Internal dynamics of intense twin beams and their coherence

Transcription

1 wwwnaturecom/cientificreort OPEN recei e : 02 Decem er 201 accete : 11 e ruar 2016 Pu i e : 2 e ruar 2016 Internal dynamic of intene twin beam and their coherence Jan Peřina Jr 1, Ondřej Haderka 2, Aleia Allevi 3,4 & Maria Bondani 5,4 The dynamic of intene twin beam in um-deleted arametric down-converion i tudied A generalized arametric aroximation i uggeted to olve the quantum model It comarion with a emiclaical model valid for larger twin-beam intenitie confirm it alicability The exerimentally oberved maxima in the ectral and atial intenity auto- and cro- correlation function deending on um ower are exlained in term of different eed of the (back-) flow of energy between the individual down-converted mode and the correonding um mode Thi effect i alo reonible for the gradual relacement of the initial exonential growth of the down-converted field by the linear one Furthermore, it form a minimum in the curve giving the effective number of twin-beam mode Thee effect manifet a tight relation between the twin-beam coherence and it internal tructure, a clearly viible in the model Multile maxima in the intenity correlation function originating in the ocillation of energy flow between the um and down-converted mode are theoretically redicted Nowaday, arametric down-converion (PDC) decribing three mutually interacting otical field 1 rereent the mot common ource of nonclaical light 2 Thi i due to the natural airwie character of the nonlinear interaction generating one hoton air at the exene of an annihilated um hoton A the ignal and idler hoton in a air are emitted together 3, their roertie, including olarization, frequency and wave-vector, exhibit trong correlation Thee correlation occur at the level of the amlitude of their wave function o that they have no claical counterart Entanglement give unuual roertie to hoton air that lead to the violation of the law of claical hyic 4, to quantum teleortation 5 and to many other urely quantum effect Photon air may alo form intene beam, the o-called twin beam (TWB) 6,7 In the TWB, the quantum feature of hoton air are artly concealed due to their macrocoic character However, tight correlation in the number of ignal and idler hoton remain They rereent the origin of ub-hot-noie intenity correlation 8 11, which are ueful, eg, in ghot 12 and quantum 13 imaging Thee intene TWB are tyically multimode and dilay an intereting internal tructure 14,15 We note that alo ingle-mode intene TWB can be obtained alying trong ectral and atial filtering 16 During the generation of intene TWB, the dynamic of PDC i more comlex 15,17 and naturally allow for the back-flow of energy from the mode of the down-converted field into the um-field mode A it ha been exerimentally hown in 18 21, thi reult in a artial lo of coherence both inide the ignal and idler field and between them A we how here, thi artial lo of coherence reflect the change in the internal intenity tructure of an intene TWB affected by the back-flow of energy To rovide a detailed inight into uch a behavior we develo two model aroriate for intene TWB endowed with multithermal tatitic They are baed on the Schmidt mode decomoition 22,23 alied to both ectral and tranvere wave-vector domain where it rovide the natural hyical bae for TWB Thee model rereent an alternative to the reviouly develoed theorie baed on quai-monochromatic and quai-lane-wave aroximation 24 a well a to model exloiting numerical olution of the Maxwell equation and imulation 25 Wherea the former model 24 i not alicable for deleted um beam, the latter one 25 doe not rovide an inight into the internal dynamic of the PDC We note that the model develoed here for twin beam may be ueful alo for other three-mode nonlinear interaction found, eg, in acouto-otic 26, electronic or oto-mechanic 27 More general model can be develoed for other nonlinear rocee decomoable into three-mode interaction 1 oint a orator of Otic of a ac ni erit an Int 1 itoa u O omouc ec eu ic 2 Intitute of ic oint a orator of Otic 1 itoa u 0a 1 6 O omouc ec eu ic 3 Diartimento i cien a e ta ecno o ia ni erit e i tu i e Inu ria ia a e io omo Ita I omo ia a e io omo Ita Itituto i otonica e anotecno o ie oni io a iona e e e icerc e ia a e io omo Ita orreon ence an re uet for materia ou e a ree to emai : anerina r uo c Scientific ReoRt 6:22320 DOI: /re

2 wwwnaturecom/cientificreort/ In the roce of PDC, um, ignal and idler field mutually interact in a nonlinear medium with an effective nonlinear ucetibility χ (2) Auming for imlicity calar field, the aroriate nonlinear momentum oerator Ĝ int for PDC take the form 1,28,29 : Ĝ ( z) = 2 ε dxdy dt [ χ E ( r, t) E ( r, t) E ( r, t) + h c]; int 0 + (2) ( + ) ( ) ( ) i ( ) r = ( x, y, z) In Eq (1), E decribe the oitive-frequency art of the um electric-field oerator amlitude ( and E ) ( [ E ) i ] denote the negative-frequency art of the ignal [idler] electric-field oerator amlitude Symbol ε 0 i the ermittivity of vacuum and h c relace the Hermitian conjugated term The momentum oerator Ĝ int can be recat into it diagonalized form aroximately revealed by the Schmidt decomoition of weak TWB In thi cae, the o-called two-hoton amlitude 30, decribing the common tate of ignal and idler hoton, i calculated a a erturbation olution of the correonding Schrödinger equation The ubequent Schmidt decomoition rovide the orthogonal aired mode that are the genuine hyical mode of the ignal and idler field In addition, the Schmidt coefficient λ giving the quantum robability amlitude of finding a given aired mode in the tate are obtained (for detail, ee the Sulemental material 31 ) Uing thee mode and coefficient both in the tranvere wave-vector lane 23,32,33 and in the frequency domain 22, the av L momentum oerator Gˆ = dzgˆ int ( z)/ L 0 int averaged over the crytal length L i obtained a: av ˆ int mlq, mlq, imlq, m= l, q= 0 G () z = i K ^a () z ^a () z ^a () z + hc In Eq (2), the annihilation (^a) and creation ( ^a ) oerator belong to the um, ignal and idler mode that form indeendent mode trilet The common couling contant K include multilicative factor quantifying the nonlinear interaction in the tranvere wave-vector lane ( t ) and in the frequency domain ( f ) a well a the normalization to hoton number (ξ ); K = t f /( Lξ ) [for more detail, ee17 ] The overall um-field amlitude ξ i derived from the um-field ower P, it reetition rate f and central frequency ω 0 0 a ξ = P/( f ω ), where tand for the reduced Planck contant It i aumed that the overall um ower P iminging on the crytal can be divided into individual um mode indexed by ( ml) (tranvere wave-vector comonent) and q (frequency comonent) roortional to the quared roduct λmlλq 2 of the Schmidt coefficient that characterize the two-hoton amlitude in the tranvere wave-vector lane and in the ectrum Thu, an ( mlq)-th mode N of a claical trong um field ha it incident claical amlitude A mlq, (0) [correonding to the mean value of normally-ordered quantum amlitude] given a λmlλq ξ av The Heienberg equation derived from the momentum oerator Ĝ int in Eq (2) decribe the evolution inide indeendent ubace of the individual mode trilet ( mlq) In what follow, we further ay attention to an arbitrary ubace ( mlq) and omit it indice for imlicity Before we addre the nonlinear interaction in it quantum form, we firt conider it a a claical nonlinear roblem The Schmidt decomoition rovide the ignal and idler mode in air in which they hare a common Schmidt coefficient λ A a conequence, the ignal and idler field amlitude (exreed in hoton number) belonging to one air are the ame Uing thi ymmetry, the claical counterart of the Heienberg equation written for mean value of the ymmetrically-ordered ignal (A ) and um (A ) oerator amlitude attain the form [ A( z) Ai ( z), A and K are aumed real]: da ( z) da ( z) 2 = KA( z) A( z), = KA ( z) dz dz The integral of motion A 2 ( z) + A 2 ( z) A 2 in Eq (3) allow u to find their olution by direct integration of the econd equation in (3): A ( z) = A A coh( KA z) A inh( KA z) A coh( KA z) A inh( KA z), (4) (1) (2) (3) A ( z) = AA A coh( KA z) A inh( KA z) ; (5) A A (0) and A A (0) A equation (3) are written for the ymmetric ordering of field oerator, the incident um, vacuum ignal and vacuum idler amlitude are equal to A(0) = ( A )(0) + 1/2 and N 2 A(0) = Ai (0) = 1/ 2 The olution (4) and (5) are valid until the um mode i comletely deleted Thi occur at z = z 0 for which A ( z ) = A = 1/ 2 0 : 1 z0 = 2KA 2A A ln 1 + A + A A A A In the interval z0 z 2z0 the flow of energy in the analyzed mode trilet i revered, ie the um mode take back the energy from the down-converted mode In thi cae, field evolution i again decribed formally (6) Scientific ReoRt 6:22320 DOI: /re

3 wwwnaturecom/cientificreort/ Figure 1 Number N ex of exerimental ignal hoton (Δ with error bar) detected in a mall area of the emiion PDC cone meaured for different um ower P 18 Number N of emitted hoton air a obtained in GPA (olid curve) and SCM (dahed curve) are hown for comarion; N = aˆ aˆ mlq mlq, mlq, by Eq (4) and (5), however, with z relaced by 2z 0 z At z = 2z 0 the mode trilet return to the incident tate and the dynamic reeat from the beginning To include the quantum tatitical character of the ignal and idler field in multi-thermal PDC, we firt rooe a model baed on the generalized arametric aroximation (GPA) 34 In thi aroximation, the um field i taken a a claical field ariing from the claical olution written in Eq (4), ie it undergoe deletion Thi av remove nonlinearity from the original Heienberg equation derived for the momentum oerator Ĝ int in Eq (2) The reultant Heienberg equation form a imle linear oerator et of equation: da ^( z) da ^ = KA z ^ i( z) ( ) ai( z), = KA( z) ^a( z) dz dz The olution of Eq (7) i found in the uual way: (7) z g g g i g g i i ^a ( z) = U ( z) a^ (0) + V ( z) ^a (0), ^a ( z) = U ( z) a^ (0) + V ( z) ^a (0), (8) g U ( z) = coh [ ϕ( z)], V ( z) = inh [ ϕ( z)] (9) defining ϕ ( z) = dzka ( z ) For the claical um amlitude A ( z) written in Eq (5), we have: 0 ϕ = A + A A A ( z) KA z ln + 2A 2A ex(2 KA z) We note that the econd term on the rh of Eq (10) account for the um deletion and o it goe beyond the uual arametric aroximation The olution (8) and (9) reerve the canonical commutation relation A we will ee below, though the model doe not reerve the overall energy (the momentum oerator G int i nonconervative), it decribe well coherence and mode tructure of the TWB To verify the validity of GPA, we develo in arallel a emiclaical model (SCM) that reerve the overall energy In thi model, we build the oerator olution from the claical one written in Eq (5) uing the amlitude related to the normal ( A N = A 2 1/2 ) and anti-normal ( A A = A 2 + 1/2 ) ordering of field oerator Thi olution i obtained in the form of Eq (8) uing the following coefficient c A c N (10) U ( z) = A ( z), V ( z) = A ( z) (11) We recall that Ai NA, ( z) A NA, ( z) Thi model i valid for larger intenitie for which the quantum decrition coincide with the claical one Moreover, the model i aealing alo for lower intenitie a it reerve the commutation relation Pum deletion naturally limit the um ower P for which an exonential growth of the number N of emitted hoton air i oberved 18 The increae of the overall hoton-air number N low down a the um ower P increae and finally a linear increae i reached (ee Fig 1) Thi change occur for the um ower P at which the individual um mode containing the larget ortion of the incident um ower (with the larget Schmidt coefficient) become comletely deleted and begin to take their energy back from the ignal and idler Scientific ReoRt 6:22320 DOI: /re

4 wwwnaturecom/cientificreort/ Figure 2 Number n λ of hoton air in mode with the Schmidt coefficient λ for um ower P = 25 mw (yellow olid curve), 35 mw (blue), 100 mw (green) and 390 mw (red) Denity ρ λ of atioectral mode revealed by the Schmidt decomoition i alo hown (dahed curve); nλ = aˆmlq, aˆmlq, for mlq uch that λ λ = λ ml q mode with whoe they hare the common mode trilet Thi reflect the mot imortant feature of the nonlinear dynamic of individual mode trilet in PDC: the larger the mode incident um ower, the fater the nonlinear dynamic of the correonding trilet Detailed inight into the dynamic i rovided in Fig 2 howing the deendence of hoton-air number n λ on the Schmidt coefficient λ λml λq for everal um ower P forming an increaing ucceion For high ower P, the um mode with the larget Schmidt coefficient λ (correonding to the lowet-order mode with mall number l and q) are comletely deleted inide the crytal and they even take ome energy back from their ignal and idler mode contrary to the ret of um mode (with lower Schmidt coefficient λ), which are only loing their energy during the roagation in the crytal Thi reult in the lower-than-exonential growth of the ignal- and idler-field intenitie The oberved nearly-linear character of thi growth i a conequence of the large number of mode with different evolution that contitute the TWB (ee Fig 2 for the denity ρ λ of mode: the maller the λ the larger the denity ρ λ ) The comarion of curve in Fig 1 reveal that the GPA lightly underetimate the hoton-air number N, which i correctly rovided by the SCM for any ower P The exerimental mean ignal-hoton number N ex detected in a mall ortion of the emiion PDC cone 18 are linearly roortional (within the exerimental error) to the theoretical hoton-air number N of both model Pum deletion and back-flow of energy occurring in individual mode trilet qualitatively influence the coherence of TWB Both the atial and ectral intenity auto- and cro-correlation function widen at the increaing ower P (ee Fig 3) Thi i a conequence of the fact that the ignal and idler mode with the larget Schmidt coefficient λ take the energy from the um mode much fater than the remaining mode, thu becoming more and more dominant a the ower P increae (comare the curve for 25 and 35 mw in Fig 2) Since the hae variation along the atial and ectral rofile of thee highly-oulated mode with mall number l and q i mall comared to the remaining mode [for the rofile, ee the Sulementary material], coherence of the TWB naturally increae Thi i accomanied by a decreae in the number K of mode effectively contituting the TWB The number K of uch mode can be quantified, eg, by the Fedorov ratio 35 (ee Fig 4) However, at a certain um ower P th, the TWB coherence begin to decreae At thi threhold ower P th, the TWB mode with the larget Schmidt coefficient λ took all the ower from their um mode omewhere inide the crytal and o they had to return at leat art of it back Thi allow the mode with maller Schmidt coefficient λ to become the mot oulated art of the TWB (ee the curve for 100 mw in Fig 2) Thi reult in the increae of the number K of mode accomanied by a artial lo of the atial and ectral coherence We note that wherea atial and 51 ectral mode contitute the analyzed weak TWB, 1200 atial and 13 ectral mode are found in the TWB for the threhold ower Pth = 35 mw The curve lotted in Fig 1 4 have been obtained for a β barium borate (BBO) crytal cut for a nearly collinear ectrally degenerate tye-i interaction (cut angle ϑ =37 deg), having effective length Leff = 27 mm and umed by a beam 530 μm-wide with the central wavelength λ = 349 nm and ectrum 12-nm wide (ee Method for detail about the exeriment) The Schmidt mode conidered in both model have been determined in 36 (for more detailed analyi of the mode, ee 37 ) and alied for un-deleted 17 a well a deleted 34 um beam Thee curve fit the exerimental oint obtained for an 8-mm long BBO crytal meaured under the decribed condition The comarion reveal that the model redict well the behavior of ectral and atial auto- and cro-correlation function Wherea the exerimental auto-correlation function are narrower than the correonding cro-correlation function, the model rovide very imilar rofile for both of them Thi originate in the ectral and atial factorization of mode aumed in the model The effective crytal length of 27 mm ued in the model correond to the nonlinear walk-off length of the ued BBO crytal For longer crytal, the um- and TWB mode loe their atial ynchronization due to the walk-off which reult in a comlex nonlinear interaction in the tranvere lane Ditortion of the um-beam tranvere rofile occur 18 The comarion of grah in Fig 3 and 4 reveal that the model i till able to redict the TWB behavior in it ectral 0 Scientific ReoRt 6:22320 DOI: /re

5 wwwnaturecom/cientificreort/ Figure 3 (a) Satial radial and (b) ectral width A and C of intenity auto- (, black) and cro-correlation (Δ, red) function, reectively, veru um ower P; exeriment (iolated ymbol with error bar 18 ), GPA (olid curve with ymbol), SCM (dahed curve with ymbol); A( x, x ) [ aˆ( x) aˆ( x)] [ aˆ( x ) aˆ ( x )], N C( x, x ) [ aˆ( x) aˆ( x)] [ aˆi( x ) aˆi ( x )], φ ω = φ aˆ ( x) aˆ ( k,, ) t ( k, ) f ( ω ) aˆ N b b b b b mlq bml, b b bq, b b, mlq and t bml, ( f bq, ) are the atial (ectral) Schmidt mode of field b, b =, i For more detail, ee 17 All 4 curve in (a) and (b) nearly coincide Figure 4 (a,b) Number K k,ex [ K ω,ex ] of atial radial [ectral] mode ( [Δ] with error bar) exerimentally detected in a mall area inide the emiion PDC cone a function of um ower P 18 For comarion, uitably recaled ectral (atial radial) Fedorov ratio Kω C, ω/ n, ω ( Kk C, k/ n, k) obtained in GPA (olid curve) and SCM (dahed curve) are alo drawn The theoretical curve nearly coincide in both cae Scientific ReoRt 6:22320 DOI: /re

6 wwwnaturecom/cientificreort/ Figure 5 (a) Exerimental etu ued for the determination of atio-ectral intenity correlation in a TWB HWP: half-wave late; PBS: olarizing cube beam litter; BBO: nonlinear crytal; L: len, with 60-mm focal length; M j : herical mirror; G: grating; EMCCD: electron-multilying CCD camera (b) A tyical ingle-hot image of the outut of the imaging ectrometer The ignal (oitive value) and idler (negative value) atial radial emiion angle θ veru the wavelength λ i lotted The eckle-like attern with correlated grain i viible art, but it underetimate the atial coherence and, hand by hand, overetimate the number of atial mode Imortantly, thi comarion clearly how that the generalized arametric aroximation give ractically the ame intenity auto- and cro-correlation function a well a number of mode of TWB comared to the emiclaical model Thi jutifie the ue of the GPA for the determination of coherence roertie of the TWB for the conidered um ower P Detailed analyi of the model baed on the grah in Fig 2 oint out the occurrence of more threhold ower P th, i at which the TWB coherence exhibit local maxima Indeed, thee local maxima are oberved for ower P at which the mode with the larget Schmidt coefficient λ attain their maximal oulation (ee the curve for 390 mw in Fig 2) Note that, in the TWB there occur i differently-oulated grou of mode for an i-th threhold ower P th, i ( i = 2, ) A a reult, coherence maxima may be le reolved for larger value i Alo, in accord with the reviou finding, the number K of mode i ubtantially reduced for thee ower P th, i The develoed theory of intene PDC may eaily be alied to other nonlinear tructure including oled nonlinear material and waveguiding tructure Similar aroache can be develoed to decribe other nonlinear interaction including trilet of mutually interacting field Raman and Brillouin cattering can be mentioned a tyical examle For all thee rocee, the relation between the internal tructure and coherence of the interacting field would rovide a comletely new inight into the evolution of the nonlinear interaction In concluion, we have develoed a theory for intene twin beam alicable in the regime of um deletion Following the dynamic of individual mode trilet, the theory naturally exlain the exerimentally oberved increae (decreae) of atial and ectral coherence accomanied by a decreae (increae) of the number of mode oberved in different regime of um ower The comarion of the reult with the exerimental data and with the reult of emiclaical model confirm the validity of the uggeted generalized arametric aroximation for treating intene twin beam The model alo redict the occurrence of additional coherence maxima for high um ower Thi rereent a challenge for further exerimental invetigation of intene twin beam Method Exerimental etu The exeriment 18 wa erformed in a etu hown in Fig 5(a) uing a tye-i 8-mm long BBO crytal (cut angle = 37 deg) The crytal wa umed by the third-harmonic ule (349 nm, 45- ule duration) of a mode-locked regeneratively amlified Nd:YLF laer (High-Q-Laer) running at 500 Hz Radial rofile of the um beam, collimated by mean of a telecoe in front of the crytal, wa 530 μm wide (FWHM) at the lowet um ower A half-wave late followed by a olarizing-cube beam litter wa ued to change the um ower Phae-matching in the crytal for frequency degenerate down-converted beam wa reached in a lightly non-collinear configuration The down-converted light wa collected by a len with 60-mm focal length and then focued in the lane of the vertical lit of an imaging ectrometer (Lot Oriel, 600 line/mm grating) The angularly diered far-field radiation wa recorded, hot by hot, by a ynchronized EMCCD camera (ixon Ultra 897, Andor) oerated at full-frame reolution ( ixel, 16-μm ixel ize) We note that Scientific ReoRt 6:22320 DOI: /re

7 wwwnaturecom/cientificreort/ for the detection of intene TWB EMCCD camera are more convenient comared to iccd camera, which have been often exloited in thi area The overall reolution of the ytem comoed of the imaging ectrometer and the camera wa 02 nm in frequency and 0015 deg in the radial angle The reence of intenity correlation between the ignal and idler beam wa verified by oberving ymmetrically-oitioned eckle around the degenerate wavelength and the collinear direction, a hown in Fig 5(b) Wherea the ectral correlation are oberved in the horizontal direction, the atial radial correlation are viible in the vertical direction in Fig 5(b) Image roceing The intenity auto- and cro-correlation function characterizing the TWB coherence are obtained a correlation coefficient between a ingle ixel (i, j) and all the ixel (k, l) contained in a ingle-hot image: Γ k l = Iij, Ikl, ij, (, ) I I ij, kl, In (12), I tand for the intenity value of each ixel exreed in digital number and after ubtraction of the mean value of the noie meaured with the camera in erfect dark Symbol indicate the averaging over a equence of 1000 ubequent image taken in the exeriment Thi calculation reult in a matrix of the ame ize a that of the ingle-hot image In thi matrix, two eak occur One eak characterize the auto-correlation area, the other eak decribe the cro-correlation area 41 Error bar hown in Fig 1 4 in the main text were obtained by erforming the calculation for different ixel The mean hoton number have been obtained from a mall area ( ixel) cloe to frequency degeneracy and in quai-collinear interaction geometry Calibration of the camera enitivity, it quantum efficiency and all the otical loe were taken into account The number of TWB mode in the radial direction wa determined by the Fedorov ratio 35 defined a the ratio between the radial ignal-field intenity width and the width of radial intenity cro-correlation function On the other hand, a the field ectral width were not meaured, the g (2) intenity auto-correlation function 18,33,42 wa ued to quantify the number K ω of ectral mode ( Kω = 1/[1 Γij, (, i j)] ) Reference 1 Boyd, R W Nonlinear Otic, 2nd edition (Academic Pre, New York, 2003) 2 Mandel, L & Wolf, E Otical Coherence and Quantum Otic (Cambridge Univ Pre, Cambridge, 1995) 3 Hong, C K, Ou, Z Y & Mandel, L Meaurement of ubicoecond time interval between two hoton by interference Phy Rev Lett 59, (1987) 4 Weih, G, Jennewein, T, Simon, C, Weinfurter, H & Zeilinger, A Violation of Bell inequality under trict Eintein locality condition Phy Rev Lett 81, (1998) 5 Bouwmeeter, D et al Exerimental quantum teleortation Nature 390, (1997) 6 Kolobov, M I & Sokolov, I V Satial behavior of queezed tate of light and quantum noie in otical image Zh Ek Teor Fiz 96, (1989) 7 Jedrkiewicz, O, Gatti, A, Brambilla, E & Di Traani, P Exerimental obervation of a kewed X-tye atiotemoral correlation of ultrabroadband twin beam Phy Rev Lett 109, (2012) 8 Jedrkiewicz, O et al Detection of ub-hot-noie atial correlation in high-gain arametric down-converion Phy Rev Lett 93, (2004) 9 Bondani, M, Allevi, A, Zambra, G, Pari, M G A & Andreoni, A Sub-hot-noie hoton-number correlation in a meocoic twin beam of light Phy Rev A 76, (2007) 10 Blanchet, J-L, Devaux, F, Furfaro, L & Lantz, E Meaurement of ub-hot-noie correlation of atial fluctuation in the hotoncounting regime Phy Rev Lett 101, (2008) 11 Brida, G et al Meaurement of ub-hot-noie atial correlation without backround ubtraction Phy Rev Lett 102, (2009) 12 Gatti, A, Brambilla, E & Lugiato, L Quantum imaging In Wolf, E (ed) Progre in Otic, Vol 51, (Elevier, Amterdam, 2008) 13 Brida, G, Degiovanni, I P, Genovee, M, Ratello, M L & Berchera, I R Detection of multimode atial correlation in PDC and alication to the abolute calibration of a CCD camera Ot Exre 18, (2010) 14 Chrit, A, Laiho, K, Ecktein, A, Caemiro, K N & Silberhorn, C Probing multimode queezing with correlation function New J Phy 13, (2011) 15 Chrit, A, Brecht, B, Mauerer, W & Silberhorn, C Theory of quantum frequency converion and tye-ii arametric downconverion in the high-gain regime New J Phy 15, (2013) 16 Pérez, A M et al Bright queezed-vacuum ource with 11 atial mode Ot Lett 39, (2014) 17 Peřina, J Jr Coherence and dimenionality of intene atioectral twin beam Phy Rev A 92, (2015) 18 Allevi, A et al Coherence roertie of high-gain twin beam Phy Rev A 90, (2014) 19 Allevi, A & Bondani, M Statitic of twin-beam tate by hoton-number reolving detector u to um deletion J Ot Soc Am B 31, B14 B19 (2014) 20 Allevi, A, Jedrkiewicz, O, Haderka, O, Peřina, J Jr & Bondani, M Evolution of atio-ectral coherence roertie of twin beam tate in the high gain regime In Banazek, K & Silberhorn, C (ed) Proc of SPIE 9505, 95050S (SPIE, Bellingham, 2015) 21 Allevi, A et al Effect of um deletion on atial and ectral roertie of arametric down-converion In Banazek, K & Silberhorn, C (ed) Proc of SPIE 9505, (SPIE, Bellingham, 2015) 22 Law, C K, Walmley, I A & Eberly, J H Continuou frequency entanglement: Effective finite Hilbert ace and entroy control Phy Rev Lett 84, (2000) 23 Law, C K & Eberly, J H Analyi and interretation of high tranvere entanglement in otical arametric down-converion Phy Rev Lett 92, (2004) 24 Caani, L, Brambilla, E & Gatti, A Tailoring the atiotemoral tructure of bihoton entanglement in tye-i arametric downconverion Phy Rev A 81, (2010) 25 Brambilla, E, Gatti, A, Bache, M & Lugiato, L A Simultaneou near-field and far-field atial quantum correlation in the highgain regime of arametric down-converion Phy Rev A 69, (2004) (12) Scientific ReoRt 6:22320 DOI: /re

8 wwwnaturecom/cientificreort/ 26 Zhao, C et al Three-mode otoacoutic arametric amlifier: A tool for macrocoic quantum exeriment Phy Rev Lett 102, (2009) 27 Lemonde, M-A, Didier, N & Clerk, A A Nonlinear interaction effect in a trongly driven otomechanical cavity Phy Rev Lett 111, (2013) 28 Peřina, J Quantum Statitic of Linear and Nonlinear Otical Phenomena (Kluwer, Dordrecht, 1991) 29 Peřina, J Jr & Peřina, J Quantum tatitic of nonlinear otical couler In Wolf, E (ed) Progre in Otic, Vol 41, (Elevier, Amterdam, 2000) 30 Keller, T E & Rubin, M H Theory of two-hoton entanglement for ontaneou arametric down-converion driven by a narrow um ule Phy Rev A 56, (1997) 31 Bobrov, I B, Straue, S S, Kovlakov, E V & Kulik, S P Schmidt-like coherent mode decomoition and atial intenity correlation of thermal light New J Phy 15, (2013) 32 Jut, F, Cavanna, A, Chekhova, M V & Leuch, G Tranvere entanglement of bihoton New J Phy 15, (2013) 33 Dyakonov, I V, Sharaova, P R, Ikhakov, T S & Leuch, G Direct Schmidt number meaurement of high-gain arametric down converion Laer Phy Lett 12, (2015) 34 Peřina, J Jr Satial, ectral and temoral coherence of ultraintene twin beam Phy Rev A 93, (2016) 35 Fedorov, M V et al Sontaneou emiion of a hoton: Wave-acket tructure and atom-hoton entanglement Phy Rev A 72, (2005) 36 Peřina, J Jr Coherence and mode decomoition of weak twin beam Phy Scr 90, (2015) 37 Fedorov, M V et al Sontaneou arametric down-converion: Aniotroical and anomalouly trong narrowing of bihoton momentum correlation ditribution Phy Rev A 77, (2008) 38 Jot, B M, Sergienko, A V, Abouraddy, A F, Saleh, B E A & Teich, M C Satial correlation of ontaneouly down-converted hoton air detected with a ingle-hoton-enitive CCD camera Ot Exre 3, (1998) 39 Haderka, O, Peřina, J Jr, Hamar, M & Peřina, J Direct meaurement and recontruction of nonclaical feature of twin beam generated in ontaneou arametric down-converion Phy Rev A 71, (2005) 40 Peřina, J Jr, Haderka, O, Michálek, V & Hamar, M State recontruction of a multimode twin beam uing hotodetection Phy Rev A 87, (2013) 41 Machulka, R et al Satial roertie of twin-beam correlation at low- to high-intenity tranition Ot Exre 22, (2014) 42 Haderka, O, Machulka, R, Peřina, J Jr, Allevi, A & Bondani, M Satial and ectral coherence in roagating high-intenity twin beam Sci Re 5, (2015) Acknowledgement The author thank the roject P205/12/0382 (OH and JP) of GA CR and LO1305 of MSMT CR for uort The uort of MIUR (FIRB LiCHIS RBFR10YQ3H) i alo acknowledged We alo thank E Brambilla, A Gatti and O Jedrkiewicz for dicuion Author Contribution JP develoed the theory; AA, MB and OH carried out the exeriment and roceed the exerimental data All author contributed to the interretation of the exeriment and to the rearation of the manucrit Additional Information Sulementary information accomanie thi aer at htt://wwwnaturecom/re Cometing financial interet: The author declare no cometing financial interet How to cite thi article: Peřina, J et al Internal dynamic of intene twin beam and their coherence Sci Re 6, 22320; doi: /re22320 (2016) Thi work i licened under a Creative Common Attribution 40 International Licene The image or other third arty material in thi article are included in the article Creative Common licene, unle indicated otherwie in the credit line; if the material i not included under the Creative Common licene, uer will need to obtain ermiion from the licene holder to reroduce the material To view a coy of thi licene, viit htt://creativecommonorg/licene/by/40/ Scientific ReoRt 6:22320 DOI: /re