Downloaded by guest on September 26, 2021 oewih,reads weight, mode www.pnas.org/cgi/doi/10.1073/pnas.1904839116 square modulus the pattern by given is observed The density wavevector. charge diffraction the provide on information diffraction fea- useful time. X-ray in spatial recorded off-resonant be The can Far-field distributions dynamical reactions. charge small-scale monitor chemical of and tures and equilibrium processes of out relaxation quan- driven of new systems properties offer spatio-spectral tum These study to experimentally. possibilities accessible (subnanome- exciting length become and scales (femtosecond) ter) time short sources, table- high-harmonics top and lasers free- bright With world. R imaging phase-sensitive imaging by enhancement contributions. image modes for used Schmidt be the can reweighting field the of state decomposi- the Schmidt of A tion samples. delicate signal-to-noise to damage high avoiding providing ratio, possible, fields weak with imaging with compared intensity, a Asban Shahaf entangled using imaging and diffraction phase-sensitive Quantum sn cmd eopsto ftetopoo amplitude two- the of (q decomposition Φ Schmidt vacuum using to due achieved detector. where the been diffraction of informa- linear fluctuation has for phase 13 detection uncovers ref. in also photon. heterodyne-like shown and signal recently was image the as the of tion, and reveals arrival sample. scanned idler the a spatially with The is coincidence with “idler,” in interacts the measured “signal,” only counterpart, the which entangled in as Its photons denoted entangled photon, of object one scattering studied off-resonant the on of square modulus the |σ circumvent correlated reveal using mesoscopic to They analogs of classical shown . image have real-space been techniques the Such have objects. producing regime by problem visible techniques this the beam in Correlated such (3). 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Fig. in shown setup the consider we paper this In (r) aervltoie u blt oosretemicroscopic the sources observe to light ability ultrafast our short-wavelength revolutionized have in advances apid s | , 2 q 1,12). (11, i = ) S ( p P ) σ [ c ρ ¯ n ∞ (Q) tt e aoaoyo rcso pcrsoy atCiaNra nvriy hnhi206,China 200062, Shanghai University, Normal China East Spectroscopy, Precision of Laboratory Key State a,b,1 | i ] nage photons entangled √ ∝ osatnE Dorfman E. Konstantin , sntaalbe h neso eursphase requires inversion the available, not is λ Re n σ u n X (r) nm (q ∝ s eursaFuirtasom Since transform. 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Reviewers: and research; K.E.D., performed S.A., S.A. and research; data; designed analyzed S.M. S.A. and K.E.D., S.A., contributions: Author Here u eovdsga,tepaedpnetiaei oie to modified is image phase-dependent the S and signal, pulse resolved actinic in an order by prepared plane. detection owo orsodnemyb drse.Eal mkmluieu sasban@ [email protected], Email: addressed. be [email protected] may or uci.edu, correspondence whom To m ∗ o mg nacmn yrwihigteShitmodes Schmidt the reweighting by contributions. delicate enhancement image to for damage avoiding (iii samples. possible, imaging weak-field with where compared photons, sity, interacting (ii the problem.” with the “phase of the applica- state avoids experimental the several ( discuss in tions: We show imprinted Scan- detectable. we information matter, is twin. with field phase its interact not with the off did proposed, coincidence that diffracted that in photon is is the detected pair ning entangled and technique an sample of imaging a photon one diffraction whereby quantum A Significance [ I u eodmi euttclstesailrslto enhance- resolution spatial the tackles result main second Our ρ ¯ a,b,1 p (r) 3/2 i ]∝ and , lsia ifato ncnrs eurstointeractions two requires contrast in diffraction Classical . Re β i bann elsaeiae ndfrcinimaging diffraction in images real-space Obtaining ) I nm (1) σ p 1/2 P (r) ρ cmd eopsto ftefil a eused be can field the of decomposition Schmidt A ) = ¯ PNAS nm ∞ where , i o ag ifato nlsadfrequency- and angles diffraction large For . NSlicense.y PNAS R satodmninlvco ntetransverse the in vector two-dimensional a is γ σ d nm r (r) | β u √ ue1,2019 11, June n nm (1) stecag est ftetre object target the of density charge the is ∝ I (r)σ λ p n I ouae yteFuirdecomposi- Fourier the by modulated p λ steitniyo h ore hsis This source. the of intensity the is S γ m (r)u fcascldfrcin hsmakes This diffraction. classical of nm (1) v . n ∗ y m ∝ ∗ nEq. in ( spaedpnet otatto contrast dependent, phase is ρ (r), ¯ h mg clsas scales image The ) | I i p 2 )v o.116 vol. hc loapisfor applies also which , m p www.pnas.org/lookup/suppl/doi:10. 1 ( I β ρ 1 2) (1, = p ¯ a ocasclanalog; classical no has nm (2) i stesuc inten- source the is ) | where , = o 24 no. 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PHYSICS the field amplitude Φ into a product of single-photon amplitude; all of the interesting quantum optical effects discussed below are derivatives of this feature. Φ can be represented as a super- position of separable states using the Schmidt decomposition (19–21)

∞ X √ Φ(qs , qi ) = λn un (qs )vn (qi ), [5] n

where the Schmidt modes un (qs ) and vn (qi ) are the eigenvec- tors of the signal and the idler reduced density matrices, and P the eigenvalues λn satisfy the normalization n λn = 1 (20). The number of relevant modes serves as an indicator for the degree of inseparability of the amplitude, i.e., photon entangle- ment. Common measures for entanglement include the entropy P −1 P 2 Sent = − n λn log2 λn or the Schmidt number κ ≡ n λn . The latter is also known as the inverse participation ratio as it quantifies the number of important Schmidt modes or the effec- Fig. 1. Sketch of the proposed setup. A broadband (2) tive joint size of the two photons. In a maximally pump pulse with the wavevector kp propagates through a χ crystal, gen- erating an entangled photon pair denoted as signal and idler. The photons entangled wavefunction, all modes contribute equally. are distinguished either by (type II) or by frequency (type I) and The spatial profile of the photons in the transverse plane are separated by a (BS). The signal photon interacts with the (perpendicular to the propagation direction) can be expanded sample and can be further frequency dispersed and collected by a “bucket” and measured using a variety of basis functions; e.g., Laguerre– detector Ds with no spatial resolution. The idler is spatially resolved in Gauss (LG) or Hermite–Gauss (HG) has been demonstrated the transverse plane by the detector Di. The two photons are detected in experimentally (22, 23). These sets satisfy orthonormality coincidence (Eq. 12). R 2 P 0 d q un (q)vk (q)= δnk and closure relations n un (q)vn (q )= (2) 0 δ (q − q ). The deviation of λn from a uniform (flat) dis- tribution reflects the degree of entanglement. Perfect quan- Thanks to favorable scaling, weak fields can be used to study tum correlations correspond to maximal entanglement entropy fragile samples to avoid damage. and thus a flat distribution of modes. This is further clarified by the closure relations, which demonstrate the convergence Spatial Entanglement into a point-to-point mapping in the limit of perfect trans- Various sources of entangled photons are available, from quan- verse entanglement. The biphoton amplitude exhibits two lim- tum dots (14) to cold atomic gases (15) and nonlinear crystals, iting cases for the infinite inverse participation ratio which are and are reviewed in ref. 4. A general two-photon state can be demonstrated in Fig. 2. When the sinc function in Eq. 3 is written in the form

X (µs ) (µi ) † † |ψi = Φ(ks , ki )  a a |0s , 0ii , [2] ks ki ks ,µs ki ,µi ks ,ki

(ν)  †  where k is polarization, ak,ν ak,ν are field annihilation (cre- ation) operators, and Φ(ks , ki ) is two-photon amplitude. In the paraxial approximation the transverse momentum {qs , qi } and the longitudinal degrees of freedom are factorized. The trans- verse amplitude of the photon pair generated using a parametric down converter takes then the form (4, 16–18)

2 2 Φ(qs , qi ) = Γ (qs + qi )sinc L (qs − qi ) , [3]

and here Γ(q) is the pump envelope of the transverse compo- 2 lz λp nents, L = 4π , where λp is the central wavelength and lz is the length of the nonlinear crystal along the longitudinal direction. The state of field is then given by

X |ψi = |vaci + C Ap (ωs + ωi )Φ(qs , qi )

qs , qi ωs , ωi

× |qs , ωs ; qi , ωi i, [4] Fig. 2. Transverse beam amplitude profile for different Schmidt num- where C is a normalization prefactor and Ap is the pump bers. For κ3 = 1 the amplitude in Eq. 5 is separable and the photons are envelope. not entangled. As κ is increased the amplitude approaches a narrow dis- tribution. κ1 = 2,500 and κ2 = 25.5 are obtained in the σpL > 1 regime, (∞) Schmidt Decomposition of Entangled Two-Photon States. The hall- and the amplitude approaches Φ ∝ δ (qs + qi). κ4 = 25.5 and κ5 = 2,500 (∞) mark of entangled photon pairs is that they cannot be considered are taken in the σpL < 1 regime, with the asymptotic amplitude Φ ∝ as two separate entities. This is expressed by the inseparability of δ (qs − qi).

11674 | www.pnas.org/cgi/doi/10.1073/pnas.1904839116 Asban et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 o,teilrrdcddniymti sn ogrdiagonal. longer no is matrix basis. density Schmidt the reduced free- of in idler degree subspace matter the external idler an dom, with the interacts signal in the When diagonal is which the in right change a the subspace to field or corresponds the left matrix in the density space from joint interaction the single of a field first the order, To of interactions. nontrivial powers light–matter can of in number one matrix to field, density correspond which weak the a of For evolution the freedom. expand of degrees field’s the resents where by given is field electric commutator The the represents space Hilbert sa tal. et Asban by given is beam den- of reduced matrix initial state the sity initial , as the from such When structure arising internal correlations phase. nontrivial a its the contains in with field interaction the change single a a merely to due matter, exchange photon no records that such separable, is function For pump. where a in given is number Schmidt (24), form the closed Gaussian, a by approximated mltd a w iiigcss when cases, limiting by two denote has to We amplitude conjugate 2. coordinate, plane Fig. transverse in is depicted amplitude as corresponding infi- or the vanishing either and of cases extreme product the In nite 2. as photons Fig. two in the indi- shown between modes correlations Schmidt quantum relevant stronger of cates number high A entanglement). ihtevco field vector the with is coupling radiation/matter resonance where case limiting detec- opposite idler’s the The into directly tor. photon signal the by explored plane Φ Fg ) h on ih–atrdniymti nteinteraction the in by matrix given density is light–matter picture joint quan- The of 1). scheme role (Fig. detection the proposed reveals the idler in correlations the tum of matrix Basis density Schmidt reduced the The in Matrix Density Idler Reduced The sign. corresponding the nota- with plane abbreviated detector the use We whereby image. tion mirror the in photon signal results the which by monitored plane sample the maps tude  ( ∞) E k (+) (ρ Φ (ρ µ T σ (r, s p 2 , tnsfrtemte’ ere ffedmwhile freedom of degrees matter’s the for stands ρ s ersnssprprtrtm reigadteoff- the and ordering time superoperator represents t stevrac ftetases oetmo the of momentum transverse the of variance the is , = ) i ρ σ ρ Φ = ) ρ φ p σ i φ i ρ Φ = ) i ¯ = p ρ =  i 0 = (0) l µφ int E l tr eoe h apn rmtesml othe to sample the from mapping the denotes 0 h htn r aial entangled maximally are photons the k ( 1 = δ (t µφ −)  ( (ρ ∞) A = ) κ s (r, eget we , n ρ X s (r, = ,i (ρ µ,φ Φ − t ,i T 4 1 ) ( t s 0  ρ κ=1) e si = ) , λ  † −i ρ i n ) σ bandb rcn vrtesignal the over tracing by obtained = i hsapiuemp h image the maps amplitude This . v E p Φ = ) R n − ≡ ∗ κ r L (r, d σ ρ (k 1 = τ + Φ p E( φ ˙ 2π H L i t c (1) r,t = )v 0 V I σ )= ∞ → ,− ~ω δ ) p 1 n h w-htnwave- two-photon the and tr n k (q nthe on (−) subscript The . (ρ L ( P µ Φ k τ s H  k ρ ) s , ρ X ρ ( i 0 2 k µφ q ∞) I + ν µ  s sgvnb h ampli- the by given is , i σ E /i = |1 q (q Φ = ) ⊗ h aito field radiation The . ρ p k (+) (q  s i l R i /i k ( ih1 ρ stereal-space the as ν ) s → φ hsamplitude This . h real-space The . ) d , (r, a , q i rσ k,ν (ρ Φ 0 0 i |, t ) s )+ (r, e ∝ O ( Φ ) i k·r−i − ρ δ t E s )A (q ≡ k ( s κ q −ρ → −) , s φ ω i 2 ∞, → [O ρ ) k ± (r, (r, t rep- i (no , = ) [8] [9] [6] [7] , q ·] t t i i . ) ) ) xlcty ntelmto ifato osalage h idler’s the 1 section angles Appendix, small (SI by to given diffraction is matrix of density limit reduced the in Explicitly, ilpaefco hrceitct a-eddfrcin Using diffraction. far-field to the characteristic from in Eq. calculated Eq. factor one image in phase the matrix coincidence to tial density The expression reduced similar not. a the is parax- yields be scattering field field to far arbitrary scattered understood with is the field diffraction ial, incoming far-field the While to directions. turn next We matrix Diffraction Far-Field density reduced idler’s the Eq. from between in given calculated object the value Each of Eq. tion dataset. projection 1D modes. a a two each to in for resulting corresponds one index, coherence corresponding indexes, 3 the two Fig. over In by have image. the We labeled of visu- object. dimension is this spatial for an mode 4 with Fig. Each in (signal) alization. depicted twin basis, Hermite–Gauss its the of due chosen photon) interaction noninteracting the (the idler to of matrix density of reduced coherence matter. induced with interaction the its probe Schmidt to to due chosen field one the the allows on setup quantities Our matter basis. of projections the are eue est arxo h de u oteitrcinwt h object the with interaction the to due Eq. idler by pre- the given of object Eq. matrix the by density with given reduced modes, interaction basis the Hermite–Gauss before in sented the matrix to density corresponding reduced size” idler’s “spot The (B) object. jected 3. Fig. where C 0.05 0.1 i.3dsly h nue cmd-pc oeec fthe of coherence Schmidt-space induced the displays 3 Fig. A 0 4 ρ o h eu ecie nFg ,tecicdneimage coincidence the 1, Fig. in described setup the for P h pro- The (A) basis. Schmidt the in matrix density idler reduced The φ (1) nm i = 10. = 10. n n i ,m β X nm (1) ,i β ,i 1 √ nm PNAS (1) 0 P a edrvda h nest expecta- intensity the as derived be can λ nm = n λ Z | v m m n ∗ and , ue1,2019 11, June d (k r i u )v n m D (r)σ B k 0.005 10 i 0 0.01 (r)u  | | 1 iha diinlspa- additional an with C 0 i o.116 vol. ih1 m ∗ and h hnei the in change The (D) 9. (r) i 0 | ehv traced have we D, HG + | n h 00 o 24 no. . c oe (C mode. , ) | 11675 [11] [10] The ) m

PHYSICS Z ∞ X √ ∗ S [ρ¯i ] ∝ Re dωs E [ωs ]dρs λn λm un (ρs )vn (ρ¯i ) nm

Z 0 0 ∗ 0 0 −iQs ·ρ × vm (ρ¯i ) dρ um ρ σ ρ e . [14]

This shows a smooth transition from momentum to real-space imaging. For low Schmidt modes that do not vary appreciably across the charge density scale, the last term yields σ (Qs )≈ 0 R 0 ∗ 0 0 −iQs ·ρ dρ um (ρ ) σ (ρ )e . Consequently, when the Schmidt modes do not vary on the length scale of the charge density up to high order, the Fourier decomposition of the charge density is projected on un and reweights the corresponding idler modes. The resulting image given by spatial scanning of the idler is the Fourier transform of the charge density projected on the relevant idler mode. Alternately, when the Schmidt modes vary along the charge density, the exact expression for the far-field diffraction image is given by

∞ X √ ∗ S [ρ¯i ] ∝ Re γnm λn λm vn (ρ¯i )vm (ρ¯i ) [15] nm Z Fig. 4. Hermite–Gaussian modes. Modes are labeled by two indexes, each X (1) ∗ representing one dimension in the transverse plane. γnm = βkm dρs dωs E [ωs ]un (ρs )uk (Qs ), [16] k

(1) is given by the intensity–intensity correlation function (SI where βnm is defined in Eq. 11. From the definition of Qs it Appendix), is evident that its angular component of uk is identical to the corresponding one in un and therefore γnm is composed of sum- Z mation over modes with the same angular momentum in the LG ¯  ¯  S [ρ¯i ] = dXs dXi Gs Xs , Xs Gi Xi , Xi basis set. It is also possible to obtain the real-space image of the charge density when the signal is frequency dispersed. Assuming for sim- D ˆ ˆ E × T Is (rs , ts )Ii (ri , ti )UI (t) , [12] plicity perfect quantum correlations between the signal and idler, we obtain ω¯s −i c ρ¯i ˆ ˆ(−) ˆ(+) S [ρ¯i ,ω ¯s ] ∝ Reσ (ρ¯i )e . [17] where Im (rm , tm ) ≡ Em,R (rm , tm )· Em,L (rm , tm )are field inten- sity operators and m = (s, i). The gating functions Gm represent This image is phase dependent and therefore allows us to trans- the details of the measurement process (25, 26). Eq. 12 can be form freely between momentum and real space which is not calculated straight from the reduced of the idler, possible in ordinary diffraction of classical light. The phase- despite the fact that it includes the signal’s intensity operator. dependent Fourier image in this limit is also given by resolving The reason stems from the fact that the intensity operator expec- the signal photon with respect to the frequency ω¯s as well (SI tation value monitors the single-photon space. The Appendix). over a singly occupied signal state results in the same conclusion. Reweighted Modal Contributions Estimating this expression includes a 10-field operators correla- tion function which is shown explicitly in SI Appendix, section 2, The apparent classical-like form of the coherent superposition Eq. S4. In the far field, after rotational averaging we obtain (SI in the Schmidt representation, where each mode carries dis- tinct spatial matter information, suggests experiments in which Appendix, section 2) a single Schmidt mode is measured at a time (23). This bears Z Z some resemblence to the coherent mode representation of par- tially coherent sources studied in refs. 27 and 28. Moreover, it S [ρ¯ ] ∝ Re dωs E [ωs ] dρ Φ(ρ , ρ¯ )× i s s i allows the reweighting of high angular momentum modes avail- able experimentally (29) and known to have a decreasing effect Z 0 dρ0Φ ρ0, ρ¯  σ ρ0e−iQs·ρ . [13] on the image upon naive summation. Reweighting of truncated i sums is extensively used as a sharpening tool in digital sig- nal processing, especially in medical image enhancement (30). ωs This approach raises questions regarding the analysis of optimal Here Qs = c ρˆs is the diffraction wavevector, E [ωs ]= R 2 Schmidt weights, error minimization, and engineered functional dωi G (ωs )G (ωi )|A (ωs + ωi )| is a functional of the decrease of weights as done in theory for sampled signals. The S = − (S − S ) frequency, 0 is the image with the noninteracting- structure of the spatial information mapping from the signal to ¯ uniform background (S0) subtracted, and ρi is the mapping the idler takes a simpler form for small scattering angles. When coordinate onto the detector plane with the corresponding sign. P we examine the first- and second-order contributions due to a σ (ρ) ≡ α;a,b ha|σˆ (ρ − ρα)|bi denotes a matrix element of single charge distribution, the resulting image of a truncated sum the charge-density operator, traced over the longitudinal axis, composed of the first N modes is given by with respect to the eigenstates {a, b}, and ρα are positions of in the sample. The matter can be prepared initially in N (p) X √ (p) ∗ a superposition state. Substituting the Schmidt decomposition SN [ρ¯i ] ∝ Re λn λm βnm vn (ρ¯i )vm (ρ¯i ), [18] (Eq. 5) into Eq. 13 gives n, m=0

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Fig. i.5A Fig. d β egtdrcmiaino h rnae u nEq. in sum truncated the of recombination Weighted od hs nomto ftesuidojc and object studied the of information phase holds 11, nm (2) tnsfradtce oeiiilyi aumstate. vacuum a in initially mode detected a for stands |I 18 σ rsnsteShitsetu o emcharac- beam a for spectrum Schmidt the presents (ρ, = p ihafltee cmd pcrmw demonstrate we spectrum Schmidt flattened a with l σ β N R φ) 0 = p nm (1) l rtmds hscrepnst tagtowr imag- straightforward to corresponds This modes. first | d = exp r h elpr fteimage the of part real The ) = .07 7 orsodn to corresponding 0.07, u n h k X s −i (r)|σ hc yields which ,k d 18 L/3 2π u n ρ for Lower (r) i (k eegtdtuctdsmdiffrac- sum truncated Reweighted (D) . | is-re mg.Soni recom- is Shown image. First-order B) s 2 n )σ u 0 eoeigtesailphase spatial the Recovering 20. = o hw h reweighted-flattened the shows row m ∗ (k κ (r) s ≈ − sasatrn coefficient scattering a is i.5B Fig. 14. k κ d ≈ )u cmd weights Schmidt (A) 14. m I ∗ ( r (k , φ) d Upper ), ihaddspa- added with lutae the illustrates β N nm (1) rtmodes, first o) with row), defined , using 18, [19] rjc o icpieInvto 11Poet 104.W lotakNoa thank also We illustrations. B12024). graphical Project, the (111 for Asban Innovation Young Introduction Discipline Expertise Endowed Overseas for University; Zijiang Project Normal from CHE- China East support (Grant Fund, Foundation the Scholar Award Science acknowledges by National K.E.D. the supported 1663822). by were S.A.’s supported DESC0019484. Irvine Award was by California, fellowship supported visits of was Collaborative S.M. University and. acknowledged. the DEFG02-04ER15571, Science, gratefully to of is Office K.E.D. Sciences, Energy of Energy of Basic Department of US Office Division, Biosciences and ACKNOWLEDGMENTS. for topic fascinating a provides study. future and imaging dynamics in real-space results their pulses of ultrafast by prepared initially tributions as scales image coincidence overall ∝ the while sample the with Linear act densities. as fol- charge to scales perturbed us diffraction initially allows of fields weak evolution using the para- by Avoiding low by complexes (33). such crystal generated diamond of a damage been using (32). have conversion, damage down photons reduce metric X-ray to proposed hard been Entangled have pulses Ultra- X-ray sam- measurement. destructive short fresh a a assuming lasers iteration, use each to X-ray in have ple is typically free-electron strategy and One fragile, building dynamics. are multiple-timescale complex, for are force Such (31). driving major a Signal considered theory, be studies. (30). can sampling future well noise, as in for quantization used high-frequency challenge avoiding techniques a perfectly optimization is amplitude, acquisition weights Find- two-photon photons. optimal the the between ing information into of spatial converges the distribution modes transferring of delta contribution equal a relations that suggests closure This the moti- by is distribution vated modes Schmidt subwavelength the cut- discuss Reweighting natural to resolution. modes the charged quantify topologically to high importance of off practical demon- cardinal recently of is above been It numbers the have High quantum with light information. span experimentally matter of that strated the states modes refine momentum spatial can angular the the one to signal image, reweighting the measured By from transfer beams. information also idler the can for detection Eq. picture itive resolution. quantum imaging with enhanced photons achieve entangled of by ments limited is divergence. resolution beam the the low and acquires and length amplitude pump crystal the the the limit of crystal fur- spectrum short requires angular the and with In amplitude compatible study. the not ther for generate is approximation to limit paraxial required This the more is correlations. is divergence quantum factor beam strong matching phase strong the and in crystal, dominant nonlinear achieved long is vanishing a resolution or For infinite High of entanglement. limits of the degree of sensitive the is orders image to detected odd the The nonvanishing interaction. of the radiation–matter from of the functions of description resulting complete correlation diffraction distribution a no provided charge even Homodyne in have However, We imprinted in order. density. detectable. charge is results this is interaction sources in which single classical generated photons, a is the quantum to photon the of due in phase field change the the the distribution order, of charge first state the To in interaction. orders matter informa- odd phase at carries matter tion from light quantum scattered The Discussion Phase 5 image. Fig. diffracted in the demonstrated of is features measurement fine of enhancement the I h mgn fsnl oaie ilgclmlclshsbeen has molecules biological localized single of imaging The measure- diffraction coincidence that demonstrated have We p 3/2 sn ifato fetnldpoosfo hredis- charge from photons entangled of diffraction Using . PNAS h upr fteCeia cecs Geosciences, Sciences, Chemical the of support The ∝ I p 1/2 | ue1,2019 11, June ihtesga htn htinter- that photons signal the with P σ n p l u hc r adt realize. to hard are which , n | C (q)v o.116 vol. and 18 n (q D. rvdsa intu- an provides σ 0 = ) | (q) o 24 no. δ h light– the , ∼10 (2) (q | 4 − (29). 11677 q 0 ) .

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