Generalized Multiphoton Quantum Interferencehdeguise.lakeheadu.ca/pdffiles/Generalized_multiphoton_interference.pdf · Generalized Multiphoton Quantum Interference ... In our framework,

Generalized Multiphoton Quantum Interference

Max Tillmann,1,* Si-Hui Tan,2 Sarah E. Stoeckl,1 Barry C. Sanders,3,4 Hubert de Guise,5

René Heilmann,6 Stefan Nolte,6 Alexander Szameit,6 and Philip Walther11Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria2Singapore University of Technology and Design, 20 Dover Drive, 138682 Singapore

3Institute for Quantum Science and Technology, University of Calgary,Calgary, Alberta, T2N 1N4 Canada

4Program in Quantum Information Science, Canadian Institute for Advanced Research,Toronto, Ontario, M5G 1Z8 Canada

5Department of Physics, Lakehead University, Thunder Bay, Ontario, P7B 5E1 Canada6Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller Universität Jena,

Max-Wien-Platz 1, D-07743 Jena, Germany(Received 22 April 2015; revised manuscript received 21 July 2015; published 27 October 2015)

Nonclassical interference of photons lies at the heart of optical quantum information processing. Here,we exploit tunable distinguishability to reveal the full spectrum of multiphoton nonclassical interference.We investigate this in theory and experiment by controlling the delay times of three photons injected into anintegrated interferometric network. We derive the entire coincidence landscape and identify transitionmatrix immanants as ideally suited functions to describe the generalized case of input photons witharbitrary distinguishability. We introduce a compact description by utilizing a natural basis that decouplesthe input state from the interferometric network, thereby providing a useful tool for even larger photonnumbers.

DOI: 10.1103/PhysRevX.5.041015 Subject Areas: Optics, Quantum Physics,Quantum Information

I. INTRODUCTION

The recent development of quantumphotonics technology[1] allows experiments using a growing number of photonsand large, complex interferometric networks. Manipulatingsuch large Hilbert spaces requires well-adapted tools in boththeory and experiment. Although nonclassical interference isoften associated with perfectly indistinguishable photons,this represents only the simplest case of photon statesfully symmetric under permutation. Experimentally, partialdistinguishability is ubiquitous, because the generationof indistinguishable multiphoton states currently remains achallenge. Moreover, partial distinguishability is of funda-mental interest, as highlighted, for instance, by the non-monotonicity of the quantum-to-classical transition [2,3].The objective of this paper is to show how controllable

delays in multimode coincidence experiments are related tothe interference of photons of controllable partial distin-guishability. This is done by presenting a novel descriptionfor the nonclassical interference of multiple photons ofarbitrary distinguishability propagating through arbitrary

interferometers. We introduce a symmetry-adapted andtherefore natural basis that plays the role of normal coor-dinates for the description of the nonclassical interference ofphotons. In our framework, a different interferometer justdepends on a different set of normal coordinates; the degreeof nonclassical interference is determined solely by theproperties of the photons. Distinguishability, as the centralproperty, is tunable by treating temporal delay as an explicitparameter, thereby allowing access to the whole spectrumof nonclassical interference. In this perspective, our resultsdiffer from Tichy [4] and Shchesnovich [5]; these authorsalso discuss partial distinguishability but in formalismswhere time delays, which are our controllable parameters,are not immediately explicit. The work of Tamma andLaibacher [6] also tackles partial distinguishability in aperspective different from ours.Here, we treat the nonclassical interference of n ¼ 3

photons injected into different input ports of a linear-opticalquantum network and investigate the probability to detectthem as an n-fold coincidence for the case that the photonsleave the interferometer in different output ports. Ourapproach can be generalized to a higher number of photonsn; however, the case for n ¼ 3 allows an intuitive visuali-zation through a three-dimensional coincidence landscape,where two axes span the distinguishability space for the threephotons whereas the third axis quantifies the output prob-ability for a coincident detection event. The features of such

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PHYSICAL REVIEW X 5, 041015 (2015)

2160-3308=15=5(4)=041015(23) 041015-1 Published by the American Physical Society

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coincidence landscapes are usually understood as the resultof nonclassical interference. We formalize this intuitionby associating landscape features with immanants, whichare polynomial functions in the elements of the scatteringsubmatrix, havingdefinite symmetries under permutations ofrows or columns of this submatrix. The permanent and thedeterminant are special cases of immanants correspondingto fully symmetric or fully antisymmetric functions underpermutations of the rows or columns of the scatteringsubmatrix, respectively. More general immanants havemixed permutation symmetries and are related to partialdistinguishability of the input photons as in Ref. [7].Figure 1 shows how the analysis can break the rates of

the landscape in terms of contributions from immanants ofthe submatrix describing the scattering from the input to theoutput channels. In Fig. 1(b), the landscape is cut along twocharacteristic directions where landscape features are tobe found: the diagonal and antidiagonal line of pairwisetemporal delay. The six points sampled in our experiments,indicated by the pins in Figs. 3(b) and 3(e), are chosenbecause they are located on such landscape features. Detailsof the setup and data are provided in Sec. V. Although theresults for a finite number of pairs of delays are presented inFig. 3, it is clear from the experimental setup that any partof the landscape can, in principle, be reached by suitablycontrolling the time delays.Our work is motivated in part by the resurgence of

interest in interference of perfectly indistinguishable pho-tons within the paradigm of BosonSampling. Within thisformalism coincidence rates for such indistinguishable

photons are related to the modulus square of the permanentof a submatrix of the scattering matrix. Permanents areknown to be hard to compute (in the computationalcomplexity sense), and Aaronson and Arkhipov [8] haveshown that the distribution of permanents occurring whenm identical photons are made to interfere in an n-channelnetwork (n ≫ m) is also hard to compute. Some immanantsare known [9] to be in the same complexity class aspermanents, so partial distinguishability could be appli-cable to some hypothetical generalized BosonSampling ofpartially distinguishable photons if the distributions ofimmanants could also be shown to be hard to simulatewith a classical computer.

II. REVIEW OF THE QUANTUM INTERFERENCEOF TWO BOSONS

In the seminal experiment by Hong, Ou, and Mandel[10], two photons are injected into distinct input ports of abeam splitter, which is effectively an m ¼ 2 interferometer,where m is the number of modes of the interferometer. Oneelement of the output probability distribution correspond-ing to the case where the two photons exit the beam splitterin different output ports is recorded via a coincidencemeasurement. In the following, we assume the detectorsto be ideal (for further details, see Appendix A 3(a),Eq. (A24)). In Fig. 2 we show how the coincidenceprobability Pc depends on the transformation matrix B,here defined by the splitting ratio of the beam splitter, andthe distinguishability of the photons. In the prominent

FIG. 1. The coincidence landscape and its substructure. (a) Coincidence landscape which is the visual representation of thenonclassical interference of three photons of tunable distinguishability. (b) Cut along the diagonal and antidiagonal lines of pairwisedelays Δτ1 ¼ Δτ2 and Δτ1 ¼ −Δτ2, also indicated as a purple trajectory in (a). The height of the landscape corresponds to thecoincidence output probability of a three-photon scattering event. The substructure, expressed in terms of contributions from thepermanent “per”, the determinant “det”, and immanants “imm”, reveals the nature of the scattering event. Beside the plateaux (colorcoded in red), which can be explained classically, the whole landscape is governed by a genuine quantum interference of the photons.

MAX TILLMANN et al. PHYS. REV. X 5, 041015 (2015)

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example of a balanced, i.e., 50=50, beam splitter andperfectly indistinguishable photons, i.e., also zero temporaldelay, the coincidence rate vanishes. The establishedtechnique to calibrate for the point of maximal nonclassicalinterference relies on tuning the relative temporal delay Δτand therefore the distinguishability between the two pho-tons. This is described by an overlap integral that accountsfor the key properties of the photons such as spectral shape,polarization, and spatial mode, in addition to the relativetemporal delay.The results depend on the permutational symmetry of the

two interfering particles. We consider a basis vector vaccounting for the permutational symmetries as a naturalbasis vector for quantum interference. The components of vare matrix functions having definite permutation sym-metries in the entries of the unitary network matrix B.The first component is chosen to be the permanent (per) ofthe matrix, which is fully symmetric under permutation ofrows or columns; the second component is the determinant(det) and is fully antisymmetric under permutation. These

are the only two possible symmetries when permuting twoobjects. By using the basis vector v, we obtain an elegant andcompact form expression for the coincidence ratePcðΔτÞ, inwhich the rate matrix Rð2ÞðΔτÞ takes a diagonal form, withentries depending only on properties of the input state:

PcðΔτÞ ¼Z

dωZ

dω0jhψ in11 jB†a†1ðωÞa†2ðω0Þj0ij2

≔ v†½Rð2ÞðΔτÞ�v

¼�perðBÞdetðBÞ

�†�12

�1 0

0 1

�

þ 1

2jβ12ðΔτÞj2

�1 0

0 −1��

perðBÞdetðBÞ

�; ð1Þ

where jβ12ðΔτÞj2 is a distinguishability function originatingfrom the spectrotemporal overlap of two photons,

βij ¼Z

dωαiðωÞαjðωÞeiωðτi−τjÞ; ð2Þ

and is equal to 12ζe−ξΔτ2 for single photons of Gaussian

spectral shape as used in our experiment. Here, 0 ≤ ζ ≤ 1 isderived from the mode-overlap integral above, jψ in11i ¼A†1ðα1ÞA†

2ðα2Þeiðω1τ1þω2τ2Þj0i is the state impinging on thebeam splitter, and ξ is a factor describing the shape of theinterference feature (see Appendix A 1 for further details).Note that the distinguishability parameter βij can be gen-eralized to account formismatch in any degree of freedomofthe photons.The ratio of the two nonzero entries of the rate matrix

Rð2Þ11 and Rð2Þ

22 reveals the nature of the nonclassicalinterference of two photons of arbitrary coherence. Forindistinguishable photons (where ζ ¼ 1) and zero temporal

delay Δτ, Rð2Þ22 is also zero and the output probability is

proportional to the permanent of B only. The permutation-ally symmetric nature of identical bosonic particles, e.g.,photons, is reflected in transition amplitudes determined bya permutationally symmetric function—the permanent.Temporal delays larger than the coherence time of the

photons, Δτ ≫ τc, result in complete loss of coherence asthe mode-overlap integral jβ12ðΔτÞj2 of Eq. (1) convergesto zero. In this case, often characterized as classical

behavior of two photons, Rð2Þ11 ¼ Rð2Þ

22 ¼ ð1=n!Þ ¼ð1=2!Þ ¼ 0.5. The state is now an equal mixture ofsymmetric and antisymmetric parts and does not exhibitany of the indistinguishability features associated withquantum interference.This analysis can be generalized to the quantum inter-

ference of two photons in larger interferometric networks:the two input ports and the two output ports of such anetwork define 2 × 2 scattering submatrices ~B, the vector vnow contains matrix functions of ~B and the rate matrixRð2ÞðΔτÞ stays identical, independent of ~B. Figure 2(b)highlights how this natural basis cleanly separates effects

FIG. 2. Two-photon nonclassical interference. Two photons oftemporal coherence τc enter a beam splitter through differentinput ports. (a) The coincidence output probability Pc that theyleave in two different output ports is plotted with respect to arelative temporal delay Δτ. This delay is used to tune thedistinguishability of the otherwise identical photons. The bluecurve shows Pc for a 50=50 beam splitter, and the green one for a67=33 beam splitter. (b) Contribution of the permanent (per) anddeterminant (det) to the output probability for a coincidentdetection event Pc. It is the same for both beam splitters becausethis description is independent of the interferometer. In the casefor zero delay (Δτ ¼ 0), only the permanent contributes. Byexplicitly calculating the permanent, which is zero for a 50=50beam splitter, the vanishing coincidence probability Pc (a) forzero delay is obtained.

GENERALIZED MULTI-PHOTON QUANTUM INTERFERENCE PHYS. REV. X 5, 041015 (2015)

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arising from distinguishability of the input state fromeffects of the interferometric network: whereas the numeri-cal values of the permanent and the determinant depend onthe matrix B, the ratio of their contributions depends onlyon partial distinguishability of the photons, irrespective ofthe details of the interferometer.

III. QUANTUM INTERFERENCEOF THREE BOSONS

Consider a scenario where two photons are nearlyindistinguishable and the third is delayed significantly.Adding a third photon leads to situations that can no longerbe understood by the weighted sum of the permanent anddeterminant. In order to describe such a behavior, a moregeneral matrix function, the immanant, is required [7,11].The immanant [12] expands the concept of the permanentand determinant to mixed permutation symmetries; theimmanant is defined as

immfλgðMÞ ¼Xσ

χfλgðσÞYi

MiσðiÞ; ð3Þ

forMij matrix elements ofM, with χfλgðσÞ of the element σin the representation fλg, and σðiÞ the result of permutingcolumn i to column j ¼ σðiÞ. The permanent, for whichevery χfngðσÞ ¼ 1, and the determinant, for whichχf1ngðσÞ ¼ sgnðσÞ, are special cases of the immanant forthe fully symmetric representation (conventionally labeledfng) and alternating representation (conventionally labeledf1ng) of Sn, the permutation group of n objects. Thecharacter table of S3 is provided in Table I.For three photons there is, in addition to the

permanent and the determinant, one immanant correspond-ing to the representation f2; 1g or Young diagram of S3.For simplicity, we henceforth refer to this immanant as“the immanant” and refer explicitly to the permanent anddeterminant as needed. The immanant of M is given by

immðMÞ ¼ 2M11M22M33

− ðM12M23M31 þM13M21M32Þ: ð4ÞUnlike the permanent or the determinant, an immanantdoes not come back to a multiple of itself if we permute,say, columns 1 and 2. Indeed, one finds, by permutingcolumns of M, a total of four linearly independentimmanants. They can be organized in two pairs of linearcombinations of immanants so that, upon permutations, theelements of each pair mix among themselves but not acrosspairs. Although such behavior under permutation mayappear unseemly, it is not at all uncommon:

jψi ¼ a†1ðω1Þ½a†2ðω2Þa†3ðω3Þ þ a†3ðω2Þa†2ðω3Þ�j0i ð5Þtransforms back to itself upon the permutation ω2 ↔ ω3,but not under ω1 ↔ ω2. Such states of mixed permutationsymmetries are common in the many-body physicsproblem.In the smallest instance of a three-photon quantum

interference, the photons are injected into an m ¼ 3 modeinterferometric network and measured as threefold coinci-dences at the three output ports. The optical transformationimplemented by the interferometer can be any 3 × 3 linearoptical transformation T, and the distinguishability of thethree photons is arbitrarily tunable by setting the relativetemporal delays: Δτ1 between the first and second photonand Δτ2 between the second and third photon. Thecoincidence rate P111ðΔτ1;Δτ2Þ is given by

P111ðΔτ1;Δτ2Þ ¼Z

dωZ

dω0Z

dω00jhψ in111 jT†a†1ðωÞa†2ðω0Þa†3ðω00Þj0ij2 ð6Þ

¼ v†3½Rð3ÞðΔτ1;Δτ2Þ�v3 ð7Þ

¼ ðP S v3Þ†½1þ ρ12jβ12j2 þ ρ23jβ23j2 þ ρ13jβ13j2 þ ρ132β�12β

�23β13 þ ρ123β12β23β

�13�ðP S v3Þ ð8Þ

¼ ðP S v3Þ†½1þ ρ12ζ12e−ξ12Δτ21 þ ρ23ζ23e−ξ23Δτ

22 þ ρ13ζ13e−ξ13ðΔτ1−Δτ2Þ

2

þ ζ123ðρ132eξ�123ðΔτ1;Δτ2Þ þ ρ123eξ123ðΔτ1;Δτ2ÞÞ�ðP S v3Þ; ð9Þ

where v3, P, and S are given explicitly in Eq. (A19). The terms β12, β23, and β13 are distinguishability parameters definedanalogously to Eq. (2). In Eq. (9) these parameters are computed for single photons of Gaussian spectral shape as used inour experiment. Our key point is that, in Eq. (7), the rate matrix Rð3ÞðΔτ1;Δτ2Þ can always be brought to block-diagonalform, containing two 1 × 1 blocks and one 4 × 4 block. Indeed, the matrices in Eq. (9) can always be further refined so as totake a final block-diagonal form:

TABLE I. The character table for S3.

Elements 1 fP12; P13; P23g fP123; P132girrep λ χλð1Þ χλðPabÞ χλðPabcÞ Dimension

1 1 1 12 0 −1 2

1 −1 1 1


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1 ¼

0BBBBBBBBB@

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

1CCCCCCCCCA; ρ12 ¼

0BBBBBBBBB@

1 0 0 0 0 0

0 −1 0 0 0 0

0 0 1 0 0 0

0 0 0 −1 0 0

0 0 0 0 1 0

0 0 0 0 0 −1

1CCCCCCCCCA;

ρ23 ¼

0BBBBBBBBB@

1 0 0 0 0 0

0 −1 0 0 0 0

0 0 − 12

− ffiffi3

p2

0 0

0 0 − ffiffi3

p2

12

0 0

0 0 0 0 − 12

− ffiffi3

p2

0 0 0 0 − ffiffi3

p2

12


0BBBBBBBBB@

1 0 0 0 0 0

0 −1 0 0 0 0

0 0 − 12

ffiffi3

p2

0 0

0 0ffiffi3

p2

12

0 0

0 0 0 0 − 12

ffiffi3

p2

0 0 0 0ffiffi3

p2

12

1CCCCCCCCCA;

ρ123 ¼

0BBBBBBBBB@

1 0 0 0 0 0

0 1 0 0 0 0

0 0 − 12

− ffiffi3

p2

0 0

0 0ffiffi3

p2

− 12

0 0

0 0 0 0 − 12

− ffiffi3

p2

0 0 0 0ffiffi3

p2

− 12


0BBBBBBBBB@

1 0 0 0 0 0

0 1 0 0 0 0

0 0 − 12

ffiffi3

p2

0 0

0 0 − ffiffi3

p2

− 12

0 0

0 0 0 0 − 12

ffiffi3

p2

0 0 0 0 − ffiffi3

p2

− 12

1CCCCCCCCCA: ð10Þ

The vector P S v3 of Eq. (9) is given explicitly by

P S v3 ¼

0BBBBBBBBB@

perðTÞdetðTÞ

1

2ffiffi3

p immðTÞ þ 1

2ffiffi3

p immðT312Þ16immðTÞ − 1

3immðT132Þ − 1

6immðT213Þ þ 1

3immðT312Þ

16immðTÞ þ 1

3immðT132Þ þ 1

6immðT213Þ þ 1

3immðT312Þ

− 1

2ffiffi3

p immðTÞ þ 1

2ffiffi3

p immðT213Þ; :

1CCCCCCCCCA; ð11Þ

where Tijk is the matrix T in which rows 1, 2, and 3 havebeen rearranged in order i, j, k. In addition, a†1ðωÞ, a†2ðω0Þ,and a†3ðω00Þ are the creation operators in modes 1,2,3 of Tfor photons with different spectral shape functions depen-dent on the frequency variables ω;ω0;ω00. Here, jψ in111i ¼A†1ðω1ÞA†

2ðω2ÞA†3ðω3Þeiðω1τ1þω2τ2þω3τ3Þj0i is the three-

photon state impinging on the interferometer, withA†kðωkÞ defined in Eq. (A2).The form of the matrices in Eq. (10) is fully dictated by

the theory of the group of permutation of three objects: theblocks in Eq. (10) correspond to irreducible representationsof this group. The block diagonalization can be done usingseveral methods; one algorithm can be found in Chap. 4 ofRef. [13]. No further reduction into smaller blocks ispossible; i.e., the vector P S v3 is optimal, in the sense

that it is the vector (up to a choice of which matrices ρare diagonalized) that will produce the simplest form ofthe rates. Indeed, an expression for the coincidence rateof Eq. (6), expanded in terms of immanants, determi-nants, and permanents, is also given in Eq. (A12); it isa linear superposition of 60 terms. However, utilizing asymmetry-adapted basis allows for the compact expres-sion given in Eqs. (9) and (7) (see Appendix A 2 forfurther details). Here, the four linearly independentimmanants, the permanent, and the determinant of Tconstitute the components of a six-dimensional basisvector P S v3.Much like Eq. (1), the various ζ terms of Eq. (9) are

derived from the mode-overlap integral while the ξ termsdescribe the shape of the interference feature. In this


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notation the overlap terms weight a sum of six matrices: theidentity matrix and five permutation matrices ρ12, ρ13, ρ23,ρ123, and ρ132, the subscripts of which label the permutationoperation.

IV. NATURAL BASIS FOR THREE PHOTONS

Components of the natural basis vector P S v3 intro-duced to yield the fully block-diagonal form of Eq. (9)have specific permutation properties: components of onesymmetry type transform to components of the same typeunder permutation; i.e., they are decoupled under per-mutation. Components of P S v3 thus play the role ofnormal coordinates for the nonclassical interference ofphotons.Equation (9) highlights the six different permutational

possibilities for three photons. Summing the matricesinside the square brackets yields the 6 × 6 rate matrixRð3ÞðΔτ1;Δτ2Þ of Eq. (7). This Rð3ÞðΔτ1;Δτ2Þ contains allthe information regarding the input state, e.g., modemismatch and temporal delay, to specify the nonclassicalinterference of three photons independent of the scatteringtransformation T.Two entries of the block-diagonal rate matrix are

sufficient for an interpretation. Fper ¼ jRð3Þ11 ðΔτ1;Δτ2Þj2

quantifies the fraction of the output probability distributionproportional to the modulus square of the permanent; thecorresponding basis component is fully symmetric under

permutation. Fdet ¼ jRð3Þ66 ðΔτ1;Δτ2Þj2 quantifies the frac-

tion of the output probability distribution proportional tothe modulus square of the determinant; the correspondingcomponent is antisymmetric under permutation. Thecontribution proportional to moduli square of the imman-ants can also be explicitly calculated. When only inter-ested in their overall contribution, it is given asFimm ¼ 1 − Fper − Fdet. In the case of perfectly overlap-ping photons, Fper ¼ 1, and therefore only the permanentof the scattering matrix contributes to the output probabilitydistribution. Classical behavior of the photons can beidentified for Fper ¼ Fdet ¼ 1

6.

As in the two-photon case, the input state and theinterferometer decouple in the natural basis. As a conse-quence, the treatment of the quantum interference ofthree photons in larger interferometric networks consist-ing of many modes becomes very efficient. For such aproblem it is sufficient to calculate the rate matrix Rð3Þ

only once. The scattering matrix T, necessary to calculatethe basis vector v3 for a specific element of an outputprobability distribution, is just a 3 × 3 submatrix of thelarger scattering matrix. It is specified by the input ports ofthe photons and the ports in which they exit the interfer-ometer. To obtain multiple elements of a probabilitydistribution, it is sufficient to determine their respectivecomponents in v3.

V. COINCIDENCE LANDSCAPE

In the experiment, four-photon events generated byhigher-order emission from a spontaneous parametricdown-converter are distributed to four different spatialmodes. Relying on a detection event in the trigger modeand postselection, the three-photon input state, one photonin each input mode coupled to the interferometer, isheralded. We ensure that all photons are indistinguishablein a polarization basis. The spectral properties of thesephotons are independently measured using a single-photonspectrometer. Their relative temporal delay Δτ1 and Δτ2can be set using motorized delay lines. The transformationof the femtosecond-written integrated interferometer, a5 × 5 unitary matrix, is recovered using the reconstructionmethod specified in Appendix A 3.Injecting the photons in three input ports of the inter-

ferometer and detecting them in three separate outputports uniquely selects a 3 × 3 scattering submatrix T[see Figs. 3(a) and 3(d)]. For each 3 × 3 submatrix, theuse of a precisely tunable delay allows us to reveal thefull spectrum and thereby the nature of the nonclassicalinterference. We visualize this as a three-dimensionalcoincidence landscape as shown in Figs. 3(b) and 3(e).The relief of such a landscape features distinct “landforms”which are in correspondence with distinguishability fea-tures of the photons.In the center region, Δτ1 ≈ Δτ2 ≈ 0� τc, a peak or dip

arises due to constructive or destructive interference of allthree photons. In the absence of any spectral distinguish-ability, the absolute zero position Δτ1 ¼ Δτ2 ¼ 0 corre-sponds to 100% contribution from the permanent of T.Along the three axes Δτ1 ¼ 0, Δτ2 ¼ 0, and Δτ1 ¼ Δτ2,

valleys or ridges form due to the nonclassical interferenceof two indistinguishable photons with the third one beingpartially distinguishable. Along those ridges and valleysthe contributions to the output probability come from thepermanent and the immanants of the scattering matrix.“Classical” behavior, i.e., complete distinguishability, of

the three photons is associated with plateaus for temporaldelays, Δτ1 ≈ −Δτ2 ≫ j � τcj. For such large temporaldelays along the antidiagonal axis, all mode-overlapintegrals of Eq. (9) converge to zero. Consequently, theseare the areas where determinants of the scattering matrixcontribute, accounting for the antisymmetrical part of theinput state. Coincidences for six points of pairwise differenttemporal delays, P1–P6, for two different scattering sub-matrices [see, for instance, Figs. 3(c) and 3(f)] are mea-sured. These six points are selected because they highlightthe connection between landscape features, permutationsymmetries, and partial distinguishability. Furthermore,they provide a sufficient set of experimental data for fittingthe coincidence landscapes. A reduced χ2 of 1.38 and 1.10for the two landscapes quantifies the overlap between ourtheory and the experiment. Our theory assumes a singlephoton per input mode; hence, the deviations are most


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likely due to higher-order emissions and frequency corre-lations of the input state.The landscape interpretation can be extended as needed

to the interference of larger numbers of photons n,which generate n-dimensional “hyperlandscapes.” Theseare spanned by n − 1 axes of pairwise temporal delays withthe last axis representing the actual coincidence rate. Thelandforms range from complex n-dimensional featurescorresponding to the partial indistinguishability of all nphotons to the flat plateaus associated with completelydistinguishable photons.

A. Systematic exploration of partial distinguishability

We investigate generalized nonclassical interference ofthree photons in a five-moded interferometric network intheory and experiment. This serves to illustrate the full

permutational spectrum of a generalized nonclassicalinterference for complex networks exhibiting a genericstructure. The photons exhibit slight spectral mismatchand are thus never fully indistinguishable; additionally,the level of partial distinguishability can be increasedby controlling temporal delays. Figure 4(a) illustrates theresult for a very small degree of partial distinguishability,whereas in Figs. 4(b) and 4(c) the partial distinguish-ability is increased by varying the temporal delay alonga diagonal delay axis Δτ1 ≈ Δτ2. The extreme case ofcomplete distinguishability Δτ1 ≈ −Δτ2 ≫ τc, and thusclassical behavior is shown in Fig. 4(d). As a referencewe include in all figures the ideal case of zero delayand perfect indistinguishability as gray bars. The grouptheoretical interferometer-independent contributions Fper,Fdet, and Fimm are contained as an inset in the legend ofeach figure.

FIG. 3. Three-photon coincidence landscapes. Three photons enter the interferometric network, one each in modes 1, 2, and 4(highlighted in yellow), and exit the network in modes (a) 3, 4, and 5 and (d) 1, 3, and 4, respectively (highlighted in blue). Theintersections of the inputs’ columns and the outputs’ rows uniquely select matrix entries that constitute 3 × 3 submatrices. Tuning thetemporal delay of the three photons with respect to each other (Δτ1 and Δτ2) gives rise to coincidence landscapes [(b) and (e)]. Theirtemporal distinguishability determines the degree of nonclassical interference and therefore the probability to detect such an event. Sixcharacteristic points (P1–P6) of each landscape are experimentally sampled. Theoretical prediction (left bars, shaded) andexperimentally obtained output probabilities (right bars) for the six points and both output combinations are shown in (c) and (f).The reduced χ2 is 1.38 and 1.10, respectively, and the experimental errors are calculated as standard deviations.


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The elements of each output probability distribution arerecovered by calculating the correspondingmatrix functions.Note that for each element the absolute value of these matrixfunctions, e.g., jperðTÞj2 or j detðTÞj2, can vary largelydepending on the scattering submatrix T. This is pronouncedfor the output event 123,where jperðT123Þj2 ≈ 1

5j detðT123Þj2.

In general, the fraction of the output probability distributionproportional to the permanent drops rapidly with increasingdistinguishability. Instead, contributions from immanantsbecome dominant and reflect cases where two of the threephotons interfere nonclassically.For large delays along the diagonal axis Δτ1 ≈ Δτ2 ≫ τc,

two photons stay nearly indistinguishable and the contri-bution from the determinant is suppressed to Fdet ≈ 0[see Fig. 4(c)]. For comparably large delays along theantidiagonal axis, Δτ1 ≈ −Δτ2 ≫ τc, the three photons’wave functions do not overlap anymore and the determinant

contributes with Fdet¼ 16[see Fig. 4(d)]. This is the case of

classical behavior of the three photons [see Fig. 4(d)] andcan always be identified by an equal contribution from thepermanent and determinant: Fper ¼ Fdet ¼ ð1=3!Þ in thecase of a three-photon interference and Fper ¼Fdet¼ð1=n!Þfor the nonclassical interference of n photons.Our theory emphasizes the permutation symmetries of n

photons using the representation theory of the symmetricgroup Sn. The theory is thus independent of the number ofmodes m in the interferometer, a feature that is extremelyconvenient for large-scale networks where m ≫ n, even ifthe number of permutations of the output increases as n!.

B. From permanents to immanants

Quantum computing leverages quantum resources toefficiently perform certain classically hard computations[14]. Whereas many quantum algorithms solve a certain

FIG. 4. Experimental output distribution for indistinguishable, semidistinguishable, and distinguishable photons. Various temporaldelays for three photons lead to different contributions of permanents, immanants, and determinants. The normalized output probabilitydistributions for the three photons is measured as coincidences from different spatial modes, resulting in ten elements. For all temporaldelays the three photons exhibit slight spectral mismatch. Panel (a) depicts the case for a small temporal offset Δτ1 ≪ τc ≫ Δτ2 (P1 ofFig. 3), whereas for (b) and (c) this delay is increased along a diagonal axis Δτ1 ≈ Δτ2 (P3 and P4 of Fig. 3). The extreme case ofcomplete distinguishability and therefore classical behavior is shown in (d) (P6 of Fig. 3). As a reference, the gray bars illustrate the casefor perfect indistinguishability and therefore only contribution from the permanent of the scattering submatrix. The interferometerindependent contribution Fper, Fdet, and Fimm is shown in the figure legend. The error bars of the experimental data are standarddeviations over 19 independent runs.


041015-8

decision problem, BosonSampling introduces a new para-digm: it seeks efficient sampling of a distribution ofpermanents of matrix transformations, which is a task thatis hard to implement efficiently on classical computers.In order to scale BosonSampling to larger instances, two

main issues need to be addressed. The first issue is thetechnology [15–17] needed to increase the size of theinstances implemented. The second issue is the handling ofpossible errors [18–20]. BosonSampling is a purely passiveoptical scheme and therefore lacks error-correction capa-bilities [21]. Only in the ideal case where the interferingphotons are indistinguishable in all degrees of freedom isthe resulting output probability distribution proportional tothe permanent only. Our analysis exposes that this con-dition is rather fragile and therefore distinguishability mustbe regarded as the dominant source of error. Remarkably,large classes of immanants are known to be in the samecomplexity class as permanents [22,23]. Thus, it is naturalto ask if the output probability distributions dependinglargely on immanants rather than just the permanent arealso computationally hard. Whether this holds for samplingfrom these distributions is an active field of research.

C. Generalization to higher number of photons

The theory presented here in detail for the case of n ¼ 3photons can be generalized to a larger number of photons n.The rate matrix becomes a matrix of dimension n! × n! andcarries the so-called regular representation of Sn. It can beblock diagonalized exploiting the same methods used toblock diagonalize the regular representation. The algorithmto obtain this block-diagonal form for the rate matrix is

described in Table 2 for any n-photon input. As highlightedin Sec. VA, the rate matrix is independent of the number ofmodes in the interferometer. We demonstrate the use of thisalgorithm for a five-photon input into a nine-mode inter-ferometer. In this specific case of n ¼ 5, the rate matrix is ofdimension 5! ¼ 120 and can be expressed as

Pσ∈S5ρσOσ ,

where ρσ spans a 120 × 120 representation Γ of S5,and where Oσ contains various delay-dependent factors.The representation Γ is reducible and decomposes as

ð12ÞThe terms in Eq. (12) are respectively associated with

photons of different partial distinguishability: The first termis associated with fully indistinguishable photons, thesecond term with four photons arriving with no delay anda last photon fully distinguishablewith respect to the quartet,and the third term with three photons arriving with no delayand a second pair of photons with no relative delay betweenthemselves. The pair and the trio are fully separable in thetemporal domain. The remaining terms can be interpreted inthe same fashion, with the last term covering the case of allfive photons being fully distinguishable with respect to eachother. Each term is labeled by the Young diagram of itspartition and associated with an immanant. In general, thecoincidence rate is a sum of contributions from modulussquared of linear combinations of these immanants, whichcan be constructed using the character table for S5, as can befound in for instance in Ref. [12]. These linear combinations

TABLE II. Algorithm to block diagonalize the rate matrix.

Inputn Number of input photonsU A n × n unitary matrixαi Spectral function of the ith photon

OutputRðnÞ Rate matrix of n-photon coincidence that is

block diagonalized in blocks labeled by the partitions of SnRoutine

1. Find ρσ , the regular representation of the element σ ∈ Sn.2. Calculate the overlap functions between the n photons forall σ ∈ Sn:Oσ ¼

Qni¼1½

RdωiαiðωiÞαiðωσðiÞÞ expð−iωiτi þ iωσðiÞτiÞ�.

3. Make xlist, a list ofQ

ni¼1 UσðiÞ;i for all σ ∈ Sn.

4. Let ½Ui;σðjÞ� be the matrix U with columns j permuted to σðjÞfor σ ∈ Sn, and define the function immλðUÞ to beimmλðUÞ ¼ P

σ∈SnχλðσÞQ

ni¼1 Ui;σðiÞ,

where λ is a partition of Sn and χλðσÞ is the character of σ in thepartition λ. Make immlist, a list of n! elements that comprisesimmλð½Ui;σðjÞ�Þ that are linearly independent of one another.

5. Find the n! × n! matrix S in the linear equationxlist ¼ S · immlist.

6. Return RðnÞ ¼ S† · ðPσ∈SnρσOσÞ · S.


041015-9

of immanants can be found using, for instance, classoperator methods [13].The algorithm described in Table II can be used to

compute the rate matrix that is block diagonalized in thepartitions listed in Eq. (12) for five photons propagating

through an interferometer. Note that the specific inputchosen and the size of the interferometer, in Fig. 5 a 9 × 9interferometer, affect only the basis vectors. For this

exemplary case we look at the�9

5

�¼ 126 different ways

FIG. 5. Five-photon BosonSampling including distinguishability. Simulation of a BosonSampling instance of five photonspropagating through an interferometric network of nine modes. The output probability distribution of five photons exiting theinterferometer in five different modes is normalized and contains 126 elements. Owing to the large size of this probability distribution,we choose not to list them individually, and we label the horizontal axes by “elements.” Panel (a) depicts the close-to-the-ideal casewhere realistic errors such as slight spectral mismatch and temporal delay (Δτi ≤ 1

20τc) of the photons lead to a small degree of partial

distinguishability. Panel (b) shows a case where the interfering photons exhibit increased partial distinguishability (Δτi ≤ 15τc). In these

exemplary output probability distributions [(a) and (b)], contributions from permanents (per) and immanants ( , , ) arise. Theimmanant contributions cover physical scenarios with different symmetries under exchange of five photons. The labels of the immanantsdescribe the number of photons that are distinguishable. , for instance, is the contribution from the case when four photons areindistinguishable from one another but distinguishable from the fifth photon.


041015-10

in which five photons can exit a nine-mode interferometer,each photon in a separate mode. Permanents, , and

immanants of the partitions, , , and , con-

tribute to this 126-element probability distribution and arecomputed as the trace of their respective block matricesgiven by the algorithm of Table II. In Fig. 5, this outputprobability distribution and its substructure is shown fortwo different degrees of partial distinguishability.

VI. DISCUSSION

We present a novel analysis of multiphoton quantuminterference revealing the full permutational spectrum ofinput states with arbitrary distinguishability. A compre-hensive physical interpretation is achieved by establishinga correspondence between matrix immanants and thesemixed symmetry input states. We introduce a rate matrixcontaining all the information on the nonclassical interfer-ence and basis vectors containing the information on theinterferometric network. Output probabilities are recoveredas an inner product of these vectors with the rate matrixserving as a metric. The rate matrix is block diagonalizedand each block corresponds to a different physical scenarioof nonclassical interference. This indicates that this blockdiagonalization and consequent interpretation are not onlyfundamental but also universal features of multiphotoninterferometry. We experimentally confirm our theory by

recovering the full coincidence landscape of three arbitrar-ily distinguishable photons. We show that the theory can beapplied to higher numbers of photons with an exemplarysimulation of the nonclassical interference of five photonsthrough a 9 × 9 network. Our approach thus provides adeeper understanding of the rich spectrum of multiphotonnonclassical interference. While passive schemes likeBosonSampling benefit most from this approach, it appliesanalogously to pivotal building blocks of linear opticalquantum computing [24,25] as crucial nondestructivetwo-qubit gates exploit ancillary photons and thus relyon multiphoton interference [26,27].

VII. METHODS

A. State generation

ATi:sapphire oscillator emitting 150-fs pulses at 789 nmand a repetition rate of 80 MHz is frequency doubled in aLiB3O5 (LBO) crystal (see Fig. 6 for a schematic of theexperimental setup). The output power of this secondharmonic generation can be controlled by a power regu-lation stage consisting of a half-wave plate (HWP) and apolarizing beam splitter (PBS) placed before the LBOcrystal. The resulting emission at 394.5 nm is focused into a2-mm-thick β-BaB2O4 (BBO) crystal cut for degeneratenoncollinear type-II down-conversion [28]. A compensa-tion scheme consisting of HWPs and 1-mm-thick BBO

FIG. 6. Experimental setup. Four photons are generated via spontaneous parametric down-conversion and distributed to four spatialmodes with two PBSs. A fourfold coincidence event consisting of three photons exiting the network and a trigger event postselects thedesired input state. The delay lines allow us to tune the distinguishability and therefore the quantum interference of the three photonspropagating through the waveguide. The integrated circuit is shown in a Mach-Zehnder decomposition and consists of eight beamsplitters and 11 phase shifters.


041015-11

crystals is applied to counter temporal and spatial walk-off.The two spatial outputs of the down-converter pass throughnarrow band interference filters (λFWHM ¼ 3 nm) toachieve a coherence time greater than the birefringentwalk-off due to group velocity mismatch in the crystal(jvge − vgo j × half-crystal thickness). Additionally, this ren-ders the photons close to spectral indistinguishability. Thedown-conversion source is aligned to emit the maximallyentangled Bell state jϕþi ¼ ð1= ffiffiffi

2p ÞðjHHi þ jVViÞ when

pumped at 205-mW cw-equivalent pump power. The stateis coupled into single-mode fibers (Nufern 780-HP)equipped with pedal-based polarization controllers tocounter any stress-induced rotation of the polarizationinside the fiber. Each of these spatial modes is thencoupled to one input of a PBS while its other input isoccupied with a vacuum state. The outputs pass HWPsand are subsequently coupled to four polarization-maintaining (PM) fibers (Nufern PM780-HP). Temporaloverlap is controlled by two motorized delay lines thatexhibit a bidirectional repeatability of �1 μm. Temporalalignment precision is limited by other factors in thesetup to approximately �5 μm and is therefore within aprecision of 2.5% of the coherence length of the photons.The polarization-maintaining fibers are mated to a single-mode fiber v-groove array (Nufern PM780-HP) with apitch of 127 μm and butt coupled to the integratedcircuit. The coupling is controlled by a manual six-axisflexure stage and is stable within 5% of the total single-photon counts over 12 h. The output fiber array consistsof a multimode (MM) v–groove array (GIF-625) and thephotons are detected by single-photon avalanchephotodiodes that are recorded with a home-built field-programmable gate array logic. The coincidence timewindow is set to 3 ns. In order to measure the six pointsof the coincidence landscapes, a three-photon input stateis injected into the integrated network (see Appendix A 5for further details). Therefore, the BBO is pumped withcw-equivalent power of 700 mW and the ratio of the six-photon emission over the desired four-photon emission ismeasured to be below 5%.

B. Integrated network fabrication

The integrated photonic networks are fabricated usinga femtosecond direct–write writing technology [29,30].Laser pulses are focused 370 μm below the surface of ahigh-purity fused silica wafer by a NA ¼ 0.6 objective.The 200-nJ pulses exhibit a pulse duration of 150 fs at100 KHz repetition rate and a central wavelength of800 nm. In order to write the individual waveguides, thewafer is translated with a speed of 6 cm=s. The wave-guide modes exhibit a mode field diameter of 21.4 ×17.2 μm2 for a wavelength of 789 nm and a propagationloss of 0.3 dB=cm. This results in a coupling loss of−3.5 dB with the type of input fibers used in this

experiment. Coupling to the output array results innegligible loss due to the use of multimode fibers.

ACKNOWLEDGMENTS

The authors thank I. Dhand and J. Cotter for helpfuldiscussions, M. Tomandl for assistance with the illustra-tions, and J. Nielsen and J. Kulp for computationalassistance. M. T., S. E. S., and P. W. acknowledge supportfrom the European Commission with the project EQuaM–Emulators ofQuantumFrustratedMagnetism (No. 323714),GRASP–Graphene-Based Single-PhotonNonlinear OpticalDevices (No. 613024), PICQUE–Photonic IntegratedCompound Quantum Encoding (No. 608062), QuILMI–Quantum Integrated Light Matter Interface (No. 295293)and QUCHIP–Quantum Simulation on a Photonic Chip(No. 641039), the Vienna Center for Quantum Science andTechnology (VCQ), and the Austrian Science Fund (FWF)with the projects PhoQuSi Photonic Quantum Simulators(Y585-N20) and the doctoral programme CoQuS ComplexQuantum Systems, the Vienna Science and TechnologyFund (WWTF) under Grant No. ICT12-041, and theAir Force Office of Scientific Research, Air ForceMaterial Command, United States Air Force, under GrantNo. FA8655-11-1-3004. B. C. S. acknowledges supportfrom AITF (Alberta Innovates Technology Futures),NSERC (Natural Sciences and Engineering ResearchCouncil), and CIFAR (Canadian Institute for AdvancedResearch). The work of H. deG. is supported in part byNSERC of Canada. This material is also based on researchsupported in part by the Singapore National ResearchFoundation under NRF Award No. NRF-NRFF2013-01(S.-H. T.). R. H., S. N., and A. S. acknowledge supportfrom the German Ministry of Education and Research(Center for Innovation Competence programme, GrantNo. 03Z1HN31), the Deutsche Forschungsgemeinschaft(Grant No. NO462/6−1), and the Thuringian Ministry forEducation, Science andCulture (Research group Spacetime,Grant No. 11027-514).

APPENDIX A SUPPLEMENTARY INFORMATION

1. Two-photon nonclassical interference

Two photons injected into different inputs of an arbitrarybeam splitter or a network built from arbitrary beamsplitters and phase shifters will interfere nonclassically[10,31]. This input state can be expressed as

jΨin11i ¼ ½A†1ðα1Þeiω1τ1 �½A†

2ðα2Þeiω2τ2 �j0i; ðA1Þ

with

A†i ðαiÞ ¼

Z∞

0

dωiαiðωiÞa†i ðωiÞ; ðA2Þ


041015-12

for A†i ðαiÞ, a creation operator for a photon with spectral

function

jαðωiÞj2 ¼1ffiffiffiffiffiffi2π

pσiexp

�− ðωi − ωc;iÞ2

2σ2i

�ðA3Þ

centered at time τi. The frequency-mode creation operatorson the rhs of Eq. (A2) satisfy the commutator relation

½aiðωÞ; a†jðω0Þ� ¼ δijδðω − ω0Þ1; ðA4Þ

with 1 the identity operator. This commutation relation alsodefines the photons’ symmetry under permutation opera-tions. For two photons it is sufficient to define their relativetemporal delay as Δτ ¼ τ1 − τ2. Only in the case of idealbosonic particles exhibiting no modal mismatch and perfecttemporal overlap, i.e., Δτ ¼ 0, does the rhs of Eq. (A4)become the well-known bosonic commutator relationdescribing perfect symmetry under exchange. When thetwo-photon input state [see Eq. (A1)] is mixed via atransformation matrix B ¼ U2×2 and projected on an outputwhere the two photons exit in different modes, the outputprobability becomes

PcðΔτÞ ¼Z

dω1

Zdω2jhΨin11 jB†a†1ðω1Þa†2ðω2Þj0ij2

ðA5Þ

¼�perðBÞdetðBÞ

�†�12

�1 0

0 1

�þ1

2ζe−ξΔτ2

�1 0

0 −1��

×

�perðBÞdetðBÞ

�ðA6Þ

¼ v†2½Rð2ÞðΔτÞ�v2; ðA7Þ

with

ζ ¼ 2σ1σ2σ21 þ σ22

exp

�− ðωc;1 − ωc;2Þ2

2ðσ21 þ σ22Þ�; ξ ¼ σ21σ

22

σ21 þ σ22ðA8Þ

denoting factors arising from the spectral overlap integraland

v2 ¼1ffiffiffi2

p�perðBÞdetðBÞ

�ðA9Þ

the new basis vector constituted by matrix functions of thescattering submatrix T. As a second-order correlation effect,this nonclassical interference is dependent on the permuta-tional symmetry of the interfering wave functions alsoreflected in the basis vector v2. For the case of indistinguish-able photons (ωc;1 ¼ ωc;2, σ1 ¼ σ2, or Δτ ¼ 0), the outputprobability is only proportional to the permanent. This is afunction symmetric under permutation of rows of the trans-formation matrix arising in photon interferometry due tobosonic exchange symmetry.However,with loss of completeindistinguishability (ωc1 ≠ ωc2, σ1 ≠ σ2, and Δτ ≠ 0),Eq. (A6) becomes proportional to a combination of thedeterminant and the permanent. This is a consequence ofthe input state losing its symmetry under exchange.Equation (A7) decouples the influence of the interferometerfrom the influence of the input state. The latter is contained inthe diagonal 2 × 2 rate matrix Rð2ÞðΔτÞ, whereas thedescription of the interferometer is absorbed in the newbasis vector v2. The two nonzero entries of the rate matrix,

Rð2Þ11 and Rð2Þ

22 , are sufficient to reveal the nature of thenonclassical interference of two photons of arbitrary coher-

ence. Where Rð2Þ11 quantifies the contribution from the

permanent of the scattering submatrix, Rð2Þ22 quantifies the

contribution from the determinant of the scattering subma-trix. The output probability Pc is recovered by calculatingthose matrix functions.

2. Three-photon nonclassical interference

Nonclassical interference of photons depends on indis-tinguishability of the interfering photons and transforma-tions mixing the modes. Adding a third photon noticeablyincreases the complexity. An input state corresponding tothree photons in three different transverse spatiotemporalmodes can be described as

jΨin111i ¼ ½A†1ðα1Þeiω1τ1 �½A†

2ðα2Þeiω2τ2 �½A†3ðα3Þeiω3τ3 �j0i:

ðA10Þ

For three photons it is sufficient to define two relativetemporal delays, Δτ1 ¼ τ1 − τ2 and Δτ2 ¼ τ3 − τ2. Whenthis input state is transformed via a submatrix T ¼ U3×3and projected on an output where the three photons exit indifferent modes, the fully expanded output probability canbe written as


041015-13

P111ðΔτ1;Δτ2Þ ¼Z

dωZ

dω0Z

dω00jhΨin111 jT†a†1ðωÞa†2ðω0Þa†3ðω00Þj0ij2 ðA11Þ

¼ 1

6j detðTÞj2 þ 2

9jimmðT132Þj2 þ

1

9imm�ðT132ÞimmðT213Þ þ

1

9immðT132Þimm�ðT213Þ

þ 2

9jimmðT213Þj2 þ

2

9jimmðT231Þj2 þ

2

9jimmðTÞj2 þ 1

9immðT231Þimm�ðTÞ

þ 1

6jperðTÞj2 þ 1

9immðTÞimm�ðT231Þ

þ ζ13 expð−2ξ13ðΔτ1 − Δτ2Þ2Þ�− 1

6j detðTÞj2 − 2

9immðTÞimm�ðT132Þ − 1


−1


1

9imm�ðT213ÞimmðT231Þ − 1


þ 1

9immðT213Þimm�ðT231Þ − 2

9immðT132Þimm�ðRÞ − 1

9immðT213Þimm�ðTÞ þ 1

6jperðTÞj2

�

þ ζ12 expð−2ξ12Δτ21Þ�− 1

6j detðTÞj2 þ 1

9immðTÞimm�ðT132Þ þ

2


þ 2


1


2


þ 1

9immðT213Þimm�ðT231Þ þ

1



6jperðTÞj2

�

þ ζ23 expð−2ξ23Δτ22Þ�− 1

6j detðTÞj2 þ 1



−1




−2

9immðT213Þimm�ðT231Þ þ

1

9immðT132Þimm�ðTÞ − 1


6jperðTÞj2

�

þ ζ123 expð−Ia þ iIsÞ�1

6j detðTÞj2 − 1

9jimmðT132Þj2 − 2

9imm�ðT132ÞimmðT213Þ

þ 1

9immðT132Þimm�ðT213Þ − 1

9jimmðT213Þj2 þ

1


9jimmðT231Þj2

−1

9jimmðTÞj2 − 2


6jperðTÞj2

�

þ ζ123 expð−Ia − iIsÞ�1

6j detðTÞj2 − 1

9jimmðT132Þj2 − 2


þ 1


9jimmðT213Þj2 þ

1

9imm�ðTÞimmðT231Þ − 1

9jimmðT231Þj2

−1

9jimmðTÞj2 − 2

9imm�ðT231ÞimmðTÞ þ 1

6jperðTÞj2

�; ðA12Þ

with

ζ123 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiζ12ζ23ζ13

p;

Ia ≡ IaðΔτ1;Δτ2Þ ¼ −ðΔτ1Þ2 ξ122

− ðΔτ1 − Δτ2Þ2ξ132

− ðΔτ2Þ2ξ232

;

Is ≡ IsðΔτ1;Δτ2Þ ¼ Δτ1ν12 − ðΔτ1 − Δτ2Þν13 − Δτ2ν23;

ζij ¼2σiσjσ2i þ σ2j

exp

�− ðωc;i − ωc;jÞ2

2ðσ2i þ σ2jÞ�; ðA13Þ


041015-14

ξij ¼2σ2i σ

2j

σ2i þ σ2j; νij ¼

ωc;iσ2j þ ωc;jσ

2i

σ2i þ σ2j: ðA14Þ

The subscripts denote the mode labels for the submatrix T. Tijk is the matrix T with the rows permuted according to 1 → i,2 → j, and 3 → k.For a more elegant expression, Eq. (A12) can be simplified introducting six matrices, 1, ρ12, ρ13, ρ23, ρ123, and ρ132:

P111ðΔτ1;Δτ2Þ ¼ðP S v3Þ†½1þ ρ12ζ12e−ξ12Δτ21 þ ρ23ζ23e−ξ23Δτ

22 þ ρ13ζ13e−ξ13ðΔτ1−Δτ2Þ

2

þ ζ123ðρ132eξ�123ðΔτ1;Δτ2Þ þ ρ123eξ123ðΔτ1;Δτ2ÞÞ�ðP S v3Þ ðA15Þ

¼ v†3½Rð3ÞðΔτ1;Δτ2Þ�v3; ðA16Þwhere

ξ123ðΔτ1;Δτ2Þ ¼ Ia þ iIs: ðA17Þ

The vector P S v3 contains all the immanants and the determinant and permanent of T:

P S v3 ≡

0BBBBBBBBBB@

1ffiffi6

p perðTÞ1ffiffi6

p detðTÞ1

2ffiffi3

p immðTÞ þ 1

2ffiffi3

p immðT213Þ16immðTÞ − 1

3immðT132Þ − 1

6immðT213Þ þ 1

3immðT312Þ

16immðTÞ þ 1

3immðT132Þ þ 1

6immðT213Þ þ 1

3immðT312Þ

− 1

2ffiffi3

p immðTÞ þ 1

2ffiffi3

p immðT213Þ

1CCCCCCCCCCA; ðA18Þ

with

v3 ¼

0BBBBBBBBB@

perðTÞimmðTÞ

immðT132ÞimmðT213ÞimmðT312ÞdetðTÞ

1CCCCCCCCCA;

P ¼

0BBBBBBBBBBBB@

1ffiffi6

p 1ffiffi6

p 1ffiffi6

p 1ffiffi6

p 1ffiffi6

p 1ffiffi6

p

1ffiffi6

p − 1ffiffi6

p − 1ffiffi6

p 1ffiffi6

p 1ffiffi6

p − 1ffiffi6

p

1ffiffi3

p − 1

2ffiffi3

p 1ffiffi3

p − 1

2ffiffi3

p − 1

2ffiffi3

p − 1

2ffiffi3

p

0 − 12

0 − 12

12

12

0 12

0 − 12

12

− 12

− 1ffiffi3

p − 1

2ffiffi3

p 1ffiffi3

p 1

2ffiffi3

p 1

2ffiffi3

p − 1

2ffiffi3

p

1CCCCCCCCCCCCA

; S ¼

0BBBBBBBBBBBB@

16

13

0 0 0 16

16

0 13

0 0 − 16

16

0 0 13

0 − 16

16

− 13

0 0 − 13

16

16

0 0 0 13

16

16

0 − 13

− 13

0 − 16

1CCCCCCCCCCCCA

: ðA19Þ

Here, P is a basis transformation and S is a matrix mapping matrix elements to matrix functions. The six matrices ρ are, infact, permutation matrices reduced to the block-diagonal form of Eq. (10).Equation (A15) describes the same features as Eq. (A12) but highlights the permutational options for three photons.

It is given in a maximally decoupled basis, which allows for a compact notation. The terms originating from theoverlap integrals (ζ terms and ξ terms) contain all the information on the physical properties of the interfering photons. Theeffect of the permutation symmetry of the photons is included in the permutation matrices ρ. Equation (A16) features aneven further compressed notation and allows for an elegant interpretation. Whereas the block-diagonal 6 × 6 rate matrix


041015-15

Rð3ÞðΔτ1;Δτ2Þ contains all the information on the permuta-tional symmetry and nonclassical interference itself, thebasis vector v3 contains the information on the interfer-ometer. As alluded in Sec. IV two entries of this rate matrix

are sufficient for an interpretation. Fper ¼ Rð3Þ11 ðΔτ1;Δτ2Þ

quantifies the fraction of the output probability distribution

proportional to the permanent and Fdet ¼ Rð3Þ66 ðΔτ1;Δτ2Þ to

the determinant of the submatrix T. The contributionproportional to immanants can also be explicitly calculated.When interested in only their overall contribution, this isgiven as Fimm ¼ 1 − Fper − Fdet. In the extremal case whenall the photons are indistinguishable, i.e.,

ωc;1 ¼ ωc;2 ¼ ωc;3 ¼ ωc; σ1 ¼ σ2 ¼ σ3 ¼ σ;

Δτ1 ¼ Δτ2 ¼ 0; ðA20Þ

we have ζij ¼ 1, ξij ¼ σ2, and νij ¼ ω, so the outputprobability reduces from a superposition of 60 terms to justP111 → jperðTÞj2. The rate matrix for the case of com-pletely indistinguishable photons, Rð3ÞðΔτ1 ¼ 0;Δτ2 ¼ 0Þ,describing the perfect quantum interference, and the ratematrix for the case of completely distinguishable photons,Rð3ÞðΔτ1 ¼ ∞;Δτ2 ¼ −∞Þ, describing the classical case,are given as

Rð3ÞðΔτ1 ¼ 0;Δτ2 ¼ 0Þ ¼

0BBBBBBBBB@

1 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1CCCCCCCCCA;

Rð3ÞðΔτ1 ¼ ∞;Δτ2 ¼ −∞Þ ¼

0BBBBBBBBB@

16

0 0 0 0 0

0 29

0 0 19

0

0 0 29

19

0 0

0 0 19

29

0 0

0 19

0 0 29

0

0 0 0 0 0 16

1CCCCCCCCCA:

3. Matrix reconstruction

The fabrication of integrated photonic networks using afemtosecond-laser direct-writing technology works withhigh precision and high stability. Discrete unitary operatorsacting on modes can be realized solely from beam splittersand phase shifters [32]. These networks are arranged likecascaded Mach-Zehnder interferometers, shown in Fig. 7.Notably though, even advanced writing precision canintroduce small deviations from the initially targeted valuesof individual elements. In our case this writing precision is

limited to around 50 nm over the whole length of thewaveguide (in this experiment, 10 cm). In a cascadedinterferometric arrangement small deviations of individualelements may add up to a noticeable deviation in the overalltransformation. The splitting ratio of individual directionalcouplers is set by their mode separation and couplinglength. Both characteristic variables are 3 orders of mag-nitude bigger than the positioning precision and thereforeunaffected by it. Unfortunately, small length fluctuationsdue to the positioning precision can introduce unintendedphase shifts. In the worst case, i.e., a phase shifter spanningthe whole length of a waveguide, the resultant phase shiftscan even reach π=8. The layout used for the interferometricnetworks reported here (see Fig. 7) circumvents this worstcase. Even if the unintended phase shifts are decreased by afactor of 3 at least, their influence needs to be evaluated andthe actually implemented unitary needs to be reconstructed.The characterization procedure we use builds on the oneintroduced in Refs. [33,34]. Two-photon states from adown-conversion source are injected into different modesof the optical network to be characterized. This in situmethod allows for a characterization with states having thesame physical properties, e.g., frequency and spectralshape, as used later in the experiment.

a. Estimating the visibilities of submatrices

We assume the optical interferometer can be describedby a 5 × 5 unitary matrix and we reconstruct its

FIG. 7. Integrated photonic network. Schematic drawing of theoptical network. The circuit consists of eight directional couplers(η1;…; η8), 11 phase shifters (ϕ1;…;ϕ11), five input modesð1;… ; 5Þ, as well as five output modes (10;…; 50). To allowcoupling to the waveguide with standard fiber arrays, the inputand output modes are separated 127 μm and the total length of thechip is 10 cm.


041015-16

transformation via visibilities measured by injecting twophotons into any combination of two of its five inputs. Thevisibility for two photons entering input modes i; j andexiting in the output modes k; l can be calculated from the2 × 2 submatrix Ui;j;k;l. For five input and output modesthis results in ð5

2Þ × ð5

2Þ ¼ 100 possibilities. Owing to the

structure of the interferometer (see Fig. 7), a photoninjected into port 5 cannot exit from output 10. This leadsto a visibility of zero for the four input pairs ij ¼15; 25; 35; 45 and the output pairs kl ¼ 15; 25; 35; 45.These visibilities are omitted from this reconstructionalgorithm, so the unitary transformation is reconstructedfrom 84 nonzero visibilities.Our interferometric network consists of eight beam

splitters and 11 phase shifters. Each beam splitter imple-ments a SU(2) transformation with matrix representation:

�cos β

2i sin β

2

i sin β2

cos β2

�; ðA21Þ

where β is the Euler angle associated with the transmittivityη via the relationship η ¼ cos2ðβ=2Þ. Note that in Eq. (A21)the beam splitter also implements a relative phase shift of πbetween the first and second mode.The 11 phase shifters produce additional phases in their

respective modes. Each phase shifter has a matrix repre-sentation of

�eiα1 0

0 eiα2

�; ðA22Þ

with αi the phase shift in mode i.The spectral shape of the photons is measured with a

single-photon spectrometer (Ocean Optics QE6500) and to agood approximation is of Gaussian shape. Such Gaussiansare defined by only two parameters, namely, their centralfrequency and the variance, which for the ith photon of theinput pair is given by Eq. (A3), and expressed here as

jϕiðωÞj2 ¼1ffiffiffiffiffiffi2π

pσiexp

�− ðω − ωc;iÞ2

2σ2i

�; i ¼ 1; 2:

ðA23ÞAssuming both photons exhibit identical spectral function,i.e., jϕ1ðωÞj2 ¼ jϕ2ðωÞj2, and the detectors are modeled bythe detection positive-operator valued measure (POVM)withtwo elements fΠ0;Π1g satisfying completeness,

PiΠi ¼ I,

Π1 ¼Z

dωa†ðωÞj0ih0jaðωÞ; Π0 ¼ I − Π1; ðA24Þ

then the visibility is

V ¼ − h1h�2 þ h�1h2jh1j2 þ jh2j2

; ðA25Þ

with

h1 ¼ U11i;j;k;lU

22i;j;k;l; h2 ¼ U12

i;j;k;lU21i;j;k;l; ðA26Þ

andUa;bi;j;k;l denotes the element in the ath row and bth column

of the matrix Ui;j;k;l. In an experiment the two photonswill always have slightly different spectral functions whosemismatch needs to be accounted for. The central wavelengthsand spectral bandwidths of the photons used in this charac-terizationmeasurement are λc;1¼ 789.05 nm,Δλ1 ¼ 2.9 nmand λc;2 ¼ 788.60 nm, Δλ2 ¼ 2.9 nm, respectively. Thecoincidence counts Nc as a function of time delay t andspectral mode mismatch are

NcðtÞ ¼ ð1þ T � tÞ�Y0 þ A

2σ1σ2σ21 þ σ22

× exp

�− ðωc;1 − ωc;2Þ2 þ 4σ21σ

22ðt − tcÞ2

2ðσ21 þ σ22Þ�

− ðHO1 þHO2 − dÞ�; ðA27Þ

where Y0, A, tc, and T are parameters to be fitted to theexperimental data. The experimental data for a given input-output combination i; j; k; l is typically recorded for 30increments with a step width of 66 fs and integrated over800 s each step. The coincidences are readout by a field-programmable gate array logic. As individual delays areset by translating a fiber coupler with a motorized screw(Newport LTA-HL), there can be a small drift in couplingefficiency over thewhole delay range of 2000 fs.Without thisdrift, the background of the visibility would be a horizontalstraight line. For drifts smaller than 5% of the two-photonflux, the drift is in good approximation linear and can bemodeled with an additional parameter T. The positioningprecision of the delay lines is limited to approximately�5 μm, which is within 2.5% of the coherence time ofthe interfering photons. When the two-photon input state isgenerated via down-conversion pumped by a pulsed lasersystem, higher-order emission can lead to unwanted con-tribution to the input state. The first higher order, which is afourfold emission, causes a small contribution of two photonsin each input mode during the characterization of a 2 × 2submatrix. This can add a constant background to the twofoldcoincidences in the following scenario: two photons in oneinput mode are lost and the two photons in the other inputmode leave the network in different output ports.Wemeasuresuch contributions by blocking one of the two input modesand recording the two-photon coincidences at the output.These signals are labeled HO1 and HO2, respectively, andsubtracted from the data. The background coincidence rate dmay be interpreted as a contribution to Nc stemming fromdark counts due to electrical noise and background light. Thisrate d is also present in HO1 and HO2. Therefore, it has to beadded to Eq. (A27) to account for all unwanted coincidences


041015-17

only once. The error for the raw data is verified to bePoissonian. For the data processing, the error of the higher-order term ðHO1 þ HO2 − dÞ and the abscissa error causedby the limited alignment precision of the delay lines need tobe taken into account additionally. These errors provideweighting in the minimization algorithm and influence thestandard errors of the fitted parameters. The visibility,

V ¼ 1 − Y0 þ AY0

; ðA28Þ

is finally calculated from the parameters Y0 and A, whereasthewidth of the dip or peak is fixed by the spectral function ofthe two photons. Only 84 out of 100 visibilities are nonzero,

and their value fVðexptÞi ; i ¼ 1;…; 84g and standard

deviation fσi; i ¼ 1;…; 84g are extracted via the procedureoutlined above. The resultant data fits with theory exhibitχ2red ¼ 1.74 [35]. An example for one of the 84 data sets isshown in Fig. 8.

b. Parameter estimation and reconstructionof the unitary matrix

A unitary transformation of a linear optical networkcan be reconstructed from single-photon transmission

probabilities and two-photon interference visibilities[34]. A technique using coherent states [36] follows asimilar approach. Both techniques reconstruct the unitarydescription in a dephased representation where the single-photon or single-input coherent state data are used toestimate the real parts of the matrix entries. The imaginaryparts of the matrix–entries are reconstructed from the two-photon interference visibilities or directly from the relativephase shifts. When the layout and initially targeted param-eters of the building blocks (see Fig. 7) of the interfero-metric network are known, their actual parameters can befitted alternatively. Our technique uses an overcompleteset of visibilities, and the parameters of the interferometerthat give an optimal fit to the experimentally measuredvisibilities are obtained using a least-squares optimizationweighted with the standard errors of the experimentalvisibilities (see Appendix A 3 a for details). Eight of the19 parameters are transmittivities, β1; β2;…β8, and 11 arephases, ϕ1;ϕ2;…ϕ11. To find the best-fit set of parameters,the data are processed with a Matlab program that usesfmincon to minimize the function Vopt,

Vopt ¼X84i¼1

ðVðexptÞi − VðthÞ

i Þ2σ2iΓ

; ðA29Þ

FIG. 8. Example for one data set used for the reconstruction of U5. The best fit of Eq. (A27) to the data set is shown in blue. Here, thevisibility is calculated from the best fit parameters Y0 and A. The reduced χ2 resulting from the fit shown in blue is χ2blue ¼ 2.02.Fluctuations in the count rate for values of jΔτj > 500 fs drive the reduced χ2 away from 1. These fluctuations can be interpreted as therandom noise background in the lab and the increased reduced χ2 reflects that. However, the precision of the fitted parameters Y0 and Aand ultimately the extracted visibility V is only marginally affected by these fluctuations. The 5 × 5 unitary description of theinterferometric network is reconstructed from 84 of these visibilities. The curve in green, NðthÞ

c ðtÞ [see Eq. (A33)] is calculated from fourmatrix entries of the reconstructed unitary and results in an overlap with the data of χ2green ¼ 2.70. The agreement of these two curves andcorresponding reduced χ2’s is a qualitative measure for the precision of the reconstruction.


041015-18

where VðthÞi is the theoretical value of the visibility calcu-

lated from our special unitary model of the interferometerusing Eq. (A25) for the ith data set, and Γ is a constantvalue equal to ðnumber of data sets in visibilities−number of parameters− 1Þ ¼ 2522− 19− 188− 1¼ 2314.Equation (A29) looks similar to a reduced χ2 but has to be

interpreted differently. A value close to 0 is desirable andindicates good agreement between experimentally extractedand theoretically predicted visibilities. In our case, the resultis Vopt ¼ 0.351.The 5 × 5 reconstructed matrix U5 using the procedure

outlined above is

U5 ¼

0BBBBBB@

0.0320− 0.3370i 0.07239þ 0.8203i −0.2780 − 0.1060i 0.1228 − 0.3220i 0

0.0114þ 0.2751i −0.3863þ 0.1860i −0.1353þ 0.2073i −0.7842 − 0.1502i 0.0124 − 0.2036i

−0.7757 − 0.2328i −0.2937þ 0.0018i −0.2677 − 0.0162i 0.0267þ 0.3517i −0.2476− 0.0151i

0.1444− 0.2611i −0.1518 − 0.0840i −0.1392þ 0.0839i −0.1327 − 0.0092i 0.0203þ 0.8449i

0.2225þ 0.1231i 0.0715 − 0.1293i −0.7929 − 0.0268i 0.0871þ 0.3067i 0.4123 − 0.1121i

1CCCCCCA:

ðA30Þ

4. Quality of the reconstructed description

Using this matrix, the probability of coincidence counts

PðthÞ11 can be predicted for any two-photon inputs and

outputs. For the inputs i and j, i < j, and outputs k andl, k < l, this reads as

PðthÞ11 ðt − tcÞ ¼ jUki

5 Ujl5 j2 þ jUli

5Ukj5 j2

þ ðUli5U

kj5 U

ki5�Ujl

5�

þ Uli5�Ukj

5�Uki

5 Ujl5 Þfðt − tcÞ; ðA31Þ

where

fðtÞ≡ ½2σ1σ2=ðσ21 þ σ22Þ�

× exp

�− ðωc;1 − ωc;2Þ2 þ 4σ21σ

22t

2

2ðσ21 þ σ22Þ�; ðA32Þ

and Uab5 is the element in the ath row and bth column

of U5. The actual coincidence count is then

NðthÞc ðtÞ ¼ N0ð1þ TÞPðthÞ

11 ðt − tcÞ; ðA33Þ

where N0, tc, and T are parameters used to find the best fitto the experimental data. The exact χ2red is calculated using

χ2red ¼Xmi¼1

ðNðexptÞc;i − NðthÞ

c;i Þ2νϵ2i

; ðA34Þ

where m ¼ 3030, ν ¼ m − 20− 100 − 1 ¼ 2909, ϵi is the

error for the corresponding data point, and NðexptÞc;i denotes

the experimental data corrected for higher-order emissions.The sum is taken over the data set and the index labels thedata. The obtained χ2red between the data and the predictedcoincidence counts using U5 is

χ2red ¼ 2.086: ðA35Þ

This value should be compared to reduced χ2expt ¼ 1.74obtained by fitting the primary data to extract the 84visibilities in the beginning (see Appendix A 3 a). Thedifference between those two reduced χ2’s can be attributed

FIG. 9. State generation. A pump beam is focused into a 2-mmβ-BaB2O2 crystal cut for noncollinear, degenerate, type-II down-conversion. The generated state is emitted into the spatial modesa and b. A compensation scheme consisting of half-waveplates and 1-mm-thick BBO crystals is applied for counteringtemporal and spatial walk-off. Narrow band interference filters(λFWHM ¼ 3 nm) are applied to increase the temporal coherenceof the photons and render them close to spectral indistinguish-ability. The modes a and b are subsequently split by polarizingbeam splitter cubes and two half-wave plates in their reflectedports are set to 45° to ensure the same polarization in all fouroutput modes (a00, a0, b0, and b00). With this scheme, threeindistinguishable photons in modes a0, b0, and b00 each can beheralded from a fourfold emission by a successful trigger event inmode a00.


041015-19

to accuracy of the reconstructed unitary matrix U5. Whilefluctuations in the count rate for values of jΔτj > 500 fsdrive both reduced χ2 away from 1, the difference betweenthe two reduced χ2 is relatively small (≈0.35). An examplefor one of the 100 data sets is shown in Fig. 8.

5. State generation

We use an 80-MHz Ti:sapphire oscillator emitting 150-fspulses at a wavelength of 789 nm, which get frequencydoubled via a LiB3O5. The up-converted beam is focusedinto a 2-mm-thick β-BaB2O2 crystal cut for degeneratenoncollinear type-II spontaneous parametric down-conversion. To achieve near-spectral indistinguishabilityand enhance temporal coherence of the down-convertedwave packets, the photons are filtered by λFWHM ¼ 3 nminterference filters. The source is aligned to emit themaximally entangled state

jϕþi ¼ 1ffiffiffi2

p ðjHiajHib þ jViajVibÞ; ðA36Þ

when pumped with low pump power (200 mW cwequivalent). H and V denote horizontal and verticalpolarization and a and b are the two spatial emissionmodes. When pumped with higher pump powers (700 mWcw equivalent), noticeable higher-order emission occurs:

jψia;b ¼1ffiffiffi3

p ðjHHiajHHibþjHViajHVibþjVViajVVibÞ:

ðA37Þ

This state is guided to two PBS cubes. A detection eventin the trigger mode a00 heralds the generation of either thestate jVia0 jVib0 jHib00 or jHHib00 (see Fig. 9). Only in thefirst case are the three modes a0, b0, and b00 occupied withone single photon, whereas in the latter case mode b00 isoccupied with two photons and mode b0 with vacuum.Postselection on a fourfold coincidence between modes a00,a0, b0, and b00 allows for the heralding of the desired inputstate where only one photon enters each input mode.The half-wave plates in mode a0 and b0 are set to 45° torender them indistinguishable in polarization from the otherphotons. This heralding scheme holds independently ofany transformation for the photons in modes a0, b0, and b00as long as it acts on spatial modes, e.g., consisting ofbeam splitters and phase shifters only.

6. Analysis of the threefold coincidence data

Three photons are inserted into input modes 1, 2, and 4of the interferometric network. The spectral characteristicsof these photons are measured using a single-photonspectrometer (Ocean Optics QE6500) and are in goodapproximation of Gaussian shape. Note that these spectraldata differ slightly compared to the characterization mea-surements (see Appendix A 3 a).

λc ΔλFWHM

In1 789.35 nm 2.85 nmIn2 789.52 nm 2.79 nmIn4 789.41 nm 2.72 nm

TABLE III. Data for the coincidence landscapes of Fig. 3. Three photons are injected into input ports 1, 2, and 4 of the interferometricnetwork. They are recorded as a fourfold coincidence measurement between output modes 1, 3, and 4 and the trigger mode, labeled “Out134,” and output modes 1, 3, and 4 and the trigger mode, labeled “Out 345.” The theoretical prediction, “theoretical,” for the outputprobability to record a certain event for a pair of temporal delays Δτ1 and Δτ2 is calculated using Eq. (A16). The experimentallyacquired count rates, “experimental,” are normalized to unit vectors and fitted to the theoretical data resulting in a goodness of fit ofχ2red ¼ 1.38 and χ2red ¼ 1.10, respectively. The experimental errors are standard deviations over 19 independent measurements.

Δτ1 Δτ2 Theoretical Experimental χ2red Count rate

Out 134

0 fs 130 fs 3.41% 3.17%� 0.26%0 fs −870 fs 1.89% 2.18%� 0.19%

−300 fs −170 fs 3.13% 2.99%� 0.25% 1.38 ≈10 mHz−1000 fs −870 fs 2.95% 2.96%� 0.26%−1000 fs 130 fs 2.20% 2.51%� 0.21%−1000 fs 1130 fs 2.73% 2.72%� 0.31%

Out 345

0 fs 130 fs 14.19% 14.73%� 0.93%0 fs −870 fs 23.69% 24.01%� 0.84%

−300 fs −170 fs 17.67% 19.10%� 0.98% 1.10 ≈80 mHz−1000 fs −870 fs 25.09% 24.01%� 0.85%−1000 fs 130 fs 21.14% 21.32%� 0.80%−1000 fs 1130 fs 31.40% 30.85%� 1.40%


041015-20

These spectral data allow us to express the mode-overlap integrals as a function of the time delays Δτ1 and Δτ2 betweenthe first and second photon and the second and third photon, respectively. The theoretical prediction for the outputprobability in any of the ten threefold output ports is then calculated using Eq. (A15). Consequently, each 3 × 3 submatrix Ris constituted by matrix elements selected by the input and output ports. The output probability [see Eq. (A15)] of anylandscape contains a constant term and four terms proportional to different mode-overlap functions. By sampling six pointsof pairwise temporal delay of Δτ1 and Δτ2, P1–P6 in the table below, contributions of each of these terms can be assessed.These six points are shown here.

TABLE IV. Data for the normalized output probability distributions of Fig. 4. The column “Figure” specifies the temporal delays ofeach part of Fig. 4, whereas the column “Tijk” refers to the scattering submatrix of Eq. (A12). The experimentally acquired count ratesare normalized to unit vectors, then fitted to the theoretical output probability and given under column “Experiment in %.” The errors arestandard deviations over 19 independent measurements. The last column, “Theory in %,” shows the theoretical output probabilityaccording to Eq. (A16). The contribution of the permanent, the immanant, and the determinant to this theoretical value is given in thecolumns “per in %,” … imm in %,” and “det in %,” respectively.

Figure Tijk Experiment in % per in % imm in % det in % Theory in %

4(a) 245 1.46� 0.39 1.72 0.13 0.00 1.86235 10.02� 0.83 11.32 0.44 0.00 11.76123 46.8� 3.00 33.38 11.97 0.00 45.36345 0.47� 0.20 0.03 0.16 0.00 0.19

Δτ1 ¼ 0 fs 234 7.24� 0.80 7.08 0.77 0.00 7.85134 6.69� 0.71 6.30 1.54 0.00 7.85

Δτ2 ¼ 130 fs 125 7.96� 0.89 5.21 2.87 0.00 8.08145 1.69� 0.40 1.41 0.13 0.00 1.55135 8.01� 0.77 3.98 1.10 0.00 5.08124 9.71� 0.94 8.95 1.50 0.00 10.45

4(b) 245 1.35� 0.24 0.93 0.59 0.01 1.53235 7.93� 0.68 6.11 2.45 0.11 8.67123 50.86� 2.60 18.02 29.62 1.19 48.82345 0.78� 0.16 0.02 0.64 0.01 0.66

Δτ1 ¼ −300 fs 234 6.00� 0.41 3.82 3.04 0.03 6.89134 5.12� 0.58 3.40 1.21 0.01 4.61

Δτ2 ¼ −170 fs 125 8.07� 0.74 2.81 3.69 0.03 6.53145 1.86� 0.26 0.76 1.20 0.02 1.98135 8.64� 0.68 2.15 8.86 0.14 11.15124 9.39� 0.59 4.83 4.16 0.16 9.15

4(c) 245 1.17� 0.26 0.37 0.77 0.05 1.19235 6.59� 0.58 2.46 3.35 0.59 6.40123 53.64� 1.90 7.26 40.72 6.54 54.52345 0.92� 0.20 0.01 0.96 0.03 1.00

Δτ1 ¼ −1000 fs 234 5.29� 0.43 1.54 3.63 0.15 5.32134 4.06� 0.40 1.37 2.43 0.04 3.84

Δτ2 ¼ −870 fs 125 8.19� 0.64 1.13 6.26 0.19 7.58145 1.57� 0.22 0.31 1.24 0.12 1.67135 9.91� 0.74 0.87 8.03 0.78 9.68124 8.67� 1.00 1.95 5.97 0.89 8.80

4(d) 245 0.92� 0.23 0.19 0.51 0.17 0.86235 5.12� 0.58 1.21 1.91 1.97 5.09123 58.26� 2.70 3.58 33.36 21.69 58.63345 0.60� 0.09 0.00 0.56 0.11 0.68

Δτ1 ¼ −1000 fs 234 4.17� 0.39 0.76 2.85 0.50 4.11134 4.03� 0.51 0.68 2.79 0.12 3.59

Δτ2 ¼ 1130 fs 125 7.38� 0.62 0.56 6.00 0.64 7.19145 1.32� 0.28 0.15 0.91 0.39 1.45135 10.00� 0.38 0.43 7.15 2.57 10.15124 8.18� 0.71 0.96 4.33 2.95 8.25


041015-21

Δτ1 Δτ2P1 0 fs 130 fsP2 0 fs −870 fsP3 −300 fs −170 fsP4 −1000 fs −870 fsP5 −1000 fs 130 fsP6 −1000 fs 1130 fs

An offset of Δτoff ¼ 130 fs is introduced in the temporaldelay mode Δτ2; other than that the delays are set tocombinations of 0, −300, and �1000 fs. The precision ofthe temporal alignment is estimated to be �16 fs. In onemeasurement run the points P1–P6 are recorded consecu-tively for 2 h each. To account for effects of drift, this orderis reversed in the next measurement run; therefore, thepoints are recorded in the order P6–P1. The fourfold countrates range from 1 to 100 mHz, dependent on the outputcombination. In between each measurement run the setup isrealigned to optimize for maximal count rates. In order toobtain sufficient statistics, the whole data acquisition isrepeated over 19 measurement runs for a total of 228 h.As Poissonian error modeling results in too optimistic

error bars in the case of long data acquisition due tomultiple sources of error, we adapt the error modeling. The19 measurements are independent runs; therefore, theirmean and standard deviation of the mean provide moreuseful information. Each individual measurement run isrepresented as a six-dimensional vector, with the ith entryof the vector containing the fourfold counts of the Pithdelay point integrated over 2 h. These vectors can then benormalized to unit vectors thereby obtaining relative outputprobabilities. The mean and the standard deviation of themean can now be calculated for each of the six delay points.Ultimately, the overlap with the theoretical prediction isobtained by a least-squared minimization weighted with thestandard deviations. Here, a linear scaling factor is intro-duced relating the relative experimental probabilities to theabsolute theoretical ones (see Table III). The goodness of fitis calculated using the reduced χ2. The number of degreesof freedom in this case is ν ¼ 6–2 ¼ 4.The experimental data for the four different scenarios of

BosonSampling affected by distinguishability, shown inFig. 4, are recorded using the same method as above.Table IV shows the experimental data and theoreticalprediction for Figs. 4(a)–4(d).

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