state of Schmidt rank M, in terms of the reduced single-photon density matrix M ρˆpure =|ΨMihΨM|, |ΨMi=M−1/2 a†qma†−qm|0i, ρ(q1q)′ = ρqq2,q′q2. (11) mX=1 Xq2 (4) Thecoefficientα isthesingle-photondetectionefficiency and its fully mixed counterpart 1 (which may or may not be different from α ). 2 M The next step is to calculate the statistical distri- ρˆ =M−1 a† a† 0 0a a . (5) butions P , P of I , I . Following the framework of mixed qm −qm| ih | qm −qm 1 2 1 2 m=1 random-matrix theory [21, 22], we make use of the fact X thatthematrixelementsS haveindependentGaussian kq Both states (4) and (5) describe a pair of photons with distributionsfor 1. (Correctionsareoforder1/ .) anticorrelated transverse wave vectors [20]: If one pho- N ≫ N The first moment vanishes, S = 0, while the second kq ton has wavevector q , then the other photon has wave h i m momentdepends on whether the radiationis detected in vector q . (We assume q = 0 for each m.) The dis- m m transmission or in reflection. In transmission through a − 6 tinction between the two states is that the two photons random medium of length L with mean free path l, one in state (4) are quantum mechanically entangled, while has the correlation in state (5) is entirely classical. We will 2l see how this difference shows up in the statistics of two- S 2 = σ2. (12) kq photon speckle. h| | i L ≡ N Scattering by the random medium (in the absence of Let us begin by calculating the first two moments of absorption) performs a unitary transformation on the P , P . Carrying out the Gaussian averages, we find for 1 2 creation and annihilation operators. If we collect the the mean values: operators for the incident radiation in the vector a and the operators for the scattered radiation in the vector b, I =α σ2, I =α σ4. (13) 1 1 2 2 h i h i 3 (We omit the arguments k and k′ for notational simplic- ity.) Neithermeanvaluecontainsanyinformationonthe nature of the two-photon state. This is different for the variances VarI I2 I 2, for which we find i ≡h ii−h ii VarI =α2σ4Tr ρ(1) 2, (14) 1 1 VarI2 =α22σ8 Tr(cid:0)ρ2+(cid:1)2Tr ρ(1) 2 . (15) We conclude that the pu(cid:2)rity (3) of th(cid:0)e two(cid:1)-p(cid:3)hoton state can be obtained from the visibilities (VarI )/ I 2 i i i V ≡ h i of the single-photon and two-photon speckle patterns, = 2 . (16) 2 1 P V − V This is the first key result of our work. To make contactwith some of the literatureon biphoton interferometry, we note that in the case of a pure FIG. 2: Probability distribution (20) of the two-photon two-photon state (when =1) knowledge of the single- P speckleforthemaximallyentangledpurestate(4)ofSchmidt photon visibility fixes the two-photon visibility . 1 2 rankM. Theexponentialdistribution(19)(blacksolidcurve) V V The same holds (with some restrictions on the class of isreachedinthelimitM →∞. Theblackdashedcurveshows pure states and with a different definition of visibility) the large-M Gaussian distribution for the fully mixed state forthe complementarityrelationsofRefs.[6–9]. No such (5) (plotted for M =50). one-to-one relationship between and exists, how- 1 2 V V ever, for a mixed two-photon state. We can actually give a closed form expression for P We next turn to the full probability distribution P 2 2 in terms of the eigenvalues of the matrix product cc† of the two-photon speckle. Notice first that, if ρ is far from a pure state, the ratio √ of the width of the (see App. B), but it is rather lengthy. A more compact 2 V expressionresults for the specialcase ofa maximallyen- distribution and the mean value is 1. Indeed, for the fully mixed state (5) one has Trρ≪ˆ2 = 1/M and tangledpurestateofSchmidtrankM [Eq.(4)]. Thenall mixed eigenvaluesofcc† arezeroexceptasingle2M-folddegen- Tr(ρˆ(1) )2 =1/2M,so =2/M 1forM 1. The mixed V2 ≪ ≫ erate eigenvalue [23] equal to 1/2M. The single-photon relative magnitude of higher order cumulants is smaller speckle distribution P I2M−1exp( 2MI /α σ2) is a byadditionalfactorsof1/M,henceP2 tendstoanarrow chi-squaredistribution1w∝ith14M degree−soffre1edo1m(since Gaussian for a fully mixed state with M 1. ≫ I M (S 2+ S 2) is the sum of 2M Gaus- The situation is entirely different in the opposite limit 1 ∝ n=1 | k,qn| | k,−qn| sian complex numbers squared). Substitution into Eq. of a pure state. The density matrix of a pure state fac- P (18)leadstothefollowingdistributionofthetwo-photon torizes, speckle: ρq1q2,q1′q2′ =cq1q2c∗q1′q2′, (17) 4M 2MI M−1/2 2 withcasymmetric matrixnormalizedbyTrcc† = P2(I2)=Θ(I2)α σ4(2M 1)! α σ4 1. The correspondinNg×reNduced single-photondensity ma- 2 − (cid:18) 2 (cid:19) trix is ρ(1) = cc†. The probability distributions P2 and K 2 2MI2 . (20) P1 in this case of a pure two-photonstate are related by × 2M−1" rα2σ4 # an integral equation, which we derive in App. A: The function K is a Bessel function. This distribu- 2M−1 α1 ∞ P1(I1) α1 I2 tion has appeared before in the context of wave propa- P (I )=Θ(I ) dI exp . 2 2 2 α σ2 1 I −α σ2I gation through random media [2] (where it is known as 2 Z0 1 (cid:18) 2 1(cid:19) (18) the “K-distribution”),but there the parameter M has a Here Θ(I) is the unit step function [Θ(I) = 1 if I > 0, classical origin (set by the number of scattering centra) Θ(I)=0 if I <0]. — rather than the quantum mechanical origin which it Without further calculation, we can conclude that hasinthepresentcontext(beingtheSchmidtrankofthe when P is narrowly peaked around the mean I , the entangled two-photon state). 1 1 h i corresponding two-photon speckle distribution is the ex- We haveplottedthe distribution(20)for different val- ponential distribution, ues of M in Fig. 2. The limiting value for I 0 equals 2 → α I 2M P (I ) exp 1 2 , if 1. (19) lim P (I )= . (21) 2 2 ∝ (cid:18)−α2σ2hI1i(cid:19) V1 ≪ I2→0 2 2 (2M −1)hI2i Thelimitingexponentialformisreached,forexample,in The exponential form (19) is reached quickly with in- the pure state (4) for M 1 (when = 1/2M 1). creasing M (black solid curve in Fig. 2). For compari- 1 ≫ V ≪ This is the second key result of our work. son,weshowinthe samefigure(blackdashedcurve)the 4 Gaussian distribution reached for large M in the case of we are considering the density matrix ρ of a pure state, the fully mixed two-photon state (5). The striking dif- which means that factorizes as M ference with the entangled case is the third key result of our work. Mqq′ =(c·v)q(c·v)∗q′. (A6) In conclusion, we have presented a statistical descrip- All eigenvalues of this matrix of rank 1 vanish, except tion of the biphoton analogue of optical speckle. For an one nonzero eigenvalue µ given by 1 arbitrary pure state of two photons, the distribution P 2 of the two-photonspeckle is related to the single-photon 1 µ = (c v) 2 =v ρ(1) v∗ = I (k), (A7) speckle distribution P1 by an integral equation. A nar- 1 | · q| · · α1 1 q row Gaussian distribution P maps onto a broad expo- X 1 nential distribution P2. If the two-photon state is not inviewofthedefinition(10)ofthesingle-photoncurrent pure, there is no one-to-onerelationshipbetween P1 and I1. P2. For that case we show that knowledge of the visi- Becauseofrelation(A7)theaverageovervinEq.(A5) bilities of the single-photonand two-photon speckle pat- may be replaced by an average over I , 1 ternsallowsoneto measurethepurityofthe two-photon ∞ state,therebydiscriminatingbetweenclassicalandquan- F (ξ)= dI P(I ) 1 iξ(α /α )σ2I −1. (A8) 2 1 1 2 1 1 tum correlations of M degrees of freedom. All together, Z0 − this theory provides a framework for the interpretation (cid:0) (cid:1) Inverse Fourier transformation gives of ongoingexperiments [24] onthe propagationofmulti- dimensionally entangled radiation through random me- 1 ∞ P(I )= dξe−iξI2F (ξ) dia. 2 2π 2 We acknowledge discussions with W. H. Peeters and Z−∞ α ∞ P (I ) α I J. P. Woerdman. This research was supported by the =Θ(I ) 1 dI 1 1 exp 1 2 . 2 α σ2 1 I −α σ2I Dutch Science Foundation NWO/FOM. 2 Z0 1 (cid:18) 2 1(cid:19) (A9) This is the required relation (18) between the distribu- APPENDIX A: DERIVATION OF THE INTEGRAL RELATION BETWEEN P2 AND P1 tionsP2 andP1 oftwo-photonandsingle-photonspeckle. FOR A PURE STATE APPENDIX B: CALCULATION OF P FOR A 2 We startfromthe expression(8) for the biphotoncur- PURE STATE rent, which we write in the more compact form I2(k,k′)=α2v′ v′∗, (A1) A general expression can be obtained for the distribu- ·M· in terms of a vector v′ with elements vq′ = Sk′q, and a tfeiownsPte1psofofsiAngplpe.-pAh:oton speckle, by repeating the first matrix with elements M Mqq′ = vq2ρqq2,q′q2′vq∗2′. (A2) F1(ξ)= ∞ dI1eiξI1P1(I1) qX2,q2′ Z−∞ (Wehavealsodefinedavectorvwithelementsvq =Skq.) =hexp(iξα1v·ρ(1)·v∗)iv The probability distribution P2 of I2 is defined by 1 = . (B1) P2(I2)=hδ(I2−α2v′·M·v′∗)iv,v′, (A3) Det(1−iξα1σ2ρ(1)) with Fourier transform The reduced single-photon density matrix ρ(1) has dis- ∞ tinct positive eigenvalues γ , each with multiplicity µ F (ξ)= dI eiξI2P (I ) m m 2 2 2 2 and satisfying the sum rule µ γ = 1. In terms of Z−∞ m m m these eigenvalues, Eq. (B1) can be written as = exp(iξα2v′ v′∗) v,v′. (A4) P h ·M· i Theaverage v,v′ consistsofanaverageoverthescat- F1(ξ)= 1 iξα1σ2γm −µm h···i − teringmatrixelementsSkq containedinthe vectorv and Ym (cid:0) (cid:1) an averageover the scattering matrix elements Sk′q con- ( 1)µm−1 tained in the vector v′. These two averages can be per- = − (µ 1)! formed independently, since we have assumed k =k′. m m− Y The Gaussian average over v′ can be carried6 out di- dµm−1 (x iξα σ2γ )−1 . rectly, × dxµm−1 m− 1 m (cid:20) m (cid:21)xm→1 1 (B2) F (ξ)= . (A5) 2 Det(1 iξα σ2 ) (cid:28) − 2 M (cid:29)v Inverse Fourier transformation gives the probability dis- To carry out the remaining average over v we need to tribution, first evaluate the determinant. At this point we use that 5 1 ( 1)µm−1 dµm−1 x I P1(I1)=Θ(I1) m α1σ2γm (−µm−1)! dxmµm−1 exp(cid:18)−α1σm2γ1m(cid:19)m′6=m(1−xmγm′/γm)−µm′ . (B3) X Y xm→1   This resultfor P holds for any state ofthe radiation,but if we nowassume that the state is pure, then we canuse 1 the relation(18) to obtain the two-photonspeckle distribution P from P . (Note that, for a pure state, the γ ’s are 2 1 m just the eigenvalues of the matrix product cc†.) Substitution of Eq. (B3) into Eq. (18) gives 2 ( 1)µm−1 dµm−1 x I P2(I2)=Θ(I2) m α2σ4γm (−µm−1)! dxmµm−1K0 2sα2σm4γ2m !m′6=m(1−xmγm′/γm)−µm′ . (B4) X Y xm→1   The result (20) given in the main text follows from Eq. (B4) upon taking a single positive eigenvalue γ = 1/2M 1 with multiplicity µ =2M, so that 1 4M ( 1)2M−1 d2M−1 2MxI 2 P (I )=Θ(I ) − K 2 2 2 2 α σ4(2M 1)! dx2M−1 0 α σ4 2 − " r 2 !#x→1 M−1/2 4M 2MI 2MI 2 2 =Θ(I ) K 2 . (B5) 2 α σ4(2M 1)! α σ4 2M−1 α σ4 2 − (cid:18) 2 (cid:19) r 2 ! [1] J. C. Dainty,editor, Laser Speckle and Related Phenom- Eliel, G. W. ’t Hooft, and J. P. Woerdman, Phys. Rev. ena (Springer, 1984). Lett. 95, 240501 (2005). [2] L. C. Andrews and R. L. 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