Configurable spatio-temporal properties in a photon-pair source based on spontaneous four wave mixing with multiple transverse modes
Daniel Cruz-Delgado, Jorge Monroy-Ruz Angela M. Barragan, Erasto Ortiz-Ricardo, Hector Cruz-Ramirez, Roberto Ramirez-Alarcon, Karina Garay-Palmett, Alfred B. U'Ren
CConfigurable spatio-temporal properties in a photon-pair source basedon spontaneous four wave mixing with multiple transverse modes
Daniel Cruz-Delgado , Jorge Monroy-Ruz , Angela M. Barragan-Diaz , Erasto Ortiz-Ricardo ,Hector Cruz-Ramirez , Roberto Ramirez-Alarcon , Karina Garay-Palmett and Alfred B. U’Ren ∗ Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, apdo. postal 70-543, 04510 D.F., Mexico Departamento de ´Optica, Centro de Investigaci´on Cient´ıfica y de Educaci´on Superior de Ensenada, Apartado Postal 360Ensenada, BC 22860, Mexico.
Compiled October 16, 2018We present an experimental and theoretical study of photon pairs generated by spontaneous four wave mixing(SFWM), based on birefringent phasematching, in a fiber which supports more than one transverse mode.We present SFWM spectra, obtained through single-channel and coincidence photon counting, which exhibitmultiple peaks shown here to be the result of multiple SFWM processes associated with different combinationsof transverse modes for the pump, signal, and idler waves. c (cid:13)
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The generation of photon pairs with configurablespatio-temporal entanglement is currently an importantgoal of quantum optics [1]. In this context, transverseoptical confinement is a valuable resource for source de-sign based, both, on spontaneous parametric downcon-version (SPDC) and on spontaneous four wave mixing(SFWM). In this work we rely on the latter process, in amedium with a third-order nonlinearity, for which pairsof pump photons are annihilated leading to the emissionof signal and idler photon pairs. While the use of mul-tiple transverse waveguide modes has been studied forSPDC [2–5] and for stimulated four wave mixing [6], itremains unexplored for SFWM. In the limit of no con-finement, as in bulk-crystal SPDC, the emission anglesare strongly correlated to the frequencies leading to com-plex spatio-temporal entanglement characteristics [7]. Inthe opposite limit, where generation occurs in a singletransverse mode, spectral properties become decoupledfrom the waveguide-determined spatial structure of thephoton pairs, also resulting in spatial factorabilty. In thisletter we the study intermediate regimes where the con-trolled presence of more than one transverse mode be-comes an effective tool for tailoring the spatio-temporalphoton-pair entanglement.We study SFWM in a fiber which supports more thanone transverse mode for the pump and the signal/idlerwaves. Under these circumstances, it is possible to obtainthe coherent addition of distinct SFWM processes corre-sponding to different combinations of transverse modesfor the four waves involved. Interestingly, phasematchingcouples the emission frequencies to specific combinationsof transverse modes so that appropriate spectral filter-ing may be used to post-select specific SFWM processes.The experimenter can then ensure spatial factorability,if a single process is post-selected, or can permit a con-trolled type and degree of spatio-temporal entanglement,if several processes are post-selected.Here we study, both experimentally and theoretically, cross-polarized SFWM in a birefringent fiber [8–12],where we concentrate on the as yet unexplored effects ofthe presence of more than one transverse mode. We mea-sure the spectral structure of the photon pairs by single-channel and coincidence photon counting. A careful com-parison with simulations allows us to for the first timeidentify the distinct spectral peaks observed with par-ticular combinations of transverse modes for the pumpand signal/idler photons; we thus determine the SFWMphoton-pair spatio-temporal structure. In contrast withrelated guided-wave SPDC results [2] which resort to nu-merical mode analysis, the transverse mode structure inour fibers can be closely approximated by well-known lin-early polarized ( LP ) modes. Also, our spectral emissionpeaks do not overlap one another (with one exception,see below) ensuring mode orthogonality and the abilityto fully separate the processes by frequency.The two-photon component of the SFWM state, whichexhibits entanglement in the transverse mode and spec-tral degrees of freedom, may be written as | Ψ (cid:105) = (cid:88) m (cid:88) n (cid:90) dω s (cid:90) dω i G mn ( ω s , ω i ) | ω s ; m (cid:105)| ω i ; n (cid:105) , (1)where | ω ; m (cid:105) µ ≡ a † ( ω ; m ) | (cid:105) µ is a single-photon Fockstate with frequency ω , transverse mode m , and for wave µ (with µ = s, i for signal and idler). G mn ( ω s , ω i ) is thejoint amplitude involving a signal photon in transversemode m and an idler photon in mode n . It is written as G mn ( ω s , ω i ) = (cid:88) p (cid:88) q W p W q γ pqmn (cid:90) dωα ( ω ) × α ( ω s + ω i − ω )sinc (cid:20) L k pqmn (cid:21) , (2)with sums over the pump transverse modes [13], andwhere W p is the fraction of the pump power coupled into1 a r X i v : . [ qu a n t - ph ] M a y he fiber, in mode p ; the sum over all W ’s is unity. α ( ω )is the pump spectral amplitude, L is the fiber length and∆ k pqmn = k p ( ω )+ k q ( ω s + ω i − ω ) − k m ( ω s ) − k n ( ω i ), with k µ ( ω ) the wavenumber for mode µ ; note that a nonlinearterm proportional to pump power has been disregarded. γ pqmn is the nonlinearity for a SFWM process involvingmodes p , q , m , and n , given by γ pqmn ∝ (cid:90) dx (cid:90) dy f p ( x, y ) f q ( x, y ) f ∗ m ( x, y ) f ∗ n ( x, y ) , (3)in terms of f m ( x, y ), the transverse field distribution formode m . The single-photon spectrum, corresponding tofrequency-resolved single-channel detection, is then S s ( ω ) ≡ (cid:88) µ (cid:104) Ψ | a † s ( ω ; µ ) a s ( ω ; µ ) | Ψ (cid:105) = (cid:88) m (cid:88) n (cid:90) dω i | G mn ( ω, ω i ) | . (4)Our experimental setup is sketched in Fig. 1. We em-ploy as pump for the SFWM process a picosecond mode-locked Ti:sapphire laser (76MHz repetition rate and 2nmbandwidth centered at 692nm, uncorrected for possiblechirp). A prism-based band-pass filter (PBPF) with a slitconfigured to transmit the entire laser spectrum whileblocking spurious photons at other frequencies filters thelaser output. The filtered pump beam is coupled into a12cm length of bow-tie birefringent fiber (BRF; HB800Gfrom Fibercore) using an aspheric lens with 8mm focallength; the polarization is adjusted with a half-waveplate(HWP1) so that the polarization becomes parallel tothe slow axis of the fiber. The pump power, measuredonce out-coupled from BRF, is ∼ . Note that POL suppresses any SFWMphoton pairs resulting from processes with the same po-larization for all four waves.The signal and idler photon pairs are frequency non-degenerate, emitted in spectral bands placed symmetri-cally around the pump. Thus, they may be split usinga dichroic mirror (DM; >
98% reflectivity within 350-676nm and >
93% transmissivity within 695-950nm). Inorder to suppress any remaining pump photons, the low-wavelength arm is transmitted through a bandpass filter slow axis fast axispump signalidlerfiber profilePBPFMONBRFAPD1APD2 HWP1HWP2POLBP2BP1MMF1 MMF2DM −40 −20 0 20 40−0.0100.01 cladding SAP c o r e claddingSAP i n d e x d i ff e r e n c e transverse position ( μ m)inner claddingfiber index pofile Fig. 1. Top: experimental setup, bottom: fiber index pro-file.(BP1; >
92% transmissivity within 584-676nm and > >
93% transmissivity within768-849nm and > c o i n c i den c e c oun t s pe r s
590 610 630 65005001000150020002500 idler wavelength (nm) c o i n c i den c e c oun t s pe r s
770 790 810 830 850signal wavelength (nm) c oun t s pe r s A(01)B(11)G(11)C(01)F(01)A(01)B(11) G(21) C(11)F(21) a bc de fIIIIIIIVIVIIIIII coincidence countsaccidental coincidence counts
Fig. 2. Spectrally-resolved count rates for: single-channeldetection (1st row) and coincidence detection (2nd row).Accidentals-subtracted coincidence count rate (3rd row).Note that in the coincidence data (second row ofFig. 2), the broadband background attributed to sponta-neous Raman scattering is essentially suppressed. In thedata with accidental coincidences subtracted (third rowof Fig. 2), the background is further suppressed resultingin four remarkably clean SFWM peaks. As we will show,these four peaks are related to distinct SFWM processesfor different phasematched combinations of transversespatial modes for the pump, signal and idler modes.We have used as non-linear medium a “bow-tie” fiberwith a cross-section shown in the inset of Fig. 1, forwhich the GeO -doped SiO core is surrounded by aninner SiO cladding, and flanked by two low-index B O -doped SiO stress applying parts (SAP’s) in the charac-teristic shape of bows. In order to obtain a theoreticaldescription of the SFWM process, we first model thebow-tie fiber as a step-index fiber, determined by twoparameters: the core radius r and the numerical aperture N A . We solve the characteristic equations to obtain theeffective index of refraction n lm ( ω ) for transverse mode LP lm . Second, we model the birefringence resulting fromthe SAP’s by adding a constant offset ∆ to the index re-fraction for light guided in the fiber and polarized alongthe x direction. Thus, we obtain n lm,x ( ω ) = n lm ( ω ) + ∆and n lm,y ( ω ) = n lm ( ω ) for the x / y polarizations [14](see inset in Fig. 1). For a given fiber, as specified by thethree parameters { r, N A, ∆ } , we can then determine thewavenumber for the µ polarization (with µ = x, y ) and the LP lm mode: k lm,µ ( ω ) = n lm,µ ( ω ) ω/c . Note that theactual fiber is characterized by a complex, azimuthally-asymmetric index of refraction gradient, so that the fibermodel used represents a considerable simplification.For the specific values of the parameters { r, N A, ∆ } which characterize our bow-tie fiber, the following trans-verse modes are supported at the operating frequencies: LP , LP and LP . A given SFWM process involvesa particular phasematched combination of transversemodes for each of the four waves involved. Adopting theconvention that the high-wavelength ( λ > λ p ) peak cor-responds to the signal photon and the low-wavelength( λ < λ p ) peak corresponds to the idler photon, we haveidentified 7 distinct processes which may occur (i.e. forwhich ∆ k pqmn ≈ γ pqmn has appreciablevalues), as summarized in Table 1. Note that while inthese processes the two pumps are frequency-degenerate,they can be non-degenerate in transverse mode.process p1 p2 i( λ < λ p ) s( λ > λ p )A 01 01 01 01B 11 11 11 11C 01 11 11 01D 01 11 01 11E 21 21 21 21F 01 21 21 01G 11 21 21 11Table 1. Allowed SFWM processes, involving the pumps(p1 and p2), signal(s), and idler(i) waves. Two-digitnumbers are the LP lm mode labels.For a theoretical description, our aim is to determinevalues for the parameters { r, N A, ∆ } , desirably close tothose provided by the fiber manufacturer, which lead tosimulation results which best fit the measured SFWMspectra in Fig. 2(e) and (f). We test the possibility ofeach of the four matched peaks (I through IV) beingexplained by any of the seven processes in Table 1. Wevary the birefringence within the range 4 . < ∆ < . . µ m < r < . µ m and 0 . < N A < . k = 0 (one foreach pair of peaks) in { r, N A } space, where a quadruplecontour intersection indicates the desired solution. Thisleads, on the one hand, to the conclusion that only com-binations of processes A,B,G, and C or A,B,G,and F canbe responsible for peaks I, II, III and IV, respectively.On the other hand this also gives us a prediction for thevalues of the three parameters: ∆ = 4 . ± . × − , r = 1 . ± . µ m, and N A = 0 . ± .
02. It is interest-ing to note that the core diameter 2 r obtained (3 . µ m) iscompatible with the mean field diameter provided by themanufacturer as 3 . µ m < d MF D < . µ m; likewise, the3irefringence ∆ is compatible with the value provided,as ∆ > . × − .The numerical aperture obtained, however, is outsidethe range provided of 0 . < N A < .
18. In order tounderstand this difference, we present at the bottom ofFig. 1 the index difference (between the index of re-fraction and a reference value for SiO ) profile alongthe bow-tie as measured by Fibercore Ltd. at a wave-length of 670nm for the specific batch of fiber thatwe used. The N A is given by (cid:112) n − n , where n /n are the core/cladding indices of refraction; the value of N A provided was obtained by taking a representativevalue of the core index as n (see bottom of Fig. 1),and the inner cladding index as n . It is interesting tonote that taking the SAP index as n leads to a valueof N A ≈ .
24 much closer to that obtained from oursimulation-experiment comparison; this suggests that infact the guided mode is large enough to reach the innerportions of the SAP’s [15].In Fig. 3a and b we show phasematching contours (de-fined by the condition ∆ k = 0), assuming the fiber pa-rameters found above for each of the combinations ofprocesses A through G of table 1, as a function of pumpand signal/idler wavelengths. For our pump wavelength( λ p = 692nm; note the horizontal dotted line), we mayread out from panel a the idler ( λ < λ p ) and from panelb the signal ( λ > λ p ) generation wavelengths. Panels cand d show the experimental data in magenta (similarto panels e and f in Fig. 2), overlapped with our simu-lation results in black (based on Eq. 4). We have addedto each peak (also in Fig. 2 e and f) a label indicat-ing with a capital letter the associated SFWM process(from Table 1) and with a two-digit number the associ-ated LP mode. Note that in order to compute the the-ory curves we need values for the pump power fraction W lm coupled into mode LP lm for the three modes in-volved. Three equations are obtained from: i) the ratioof the heights of low-wavelength peaks I and II is propor-tional to ( W /W ) , ii) the ratio of the height of low-wavelength peaks II and III is proportional to W /W ,and iii) W + W + W = 1. Note that while the heightsof peaks I through III are used as inputs in this calcu-lation, the height of peak IV can be predicted from theother heights leading to a useful self-consistency check.Note that transverse-mode entanglement can occur forcertain combinations of SFWM processes. For example,a hyper-entangled state with a Bell state embedded intransverse mode would result if processes C and D werephasematched at identical signal/idler frequencies.In summary, we have demonstrated the generation ofphoton pairs in a birefringent bow-tie fiber by multipleSFWM processes resulting from different phasematchedcombinations of transverse modes for the pump, signal,and idler waves. We associate the measured spectralpeaks with distinct SFWM processes. Because the al-lowed combinations of modes are correlated to emissionfrequency, spectral filtering can be used to enable or dis-able specific processes and thus configure the resulting pu m p w a v e l eng t h ( μ m ) A F F A D a bd μ m) e m i tt ed f l u x ( a . u . ) c experiment theory signal wavelength ( μ m) LP LP LP A(01)B(11) G(21) C(11)F(21) C(01)F(01)G(11) B(11)A(01)
C CAB G CF F CABG
Fig. 3. Idler-pump and signal-pump phasematching di-agrams for processes A-G (1st row). Comparison of ex-perimental and simulated SFWM spectra (2nd row).spatio-temporal entanglement to specific needs.This work was supported by CONACYT, Mexico andby DGAPA, UNAM.
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