Four-Photon (In)Distinguishability Transition
Malte C. Tichy, Hyang-Tag Lim, Young-Sik Ra, Florian Mintert, Yoon-Ho Kim, Andreas Buchleitner
FFour-Photon (In)Distinguishability Transition
Malte C. Tichy, Hyang-Tag Lim, Young-Sik Ra, Florian Mintert,
1, 3
Yoon-Ho Kim, and Andreas Buchleitner Physikalisches Institut, Albert-Ludwigs-Universit¨at,Hermann-Herder-Str. 3, D-79104 Freiburg, Germany Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang, 790-784, Korea Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universit¨at, Albertstr. 19, D-79104 Freiburg, Germany (Dated: November 21, 2018)We demonstrate the conspiration of many-particle interferences of different degree to determinethe transmission of four photons of tunable indistinguishability through a four-port beam splitterarray. The probability of certain output events depends non-monotonically on the degree of distin-guishability, due to distinct multi-particle interference contributions to the transmission signal.
PACS numbers: 03.65.-w, 05.30.Jp, 05.30.-d
The symmetrization postulate imposed by the indis-tinguishability of particles is a fundamental quantumconcept with no classical counterpart, which stronglyinfluences the behavior of matter at any energy scale.In quantum optics, the most prominent manifestationthereof is the Hong-Ou-Mandel (HOM) effect [1]: twoindistinguishable photons falling on the opposite inputports of a balanced beam splitter always leave the setuptogether, i.e. all amplitudes with two photons in dif-ferent output modes interfere destructively, while eventswith both photons in the same mode are enhanced dueto the photons’ bosonic nature . When tuning the transi-tion from distinguishability to indistinguishability of thephotons, which can be described by a single parameter[2], the visibility of the HOM dip in the probability todetect one photon per mode increases monotonically.Signatures for the full indistinguishability of more thantwo particles can be observed, e.g. , when many indistin-guishable photons bunch at one output mode of a beamsplitter [3–5]. In addition, when several particles enter amultiport beam splitter [6] simultaneously, many-particleinterferences strongly influence the probability of indi-vidual counting events, such that many distinct eventswith a given number of particles per output port are to-tally suppressed [7–9]. The transition between fully dis-tinguishable and fully indistinguishable particles has notreceived attention beyond its impact on bosonic bunch-ing [4]. Its role for the behavior of other events, e.g. theones that are suppressed for indistinguishable particles[8, 9], is widely open. A thorough understanding of par-tial indistinguishability is, however, mandatory for theexperimental characterization of the degree and natureof many-particle interferences.In the present Letter, we consider four photons thatpropagate through a four-port beam splitter array. Incontrast to an intuitive extrapolation of the features ofthe well-known two-photon case [1], and of the bunchingbehavior of many photons [4], we show that (i) the degreeof distinguishability manifests itself in non-monotonic event probabilities, and that (ii) events with large occu-pation numbers are not necessarily more likely with in- creasing indistinguishability, despite the bosonic natureof photons. These results are established by quantitativepredictions on experimentally directly accessible quanti-ties.We study many-particle interference within a setupwith four ports [10], see Fig. 1(a). In this arrangement,no single-particle (Mach-Zender-like) interference can oc-cur, what makes it perfect to study the genuine manifes-tation of many-particle interference. ˆ ,† a t ˆ ,† a t ˆ ,† a t ˆ ,† a t ˆ ,† a t ˆ ,† a t †4 ˆ b †3 ˆ b †2 ˆ b †1 ˆ b αφ ( ) a ( ) b ( ) χ Laser x x x x ˆ ,† a t ˆ ,† a t FIG. 1: (color online) (a) Diamond-shaped multiport (bal-anced) beam splitter array. Four ingoing modes ˆ a † i are redis-tributed onto the output modes ˆ b † j . The path length x j ofeach incoming mode controls the mutual distinguishability ofthe particles in the setup. (b) Creation of four photons byspontaneous parametric down conversion, by double-passageof a laser pulse through a non-linear crystal. Single-particle evolution is here described by a unitarymatrix which relates particle creation operators of inputand output ports, ˆ a † i,ω and ˆ b † k,ω , respectively, viaˆ a † i,ω → (cid:88) k =1 U ik ( α, φ )ˆ b † k,ω , (1)with the photon frequency ω unchanged, and U ( α, φ )given by U ( α, φ ) = 12 e iφ e iφ e i ( φ + α ) e i ( φ + α ) − e iα − e iα − − − − . (2) a r X i v : . [ qu a n t - ph ] M a r For any value of α and φ , the matrix U ( α, φ ) is a complexunitary matrix with | U jk | = 1 / i.e. a Hadamard matrix[11]. The phases α, φ have two rather distinct physical in-terpretations: φ absorbs all relative phases between inputmodes, and is well-known from the two-mode interfer-ence of many particles, e.g. in N00N-state interferometry[12]. α is of distinct origin: It corresponds to the phaseenclosed by the interfering modes (similar to a Sagnacinterferometer [13]), emerges only in the four-mode case[11], and effectively controls the relative phases of the output components, conditioned on the input modes.Four photon states are created by the double passage ofa laser pulse through a non-linear crystal, through spon-taneous parametric down conversion, see Fig. 1(b). Suchstates consist of the coherent superposition of a quadru-plet part, with one photon in each mode, created whenthe laser pulse induces one pair of photons at each timeit passes the crystal, with two double-twin parts, wherefour photons are distributed among two modes, result-ing from events where the pulse generates two pairs inthe one or in the other passing direction [16]. Since allthree processes occur with the same probability, the ini-tial state reads | Ψ (cid:105) = 1 √ (cid:89) j =1 ˆ a † j,t j + ˆ a † ,t ˆ a † ,t a † ,t ˆ a † ,t | (cid:105) . (3)The operatorˆ a † j,t j = (cid:90) ∞−∞ d ω √ π ∆ ω e − ( ω − ω ω e iωt j ˆ a † j,ω (4)creates a single photon with central frequency ω andspectral width ∆ ω at the input port j , at time t j . Thearrival times t j of the photons can be tuned through vari-able path lengths x j = c · t j depicted in Fig. 1(a), andhence their overlap, or indistinguishability, by virtue of | (cid:104) | ˆ a i,t j ˆ a † i,t k | (cid:105) | = exp( − ∆ ω ( t j − t k ) ) A change ofrelative path lengths induces a phase-shift between theinput-modes, which can be accounted for in φ : φ → φ + ω ( t + t − t − t ) (5)All possible final states that emerge from the multi-port can be characterized in terms of the photon numberdetected in each port. For four photons, simple combina-torics yields 35 distinct events which can be labelled withvectors (cid:126)s = ( s , s , s , s ), where s j is the number of par-ticles in port j . Their order is hereafter given by their rel-ative abundance in the fully distinguishable case, eq. (6)below, and such that vectors (cid:126)s with large s come first,i.e., (cid:126)s = (4 , , , , (cid:126)s = (0 , , , , . . . , (cid:126)s = (3 , , , , and, finally, (cid:126)s = (1 , , , j th particleneeds up to j non-vanishing components. Each termcorresponds to a certain distinguishability setting thatwe will denote by { i , i , i , i } , where photons in port k and l are indistinguishable if i k = i l . Fully in-distinguishable/distinguishable particles correspond to { , , , } / { , , , } , and only indistinguishable parti-cles interfere. Hence, the distinguishability setting deter-mines the degree of many-particle interference. In gen-eral, a setup with given arrival times for all photons cor-responds to a situation in which several distinguishabil-ity settings contribute to the initial state, with differentdegrees of multi-particle interference occurring simulta-neously.The extreme case of full distinguishability { , , , } isrealized when all photons have pairwise delays | t i − t j | (cid:29) / ∆ ω . In this case, we can safely neglect the expo-nentially suppressed components that still exhibit many-particle interference, and simple combinatorics can beapplied to yield the output event probabilities (remem-ber that no single-particle interference can occur in oursetting): P dist ( (cid:126)s ) = 4!4 (cid:81) j s j ! . (6)The opposite, fully indistinguishable limit { , , , } isrealized for t = t = t = t , when all photons caninterfere perfectly.The resulting probabilities for five representativeevents are compared in Tab. I, for the fully distinguish-able and indistinguishable case, respectively. The event (cid:126)s = (0 , , ,
3) is fully suppressed for indistinguishablephotons, for any choice of the phases α, φ , while (cid:126)s , (cid:126)s and (cid:126)s exhibit an intricate α, φ -dependence. Most im-portantly, there is no unambiguous correlation betweenthe event probability and the photon distribution on theoutput modes: as anticipated above, s = (4 , , ,
0) maybe enhanced (as expected for bosonic bunching) as wellas strictly suppressed (for the experimentalist’s choice φ = π ). Unexpectedly, (cid:126)s = (1 , , ,
1) may be en-hanced up to a weight 32 /
9. In stark contrast to thetwo-photon HOM effect, the manifestation of perfect in-distinguishability is not unique in the present, multi-particle interference scenario, and has no intuitive inter-pretation in terms of the occupation of modes. This ishighlighted by Fig. 2, which displays all event probabil-ities, for fully distinguishable and indistinguishable par-ticles, and three different choices of α and φ , and thusdemonstrates the loss of any indistinguishability-inducedhierarchy in the event probabilities. Furthermore, not Event P dist P id ( (cid:126)s ) (cid:126)s = (4 , , ,
0) 1/256 cos ( φ/ / (cid:126)s = (0 , , ,
3) 1/64 0 (cid:126)s = (0 , , ,
2) 3/128 [cos( α ) + cos( α + φ )] / (cid:126)s = (0 , , ,
2) 3/128 [1 − α + φ )] / (cid:126)s = (1 , , ,
1) 3/32 [cos( α ) − cos( α + φ )] / P dist )and indistinguishable particles ( P id ). only are the event probabilities no unambiguous wit-nesses of (in)distinguishability any more, but they evenevolve non-monotonically with decreasing distinguisha-bility of the particles, as can be demonstrated in our set-ting, by subsequently rendering different pairs of photonsindistinguishable, and thus adding interference terms be-tween an increasing number of photons. P r ob a b ilit y s j FIG. 2: (color online) Event probabilities of all 35 possibleoutput configurations (cid:126)s j , for distinguishable (black diamonds)and fully indistinguishable particles, with φ = α = 0 (bluesquares), φ = 0 , α = π/ φ = π/ , α = 0(brown triangles). Let us focus on the effects of partial interference onthe three events (cid:126)s = (1 , , , , (cid:126)s = (0 , , , , (cid:126)s =(0 , , ,
2) (all listed in Tab. I), within two exemplarydistinguishability transitions: For specificity, we assumethat the single photons have a central wavelength of λ = 780 nm, and a full width at half maximum (FWHM)of ∆ λ = 5 nm, what corresponds to a coherence lengthof l c ≈ µ m. (I) We first adjust the path lengths in a continuousway, by parametrizing x = 0, x = y , x = − y and x = 2 y . Hence, for y = − µ m, the particles arefully distinguishable { , , , } , while for y = 0 µ m, theirwave functions fully overlap, the particles are fully indis-tinguishable { , , , } . During the transition, severaldistinct contributions to the output signal arise simul-taneously (see Fig. 3a). We initially choose α = 0 and φ = 0. The latter parameter remains constant for allvalues of y with our chosen parametrization (cf. (5)).Consequently, only variations on length-scales of the or-der of the coherence length l c appear. The signal evolution during the (controlled) indistin-guishability transition is shown on the bottom left ofFig. 3: The event (cid:126)s = (1 , , ,
1) is progressively sup-pressed as higher order interference contributions, fromtwo- over three- to four-photons, kick in. This contrastswith the clearly non-monotonic event probability of (cid:126)s =(0 , , , (cid:126)s = (0 , , , (II) Another remarkable impact of the distinguisha-bility transition is born out when we adjust the pathlengths step-by-step: Starting from fully distinguishablephotons, with x = 0, x = 220 µ m, x = 440 µ m, x = 660 µ m, we first tune down x , and subsequently x and x , as indicated in Fig. 3(c), with α = π .The thus chosen, dominant distinguishability settings are { , , , } → { , , , } → { , , , } → { , , , } , andthis step-wise transition is reflected by the event prob-abilities in Fig. 3(d). In particular, the probability of (cid:126)s = (0 , , ,
3) is again non-monotonic, due to a destruc-tive two-photon contribution, followed by a constructive three-photon contribution for the { , , , } setting.An additional effect manifests when four-photon inter-ference sets in: the event probabilities start to dependon the phase φ . This leads to fast oscillations on thescale of the photon wavelength λ , an intrinsic feature ofthe interference of four or more photons, shown in theplots as shaded area between minima and maxima. Thedependence of the event probability on φ for fully indis-tinguishable photons (Tab. I) explains the onset of fastoscillations of the event probabilities for (cid:126)s and (cid:126)s , andthe growth of their amplitudes with increasing { , , , } contribution. Unexpectedly, also (cid:126)s = (0 , , ,
3) ex-hibits such oscillations which, however, vanish when four-photon interference fully dominates, since this event isthen strictly suppressed (Tab. I). It is the interplay of the { , , , } and { , , , } settings that leads to such de-pendence on φ : the double-twin part of the wave functionin the setting { , , , } interferes with the quadrupletpart in the { , , , } setting, resulting in four-photoninterference that depends on φ , and a local maximum ofthe event probability at the point where the contribu-tions of { , , , } and { , , , } are equal. This featureis specific to the partial distinguishability of the photons.The discussed phenomena constitute show-case exam-ples for the intricate effects that manifest when quantify-ing or controlling the particles’ degree of indistinguisha-bility in an experiment. Despite the mere doubling of thenumber of particles with respect to the HOM setup, the(in)distinguishability transition cannot be explained anymore by extrapolation of the two-photon effect: interfer-ence dominates over bosonic bunching, and events with (a) all distinguishable,i.e. {1,2,3,4}two identical, e.g. {1,1,2,3} three identical, e.g. {1,1,1,4}all identical,{1,1,1,1}two pairs, e.g. {1,1,2,2}
45 0 C on t r i bu ti on Transition Parameter y Μ m Μ m x2 Μ m x3 Μ m x4 Μ m C on t r i bu ti on Transition Parameter y Μ m (c) {1,2,3,4} {1,1,3,4} {1,1,1,4} {1,1,1,1}{1,1,2,2}(0,1,0,3) (0,2,0,2) (d) (0,1,0,3)(0,2,0,2) (1,1,1,1) (b) (1,1,1,1) x = 220 µ mx = 440 µ mx = 660 µ m x = 660 µ mx = 440 µ mx = 660 µ mx = 0 µ m x = 0 µ m x = 0 µ mx = 0 µ m x = 0 µ m P
45 00.000.020.040.060.080.10 Μ m y P y [ µ m] x = 0 µ m FIG. 3: (color online) Evolution of event probabilities during the transition from distinguishable to indistinguishable particles(bottom row). Left: continuous transition, parametrized by y , with α = 0. Right: step-wise transition, with correspondinginterferometer path lengths, Fig. 1, on top of panel (c), with x = 0 µ m, α = π . (a,c): Distinguishability type contributionsto the output signal: While for the step-wise transition, at most two distinguishability settings are relevant at a time, severalmay contribute in the continuous transition. (b,d): Event probabilities of (cid:126)s = (1 , , ,
1) (blue), (cid:126)s = (0 , , ,
3) (red) and (cid:126)s = (0 , , ,
2) (green). The shaded areas in (d) represent four-photon interference fringes oscillating as a function of φ . large occupation numbers are not necessarily enhancedwhen approaching indistinguishability. The phase α which quantifies the phase enclosed by the setup con-stitutes an input- and output-mode dependent parame-ter with measurable impact, as a physical consequenceemerging from the formal definition of four-dimensionalcomplex Hadamard matrices [11]. Finally, not only thedegree, but also the quality of interference changes withthe number of interfering particles.Our results impressively demonstrate that we need tofully abandon the idea that many-particle interferencemanifests itself in a unique fashion that can be predictedfrom the bosonic nature of particles alone, as usuallystated in the two-particle case [15]. Constructive anddestructive interference effects occur for all final events,and they can be turned into one another by variation ofthe phases. Our decomposition of the initial state intodistinguishability settings does not only offer a powerfultool for the computation of event probabilities, necessaryfor the experimental implementation of many-particle in-terference, but also allows us to understand and clas-sify the occurring phenomena. Due to the combinatorialexplosion of possible many-particle paths, a descriptionof the indistinguishability transition becomes prohibitivefor many more than four particles, and it remains to bestudied whether one can find other, coarse-grained ob-servables [14] with a monotonic dependence on a singleparameter, to quantify many-particle distinguishability.We thank Jaewan Kim and Joonwoo Bae for fruit-ful discussions and hospitality, during the KIAS Work-shop on Quantum Information Sciences 2009, where this work was initiated. Financial support by the DAAD-GEnKO programme (M.C.T., F.M., A.B.), by the Na-tional Research Foundation of Korea (2009-0070668 andKRF-2009-614-C00001; H.-T.L., Y.-S.R., Y.-H.K.), bythe German Academic National Foundation (M.C.T.),and through DFG grant MI 1345/2-1 (F.M.) is gratefullyacknowledged. [1] C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. , 2044 (1987).[2] M.C. Tichy et al. , arXiv:0902.1684.[3] Z. Y. Ou, Phys. Rev. A , 043829 (2008).[4] G. Y. Xiang et al. , Phys. Rev. Lett. , 023604 (2006).[5] X.-L. Niu et al. , Opt. Lett. , 1297 (2009).[6] A. Vourdas and J. A. Dunningham, Phys. Rev. A ,013809 (2005).[7] R. A. Campos, Phys. Rev. A , 013809 (2000).[8] Y. L. Lim and A. Beige, N. J. Phys. , 155 (2005).[9] M.C. Tichy et al. , Phys. Rev. Lett. , 220405 (2010).[10] M. ˙Zukowski, A. Zeilinger, and M. A. Horne, Phys. Rev.A , 2564 (1997).[11] W. Tadej and K. ˙Zyczkowski, Open. Sys. & Inf. Dyn. , 133 (2006).[12] I. Afek, et al. , Science , 879 (2010).[13] G. Sagnac, Comptes Rendus , 708[14] K. Mayer et al. , arXiv:1009.5241.[15] T. Jeltes et al. , Nature , 402 (2007).[16] The relative phase between double-twin and quadrupletcontribution is absorbed in the phase φφ