Downloaded by guest on October 2, 2021 www.pnas.org/cgi/doi/10.1073/pnas.2010827117 unu interference quantum . quantum in implications transformation, wide Bogoliubov anticipate bosonic we any so to generic fundamental underly- is This the mechanism indistinguishability. identify time-like and interferometric as 2) two- gain mechanism (of ing unsuspected amplifier quantum an a in of effect existence dual- this the from infer beam ity We matter. amplification. the into transforms even coupling reversal linear as or time splitter partial ports, light that output establish with both we experiments Here, at numerous coincidence transmittance in in (of confirmed detected splitter be beam cannot a 1/2) on Two impinging principle. of Pauli symmetry the identical by the dictated in as roots particles, 2020) two- its quantum 27, identical of has May review It paradigm for interference. the (received quantum 2020 is particle 19, effect October approved Hong–Ou–Mandel and celebrated TX, Station, The College University, A&M Texas Scully, O. Marlan by Edited Kingdom United and a Cerf J. time Nicolas in interference quantum Two-boson hsc,i sepce htti w-oo nefrneefc in effect interference two-boson this that expected is it at physics, is that indistinguishability effect. HOM “space-like” the in time-reversed usual work partial coin the the we of is which argue version future, it We and as 1/2). past indistinguishability indistinguisha- the transmittance “time-like” the from of from photons BS originates between a bility effect to unsuspected of dual this amplifier a is that parametric predict (which linear a we 2 optical the in considerations, gain nonlinear these effect between of interferometric the duality implication two-photon and striking a amplifier. a BS parametric exhibit As a a by cen- to effected by transformation nevertheless us (Bogoliubov) effected two par- but allows coupling the dub unphysical it optical of we is one as which (PTR), in operation, tral reversal time This time of 2A). tial arrow (Fig. the modes of bosonic reversal under forms light both encompasses (19). been matter mechanism even and two-boson has this it that Remarkably, strating with 18). within simulta- observed (17, or are chip experimentally (14–16) photons photonic sources single silicon independent (see, the a two y case by 30 in emitted last even neously the has 10–13), over It refs. be 9). experiments (8, e.g., numerous cannot interference in it wave verified classical as been of interference terms the in quantum as understood viewed two-particle sin- often the of is and highlights it gularity it 5) feature: 7), quantum (4, (6, genuinely effect prominent inequalities most Twiss 1B). Bell and (Fig. of Brown violation BS the Hanbury the the at with reflection destructive Together double the and for from transmission amplitudes input probability double follows two the effect between its HOM interference on two-photon This 1A). impinge (Fig. photons cancel- ports (BS) indistinguishable splitter the by beam two observed experiment 50:50 a who when seminal behind (3), detections a coincident (HOM) iden- of in Mandel lation of 1987 and pair in Ou, A a achieved Hong, bosons. for was wave statistics identical bosons bosonic bosonic of tical the of bunching of demonstration so-called symmetry striking the the since known that favors been (2) has function it Einstein particular, and In (1). Bose principle Pauli the by T etefrQatmIfrainadCmuiain cl oyehiu eBuels Universit Bruxelles, de polytechnique Ecole Communication, and Information Quantum for Centre ic ooibvtasomtosaeuiutu nquantum in ubiquitous are transformations Bogoliubov Since trans- interference quantum two-boson how explore we Here, b atce i h ymtyo h aefnto,a dictated as function, wave the identical of of symmetry behavior the the govern via physics particles quantum of laws he eateto ple ahmtc n hoeia hsc,Cnr o ahmtclSine,Uiest fCmrde abig B 0WA, CB3 Cambridge Cambridge, of University Sciences, Mathematical for Centre Physics, Theoretical and Mathematics Applied of Department a,1 n ihe .Jabbour G. Michael and | oo bunching boson 4 emtsal tm,demon- atoms, metastable He | iereversal time b 1 term term double-transmission double-reflection the since two of state as the transforms mode) Hence, each reflected). in (one for photons indistinguishable “ref” mode and in transmitted state single-photon a oeiaie ies . C BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This Submission. y Direct performed PNAS a M.G.J. is article This and interest.y financial competing N.J.C. paper.y no the declare wrote and authors and The results, research; the discussed formulas, designed the derived N.J.C. research, contributions: Author where (for transformations single-photon the PDC) and effects reflected. being see BS photons details, both 50:50 is of it amplitude A probability cancels as transmitted a the being is optics out photons the both It where quantum of bunching. effect amplitude in probability interference boson quantum landmark of intrinsically manifestation a two-photon spectacular is most effect the HOM The Effect Hong–Ou–Mandel in property particles. help quantum fundamental would identical overlooked it of as heretofore platform some atomic or elucidating photonic a indistin- trans- be in time-like bosonic guishability would of of consequence it the range viewpoint, demonstrate wide to deeper a fascinating a for from bed Furthermore, test a formations. as serve could time owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To sagniemnfsaino unu hsc n may in facets many participate and in physics. role bosons of physics a play identical which quantum two transformations, Bogoliubov of whenever manifestation observed be genuine and a past effect unknown ascribe the hitherto is an This we from distinguished). into be that (bosons cannot reversal, future amplifier indistinguishability time quantum time-like partial a establish result- to We in under splitter function. effect turns, wave beam interference effect the half-transparent this of Hong– a symmetry that the the of from on output ing build the bosons we identical of at Specifically, “bunching” the time. namely effect, iden- Ou–Mandel in of mecha- indistinguishability bosons the interference tical from quantum originates which unsuspected nism, an uncover We Significance |1i a a → and √ aeil n ehd,Gusa ntre o BS a for Unitaries Gaussian Methods, and Materials 1 2 b |1i ( ae h ooi oe and modes bosonic the label trans |{z} ir eBuels 00Buels Belgium; Bruxelles, 1050 Bruxelles, de libre e |1i ´ a | 1i a |1i − b → |{z} |1i ref √ b b 1 |1i ), 2 . y (|1i a raieCmosAttribution-NonCommercial- Commons Creative oegnrly h probability the generally, More . |1i a a |1i /b b → a hr,“rn”sad for stands “trans” (here, |1i |1i − √ NSLts Articles Latest PNAS 1 a 2 |1i ( |{z} b |1i ref | b 1i a |1i b acl u the out cancels + ) a trans |{z} |1i /b b tnsfor stands ), | f10 of 1 [2] [1]

PHYSICS where aˆ and bˆ are mode operators. It is striking that a simple A † swap bˆ ↔ bˆ transforms HBS into HPDC, suggesting a deep dual- ity between a BS and a PDC by reversing the arrow of time in mode bˆ (keeping mode aˆ untouched). The underlying concept of PTR will be formalized in Eq. 7, but we first illustrate this duality between a BS and PDC with the simple example of Fig. 2A, where n photons impinge on port B bˆ of a BS (with vacuum on port aˆ), resulting in the binomial output state

n !1/2 BS X n k n−k U |0, ni = (sin θ) (cos θ) |k, n − ki , [5] η k k=0

with η = cos2 θ (see Gaussian Unitaries for a BS and PDC). The path where all photons are reflected (k = n) is associated with BS n the transition hn, 0| Uη |0, ni = sin θ. C Reversing the arrow of time on mode bˆ (Fig. 2A) leads us to consider the transition probability amplitude for a PDC of gain g = cosh2 r with vacuum state on its two inputs and n photon PDC n pairs on its outputs, that is, hn, n| Ug |0, 0i = tanh r/ cosh r

A

Fig. 1. (A) If two indistinguishable photons (represented in red and green for the sake of argument) simultaneously enter the two input ports of a 50:50 BS, they always exit the same output port together (no coinci- dent detection can be observed). (B) The probability amplitudes for double transmission (Left) and double reflection (Right) precisely cancel each other when the transmittance is equal to 1/2. This is a genuinely quantum effect, which cannot be described as a classical . (C) The correla- tion function exhibits an HOM dip when the time difference ∆t between the two detected photons is close to zero (i.e., when they tend to be indistinguishable).

for coincident detections is given by (for details, see Materials and Methods, Two-Photon Interference in a BS and PDC)

BS 2 2 B Pcoinc(η) = | h1, 1| Uη |1, 1i | = (2η − 1) , [3]

BS where Uη denotes the unitary corresponding to a BS of transmittance η. In a nutshell, the HOM effect boils down to BS h1, 1| U1/2 |1, 1i = 0. Its experimental manifestation is the pres- ence of a dip in the correlation function, witnessing that two photons cannot be coincidently detected at the two output ports when η = 1/2 (Fig. 1C). Fig. 2. (A) BS under PTR, flipping the arrow of time in mode bˆ. The PTR duality is illustrated when n photons impinge on port bˆ (with vacuum Partial Time Reversal on port ˆa), and we condition on all photons being reflected. The retro- ˆ0 Bogoliubov transformations on two bosonic modes comprise dicted state of mode b (initially the vacuum state |0i) back propagates passive and active transformations. The BS is the fundamen- from the detector to the source (suggested by a wavy arrow). This yields the same transition probability amplitude (up to a constant) as for a PDC tal passive transformation, while parametric down conversion of gain g = 1/η with input state |0, 0i and output state |n, ni. PDC is an (PDC) gives rise to the class of active transformations (also active Bogoliubov transformation, requiring a pump beam (represented in called nondegenerate parametric amplification). Although the blue). Note that the PTR duality is rigorously valid when this pump beam involved physics is quite different (a simple piece of glass makes is of high intensity (i.e., treated as a classical light beam) since the Hamil- a BS, while an optically pumped nonlinear crystal is needed to tonian HPDC of Eq. 4 holds in this limit only. (B) Operational view of the effect PDC), the Hamiltonians generating these two unitaries are PTR duality. As noted in ref. 20, if we prepare the entangled (EPR) state P∞ |Ψi 00 ∝ |n, ni and send mode bˆ in the BS, we get the output state amazingly close, namely b,b n=0 P∞ n |Ψi 00 ∝ sin θ |n, ni, which is precisely the two-mode squeezed vac- a0,b n=0 uum state produced by PDC when the signal and idler modes are initially in † † † † HBS = i(ˆa bˆ − aˆbˆ ), HPDC = i(ˆa bˆ − aˆbˆ), [4] the vacuum state.

2 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.2010827117 Cerf and Jabbour Downloaded by guest on October 2, 2021 Downloaded by guest on October 2, 2021 t.Mr eeal,w hwin show we generally, More ity. by divide and (n number pair photon 3A. the the Fig. of conserve in distribution illustrated not with The is terms does mode. each observe BS) on we the tons hence, (unlike number; PDC photon the total as here cated gain of PDC a of output the at (see namely state related the 2, examining gain a by ified of of PDC transmittance existence a 1 of h1, in possible BS effect a the suppression for interferometric suggests effect HOM immediately the 1/2 duality, this to Due second Amplifier an the in Interference on Two-Boson state retrodicted output the mode entangled access we that so ftegain the vanishes detections if coincident for probability the that confirming PDC) and BS a in Interference of structure the out- However, each effect. on HOM photon the single (n a with mode component put the appears, it As mode so-called a on of state half PTR entangled sending the by (EPR) 2B, operational Fig. Einstein–Podoslky–Rosen made in be shown can As duality Mechanics). Quantum of in precise Picture made is mode (this of fact time state the mode the as while of here time, state understood “retrodicted” be the must that PTR line, mechanics this quantum Along of (21). picture “retrodictive” so-called the using satisfy PDC expected. a for elements matrix photon nonzero only the satisfying between those difference are Eq. the In conserves numbers. PDC latter number, and photon the BS total the while the by conserves former exhibited the rules transformations: conservation the by Eq. denced as expressed be indices where Fock duality in PTR in transposition to rise partial demonstrated gives Duality, as basis PTR of Indeed, Proof PDC. Methods, and and BS a between same the gain obtain of we PDC time, a of for arrow as the output reversing port output and the vacuum again on Conditioning PTR. of Example Methods, and of case The constant a to (up set equal made be can amplitudes two (see efadJabbour and Cerf h eedneo h rbblt fdtcigasnl pair single a detecting of probability the of dependence The h oino iervra a ecneinl interpreted conveniently be can reversal time of notion The duality general a of existence the reflect examples These 1 = sin n | w-htnItreec naB n PDC): and BS a in Interference Two-Photon .Srknl,teabove the Strikingly, PDC). and BS a for Unitaries Gaussian U ntegain the on ) htn mign nport on impinging photons θ 2 PDC P tanh = hn coinc hn 0 g ≡ |Ψi m , ,1 |1, , j 2 = 1 = n j (g | g htn isedo aum mign nport on impinging vacuum) of (instead photons | U and egteatyEq. exactly get we , r U i = ) b ˆ oeta fw substitute we if that Note . side upesd hc srmnsetof reminiscent is which suppressed, indeed is ) g h nynneomti lmnsfraBS a for elements matrix nonzero only the 7, PDC requivalently, or 0 = g 00 U PDC . 2 i m h | PDC hssrkn rdcincnide ever- be indeed can prediction striking This . + 24 g ,1 1, |i aebe wpe ti a alternatively can (this swapped been have ,0 |0, j , fPCi ie y(see by given is PDC of or ,1 |1, m = | U i i n .TePRdaiyi ieyevi- nicely is duality PTR The 25). g = = g PDC i aeil n ehd,Retrodictive Methods, and Materials + 1 = a ˆ = g g aeil n ehd,Extension Methods, and Materials m −1/2 −1/2 omlypoaae owr in forward propagates normally 1 2 /η ,1 |1, b ˆ hsdrcl mle htthe that implies directly This . η n X 1 = silsrtdin illustrated is ∞ =0 ihiptstate input with sipidb T dual- PTR by implied as 9, hn hn | i b ˆ /g n , 0| , 2 2 rpgtsbcwr in backward propagates m n − (2 = hs ehave we Thus, . /2 | U 1 U |Ψi 1/g BS 1/g η |n BS − i by , |0, g smr compli- more is − |i n ) 2 i , m aeil and Materials n 1/g /g j cosh . Two-Photon |m i i n = . 3 , Materials , ≥ 0i. , n nEq. in r 2 − fwe if ) g pho- j 2 = [9] [8] [7] [6] as , b ˆ b ˆ a ˆ 3 0 , htn mig naB ftransmittance viewed of be amplitude BS probability can a a what recall 4, on has from first (Fig. impinge We originates version indistinguishability photons future). time-like effect space-like a and HOM as in past indistinguishability, the albeit the boson effect that from to HOM bosons back the (involving that traced for effect be as interference similarly can quantum predict two-boson the we of origin The Indistinguishability Time-Like vs. Space-Like effect. HOM extended an to applied gain the ing (e.g., gain the of value integer g larger any to extends Gain actually Integer effect with PDC a to blt mltd frflcinis reflection of amplitude ability is amplitude double-transmission in o h w htn,s h oberflcinampli- double-reflection the so photons, two is the tude for signs itiuin ftepoo arnme o gain a for number n pair photon the of distributions terms the as well to as 2 vacuum with the are components significant The suppression. vanishing a has 3. Fig. otnoscre nodrt ud h y,btol nee ausof values integer only for but curve eye, The the relevant. guide to order in curves continuous state output The ports. input idler gain of PDC = ,4, 3, = )and 2) n htn iutnosyo ahotu otvnse when vanishes port output each on simultaneously photons B A Probability (A) Corresponding (B) zero). to decay quickly terms next (the pairs ∼10 A g · · · U g = dr 2 PDC g = h rbblt fdetect- of probability the 3B, Fig. in illustrated As ). = n = soigadpat dip a (showing 4 ,1i |1, ,1i |1, 1 + hnasnl htnipne nbt h inland signal the both on impinges photon single a when 2 −(1 gi,ti stecneuneo T duality PTR of consequence the is this Again, . = opnn,oigfo w-htninterferometric two-photon from owing component, − g P 2 1 = n η

ic obetasiso vn is event double-transmission a Since ). fobserving of sas lte o comparison. for plotted also is 2 ,0i −|0, + s htti nefrmti suppression interferometric this that 16 9 + √ n ,4i |4, r η = A 4 1 n fbigtasitd othe so transmitted, being of ) h itiuin r hw as shown are distributions The 3). dt ,2i |2, htnpisa h upto a of output the at pairs photon √ + = r 1 NSLts Articles Latest PNAS − η + 2 1 ncnrs,teprob- the contrast, In . η ,5i |5, r g Upper 2 1 u ihopposite with but = ,3i |3, + soigadpat dip a (showing 3 η ahphoton each , · · · .We two When ).   | f10 of 3 n are

PHYSICS indistinguishable from a double-reflection event, we must add reversal has been exploited in the context of defining separability 2 probability amplitudes, leading to |Adt + Adr| = Pcoinc(η). The criteria (23, 24), but this seems to be unrelated to PTR duality. double-transmission and double-reflection amplitudes exactly Further, the link between time reversal and optical-phase con- cancel out for η = 1/2, which originates from the fact that an jugation has been mentioned in the literature exchange of the two indistinguishable photons in space (which (see, e.g., ref. 25), but it exploits the fact that the complex con- turns a double-transmission into a double-reflection event) jugate of an electromagnetic wave is the time-reversed solution cannot lead to any consequence. of the wave equation (the phase conjugation time-reversal mir- We now argue that it is the exchange of indistinguishable pho- ror concerns one mode only). The PTR duality introduced here tons in time that is responsible for the interference effect in bears some resemblance with an early model of (26) based an amplifier (Fig. 4, Lower). When two photons impinge on a on the coupling of an “inverted” harmonic oscillator (having a PDC with gain g, they can be both transmitted without trigger- negative frequency ω) with a heat bath. The inverted harmonic ing a stimulated event, which is dual to the double transmission oscillator (e−iωt → eiωt ) can indeed be viewed as a time-reversed in a BS (where η is substituted by 1/g). Hence, the double- harmonic oscillator. Quantum amplification in this model occurs 0 −1/2 3/2 transmission amplitude in a PDC is Adt = g Adt = 1/g . from a PDC-like coupling of this inverted harmonic oscillator, Another possible path giving rise to the coincident detection of whereas quantum damping follows from the BS-like coupling of two single photons is the combination of the stimulated anni- a usual harmonic oscillator with the bath. The PTR duality is hilation and emission of a pair of photons, which is dual to also reminiscent of Klyshko’s so-called “advanced-wave picture” the double reflection in a BS. This double-stimulated event in PDC (27), which provides an interpretation of coincidence- 0 −1/2 3/2 admits a probability amplitude Ast = g Adr = −(g − 1)/g , based two-photon experiments: the wave that is detected by one where the minus sign results from the fact that the probability of the detectors behind PDC can be viewed as resulting from amplitude that the input pair disappears by stimulated annihi- an “advanced wave” emitted by the second detector, so that lation and the probability amplitude that a new pair is created PDC acts as a mirror (28). This picture may indeed be inter- PDC by stimulated emission have opposite signs. Again, since the preted as a special case of Eq. 25, namely h0, 0| Ug |ψ, φi = double-transmission and double-stimulated events are indistin- −1/2 ∗ BS g h0, φ | U1/g |ψ, 0i. In the limit where the gain g → ∞, the guishable, we must add their probability amplitudes and get two outputs of PDC can be viewed as the input ψ and output φ∗ |A0 + A0 |2 = P 0 (g) g = 2 dt st coinc , which vanishes when . Roughly of a fully reflecting (phase-conjugating) mirror with η → 0. speaking, we cannot know whether the “old” photons have been In this work, we have promoted PTR as the proper way to replaced by “new” photons or have been left unchanged, which approach the duality between passive and active bosonic trans- we dub time-like indistinguishability. formations. As a compelling application of PTR duality, we Discussion and Conclusion have unveiled a hitherto unknown quantum interference effect, which is a manifestation of quantum indistinguishability for The role of time reversal in quantum physics has long been a identical bosons in active transformations (space-like indistin- fascinating subject of questioning (see, e.g., ref. 22 and refer- guishability, which is at the root of the HOM effect, transforms ences therein), but the key idea of the present work is to consider under PTR into time-like indistinguishability). The interferomet- a bipartite quantum system (two bosonic modes) with counter- ric suppression of the coincident |1, 1i term is induced by the propagating times. Incidentally, we note that the notion of time indistinguishability between a photon pair originating from the past and a photon pair going to the future. Stated more dramati- cally, while the two photons may cross the amplification medium and be detected, the sole fact that they could instead be anni- hilated and replaced by two other photons makes the detection probability drop to zero when g = 2. The experimental verification of this effect can be envisioned with present technologies (see Experimental Scheme). A coinci- dence probability lower than 25% would be sufficient to rule out a classical interpretation, which could in principle be reached with a moderate gain of 1.28 (see Classical Baseline). Observ- ing time-like two-photon interference in experiments involving active optical components would then be a highly valuable metrology tool given that the HOM dip is commonly used today as a method to benchmark the reliability of single-particle sources and mode matching. More generally, the interference of many photons scattered over many modes in a linear optical net- work has generated a tremendous interest in the recent years, given the connection with the “boson sampling” problem [i.e., the hardness of computing the permanent of a random matrix Fig. 4. The HOM effect (Upper) is due to space-like indistinguishability: (29)], and technological progress in integrated optics now makes the double-transmission path (of amplitude Adt) where the two photons it possible to access large optical circuits (see, e.g., ref. 30). In are transmitted interferes destructively with the double-reflection path (of this context, it would be exciting to uncover new consequences of amplitude Adr) where the two photons are reflected. Exchanging the two PTR duality and time-like interference. photons in space leads to the HOM effect in a BS of transmittance η = 1/2. Finally, we emphasize that our analysis encompasses all According to PTR duality (by reversing the arrow of time in the second bosonic Bogoliubov transformations, which are widespread in mode; Lower), the two interfering paths in a PDC correspond to the double- 0 physics, appearing in quantum optics, quantum field theory, or transmission event associated with amplitude Adt (the two photons simply 0 solid-state physics, but also in black hole physics or even in the cross the PDC) and double-stimulated event of amplitude Ast (the input photon pair is up converted into the pump beam, and another pump pho- Unruh effect (describing an accelerating reference frame). This ton is down converted into a new photon pair). Exchanging the photon suggests that time-like quantum interference may occur in vari- pairs in time induces a time-like interferometric suppression in a PDC of ous physical situations where identical bosons participate in such gain g = 2. a transformation. Beyond bosons, let us point out an intriguing

4 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.2010827117 Cerf and Jabbour Downloaded by guest on October 2, 2021 Downloaded by guest on October 2, 2021 ˆ efadJabbour and Cerf formula, disentanglement the of action The or binomial the get we so reflected, or transmitted state be may photon transforma- each canonical ports, the using by Eq. computed tion, be easily can it Alternatively, unitary, Gaussian two-mode passive amplification, unitary BS fundamental parametric the namely or The interferometry (34). optical respectively linear PDC. by and effected BS are a for Unitaries Gaussian Methods and connec- Materials this broader even that up hope open We may the perspectives. diagram. electrodynamics connecting quantum PDC rule antiparticle, with substitution the own tion a to its as is diagram duality boson) BS PTR This here: neutral view described 33). truly may duality ref. (or we PTR e.g., photon the (see, to to a analogous rule related since senses substitution are many a that in such processes is by are photons other electron– two each an of by creation pair an the by positron photon and a scattering) of going (Compton scattering particles the by electron example, into related For turned time). are being in backward (antiparticles that rotation elements rule Wick substitution matrix a a scattering to two refers electrody- symmetry connecting quantum Crossing in 32). “crossing” (31, of namics notion the with connection ) iial,teunitary the Similarly, 1). where as picture modes Heisenberg the between in coupling acts linear energy-conserving an effects pia upn fanniercytl ttasom h oeoperators mode the the to transforms due It PDC transformation Bogoliubov crystal. by the nonlinear photons to according a entangled of of pumping pairs optical of generation the models htnnme ifrnei osre ytePCtasomto,namely transformation, PDC the by a conserved is difference number photon where eopsto fteexponential the of decomposition 0† h cinof action The ˆ a 0 − g ˆ a b ˆ steprmti an(g gain parametric the is and 0† o xml,when example, For 11. U b ˆ U η BS 0 η BS b ˆ = U 0i |n, |0, r h oeoeaosand operators mode the are b ˆ g PDC ˆ a ˆ a † U → b → ˆ ˆ a U ˆ a ni U g PDC U → → = nFc ttscnb ovnetyepesdb s of use by expressed conveniently be can states Fock on η BS η − BS = b ˆ ˆ a = η BS = 0 0 = = b b ˆ ˆ X ˆ a k = ( = = =0 X ˆ a k 0 0 † n ( nFc ttscnb xrse yuigthe using by expressed be can states Fock on × =0 ˆ a exp exp † n b. = = ˆ exp U U †  exp cos g g PDC† PDC† U U  sin n k  h η η BS† BS† n h k  θ ˆ a r  θ 1/2  θ ( ( √ † ˆ U a U b ˆ 1/2 ˆ a ˆ a − ˆ U b U a ˆ √ − b ˆ + † † n! ≥ (cos b ˆ a tan ˆ b ˆ n! b n ˆ g PDC (sin g PDC b ˆ η BS b η ˆ BS † † )and 1) − † htn mig noeo h input the of one on impinge photons † − = sin = θ θ asv n cieGusa unitaries Gaussian active and Passive = cos ˆ a tan = θ )  ˆ a b ˆ ) k ˆ a − ˆ a k ˆ a b) ˆ † θ (−  (cos θ † ˆ cos a ) ) θ cosh i r i n ) cos  n sinh , , sin sin stesueigprmtr The parameter. squeezing the is U η . 1 θ U θ η BS stetasitne(0 transmittance the is θ g η ) + θ θ η BS r n−k r )  ,0i |0, + = = + n−k b ,0i |0, ˆ + ˆ a † b cosh ˆ cos b ˆ sin ˆ a− b ˆ |k † cos |k cosh , 2 ˆ sinh b θ n † 2 , θ , ˆ b n θ , r − , , − r r k , , i k . i ˆ a and ≤ b ˆ [15] [16] [14] and [10] [11] [12] [13] η ≤ rbblt mltd fEq. of amplitude probability mode into impinging applying by obtained is state vacuum U squeezed two-mode the example, For h orsodn prtoa ceeis scheme operational corresponding the 6B, Fig. In basis. Fock the in tion states any for an to PTR, as under expressed dual, is is which modes transformation, bosonic Bogoliubov two (active) of coupling linear (passive) the Eqs. decompositions, sponding unitaries the comparing by mode with ciated relation mode fundamental of the state the of Eq. transposition by expressed As tons. Duality. PTR of states Proof in initially are modes idler and (Eq. signal respectively the when PDC mode a input the |Ψi if state 5B, entangled the Fig. use in shown As 21 cos and gain of PDC a for amplitude probability the have we Hence, mode of state mode output conditional on vacuum the on with BS a port of example additional the sidering PTR. of Example of emission stimulated the with to pairs, again corresponds term each where of emission the where where |φ case special The g PDC n and i a 0 = ˆ a notevcu tt,namely state, vacuum the onto ,b 00 T and sin 22 θ b ∝ = tnsfrtasoiini h okbsso h ibr pc asso- space Hilbert the of basis Fock the in transposition for stands n r uludrPR namely PTR, under dual are θ m P htr ntergthn iecrepnst h stimulated the to corresponds side right-hand the in term nth n (cosh n hφ hn yrvrigtearwo ieo mode on time of arrow the reversing By |ni. xr htn rvln nmode in traveling photons extra htn mign niptport input on impinging photons n=0 ∞ hn htnpis ntemr eea aewhere case general more the In pairs. photon hn ψ U a + , a g PDC eilsrt h T ult ewe SadPCb con- by PDC and BS a between duality PTR the illustrate We 19). m + + U φ , r m, |φ g |φ ) hsi kthdi i.6A Fig. in sketched is This b. g ˆ PDC b ψ = m, −1 U 0| m, | (Eq. = n n b n| g U PDC , i scniee ntetx Fg )adcrepnsto corresponds and 2) (Fig. text the in considered is 0 , hc seuvln to equivalent is which , h T ult silsrtdi al o e pho- few for 1 Table in illustrated is duality PTR The ˆ a 0i |m, n| × hc speieypootoa oteotu of output the to proportional precisely is which ni, exp g PDC = φ U wt aumi h te nu oe,w have we mode), input other the in vacuum (with .Nw fw aetesbttto sin substitution the make we if Now, 19). ,0i |0, exp a U U g PDC  and , |Ψi |ψ η BS g PDC  n b ˆ tcnb iwda osqec fpartial of consequence a as viewed be can it 7,  ˆ a 0 a 0i |m, = + |m,  U U m † 21 , b, 0i |m, tteotu fteB,te(unnormalized) the BS, the of output the at = − g η PDC BS b ˆ ψ m (cosh 14 † φ 00 ˆ b a ihtepoaiiyamplitude probability the with ni cosh  and i tanh b b  ˆ ∝ × 1/2 ˆ a where , and T = = = 1 tanh b 0 = b ˆ 1 P tanh r = is √  √ U r ) sin laigmode (leaving 1 r m+1  1 ngnrltrs emysythat say may we terms, general In 17. n n=0 ∞ g PDC  g X n=0 g √ n ∞ r n + 1 m   n θ g +

m hn . r m * |n, φ nEqs. in X n=0 tanh cosh cos U ∞ m b ˆ a |n +  eoe h ope conjuga- complex the denotes m 1/g BS , 1  ,w e h uptstate output the get we ni), g 1/2 ψ m NSLts Articles Latest PNAS ˆ setnld(hti,i we if is, (that entangled is a. + 0| m,  1/2 htn mign ninput on impinging photons = n n a lob interpreted be also can and * b θ r , b ˆ n

m, r  sin 1/η + |n 10 U .I econdition we If 5A). (Fig. m (cosh 1+ |n, 1/g BS ni U n m tanh + ˆ a ˆ a θ and 1/g BS ecnr htEqs. that confirm we , , † ni  mi ˆ nhne) namely unchanged), a+

cos 1/2 , r ψ ) |m, ecmaethe compare we b, n ˆ . ˆ b m+1 a 12 m † r , ˆ b φ θ. ni m * rtercorre- their or b

. |mi htn are photons θ n | = and photon f10 of 5 tanh [17] [18] [19] [20] [22] [25] [23] [21] [24] |0i, r

PHYSICS ˆ ˆaPTR = ˆa W cosh r + bPTR sinh r, [W, ˆa] = 0. [29] A ˆ We can perform a similar development starting from bPTR, resulting in

ˆ ˆ ˆ bPTR = b W cosh r − ˆaPTR sinh r, [W, b] = 0. [30]

ˆ Now, solving Eqs. 29 and 30 for ˆaPTR and bPTR, we get

ˆ ˆaPTR = ˆa W cos θ + b W sin θ, [31] ˆ ˆ bPTR = b W cos θ − ˆa W sin θ,

where we have made the substitutions cosh r = (cos θ)−1 and sinh r = tan θ.

Similar equations can be derived starting from V Ta ˆa† V Tb and V Ta bˆ† V Tb , which imply that the operator W also commutes with ˆa† and bˆ†. Hence, W is a scalar (proportional to 1), and it is sufficient to compute its diagonal matrix element in an arbitrary state (e.g., |0, 0i). Recalling that V † = V T , we have W = (V †)Tb V Tb , so that

∞ X † h0, 0| W |0, 0i = h0, 0| (V )Tb |k, li hk, l| V Tb |0, 0i k,l=0 B ∞ X † = h0, l| V |k, 0i hk, 0| V |0, li k,l=0 = |h0, 0| V |0, 0i|2 = 1/g. [32]

Substituting W with 1/g in Eq. 31, we get Eq. 26, which concludes the proof of Eq. 24. Note that PTR duality can be reexpressed by using the identity Fig. 5. (A) PTR duality in a more general case with m and n photons h    i impinging on modes ˆa and bˆ of a BS. By conditioning the output mode ˆ ˆ † ˆ ˆ Tr Uab Xa ⊗ Xb Uab Ya ⊗ Yb bˆ0 on vacuum and reversing the arrow of time, we get the same transi- [33] h T   T   i = b ˆ ⊗ ˆ T b † ˆ ⊗ ˆT tion probability amplitude (up to a constant) as for a PDC of gain g = 1/η Tr Uab Xa Yb Uab Ya Xb , with input state |m,0i and output state |n + m, ni.(B) Corresponding oper- ˆ ational scheme using an entangled (EPR) state at the input of mode b. This ˆ ˆ 00 which is valid for any joint unitary Uab and for any operators Xa(b) and Ya(b) makes it possible to access the output retrodicted state on mode bˆ . PDC acting on mode a (b). Plugging Uab = Ug into Eq. 33 and using Eq. 24 implies the general relation

depicted, relying on the preparation of an entangled (EPR) state at the input h PDC ˆ ˆ  PDC† ˆ ˆ i Tr U Xa ⊗ X U Ya ⊗ Y of mode bˆ and the projection onto an entangled (EPR) state at the output g b g b ˆ0 [34] of mode b . 1 h BS ˆ ˆ T  BS† ˆ ˆT i = Tr U Xa ⊗ Y U Ya ⊗ X . We now prove PTR duality by reexpressing Eq. 24 in the Heisenberg g 1/g b 1/g b picture, namely V Ta ˆa V Tb = (ˆa cos θ + bˆ sin θ)/g, This equation is needed to interpret PTR in the context of the retrodictive [26] picture of . V Ta bˆ V Tb = (bˆ cos θ − ˆa sin θ)/g, Retrodictive Picture of Quantum Mechanics. In the usual, predictive approach PDC ˆ ˆ where V ≡ Ug . Note that a and b are real operators in Fock basis; of quantum mechanics, one deals with the preparation of a quantum system hence, the same is true for V. Therefore, we have ˆa† = aT , bˆ† = bT , and followed by its time evolution and ultimately, its measurement. Specifically, V † = V T , where T stands for transposition in Fock basis. Obviously, we have (VTb )T = V Ta . By using the identity ((A ⊗ B)M)Tb = (A ⊗ 1)MTb (1 ⊗ BT ), 1 Ta T where represents the identity operator, we express ˆaPTR ≡ V ˆa V b as Table 1. Illustration of PTR duality with few photons BS PDC V Ta (ˆa V)Tb = V Ta (VV †ˆa V)Tb BS PDC √ † h0, 0| U |0, 0i = 1 h0, 0| U |0, 0i = 1/ g = Ta [ ˆ + ˆ ]Tb η g V V(a cosh r b sinh r) BS √ PDC h1, 0| Uη |1, 0i = η h1, 0| Ug |1, 0i = 1/g = V Ta V Tb ˆa cosh r + V Ta bˆ V Tb sinh r. [27] BS √ PDC h0, 1| Uη |0, 1i = η h0, 1| Ug |0, 1i = 1/g BS p PDC p h0, 1| Uη |1, 0i = − 1 − η h0, 0| Ug |1, 1i = − g − 1/g Equivalently, using ((A ⊗ B)M)Ta = (1 ⊗ B)MTa (AT ⊗ 1), we may also reex- BS p PDC p h1, 0| Uη |0, 1i = 1 − η h1, 1| Ug |0, 0i = g − 1/g press ˆaPTR as The second column (PDC) is obtained from the first column (BS) by time- √ † † † reversing mode bˆ, substituting η with 1/g and dividing by the factor g. (ˆa V)Ta V Tb = (VV ˆa V)Ta V Tb The first row explains the latter factor: vacuum is obviously conserved in a = [V(ˆa† cosh r + bˆ sinh r)]Ta V Tb BS, while PDC implies the stimulated emission of photon pairs (hence, the probability of keeping vacuum is strictly lower than one). The second and = ˆa V Ta V Tb cosh r + V Ta bˆ V Tb sinh r. [28] third rows correspond to the transmission of a single photon through the BS or PDC. The fourth and fifth rows correspond to the reflection of a single ˆ Ta ˆ T Ta T Defining the operators bPTR ≡ V b V b , W ≡ V V b , and identifying photon by the BS or the stimulated annihilation (fourth row) or emission Eq. 27 with Eq. 28, we see that (fifth row) of a photon pair by PDC.

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Θ and ˆ h nu state input The i. a b ˆ i j U 25. 0 = ercvraPCwt input with PDC a recover We . † fmode of ρ ˆ b ˆ j p olwdb measurement a by followed faB,while BS, a of peae ihprobability with (prepared j σ ˆ U ρ ˆ m P(m|j j eoebigmeasured, being before , ≥ ,Tr 0, ˆ a ) ipypropagates simply = { σ ˆ Tr[U Θ ˆ m j |ψ hs out- whose }, n = ρ ˆ b and )ado a of and 1) j i U fmode of † σ ˆ |ψ when 7, |φ Π m ˆ m m a b m , pho- with i In ]. [35] was ψ on b i where may we measurement, and state proper a Eq. as reexpress behave to seen easily are which codn to according defining where aeeprmn hr state where experiment same 7. 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Left ⊗ and , and ) n [40] [39] [38] [36] [37] |m) ρ σ ˆ ˆ of j b m b ), a ,

PHYSICS words, in Eq. 40, the predictive picture is used for subsystem a, while the retrodictive picture is used for subsystem b. PDC In our analysis of a BS under PTR, we have Uab = Ug and Vab = BS √ U1/g/ g, so that Eq. 33 reduces to Eq. 34. Hence, Vab is unitary (up to a constant) and can be interpreted as the propagation of the retrodicted state of mode bˆ backward in time through the BS, while the predictive state of mode ˆa normally propagates forward in time through the BS. According to Eq. 40, the joint state is then shown to evolve according to a PDC. Note that it is not always possible to construct an operator Vab that is proportional to a unitary operator, as it is the case here.

Two-Photon Interference in a BS and PDC. The HOM effect can be sim- ply understood by calculating the probability amplitude for coincident detection BS ˆ BS †ˆ† c ≡ h1, 1| Uη |1, 1i = h0, 0| ˆab Uη ˆa b |0, 0i [41] at the output of a BS. By using Eq. 11, it is simple to rewrite it as c = cos 2θ, Fig. 8. Extended quantum interferometric suppression in an amplifier which yields the coincidence probability where the detection of n photon pairs at the output is suppressed if the gain g = n + 1. This is the PTR dual of the extended HOM effect when a sin- 2 2 Pcoinc(η) = |c| = (2η − 1) . [42] gle photon and n photons impinge on the two input ports of an unbalanced BS of transmittance η = 1/(n + 1). The interference at play in the extended If the transmittance η = 1/2, no coincident detections can be observed as a HOM is again the cancellation between a double reflection and double result of destructive interference. transmission (with n − 1 other photons being always reflected from mode bˆ Now, we examine the corresponding quantum interferometric suppres- to mode ˆa). Applying PTR, this translates into the cancellation between the sion in a PDC and its dependence in the parametric gain g. Let us calculate amplitude for the two photons crossing the crystal and the amplitude for the probability amplitude for coincident detection the stimulated annihilation combined with stimulated emission of the two photons (accompanied, in both cases, with the stimulated emission of n − 1 0 PDC † †ˆ †ˆ† c ≡ h1, 1| Ug |1, 1i = h0, 0| VV ˆaVV bVˆa b |0, 0i , [43] pairs). In other words, the input pair may be up converted into the pump beam while n pump photons are down converted, or the input pair may sim- PDC ply be transmitted while n − 1 pump photons are down converted. Since the where V is a short-hand notation for Ug . Thus, we get the scalar product two scenarios are time-like indistinguishable, they interfere destructively. between V † |0, 0i, which is the state of Eq. 18 up to changing the sign of r, and the ket V †ˆaVV †bVˆ ˆa†bˆ† |0, 0i, which can be reexpressed as

cosh2 r |0, 0i + 3 cosh r sinh r |1, 1i + 2 sinh2 r |2, 2i [44] n − 1 photons being reflected). This is sketched in Fig. 8, Right. More gen- erally, the transition probability 1, n → n, 1 for a BS of transmittance η is by use of Eq. 13. This gives c0 = (1 − sinh2 r)/ cosh3 r, so that the probability given by 2 for coincidence is written as BS n−1 2 hn, 1| Uη |1, ni = (1 − η) [(n + 1)η − 1] , [49] 0 0 2 2 3 which is consistent with Eq. 47 under PTR, namely Pcoinc(g) = |c | = (2 − g) /g . [45]

2 1 2 If the gain g = 2, the probability for coincident detections fully vanishes. h | PDC | i = h | BS | i . n, n Ug 1, 1 n, 1 U1/g 1, n [50] More generally, the joint state at the output of a PDC of arbitrary gain g g when the input state is |1, 1i reads Experimental Scheme. The HOM effect is considered a most spectacular evi- ∞ n−1 X (sinh r)   dence of genuinely quantum two-boson interference, and we expect the UPDC |1, 1i = n − sinh2 r |n, ni. [46] g (cosh r)n+2 same for its PTR counterpart as it admits no classical interpretation. The n=0 experimental verification of our effect can be envisioned with present tech- √ nologies, as sketched in Fig. 9. We would need two single-photon sources, A parametric gain g = 2 corresponds to cosh r = 2 and sinh r = 1, so we which could be heralded by the detection of a trigger photon at the out- recover Eq. 8. put of a PDC with low gain (the single photon being prepared conditionally on the detection of the trigger photon in the twin beam). The two single Extension to a PDC with Integer Gain. We may also consider the case where photons would impinge on a PDC of gain 2, whose output modes should the gain takes a larger integer value (e.g., g = 3, 4, ··· ). A closer look at be monitored: the probability of detecting exactly one photon on each Eq. 46 reveals that the output term with g − 1 photon pairs fully vanishes mode should be suppressed as a consequence of time-like indistinguisha- when g is an integer. The corresponding distribution of the output photon bility. In principle, photon number resolution would be needed in order pair number to discriminate the output term with one photon pair (n = 1) from the −1 terms with more pairs (n ≥ 2). The ability of counting photons has become 2 (g − 1)n hn, n| UPDC |1, 1i = (n + 1 − g)2 [47] increasingly available over the last years (e.g., exploiting superconducting g n+2 g detectors), but this could also be achieved by splitting each of the two output modes into several modes followed by an array of on/off photode- is displayed in Fig. 3B for g = 3 and 4. This interferometric suppression tectors. Experiments involving PDC in three coherently pumped crystals have hn, n| UPDC |1, 1i = 0, ∀n, can be interpreted as dual, under PTR, to the n+1 already been achieved recently (36, 37), aiming at observing induced deco- h | BS | i = extended HOM effect n,1 U1/(n+1) 1, n 0 for a BS of transmittance herence, so the proposed setup here should be implementable along the 1/(n + 1) (Fig. 8, Left). The latter effect is easy to understand as the inter- same lines. The squeezing needed to reach a gain 2 amounts to 7.66 dB, ference between the amplitude with all n + 1 photons being reflected and which is high but lies in the range of experimentally accessible values (in the amplitude with one photon of each mode being transmitted (the other the continuous-wave regime). The experiment could alternatively be car- ˆ† ˆ† n n − 1 being reflected). Indeed, the operator a (b ) transforms into ried out with a lower gain (especially in the pulsed regime) provided the observed dip is sufficient to rule out a classical interpretation. As a mat- 1  √ √ n 0† − 0† 0† + 0† . ter of fact, a coincidence probability lower than 1/4 would be needed (see n+1 a n b n a b [48] (n + 1) 2 Classical Baseline), which could in principle be reached with a gain of 1.28 (i.e., a squeezing of 4.39 dB). Of course, the effect of losses should also be The term proportional to (a0†)n b0† vanishes as a result of the cancellation carefully analyzed in order to assess the feasibility of the scheme depicted of the term where the photon on mode ˆa is reflected and the n photons on in Fig. 9. mode bˆ are reflected, together with the term where the photon on mode ˆa Demonstrating this effect would be invaluable in view of the fact is transmitted and one of the photons on mode bˆ is transmitted (the other that the HOM dip is widely used to test the indistinguishability of single

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9–9 (2002). 594–597 419, ~ ~ 6 (1925). 765 31, hs e.Lett. Rev. Phys. Physics 9–0 (1964). 195–200 1, 2044–2046 59, e.Mod. Rev. hs Rev. Phys. 4 .Smn ee-ooek eaaiiycieinfrcniuu aibesystems. variable continuous canonical for and criterion separability reversal Peres-Horodecki Simon, time R. separability, Quantum 24. Vidal, G. Tarrach, R. theory. quantum Sanpera, in A. reversal time 23. of formulation Operational Cerf, J. of N. process Oreshkov, quantum O. the linear in 22. symmetry Time with Lebowitz, L. circuits J. Bergman, Gaussian G. P. arbitrary Aharonov, Y. Simulating 21. Cerf, J. N. Chakhmakhchyan, L. 20. Lopes R. 19. Agnesi C. 18. Silverstone W. J. 17. Tsujimoto Y. 13. 5 .Pr,C ak .Le .H h,Y ak iervrigamncrmtcsubwave- monochromatic a Time-reversing Park, Y. Cho, H. Y. Lee, K. Park, C. Park, J. 25. Zeilinger, A. Halder Aspelmeyer, M. 16. M. Zukowski, M. Beugnon J. Blauensteiner, 15. B. Kaltenbaek, R. 14. azlia ela naoyosrfrefrueu omns M.G.J. comments. PDR grant useful under FNRS - for Scientifique Recherche S referee la T.0224.18. de and Fonds was anonymous the work by This Schleich, an supported Foundation. Wiener-Anspach P. the as from support Wolfgang acknowledges well Pfister, as Olivier Tanzilli M Oreshkov, Dmitri Klaus Guha, Laurat, Saikat Ognyan Julien Filip, Kolobov, Radim I. Fan, Mikhail Linran Horoshko, D’Auria, Virginia Fabre, Claude ACKNOWLEDGMENTS. coincidence the reduce Availability. Data to sufficient is paths one 1/4. two the to probability (minus) the than between larger inter- destructive correspond- is ference partial the amplitude crystal but events, the probability double-stimulated through to the corresponding observing transmission gain, for the sufficient this to be With ing would effect. gain quantum lower this a with effect, achieved paths quantum two this Solving demonstrating requirements. for experimental obtain gain utilized the a be that lowering suggests may thus this two Interestingly, exclude than dip. to order the lower in of 25% interpretation than less classical probability a coincidence a have to need we with compared be to input two the where in path Similarly, the quantum. (probability distinguish is transmitted can we dip are PDC, photons the of that model that classical ensures a 50% below depletion a coincident for probability classical is the detections Then, distinguished. two be the can paths where two double-transmission effect the HOM add the to have probability of We As model distinguishable. classically. are classical photons interpreted a the input be first of consider can depletion guide, that the necessary a determine detections is to coincident it namely effect, of baseline, this probability classical of particles). verification a classical experimental establish for an to add assess would to probability order probabilities resulting probabil- In two the individual the hence, equal (whereas amplitudes; with vanishes two probability one) have opposite another but We by the ities through probability. a replaced going being where some either or pair PDC, with crystal photon of created (the model paths or indistinguishable classical annihilated possible a be within can interpreted pair be cannot fication Baseline. Classical o onietdtcin is detections coincident (probability for pair another by replaced with compared be to eopstos ri:un-h9001(1Jl 1997). July (21 arXiv:quant-ph/9707041 decompositions. Phys. Nat. measurement. optics. lasers. integrated waveguide sources. eghotclfcsb pia hs ojgto fmlil-ctee light. multiply-scattered (2017). of 41384 conjugation 7, phase optical by focus optical length Lett. Rev. sources. fiber narrowband atoms. trapped independently photons. independent of interference Experimental measurement. coincidence g hs e.A Rev. Phys. = a.Photon. Nat. tmcHn-uMne experiment. Hong-Ou-Mandel Atomic al., et 8 mligta h ata nitnusaiiybtenthe between indistinguishability partial the that implying 1.28, 76(2000). 2726 84, ,Hn-uMne nefrnebtenidpnetiivo silicon on iii–v independent between interference Hong-Ou-Mandel al., et |A 5–5 (2015). 853–858 11, ocascl2poo nefrnewt eaaeintrinsically separate with interference 2-photon Nonclassical al., et unu nefrnebtentosnl htn mte by emitted photons single two between interference Quantum al., et ihvsblt ogO–adlitreec i time-resolved a via interference Hong–Ou–Mandel visibility High al., et dt hs Rev. Phys. | 2 nci unu nefrnebtenslcnphoton-pair silicon between interference quantum On-chip al., et hr r odt neligti work. this underlying data no are There ihtedul-eeto probability double-reflection the with h w-htnqatmitreec feti ampli- in effect interference quantum two-photon The P P 634(2018). 062314 98, cl cl 0 (η P 0–0 (2014). 104–108 8, P (g) coinc coinc 0 ) etakUrkL nesn ai .Chekhova, V. Maria Andersen, L. Ulrik thank We = 11 (1964). B1410 134, = p.Express Opt. p.Express Opt. fEq. of (g) (η |A |A p.Lett. Opt. fEq. of ) dt Nature dt 0 | | 2 2 + + |A |A 7–8 (2006). 779–782 440, o an2PDC, 2 gain a For 9. o 05 BS, 50:50 a For 3. 7 (2019). 271 44, 6047 (2009). 4670–4676 17, 26–28 (2017). 12069–12080 25, dr |A st 0 |A | | 2 2 dt 0 st 0 = = | | 2 2 rmtept hr hyare they where path the from ) η .Tu,tecasclprobability classical the Thus, ). 1 2 + NSLts Articles Latest PNAS + hs e.Lett. Rev. Phys. (g g (1 Nature 3 − − le,Rmi Mueller, Romain olmer, ¨ 1) η P 2 ) cl 66 (2015). 66–68 520, 2 P (1/2) P cl 0 |A coinc 0 (2) dr 452(2006). 240502 96, = = (g) | 2 ;hence, 1/2; ;hence, 1/4; ic these since = | ebastien ,we 1/4, ´ c.Rep. Sci. f10 of 9 Phys. [52] [51]

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