Photon-pair generation in arrays of cubic nonlinear waveguides

Transcription

1 Photon-air generation in array of cubic nonlinear waveguide Alexander S. Solntev, Andrey A. Sukhorukov, Dragomir N. Nehev, and Yuri S. Kivhar Nonlinear Phyic Centre and Centre for Ultrahigh-bandwidth Device for Otical Sytem (CUDOS), Reearch School of Phyic and Engineering, Autralian National Univerity, Canberra ACT 0200, Autralia Abtract: We tudy hoton-air generation in array of cubic nonlinear waveguide through ontaneou four-wave mixing. We analyze numerically the quantum tatitic of hoton air at the array outut a a function of waveguide dierion and um beam ower. We how flexible atial quantum tate control uch a um-ower-controlled tranition between bunching and anti-bunching correlation due to nonlinear elf-focuing Otical Society of America OCIS code: ( ) Quantum otic; ( ) Nonlinear otic; four-wave mixing; ( ) Array waveguide device. Reference and link 1. J. C. F. Matthew, A. Politi, A. Stefanov, and J. L. O Brien, Maniulation of multihoton entanglement in waveguide quantum circuit, Nature Photonic 3, (2009). 2. A. Politi, J. C. F. Matthew, and J. L. O Brien, Shor quantum factoring algorithm on a hotonic chi, Science 325, (2009). 3. L. Sanoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crei, R. Ramoni, and R. Oellame, Polarization entangled tate meaurement on a chi, Phy. Rev. Lett. 105, (2010). 4. A. Peruzzo, M. Lobino, J. C. F. Matthew, N. Matuda, A. Politi, K. Poulio, X. Q. Zhou, Y. Lahini, N. Imail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thomon, and J. L. O Brien, Quantum walk of correlated hoton, Science 329, (2010). 5. A. S. Solntev, A. A. Sukhorukov, D. N. Nehev, and Y. S. Kivhar, Sontaneou arametric down-converion and quantum walk in array of quadratic nonlinear waveguide, Phy. Rev. Lett. 108, (2012). 6. A. Rai and D. G. Angelaki, Dynamic of nonclaical light in integrated nonlinear waveguide array and generation of robut continuou-variable entanglement, Phy. Rev. A 85, (2012). 7. J. E. Sharing, K. F. Lee, M. A. Foter, A. C. Turner, B. S. Schmidt, M. Lion, A. L. Gaeta, and P. Kumar, Generation of correlated hoton in nanocale ilicon waveguide, Ot. Exre 14, (2006). 8. H. Takeue, Y. Tokura, H. Fukuda, T. Tuchizawa, T. Watanabe, K. Yamada, and S. ichi Itabahi, Entanglement generation uing ilicon wire waveguide, Al. Phy. Lett. 91, (2007). 9. D. N. Chritodoulide, F. Lederer, and Y. Silberberg, Dicretizing light behaviour in linear and nonlinear waveguide lattice, Nature 424, (2003). 10. L. G. Helt, M. Licidini, and J. E. Sie, How doe it cale? comaring quantum and claical nonlinear otical rocee in integrated device, J. Ot. Soc. Am. B 29, (2012). 11. M. Grafe, A. S. Solntev, R. Keil, A. A. Sukhorukov, M. Heinrich, A. Tunnermann, S. Nolte, A. Szameit, and Y. S. Kivhar, Bihoton generation in quadratic waveguide array: A claical otical imulation, Sci. Re. 2, 562 (2012). 12. J. C. F. Matthew, K. Poulio, J. D. A. Meinecke, A. Politi, A. Peruzzo, N. Imail, K. Wrhoff, M. G. Thomon, and J. L. O Brien, Simulating quantum tatitic with entangled hoton: a continuou tranition from boon to fermion, htt://arxiv.org/ab/ (2011). 13. J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, Otical oliton in a ilicon waveguide, Ot. Exre 15, (2007). 14. A. V. Gorbach, W. Ding, O. K. Staine, C. E. de Nobriga, G. D. Hobb, W. J. Wadworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, Satiotemoral nonlinear otic in array of ubwavelength waveguide, Phy. Rev. A 82, (2010). # $15.00 USD Received 5 Se 2012; revied 19 Oct 2012; acceted 20 Oct 2012; ublihed 19 Nov 2012 (C) 2012 OSA 19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27441

2 15. C. J. Benton and D. V. Skryabin, Couling induced anomalou grou velocity dierion in nonlinear array of ilicon hotonic wire, Ot. Exre 17, (2009). 1. Introduction With the develoment of quantum otic it i now feaible to realize comlex quantum logic algorithm, however an increaing number of otical element i required for multi-te quantum imulation. Integrated hotonic circuit have been conidered a the olution to thi challenge, a they are intrinically calable and interferometrically table 1]. Integrated realization of multi-hoton entanglement 1], quantum factoring algorithm 2], and olarization entanglement 3] have already been demontrated exerimentally. One of articularly intereting device in integrated hotonic i a waveguide array (WGA). Recently WGA have been hown to generate unuual and trongly non-claical correlation of hoton air roagating in the regime of quantum walk 4]. Combining quantum walk with hoton air generation in nonlinear waveguide array further oen the oibility for enhanced atial quantum tate control and imroved clarity of atial correlation 5]. While WGA with quadratic nonlinearity have been recently tudied 5, 6], we exect that WGA with cubic nonlinearity can rovide an entirely new realm of all-otical control of the quantum hoton tatitic. With on-chi hoton-air ource baed on ontaneou four-wave mixing (SFWM) being readily available 7, 8], it become of much interet to tudy the quantum tate dynamic in WGA with cubic nonlinear reone. In thi work, we decribe the generation of correlated hoton air through SFWM in a cubic nonlinear WGA and analyze the interlay between SFWM hae-matching and WGA dierion for the generation of comlex entangled quantum tate. We alo demontrate the otential of Kerr-baed elf-hae modulation (SPM) and cro-hae modulation (XPM) for quantum tate control by invetigating a cae of tronger um with ecial ectral filtering. 2. SFWM in WGA at low um ower We conider lole near-degenerate SFWM with ignal and idler frequencie being cloe to the um frequency, with the ret of hoton-air being filtered out. In thi cae the mode rofile and couling between waveguide remain aroximately the ame for um, ignal and idler hoton. It ha been demontrated that even with le than 0.5% difference between thee frequencie, the um can be effectively filtered at the outut of the ytem, and hoton air correlation can be meaured with high ignal-to-noie ratio 7, 8]. The main difference between ontaneou four-wave mixing in bulk or ingle waveguide in comarion to WGA Fig. 1(a)] i a different atial dierion leading to modified hae-matching. We begin the analyi of SFWM in WGA by tudying the four-wave-mixing hae-matching for lane wave, Δβ = 2β β β i. Here β,,i are the roagation contant for um, ignal and idler. In a WGA they deend on normalized tranvere momenta k,,i a β,,i = β (0),,i + 2C co(πk,,i), where C i the couling coefficient between the waveguide 9]. Meanwhile the overla between interacting Bloch wave 5, 9] can be written a n exiπ(2k k ki )n], where n i a waveguide number. Since n ex(2iπk n iπk n iπki n)=2π N δ(2πk πk πki 2πN), where N Z, then the tranvere momenta k,,i for eriodical olution will atify the condition 2k = k + ki. Therefore we can write an analytical exreion for lane-wave four-wave mixing hae-mimatch in a WGA: Δβ = Δβ (0) + 4C co(π(k + k i )/2) 2C co(πk ) 2C co(πk i ). (1) The atial dierion and therefore the hae-matching condition for SFWM in WGA are qualitatively different from that in bulk, oening new oibilitie for generating hoton air with unuual quantum tatitic. # $15.00 USD Received 5 Se 2012; revied 19 Oct 2012; acceted 20 Oct 2012; ublihed 19 Nov 2012 (C) 2012 OSA 19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27442

3 Fig. 1. (a) Schematic and (b) ower rofile of linear um roagation in a WGA for inut um amlitude E () n =0 (0)=10 5. (c-f) Photon-air correlation in (c,e,g) real-ace and (d,f,h) k-ace for different grou-velocity dierion: (c,d) anomalou Δβ (0) = 18, (e,f) zero Δβ (0) = 0 and (g,h) normal Δβ (0) = 18. If um i no longer a lane wave, but intead it i initially couled to a finite number of waveguide, it then roagate in the regime of dicrete diffraction 9] a illutrated in Fig. 1(b). For develoing a model decribing SFWM baed hoton-air generation with a negligible number of higher multihoton event in a cubic WGA, we firt conider low um ower that do not lead to um atial rehaing or nonlinear hae modulation. In the abence of SPM, XPM and loe, the ytem Hamiltonian conit of linear 4] and nonlinear 10] art Ĥ = Ĥ (lin) + Ĥ (nonlin), Ĥ (lin) = h n Δβ (0) â n â n +Câ n 1ân +Câ n +1ân ] Ĥ (nonlin) = i hγ n E () + h ni Δβ (0) â n i â ni +Câ n i 1ân i +Câ n i +1ân i ], n E n () â n â n i δ n,n δ ni,n E n () E n () â n â ni δ n,n δ ni,n ]. Here n and n i are the waveguide number decribing the oition of the ignal, and the idler hoton, and E n () (z) i the um amlitude in waveguide number n. Δβ (0) i the linear four-wave mixing hae-mimatch in a ingle waveguide, γ i a nonlinear coefficient. The normalized um field rofile evolution along the roagation ditance z i defined through the claical couled-mode equation 9]: de n () (z)/dz = ic E () n 1 (z)+e() n +1 ]. (z) (2) Generation of hoton air in cubic nonlinear WGA through SFWM in the abence of multile hoton air can be characterized by the evolution of a bi-hoton wave function ψ n,n i (z) in a Schrödinger-tye equation. The equation i obtained from the Hamiltonian, and it ha a form imilar to that of quadratic media 11]: dψ n,n i (z)/dz =ic ψ n 1,n i (z)+ψ n,n i 1(z)+ψ n +1,n i (z)+ψ n,n i +1(z)] +iδβ (0) ψ n,n i + γe n () (z)e n () (z)δ n,n i. (3) We notice that the atial dierion decribed by Eq. (2) and (3) exactly agree with Eq. (1). After calculating the wave function, we obtain two-hoton correlation in real-ace a Γ n,n i = ψ n,n i (L) 2, where L i the roagation length. In order to find correlation for the ignal and idler hoton in k-ace we aly the two-dimenional Fourier-tranform, Γ k,ki = n ni ex(iπk n )ex(iπki n i )ψ n,n i (L) 2. For the examle reented below, we normalize # $15.00 USD Received 5 Se 2012; revied 19 Oct 2012; acceted 20 Oct 2012; ublihed 19 Nov 2012 (C) 2012 OSA 19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27443

4 Fig. 2. (a) Outut um ower rofile v. inut um amlitude (ingle waveguide excitation). (b) Normalized um ectrum (green olid line), normalized ignal (red dahed line) and idler (blue dotted line) ectral filter. Gray hading mark a art of the um ectrum contributing to the filtered hoton-air generation. all arameter to the WGA length L = 1 and nonlinearity γ = 1. The hyical value of the nonlinear coefficient can be determined following the aroach of Ref. 10]. We ue the couling coefficient C = 5, and conider a um beam couled only to the central waveguide (n = 0). We tudy the cae of low um amlitude E () n =0 (0) =10 5 when the inut beam exhibit linear dicrete diffraction 9], ee Fig. 1(b). We analyze three different tye of grou velocity dierion (GVD): anomalou Δβ (0) = 18, zero Δβ (0) = 0 and normal Δβ (0) = 18. In the cae of anomalou GVD, hoton in a air tend to end u motly away from the central waveguide, with higher robability to be at either the ame or the ooite waveguide, ee hoton-air robability correlation in Fig. 1(c). Thi behavior correond to weakly ronounced imultaneou atial bunching and antibunching. The quantum tatitic in thi cae i quai-anyonic, which can be intereting for quantum imulation 12]. The k-ace correlation how an ellitical hae centered at k = ki = 0 Fig. 1(d)]. Thi hae correond to the wavenumber with the mot efficient hae-matched interaction, i.e. Δβ = 0. For zero GVD the ignal and idler hoton motly leave the tructure from the ame waveguide, thu demontrating trong atial bunching behavior Fig. 1(e)]. Figure 1(f) how that the tranvere wavenumber for hoton air atify the relation k + ki ±1. We note that for zero dierion hoton air have much higher robability to arrive to the center of the WGA, becaue hae matching can now be achieved for a broader range of tranvere momenta. In the cae of normal GVD, the real-ace correlation Fig. 1(g)] are the ame a for anomalou GVD Fig. 1(c)]. Indeed for low um ower with negligible SPM and XPM, the ytem i ymmetrical with reect to the GVD ign. In k-ace the correlation form an ellitical hae centered around k = ki = ±1 Fig. 1(h)]. We note that there i a gradual tranition from ellie centered at 0 Fig. 1(d)] through linear hae Fig. 1(f)] to ellie centered at k = ki = ±1 Fig. 1(h)] when tuning the GVD from anomalou to normal. GVD tuning can achieved by changing the um wavelength 13], however uch tuning can be comlicated due to the required correonding ectral hift of outut filter for ignal and idler hoton. 3. SFWM and um elf-focuing at high um ower Next we invetigate the otential of SPM and XPM at high um ower for flexible quantum tate control. When the um ower i increaed, the beam elf-focuing reult in a har tranition from dicrete diffraction to the formation of a atial oliton, a hown in Fig. 2(a). However, the ower required for oliton formation are at leat an order of magnitude higher than thoe needed to remain in the regime with mall number of multihoton event. For examle, in a 3 mm long Si WGA (waveguide nm high and nm wide) the um eak ower required for noticeable nonlinear hae modulation i of the order of 10 20W 14]. The characteritic um eak ower P for hoton-air generation i much maller: in a 10 mm long Si waveguide P W 7, 8] (correonding to 1 2 W for a 3 mm long waveguide). To realize the influence of SPM and XPM on the hoton-air generation in WGA in the ab- # $15.00 USD Received 5 Se 2012; revied 19 Oct 2012; acceted 20 Oct 2012; ublihed 19 Nov 2012 (C) 2012 OSA 19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27444

5 Fig. 3. (a) Pum atial oliton formation with inut um amlitude E () n =0 (0) =4.5. (b-g) Photon-air correlation in (b,d,f) real ace and (c,e,g) k-ace for different grouvelocity dierion: (b,c) anomalou Δβ (0) = 18, (d,e) zero Δβ (0) = 0 and (f,g) normal Δβ (0) = 18. ence of multihoton event, we ugget aymmetric filtering aroach for a uled um beam of tranform-limited ule having a ectrum hown in Fig. 2(b) with a green olid line. In thi aroach we chooe two narrowband ectral filter for meauring the ignal (red dahed line) and the idler (blue dotted line) hoton, uch that they are located aymmetrically with reect to the central um frequency. Then only a narrow window of um frequencie with mall eak ower (indicated by gray hading) would be reonible for the detected hoton-air, due to the energy conervation ω (a) +ω (b) = ω (filtered) +ω (filtered) i. With uch filtering the um eak ower can be trong enough to induce um beam elf-focuing, while multihoton event are motly excluded from the meaurement. The comlete modeling of thi ytem hould account for the atio-temoral um dynamic. Here we invetigate a imlified teady-tate model that i valid if the um ule doe not exerience any dierion-related rehaing. Signal and idler filter in thi cae hould be far enough from the um in frequency domain, o that the hae-matching i affected by XPM and SPM, but alo cloe enough, o that the couling coefficient are till imilar for ignal, idler and um wave. Since couling dierion in WGA i uually maller than the temoral dierion (unle ecifically engineered otherwie 15]), thee aumtion hould be valid for a large variety of ytem. Since high um eak ower lead to the um beam focuing Fig. 2(a)], generated hoton air will have different atial ditribution deending on the um ower. The um ower will alo effectively change the SFWM hae-matching condition due to XPM. We incororate thee effect by adding the term reonible for XPM and SPM into the Eq. (2) and (3): ] de n () (z)/dz = ic E () n 1 (z)+e() n +1 (z) + iγ E n () (z) 2 E n () (z). (4) dψ n,n i (z)/dz =ic ψ n 1,n i (z)+ψ n,n i 1(z)+ψ n +1,n i (z)+ψ n,n i +1(z)] +γe n () (z)e n () (z)δ n,n i + i Δβ (0) + 2γ E n () (z) 2 + 2γ E () (z) 2] ψ n,n i (z), We acknowledge that thi model i only the firt-order aroximation and that imulation deigned to give recie quantitative reult hould incororate full dierion curve both for roagation and couling contant, a well a atio-temoral dynamic. However, we believe that the imlified model that we reent here i ueful to obtain a qualitative inight into the quantum tatitic control that can be achieved in cubic WGA. n i (5) # $15.00 USD Received 5 Se 2012; revied 19 Oct 2012; acceted 20 Oct 2012; ublihed 19 Nov 2012 (C) 2012 OSA 19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27445

6 6 (a) (b) 10 (c) 1 (d) ratio R um amlitude n i n n z, a.u n Fig. 4. (Media 1) (a) Photon-air bunching to antibunching ratio R v. the inut um amlitude E () n =0 (0) for the anomalou GVD Δβ (0) = 18 (red olid line) and the normal GVD Δβ (0) = 18 (blue dahed line). Red and blue circle correond to the the inut um amlitude E () n =0 (0)=2.850 for the lot (b-d). (b,c) Real-ace hoton-air correlation at (b) anomalou and (c) normal dierion. (d) Pum ower ditribution in the array. When the um ower i increaed, the um beam ditribution collae to a ingle waveguide, a illutrated in Fig. 3(a) for E () n =0 (0)=4.5 E() n =0 (0)/C = 0.9]. Thi elf-focuing dramatically change the atial hoton-air correlation. The real ace correlation in anomalou GVD regime collae to a ingle waveguide, following the um Fig. 3(b)]. Thi can be exlained by the henomenon of forward-roagation dierion comenation. Anomalou GVD allow to hae-match four-wave mixing in a ingle waveguide, which correond to a broad k-ace hae matching in the array Fig. 3(c)]. For zero GVD, the XPM lead to a haematching of angled SFWM, correonding to a mall-cale antibunching in real ace correlation Fig. 3(d)] and more ronounced electivity of hae-matched ignal and idler tranvere momenta Fig. 3(e)]. In the normal GVD cae the real ace correlation now demontrate very ronounced antibunching Fig. 3(f)], which correond to k-ace hae matching motly for tranvere momenta far away from zero Fig. 3(g)]. Thi make it oible for hoton air to ecae from the localized um beam. In thi cae the generated hoton air become atially filtered from the um beam and how trongly ronounced anti-bunching. Thi demontrate that by tuning the um wavelength and ower we can get a great degree of control on the atial hoton-air correlation, which i ueful for alication in quantum information. To further illutrate thi flexibility, we focu on two articulary intereting regime, namely bunching and antibunching. We introduce a bunching to antibunching ratio, R = ( ni =n Γ n,n i )/( ni = n Γ n,n i ) and tudy the dynamic of thi ratio with reect to um ower Fig. 4(a) (Media 1)], while alo tracking the atial hoton-air correlation for anomalou Fig. 4(b) (Media 1)] and normal GVD Fig. 4(c) (Media 1)] and the um roagation Fig. 4(d) (Media 1)]. We oberve that the um ower tuning rovide acce to a wide range of hoton-air quantum tatitic. We alo ee that in general, normal dierion can lead to tronger atial antibunching, a in thi cae only angled hae-matching can be atified. 4. Concluion We have demontrated that waveguide array with cubic nonlinearity can be emloyed a a a flexible latform for all-otical maniulation of the generated bi-hoton quantum tatitic. We have hown that interlay between atial dierion and four-wave mixing hae-matching in waveguide array lead to the generation of comlex atial quantum tate, which can be controlled by tuning the um ower and wavelength. We anticiate that our reult will oen new oortunitie for integrated quantum hotonic with all-otical control. Acknowledgment We acknowledge ueful dicuion with Sergey Kruk and Luke Helt. The work wa uorted by the Autralian Reearch Council. # $15.00 USD Received 5 Se 2012; revied 19 Oct 2012; acceted 20 Oct 2012; ublihed 19 Nov 2012 (C) 2012 OSA 19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27446