Table Of Content

Randomly poled crystals as a source of photon pairs Jan Peˇrina Jr.1 and Jiˇr´ı Svozil´ık1 1Joint Laboratory of Optics, Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic, 17. listopadu 50a, 772 07 Olomouc, Czech Republic∗ Generation of photon pairs from randomly poled nonlinear crystals is investigated using analyti- callysolublemodelandnumericalcalculations. Randomlypoledcrystalsarediscoveredassourcesof entangledultrabroad-bandsignalandidlerfields. Theirphoton-pairgeneration ratesscale linearly withthenumberofdomains. Entanglementtimesasshortasseveralfscanbereached. Comparison with chirped periodically-poled structures is given and reveals close similarity. PACSnumbers: 42.65.Lm,42.50.Dv,46.65.+g 1 1 I. INTRODUCTION range. For example, signal and idler fields with spectra 0 severalhundredsofnmwidecanbe generatedinchirped 2 The firstexperimentally observednonlinear opticalef- LiNbO3 crystals. On the other hand, domains with dif- n a fect, second-harmonic generation, was investigated al- ferent lengths can also be ordered randomly. A bit sur- J ready more than forty years ago by Franken [1]. Since prisingly, the nonlinear interaction can be efficient even 4 that,manyothernonlineareffectshavebeenrevealedand inthiscasesometimesreferredasstochasticquasi-phase- understoodevenatquantumlevel. Amongthem,sponta- matching(SQPM).Similarlyasorderedpoledstructures ] neous parametric down-conversion(SPDC) with its pro- the randomly poled structures (RPS) allow spectrally h duction of photon pairs belongs to the most fascinating. broadbandnonlinearinteraction. Itisnotsurprisingthat p - Thereasonisthattwophotonscomprisingaphotonpair efficiency of random structures is worse compared to or- t generated in one quantum event of this process are mu- dered ones. However, they usually put smaller require- n a tually strongly correlated (entangled) as was discovered mentstopolarizationpropertiesoftheinteractingoptical u by Hong, Ou and Mandel in eighties [2]. They used in fieldsaswellasorientationofthenonlinearmedium[12]. q their experiments nonlinear bulk crystals that later be- Also fabricationof RPSis mucheasier because highpre- [ comethemostcommonsourcesofphotonpairs[3]. Inor- cision is required in production of ordered periodically- 1 der to observe spontaneous parametric down-conversion poled structures. v phase-matching conditions of the interacting three opti- TheroleofSQPMin1Dhasalreadybeenaddressedfor 7 calfields havetobe fulfilled. Unfortunately, they cannot the process of second-harmonic generation [13–17] and 5 be naturally fulfilled in many highly nonlinear crystals. theprocessofdifference-frequencygeneration[18]. More- 7 However, Armstrong [4] has arrived with the concept of over full domain random structures allowing SQPM for 0 additionalperiodicmodulationofnonlinearsusceptibility transversal second-harmonic generation have been stud- . 1 thathasbeenpracticallydevelopedinperiodicalpolingof ied in [19]. 0 nonlinearcrystals[5]. Inthisconcept,wavevectorofthe 1 Here, we focus our attention to the generationof pho- additionallyintroducednonlinearmodulationisaddedto 1 ton pairs in randomly poled 1D nonlinear crystals [20]. the natural phase-matching condition and the so-called : It is shown that spectral properties of photon pairs and v quasi-phase-matching of the interacting optical fields is photon-pairgenerationratesarecomparableinRPSand i X reached this way. Highly nonlinear materials can be ex- chirped periodically-poled structures (CPPS) [21–24]. ploitedsincethen. Wenotethatshorteningofanonlinear r This is very promising for easy production of broadband a medium suchthat phase-matching conditions loose their and efficient sources of photon pairs. These broadband importance is the only alternative way to periodical pol- sourcesareimportant, e.g.,in metrology(quantumopti- ing. This approach has been applied in photonic-band- cal coherence tomography [25]) or quantum-information gap structures in which optical interference is crucial for processing[26,27]. Wenotethatbroadbandphoton-pair reaching an efficient nonlinear interaction [6–11]. sourcescanalsobeconstructedusingzerogroup-velocity Periodical poling has occurred to be extraordinarily dispersion conditions [28]. However, such conditions can useful. It has provided not only compensation for the be met only for certain pump frequencies considering a natural phase mismatch. The ability to tailor the prop- given material. erties of emitted opticalfields has been revealedsoon. It The paper isorganizedasfollows. InSec. II,a general isbasedonusingorderednonlineardomainswithvariable theory of SPDC modified to random structures is pre- lengths (chirped periodicalpoling). Presenceof domains sented. Photon-pair generationrates and intensity spec- of different lengths in an ordered structure allows an ef- tra are studied both for random and chirped structures ficient nonlinear interaction of fields in a broad spectral in Sec. III. Sec. IV is devoted to temporal properties of the generated photon pairs. Spatial properties of photon pairs are addressedin Sec. V. Temperature behavior ∗ [email protected] of the quantities characterizing photon pairs is analyzed 2 in Sec. VI. Sec. VII brings the analysis of fabrication er- for the pump-field amplitude spectrum. The stochastic rors. Theroleoforderinginchirpedstructuresisstudied function F introduced in Eq. (5) describes SQPM andis in Sec. VIII. Finally, conclusions are drawn in Sec. IX. derived in the form [20]: 0 χ(2)(z) F(∆k)= dz exp(i∆kz). (6) II. SPONTANEOUS PARAMETRIC χ(2)(0) DOWN-CONVERSION IN POLED NONLINEAR Z−L CRYSTALS Symbol∆k describesthenaturalphasemismatchforthe interacting fields, ∆k =k k k . p s i − − TheprocessofSPDCinanonlinearcrystalcanbecon- Inaperiodically-poledstructureneighbordomainsdif- venientlydescribedbythefollowinginteractionHamilto- ferinsignsofχ(2) nonlinearityandfunctionF definedin nian Hînt [29, 30]: Eq. (6) can be recast into the form: Hînt(t)=ε0B 0 dzχ(2)(z) F(∆k)= NL( 1)n−1 zn dzexp(i∆kz). (7) Z−L n=1 − Zzn−1 E(+)(z,t)Eˆ(−)(z,t)Eˆ(−)(z,t)+H.c.; (1) X × p s i SymbolN denotesthe number ofdomainsandn-thdo- L L denotes the crystal length. In Eq. (1), the positive- main extends from z = z to z = z . Positions z of n−1 n n frequencypartofthepumpelectric-fieldamplitudeisde- domain boundaries are random and can be expressed as noted as E(p+) and Es(−) (Ei(−)) stands for the negative- zn = zn−1 +l0 +δln (n = 1,...,NL, z0 = −L) in our frequency part of the signal (idler) electric-field ampli- model using stochastic Gaussian declinations δln. The tude operator. The z-dependent second-order suscepti- basiclayerlengthl0 isdeterminedsuchthatquasi-phase- bility tensor χ(2) describes poling of the nonlinear mate- matchingisreached,i.e. l0 =π/∆k0,∆k0 ≡∆k(ωs0,ωi0), rial. Vacuum permittivity is denoted as ε and means and ω0 means central frequency of field a. The random 0 a B the transverse area of the optical fields. Symbol H.c. re- declinations δln are mutually independent and can be places the Hermitian-conjugated term. described by the joint Gaussian probability distribution Electric-field amplitudes occurring in Eq. (1) can be P: convenientlydecomposedintoharmonicplanewaveswith 1 frequencies ω and wave vectors k : P(δL)= exp( δLTBδL). (8) a a (√πσ)NL − 1 Eâ(−)(z,t)= 2π dωaEâ(−)(ωa)exp(−ikaz+iωat), Covariance matrix B is assumed to be diagonal and its Z nonzero elements equal 1/σ2. Stochastic vector δL is a=s,i. (2) composedof declinations δl ; symbol T stands for trans- n The quantum spectral amplitudes Eˆ(−)(ω ) in Eq. (2) position. Characteristicfunction Gofthe distributionP a a in Eq. (8) takes the form: can then be expressed using photon creation operators aˆ†(ω ): a a N G(δK) exp(iδK δL) = G(δk ); (9) ~ω ≡h · iav j Eâ(−)(ωa)=−is2ε0cna(aωa) aˆ†a(ωa); (3) jY=1 B symbol meansscalarproduct. VectorδKofparameters cis speedoflightinvacuum, ~reducedPlanckconstant, of the ch·aracteristic function G is composed of elements and na stands for index of refraction of field a. δkj. One-dimensional characteristic function G(δk) in First-orderperturbation solution of Schrödingerequa- Eq. (9) equals exp( σ2δk2/4). − tion with the initialvacuum state vac in the signaland Inordertoobtainanalyticalformulas,weintegratethe | i idler fields results in the following two-photon quantum expression for function F in Eq. (7) domain by domain state ψ : and modify the contributions of the first and last do- | i mains in such a way that the following simple formula is ψ = dω dω Φ(ω ,ω )aˆ†(ω )aˆ†(ω )vac . (4) reached: | i s i s i s s i i | i Z Z 2i NL The two-photon spectral amplitude Φ introduced in F(∆k)= ( 1)nexp(i∆kz ). (10) n Eq. (4) is given as follows: ∆k − n=0 X Φ(ωs,ωi)=g(ωs,ωi)Ep(+)(ωs+ωi) As a typical structure contains hundreds of domains, in- F(∆k(ω ,ω )), (5) correct inclusion of fields from the first and the last do- s i × mains leads to negligible declinations. The formula in where g denotes a coupling constant, g(ωs,ωi) = Eq. (10) can be interpreted such that SPDC occurs only i√ωsωi/[2cπ ns(ωs)ni(ωi)]χ(2)(0),andEp(+)(ωp)stands in domains with positive susceptibility χ(2) at doubled p 3 amplitude and domains with negative susceptibility χ(2) F(∆k)2 . The larger the δk, the smaller the mean av play only the role of a ’linear’ filler. This interpretation vh|alue F|(i∆k)2 . Inspection of the formula for H in av h| | i elucidateswhyapairofdomainshavingonepositiveand Eq. (16) also shows that the larger the value of the ba- onenegativesignsofsusceptibilityχ(2) formsanelemen- sic layer length l the faster the decrease of mean values 0 tary unit for understanding properties of photon pairs. F(∆k)2 for given δk. According to the formula in av h| | i Eq. (16) the larger the standard deviation σ of a random structure the smaller the value of H . The de- | | III. PHOTON-PAIR GENERATION RATES crease of values of H results in an increase of the range AND INTENSITY SPECTRA of values of the ph|as|e mismatch δk in which the averaged squared modulus F(∆k)2 of phase-matching av h| | i Photon-pairgenerationrateaswellasintensityspectra function attains non-negligible values [see the formula in can be easily derived from mean spectral density of the Eq. (15)]. number of generated photon pairs n(ω ,ω ). The mean The formula for averaged squared modulus s i spectral density n corresponding to the quantum state F(∆k)2 of phase-matching function in Eq. (15) av h| | i ψ is defined by the formula can be substantially simplified under the assumption | i σ2(∆k )2N /2 1: 0 L n(ω ,ω )= ψ aˆ†(ω )aˆ (ω )aˆ†(ω )aˆ (ω )ψ , (11) ≫ s i hh | s s s s i i i i | iiav 2N F(∆k)2 = L wherethesymbolhiav meansstochasticaveragingoveran h| | iav (∆k0+δk)2 ensemble of geometricconfigurationsofanRPS. Assum- 1 G(∆k +δk) 0 ingthe quantumstate ψ writteninEq.(4)wearriveat − . (17) the formula: | i × 1−2G(∆k0+δk)cos(δkl0)+G(∆k0+δk)2 Increasing values of phase mismatch δk lead to greater n(ω ,ω )= g(ω ,ω )2 E(+)(ω +ω )2 s i | s i | | p s i | values of the denominator in the fraction onthe r.h.s. of F(∆k(ω ,ω ))2 . (12) Eq. (17) that result in the decrease of values of the av- s i av ×h| | i eragedsquaredmodulus F(∆k)2 ofphase-matching av SpectrumS of,e.g.,thesignalfieldandphoton-pairgen- h| | i s function. On the other hand, increasing values of devia- erationrateN canthenbederivedusingtheexpressions: tion σ weaken this behavior. For comparison, we consider another type of RPS de- S =~ω dω n(ω ,ω ), (13) s s i s i fined such that z = L+nl +δl where δl is a ran- n 0 n n Z dom declination of th−e n-th boundary. These ‘weakly- N = dω dω n(ω ,ω ). (14) random’ structures are more ordered compared to those s i s i Z Z considered earlier because the change of length of an n- The averaged squared modulus of the phase-matching th domain is compensated by the change in length of function F as determined by the formulain Eq.(10) can an (n+1)-th domain. The averaged squared modulus be written in the form: Fw−r(∆k)2 av of phase-matching function can be de- h| | i rived in this case as follows: 4 1 H(δk)2 F(∆k)2 = (N +1) −| | h| | iav ∆k2 L 1 H(δk)2 4 (cid:18) | − | Fw−r(∆k)2 = N +1 H(δk)[1 H(δk)NL+1] h| | iav (∆k)2( L − +c.c. , (15) −(cid:20) [1−H(δk)]2 (cid:21)(cid:19) + G(∆k +δk)2 exp(iδkl0) 0 δk(ωs,ωi)=∆k(ωs,ωi)−∆k0. Symbol c.c. replaces the | | (cid:20)1−exp(iδkl0) complex-conjugated term. Function H(δk) occurring in 1 exp[iδkl N ] 0 L Eq. (15) is defined as: NL exp(iδkl0) − +c.c. . × (cid:18) − 1−exp(iδkl0) (cid:19) (cid:21)) H(δk)=exp[iδkl ]G(∆k +δk), 0 0 (18) σ2∆k2 G(∆k)=exp . (16) Disorder of the boundary positions manifests itself as a − 4 (cid:18) (cid:19) filter for the averagedsquaredmodulus Fw−r(∆k)2 av h| | i The averagedsquaredmodulus F(∆k)2 ofphase- of phase-matching function, as evident from the expres- av h| | i matchingfunctiongiveninEq.(15)determinesthe aver- sion in Eq. (18). This leads to spectral filtering of the aged spectral density n and behaves as follows. It holds spectral density n. This behavior is qualitatively dif- that H 1 and H = 1 for a fully ordered struc- ferent from that observed in RPS as described by the | | ≤ | | ture. If δk = 0 in a fully ordered structure, H is real formulainEq.(15)indicatingbroadeningofthe spectral (H =1) and the averagedsquaredphase-matching func- density n with increasing values of the deviation σ. tion F(∆k)2 reachesitsmaximumvalue4(N +1)2. Spectral broadening is the most interesting feature of av L h| | i Nonzero phase mismatch δk shifts H into the complex orderedCPPSthatweconsiderhereforcomparison. Po- plane which results in lower values of the mean value sitionsofboundariesinthesestructuresaredescribedby 4 theformulaz = L+nl +ζ′(n N /2)2l2,ζ′ =ζ/∆k , n − 0 − L 0 0 and ζ denotes chirping parameter. The phase-matching function Fchirp(∆k) then takes the form [22]: 2√π Fchirp(∆k)= exp(i∆kN l /2) L 0 i∆k3ζ′l 0 iδk2 p exp [erf(f(N /2)) erf(f( N /2))], × −4∆kζ′ L − − L (a) (cid:18) (cid:19) (19) √ i δk f(x)= − ζ′(∆k +δk)xl + . 0 0 2 " ζ′(∆k0+δk)# p The error function erf is definepd as erf(x) = 2/√π xexp( y2)dy. Detailed inspection ofthe formula 0 − in Eq. (19) reveals that the larger the value of chirping paramReter ζ′ the broader the phase-matching function Fchirp(∆k). (b) As an example, we consider spectrally degenerate (nearly) collinear down-conversion from periodically- FIG.1. (a)Photon-pairgenerationrateN and(b)signal-field poled LiNbO pumped at the wavelength λ0 = 775 nm 3 p spectralwidth ∆Ss (FWHM)as functionsofthenumberNL by a cw laser beam. The signal and idler photons of domains for an ensemble of RPSswith standard deviation thusoccuratthefiber-opticscommunicationwavelength σ equal to 0 m (solid curve), 0.1 ×10−6m (solid curve with λs =λi =1.55µm. Thecrystalopticalaxisisperpendic- ×),0.5×10−6m(solidcurvewith△),1×10−6m(solidcurve ulartothefields’propagationdirectionandis parallelto with ◦), and 2 ×10−6m (solid curve with ⋄); a.u. stands for the vertical direction. All fields are vertically polarized arbitrary units. andsothelargestelementχ(2)ofthesusceptibilitytensor 33 is used. The natural phase mismatch for this configura- tion is compensated by the basic domain length l equal 0 to 9.51535 µm. A structure composed of N = 700 lay- L ers is roughly 6.5 mm long and typically delivers 4 106 × photon pairs per 100 mW of pumping in case of weakly randompositions ofboundaries(smallvalues ofvariance σ). The most striking feature of RPSs is that the photon- pairgenerationrateN increaseslinearlywiththenumber N of domains, independently of the standard deviation L σ of the random positions of boundaries [see Fig. 1(a)]. FIG. 2. Photon-pair generation rate N as it depends on the Standard deviation σ plays the central role in the deter- number NL of domains for an ensemble of ‘weakly-random’ structureswithstandarddeviationσequalto0m(solidcurve mination of spectral widths ∆Ss and ∆Si of the signal with ∗) and 2 ×10−6m (solid curve). and idler fields. The larger the value of deviation σ the broader the signal- and idler-field spectra S and S , as s i documented in Fig. 1b. This behavior is easily under- standable because structures generated with larger val- matching function in Eq. (18), as shown in Fig. 2. The ues of σ have statistically a broader spatial spectrum of greaterthe standarddeviationσ the smallerthe photon- the χ(2)(z)modulationwhichgivesmorefreedomforthe pair generation rate N. As for the signal-field spectral fulfillment of quasi-phase-matching conditions. It holds width ∆Ss its values do not practically depend on the thatthebroaderthesignal-andidler-fieldspectraS and variance σ. s S the smaller the photon-pair generation rate N [com- The behaviorofphoton-pairgenerationasobservedin i pare Figs. 1(a) and (b)]. It reflects the fact that con- RPSs can also be found in ordered CPPSs. Also here structive interference of fields from different domains is photon-pair generation rate N is linearly proportional enhanced in the area outside the central frequencies ω0 to the number N of domains and spectral widths ∆S s L s and ω0 whereas this interference is weaken in the area and ∆S increase with increasing chirping parameter ζ. i i around the central frequencies. The main result of our analysis is that this similarity is The photon-pair generation rate N increases roughly both qualitative and quantitative. For any value of the linearly with the number N of domains also in the chirpingparameterζ thereexistsavalue ofthe standard L case of ‘weakly-random’ structures described by the deviation σ such that spectral widths ∆S and S∆ of s i averaged squared modulus Fw−r(∆k)2 of phase- the generatedsignalandidler fields arethe same. More- av h| | i 5 over(andabitsurprisingly),alsophoton-pairgeneration rates N are comparable. This is illustrated in Fig. 3 for a chirped structure with N = 700 domains. Its signal- L field spectrum S is extraordinarily broad (larger that s 1 µm) for sufficiently largevalues of the chirping parameter ζ [see Fig. 3(a)]. Signal-field spectra S of the same s width can also be generated from RPSs with sufficiently large randomness (i.e., having large values of the deviation σ). Values of the standard deviation σ correspond- (a) ing to the values of chirping parameter ζ are plotted in Fig. 3(b). Photon-pair generation rates N for RPSs and CPPSsaredrawnforcomparisoninFig.3(c)inthiscase. WhereasCPPSsgivebetterphoton-pairgenerationrates N for larger values of chirping parameter ζ, RPSs even slightly overcome on average CPPSs for smaller values of ζ. Moreover, the signal-field spectra S of individ- s ual realizations may even be broader which results in sharper temporal features. On the other hand, these spectra are typically composed of many local peaks (see Fig. 4). RPSs thus represent an alternative broadband and efficientsource of photonpairs with properties com- (b) parable to CPPSs. We note, that histograms of domain lengths corresponding to RPSs are broader compared to those characterizing CPPSs. Alternatively, RPSs and CPPSs can be compared un- dertherequirementofequalphoton-pairgenerationrates N. Valuesofthephoton-pairgenerationrateN decrease with the increasing values of chirping parameter ζ [see Fig. 3(c)]. Transformation curve between standard deviation σ and chirping parameter ζ stemming from the (c) requirement of equal generation rates N is monotonous and is plotted in Fig. 5(a) in the considered case. The FIG. 3. (a) Signal-field spectral width ∆Ss (FWHM) as a function of chirping parameter ζ. (b) Transformation curve correspondingsignal-fieldwidths∆S plottedinFig.5(b) s between the standard deviation σ and chirping parameter reveal that CPPSs provide broader spectra for the most ofvaluesofchirpingparameterζ. However,thedifference pζaiarssguemneinragtiothnerastaemfeorscpheicrtpreadl (wNidcthhirsp,∆soSlsid. cu(rcv)ePwhiothton∗)- in spectral widths in CPPSs and RPSs is not dramatic. and random (N, solid curve) structures and their ratio rN (rN = N/Nchirp, dashed curve) as functions of chirping parameter ζ; NL =700. The abovepresentedresults for RPSs representanen- semble average. In practical applications, properties of individual realizations of a given RPS are naturally important. In Fig. 6, we show generation rates N and signal-fieldwidths∆S for10000realizationsoftheRPS. s Histograms of the generation rates N and signal-field widths ∆S plotted in Figs. 6(b) and 6(c) are close to s Gaussian distributions, in accordance with the central limiting theorem. (a) (b) Itholdsthatthelargerthedeviationσ the smallerthe relative quadratic fluctuations δN and δ∆S of photon- s FIG.4. Signal-fieldspectrumSsfor(a)onetypicalrealization pair generation rates N and signal-field spectral widths of RPS (solid curve) and (b) CPPS (solid curve with ∗) and ∆xSasv,]r(essepeecFtiigv.el7y).[δHxo=weverh,(∆wxe)s2hiaovu/ldhxniaovt,e∆thxat=thxes−e nanormenasleizmebdlesuocfhRthPaStson(esoplihdotcounrviseewmiitthte⋄d);.σS=p2ec.1tr×a1S0s−6amre, rheliative fluctuations are pquite large and can even ap- ζ =2.5×106m−2, NL =700. proach 40 %. 6 (a) N(a.u.) 246753 .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... 1 0.0 0.5 1.0 1.5 2.0 2.5 S (10-6m) (a) s 600 500 400 Nr300 (b) 200 100 FIG. 5. (a) Transformation curve between the standard deviation σ of RPSs and chirping parameter ζ assuming the 0 1 2 3 4 5 6 7 samephoton-pairgenerationratesN. (b)Signal-fieldspectral (b) N(a.u.) widths (FWHM) for random (∆Ss, solid curve) and chirped (∆Sschirp, solid curve with ∗) structures and their ratio rS (rS = Ss/Sschirp, dashed curve) as functions of chirping parameter ζ; NL =700. 600 500 IV. TEMPORAL CORRELATIONS, 400 ENTANGLEMENT TIME Nr300 200 There occurs a strong correlation between possible 100 detection times of a signal photon and its twin from one photon pair because both photons are generated in- 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 staidnecethbeentwoneelinnetahremdeedtieucmtioanttoimneesinostfabnto.thApfihnoittoensdisis- (c) Ss(10-6m) a consequence of dispersion properties of the nonlinear FIG. 6. (a) Photon-pair generation rates N and signal-field mediumthroughwhichbothphotonsatdifferentfrequen- spectral widths ∆Ss for 10000 realizations of RPSs (each re- cies propagate before they leave the crystal. Temporal alization is depicted as a point). (b), (c) Histograms of rates correlationsofthesignalandidlerphotonscanbeconve- N (b) and widths ∆Ss (c); Nr gives the number of samples nientlydescribedusingatwo-photontemporalamplitude with given properties. Solid curvesin (b) and (c) are best-fit defined as: Gaussian curves;σ=2.1 ×10−6m, NL =700. A (t ,t )= vacEˆ(+)(t )Eˆ(+)(t )ψ . (20) A s i h | s s i i | i This amplitude (t ,t ) gives the probability amplitude s i A of detecting a signal photon at time t and an idler pho- s ton at time t . between the signal and idler photons. It can be shown i The simplest experimental method for the determina- that temporal extension of the interference part in the tion of a typical constant characterizing temporal width coincidence-count rate Rn is proportional to entangle- ofthetwo-photondetectionwindow(entanglementtime) ment time under certain conditions. The coincidence- uses a Hong-Ou-Mandel interferometer. In this interfer- count rate Rn as a function of relative time delay τ is ometer, the signal and idler fields mutually interfere on described by the following formula: a beam-splitter and photons leaving the beam-splitter at different output ports are subsequently detected in a coincidence-count measurement. The coincidence-count rate R depends on a mutual time delay τ introduced R (τ)=1 ̺(τ), (21) n n − 7 this case and recast into the following form: 0.5 0.4 ~2 ξ 2 ∞ ρ(τ)= | p| Re exp(iω0τ) dω N 0.3 8ǫ20c2B2 R0 " p Z0 s S,s0.2 ωs(ωp0−ωs)|g(ωs,ωp0−ωs)|2exp(−2iωsτ) 0.1 ∆k(ω ,ω0 ω ),∆k(ω0 ω ,ω ) , ×F s p− s p− s s # 0.0 (cid:0) (cid:1) (26) 0.0 0.5 1.0 1.5 2.0 (10-6m) R = ~2|ξp|2 ∞dω ω (ω0 ω )g(ω ,ω0 ω )2 0 8ǫ2c2 2 s s p− s | s p− s | 0 B Z0 FIG. 7. Relative quadratic fluctuations δ∆Ss of the signal- ∆k(ω ,ω0 ω ),∆k(ω ,ω0 ω ) 2; field spectral width (solid curve) and relative quadratic fluc- × F s p− s s p− s tuations δN of the photon-pair generation rate (solid curve (27) (cid:12) (cid:0) (cid:1)(cid:12) with ∗) as they depend on standard deviation ζ; NL =700. (cid:12) (cid:12) g g(ω0,ω0). Function introduced in Eqs. (26) and 0 ≡ s i F (27)incorporatesphase-matchingconditionsinto thede- scription of temporal properties of photon pairs and is where defined according to the formula: (∆k,∆k′)= F(∆k)F∗(∆k′) ; (28) 1 ∞ ∞ F h iav ρ(τ)= dt dt 1 2 2R 0 Z−∞ Z−∞ Re[ (t ,t τ) ∗(t ,t τ) ], (22) phase-matching function F has been introduced in 1 2 2 1 av 1 h∞A −∞ A − i Eq. (10). R = dt dt (t ,t )2 . (23) 0 2 1 2h|A 1 2 | iav Considering RPSs with fluctuations of boundaries de- Z−∞ Z−∞ scribed by a Gaussian distribution in Eq. (8) function F in Eq. (28) takes the form: Inserting Eqs. (20) and (4) for the two-photon temporal 4 amplitude and quantum state ψ , respectively, into (∆k,∆k′)= ˜(∆k,∆k′) Eqs. (22) aAnd (23) we arrive at the| eixpressions: F ∆k∆k′F exp[ i(∆k ∆k′)N l ], (29) L 0 × − − 1 H(Dk)NL+1 H(∆k) ˜(∆k,∆k′)= − F 1 H(Dk) "H(∆k)+H(Dk) π~2 1 ∞ ∞ − ρ(τ)= 4ǫ20c2B2R0Re"Z0 dωs Z0 dωi ωsωi × −H(∆k)1−1[+−HH((∆∆kk))]NL Φ(ωs,ωi)Φ∗(ωi,ωs) avexp(iωiτ)exp( iωsτ) , 1 [H(Dk)]NL h i − # H(Dk) − − 1 H(Dk) ! (24) − R = π~2 ∞dω ∞dω ω ω Φ(ω ,ω )2 . +(∆k ∆k′) ; (30) 0 4ǫ2c2 2 s i s ih| s i | iav ←→− # 0 B Z0 Z0 (25) Dk = ∆k ∆k′. Function H occurring in Eq. (30) has been defin−ed in Eq. (16). Symbol (∆k ∆k′) in ←→ − Eq. (30) replaces the term that is obtained by the indi- For simplicity, we further assume cw-pumping with cated exchange applied to the preceded term inside the amplitude ξ at frequency ω0, i.e. E(+)(ω ) = ξ δ(ω brackets. p p p p p p− ω0). Formulas in Eqs. (24) and (25) can be simplified in Considering ‘weakly-random’structures, the following p 8 form of the function ˜ can be derived: F 1 exp[iDkl (N +1)] ˜w−r(∆k,∆k′)=G(Dk) − 0 L F 1 exp(iDkl ) 0 − G(∆k)G(∆k′) 1 exp(i∆kN l ) L 0 + − 1 exp( i∆k′l ) 1 exp( i∆kl ) " 0 0 − − − − 1 exp( iDkN l ) L 0 + − − 1 exp( iDkl ) 0 ! − − +(∆k ∆k′) . (31) FIG. 8. Entanglement time ∆τHOM (FWHM) as a function ←→− # of chirping parameter ζ for ensemble of RPSs with standard deviations σ derived from the curve in Fig. 2b (solid curve) On the other hand, function ˜chirp attains a simple and CPPSs (solid curvewith ∗); NL =700. F form in case of CPPSs: ˜chirp(∆k,∆k′)=Fchirp(∆k)Fchirp∗(∆k′), (32) of the signal and idler fields despite their complex pro- F files. We note that the entanglement time ∆τHOM is where the formula for Fchirp is written in Eq. (19). determined by a temporal extension (FWHM) of the Characteristics of temporal correlations (correlation coincidence-countinterferencepatternformedbytherate time) between the signal and idler fields can also be ob- R (τ). Entanglement time ∆τHOM thus shortens with n tained from the measurement of sum-frequency inten- increasing values of the standard deviation σ for RPSs. sity in a nonlinear medium combining the signal and The dependence of entanglement time ∆τHOM on the idler photons and having a sufficiently high value of deviation σ for an ensemble of RPSs composed of 700 χ(2)sum nonlinearity. Thisprocessallowsustodetermine domains is shown in Fig. 8. We can see in Fig. 8 that the temporal correlation function Isum of intensities of entanglement times ∆τHOM can be as short as severalfs the signal and idler fields. Intensity Isum of the sum- for sufficiently largevalues of the deviationσ. This indi- frequency field is given along the formula: catesthattemporalquantumcorrelationscanbeconfined ∞ 2 into anintervalcharacterizingone opticalcycleprovided Isum(τ)=ηsum dt vacEˆ(+)(t)Eˆ(+)(t τ)ψ , thatspectralphasevariationsinthesignalandidlerfields h | s i − | i Z−∞ (cid:12) (cid:12) arecompensated. Entanglementtimes∆τHOM ofCPPSs (cid:12) (cid:12)(33) with the same signal-field spectral widths ∆S are plot- (cid:12) (cid:12) s ted in Fig. 8 for comparisonthat revealsnearly identical where constant ηsum incorporates the value of χ(2)sum entanglement times of both types of structures. Typical nonlinearity and quantum detection efficiency. coincidence-count interference patterns given by R for The general formula in Eq. (33) can be recast into n photon pairs generated in both types of structures are the following form using the expression for function F compared in Fig. 9. They demonstrate a close similar- in Eq. (28): ity ofphoton-pairsbehaviorina Hong-Ou-Mandelinter- ηsum~2 ∞ ferometer. There occur typical oscillations at the shoul- Isum(τ)= 4ε2c2 2 dωp|Ep(+)(ωp)|2 dersoftheinterferencedips. Whereasregularoscillations 0 B Z0 characterizeCPPSs,irregularoscillationswithlargeram- ∞ ∞ dω ω (ω ω ) dω′ ω′(ω ω′) plitudes occur for individual realizations of RPSs. How- s s p− s s s p− s ever, widths of interference dips remain practically un- Z0 q Z0 q g(ω ,ω ω )g∗(ω′,ω ω′)exp[ i(ω ω′)τ] changed for different realizations of RPSs. × s p− s s p− s − s− s (∆k(ω ,ω ω ),∆k(ω′,ω ω′)). (34) Correlationtimes∆τsumemergingfromsum-frequency ×F s p− s s p− s generationareingenerallongerthanentanglementtimes When deriving Eqs.(33) and (34) we have assumedthat ∆τHOM observed in a Hong-Ou-Mandel interferometer thenonlinearmediuminwhichsum-frequencygeneration because of a strong phase modulation along the wide occursisideallyphasematchedforfrequenciespresentin signal- and idler-field spectra S and S (see Fig. 10). s i the signal and idler fields. Correlation times ∆τsum can be even an order of mag- A detailed analysis of the expression that gives the nitude greater compared to entanglement times ∆τHOM coincidence-count rate R in a Hong-Ou-Mandel inter- for structures with ultra-broadband spectra. However, n ferometer reveals an important property: the rate R phase modulation along the spectrum can be compen- n does not depend on phase variations along the signal- sated to certain extent which gives shorter correlation and idler-field spectra in cw regime. This property is times ∆τsum. CPPSs have more regular spectral phase frequently referred as nonlocal dispersion cancellation behavior (as demonstrated in Fig. 10) and quadratic [31, 32]. It follows that entanglement time ∆τHOM is phasecompensationisusuallysufficienttoprovidewave- inversely proportional to spectral widths ∆S and ∆S packets several fs long. As for individual realizations of s i 9 (a) FIG. 9. Coincidence-count rate Rn as it depends on relative time delay τ for one realization of RPS (solid curve), CPPS (solid curve with ∗), and an ensemble of RPSs (solid curve with ⋄); σ=2.1×10−6m, ζ =2.5×106m−2, NL =700. (b) FIG. 11. Sum-frequency field intensity Isum as a function of relative time delay τ for one realization of RPS (solid curve),CPPS (solid curvewith ∗), and an ensemble of RPSs (solid curve with ⋄). In (a) quadratic chirp in the signal- field amplitude spectrum is compensated for one realization of RPS and CPPS; in (b) complete spectral phase compensation is assumed. The curves are normalized such that FIG. 10. Phase ϕ of the two-photon spectral amplitude R∞ dτIsum(τ) = 1; σ = 2.1×10−6m, ζ = 2.5×106m−2, qΦu(eωnsc,yωp02ω−s/ωωs)p0 afosritondeepreeanldizsaotinonnoorfmRaPliSzed(sosliigdnaclu-firveeld) afrned- N−L∞=700. CPPS(solidcurvewith∗);σ=2.1×10−6m,ζ =2.5×106m−2, NL =700. V. CORRELATIONS IN THE TRANSVERSE PLANE Inordertodescribespatialpropertiesofthesignaland idler beams (in the transverse plane) a simple general- RPSs, quadratic compensation is less efficient because izationof the model presentedin Sec. II has to be devel- of more irregular phase spectral behavior. Despite this oped. The inclusion of phase-matching conditions also values of temporal constants typical for chirped struc- in the directions along the x and y axes and assumption tures can be approached [see Fig. 11(a)]. Provided that ofspectrally-flattransversepump-beamprofileE (x,y) an ideal phase compensation is reached, both types of p⊥ result in the following separable form of a two-photon structures give comparable results [see Fig. 11(b)] and spectralamplitudeΦthatadditionallydependsonradial are capable to generate photon pairs with wave-packets (ϑ , ϑ ) and azimuthal (ϕ , ϕ ) signal- and idler-field extending over the duration of one optical cycle. Ex- s i s i emission angles (see Fig. 12): perimentally, pulse shapers have been developed for this task and their capabilities in the area of photon pairs Φ(ω ,ω ,ϑ ,ϕ ,ϑ ,ϕ )=Φ (ω ,ω ,ϑ ,ϕ ,ϑ ,ϕ ) s i s s i i z s i s s i i havealreadybeendemonstrated[33]. Comparisonofthe Φ (ω ,ω ,ϑ ,ϕ ,ϑ ,ϕ ), (35) xy s i s s i i results obtainedwith quadratic andidealcompensations × reveals that correlation times ∆τsum are approx. two where function Φz arises from phase-matching conditions in the z direction and function Φ originates in times larger if we restrict ourselves to quadratic com- xy phase-matchingconditionsinthetransversexyplane(see pensation. Also the value of quadratic chirp that needs Fig.11). FunctionΦ canbederivedinanalogywiththe compensation differs for individual realizations of RPS. z formula in Eq. (5): This requires an adaptive phase compensator. On the other hand phase compensation in case of CPPS can be Φ (ω ,ω ,ϑ ,ϕ ,ϑ ,ϕ )=g(ω ,ω )E(+)(ω +ω ) z s i s s i i s i p s i reached in a simpler way, e.g., by inserting a peace of F(∆k(ω ,ω ,ϑ ,ϕ ,ϑ ,ϕ )), (36) suitable material of defined length [34, 35]. Despite this × s i s s i i RPSsarechallengingbothforbasicphysicalexperiments where the stochastic function F has been introduced in as well as metrology applications. Eq. (6). Phase-matching conditions in the xy plane give 10 y x φs k s ϑ s ϑ z φi i ki FIG.12. Geometricschemeforthedescriptionofspatialprop- erties. Directionofthesignal-(idler-)fieldwavevectorks(ki) isgivenbyradial ϑs (ϑi)andazimuthalϕs (ϕi) emission angles. (a) the function Φ the following form: xy ∞ ∞ Φ (ω ,ω ,ϑ ,ϕ ,ϑ ,ϕ )= dx dyE (x,y) xy s i s s i i p⊥ Z−∞ Z−∞ exp(i∆k x+i∆k y) (37) x y × that includes a pump-beam amplitude profile E (x,y) p⊥ in the transverse plane. Assuming normal incidence of the pump beam, the cartesian components of nonlinear phase mismatch in Eqs. (36) and (37) can be written as: ∆k =k (ω )sin(ϑ )sin(ϕ )+k (ω )sin(ϑ )sin(ϕ ) (b) x s s s s i i i i ∆k =k (ω )sin(ϑ )cos(ϕ )+k (ω )sin(ϑ )cos(ϕ ) y s s s s i i i i ∆k =k (ω +ω ) k (ω )cos(ϑ ) k (ω )cos(ϑ ). z p s i s s s i i i − − (38) We assume a Gaussian pump-beam transverse profile in numerical calculations: E (x,y) = p⊥ 1/(π∆x ∆y )exp[ (x/∆x )2 (y/∆y )2] and ∆x p p p p p − − (∆y ) stands for the pump-beam width along the x (y) p direction. We first pay attention to transverse properties of the signal beam only. Its spectral density s defined as s (c) s (ω ,ϑ ,ϕ )=sin(ϑ ) dω dϑ sin(ϑ ) dϕ s s s s s i i i i Z Z Z FIG.13. Mapofsignal-field spectraldensityss asit depends |Φ(ωs,ωi,ϑs,ϕs,ϑi,ϕi)|2 (39) onsignal-field radial emission angleϑs for(a) onerealization ofRPS,(b)anensembleofRPSs,and(c)CPPS;ϕs =0deg, dependsonthesignal-fieldradial(ϑs)andazimuthal(ϕs) σ=2.1×10−6m, ζ =2.5×106m−2, NL =700. emission angles. As we study photon-pair emission near the collinear geometry, the signal beam (as well as the idler beam) has rotational symmetry along the z axis. CPPSs [compare Figs. 13(b) and 13(c)]. In these cases The dependence of spectral density s on signal-field ra- s spectral splitting is observed [36]. This behavior origi- dial emission angle ϑ is shown in Fig. 13. Investigating s natesinphase-matchingconditionsalongthez direction. one realization of RPS we observe a typical ’strip-like’ behavior depicted in Fig. 13(a). Fixing the value of radial emission angle ϑ spectrum s (ω ) is composed of Integrationofspectraldensitiess overthesignal-field s s s s many peaks occurring at positions specific for the stud- frequency ω gives us densities n of photon-pair num- s s ied realization [compare also with Fig. 4(a)]. Each peak bersthatareplottedinFig.14forthestructuresstudied changes continuously its central frequency as the radial in Fig. 13. Whereas the profile of density n (ϑ ) is com- s s emissionangleϑ moves. Wenotethatthisistypicalalso plex for one realization of RPS, typical shapes with one s forlayeredstructuresthatformband-gaps[6]. Averaging maximum around a nonzero value of ϑ characterize the s over many realizations of RPSs smoothes this ’strip-like’ profiles of density n (ϑ ) for an ensemble of RPSs and s s behavior [see Fig. 13(b)] and leads to that resembling CPPS.