Generation of Polarization Entangled Photon Pairs in a Planar Waveguide Divya Bharadwaj 1,* and K Thyagarajan 1 1

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1 Generation of Polarization Entangled Photon Pair in a Planar Waveguide Divya Bharadwa,* and K Thyagaraan Deartment of Phyic, T Delhi, New Delhi 6, ndia n thi aer, we analyze ontaneou arametric down converion in a D lant tri tye eriodically oled lanar waveguide and how that uch a configuration ha efficiencie higher than in the cae of bulk crytal with the oibility of eae of collection like in channel waveguide with the additional feature of wavelength tunability. The rooed deign hould find alication in quantum roceing uing integrated quantum otic.. NTRODUCTON Sontaneou arametric down converion (SPDC) i one of the mot extenively ued rocee for the generation of entangled hoton air. Generation of entangled hoton air have been extenively tudied both in bulk crytal [, ], and channel waveguide [3-6]. Entangled hoton air generated through SPDC roce in bulk nonlinear otic crytal uffer from a number of roblem uch a low efficiency, deign comlication, low interaction length, and are hard to imlement for ractical alication. n order to overcome thee roblem SPDC roce ha been imlemented in nonlinear channel waveguide which have higher down converion efficiencie in comarion to a bulk crytal [3-6]. At the ame time, the flexibility in term of variation in um frequency or tunability in the generation of entangled air i not oible in channel waveguide. n thi aect, lanar waveguide which have confinement only along one direction can offer u better efficiency comared to bulk while at the ame time roviding u with a oibility of frequency tunability of the SPDC air by aroriately tuning the um wavelength and eae of collection of the generated hoton air. n thi aer we reent the deign and analyi of the generation of olarization entangled hoton air in lanar waveguide and how the advantage of thi vi a vi bulk and channel waveguide geometrie. Numerical imulation for otimization of the entangled hoton air generation roce and tunability of the generated hoton air i carried out uing lanar waveguide in KTP.. PRNCPLE We conider SPDC in a lanar waveguide in a z-cut, x- roagating otaium titanyl hohate (KTP) ubtrate. n order to imultaneouly realize two tye SPDC rocee we aume that the ubtrate ha two lanted eriodical oling [7, 8] a hown in Fig.. With aroriate oling eriod it i oible to achieve imultaneouly the following two down converion rocee from a horizontally olarized um hoton into a air of ignal and idler hoton: H H V i H V H i We will refer to H (, i) and V(, i) a horizontal and vertical olarization tate of um (ignal, idler) in the fundamental () mode reectively, with dominant comonent of the electric field oriented reectively along the lane of the waveguide and erendicular to the lane of the waveguide. The vector diagram howing the hae matching condition for the two SPDC rocee i hown in Fig.. The figure how the oibility of achieving air of horizontal and vertical olarized ignal and idler air along the two choen air of direction. Figure : Schematic of lanar waveguide with the two lant eriodically oled region. Figure : Wave vector diagram for the two SPDC rocee. n uch a cae the outut i exected to be a olarization entangled tate given by (ee Sec. ): C H, V C V, H i i C C where C and C are coefficient defined in Eq. (a). The hae mimatche of the two QPM condition correonding to the two rocee are given a: k k k k K (a) H iv () k k k k K (b) V ih

2 Here, k ˆ x k co ˆ x in ˆ y H ( V ) H ( V ) k co ˆ x in ˆ y ih ( V ) ih ( V ) i i K K co ˆ x in ˆ y g g (c) where it i aumed that the ignal and idler air correonding to the two orthogonal olarization aear along the ame angle a hown in Fig.. A hown in Figure, k i the horizontally olarized um wave vector along the x - axi; k and k are the wave H ( V ) ih ( V ) vector correonding to the horizontally (vertically) olarized ignal and idler mode making angle (emiion angle of ignal) and i (emiion angle of idler)reectively with x -axi; K and K are the grating vector making angle g and g with x- axi reectively. n Eq. (c), m m i the roagation contant, where α =,, i for um, ignal and idler reectively; m = H, V for the horizontal and vertical olarization reectively; (, i) i the um (ignal, idler) wavelength and n m i the effective indice at different frequencie and olarization. The hae mimatche correonding to the x and y- comonent for the two rocee are given by k co co K co (3a) x H iv i g k in in K in (3b) n y H iv i g k co co K co (3c) x V ih i g k in in K in (3d) y V ih i g n order that the outut tate defined by Eq. () i a olarization entangled tate, the ignal and idler air in the two rocee have to aear at the ame air of angle i.e. direction of emiion of horizontally and vertically olarized ignal (or idler) mut be ame (ee Fig. ).With a roer deign of waveguide geometry and QPM lant grating the coefficient C and C can be made equal, thu roviding the oibility of roducing maximally entangled tate with maximum efficiency in a lanar waveguide. We will how in Sec. V that down converion roce imlemented in lanar waveguide ha higher efficiency and enhanced air rate in comarion to bulk nonlinear crytal. We will alo how that by an aroriate choice of the angle made by the grating vector, it i oible to atify the quai hae matching condition correonding to both comonent and thu achieve high efficiency SPDC into lanar waveguide mode of the waveguide. n addition, we will how that by changing the um wavelength, it i oible to generate different frequency air of entangled hoton which will exit at different angle (θ, θ i ) thu roviding u the tunability of the SPDC roce.. ANALYSS n thi ection we will rovide a decrition of SPDC roce in a eriodically oled lanar waveguide for generation of olarization entangled hoton air. We conider the um to have a Gauian tranvere rofile of beam wait W along the y-direction and traveling along the x direction in a D lant -trie tye eriodically oled lanar waveguide (Figure ) with the otic axi along z- axi. We aume the um to be decribed by a claical field a it i aumed to be trong. Thu the electric field at um i given by: E r,t r,t (4a) ( H ) y ik xt r,t A P ( z )ex e (4b) W Here, A i the amlitude of um and i given a: A P /4 W c n where, P i the um ower, i the free ace ermittivity and c i the eed of light in vacuum. Since the length of roagation i mall, in writing Eq. (4b) we have neglected diffraction effect. The quantized electric field at ignal and idler correonding to different olarization are rereented by the following equation: Ê ˆ ( i )m ( i )m r,t ˆ ( i )m r,t (5a) ik( i )m rt,it ˆ ( i ) ( m) ( i )m r,t i ( i ) ze aˆ ( i )m (5b) n A ( i )m Here,,, i are um, ignal and idler frequencie, A=L xl y; L x and L y are the length of the quantization volume along the x- and y direction reectively, aˆ () i m and aˆ () i m rereent the annihilation and creation oerator of the generated ignal (idler) hoton correonding to the m=h, V olarization and m z i the modal field rofile for () (, i) um (ignal, idler) along the z direction of the H- and V- olarized fundamental mode. We have abbreviated r ( x, y, z) and rt ( x, y ) for denoting the three and two dimenional oition vector reectively. The interaction Hamiltonian i given by [9-]: L d Ĥint dydzdx ˆ ˆ HiV ˆ ˆ ViH h.c. (6) where, h.c. rereent Hermitian conugate. Subtituting Eq. (4b) and (5b) in Eq. (6), we get the following exreion for the interaction Hamiltonian:

3 A d Hˆ ˆ ˆ a a A nhniv i int x y z H iv n n V ih x yzaˆvaˆ ih h.c where, d i the effective nonlinear coefficient and L i the length of crytal along the roagation axi x. L ex i( k k k ) x dx x H ( V ) x iv ( H ) x kxl Linc e kxl i (7) (8a) y y ex i( k H ( V ) y kiv ( H ) y ) y dy W kw y (8b) W ex 4 i the overla integral between the um, ignal and idler of = and rocee, and are given by: z z z dz (8c) ( H ) ( H ) ( V ) z i z z z dz (8d) ( H ) ( V ) ( H ) z i Here, k(,i )H(V )x (,i )H(V ) co,i and k(,i )H(V ) y (,i )H(V ) in,i n accordance with the interaction icture, the overall outut tate i given a: ˆ int ih t,i e (9) The,i tate correond to vacuum tate i.e. no ignal and idler hoton. Thu, the overall normalized two hoton entangled tate (neglecting the vacuum tate) i given a H, Vi V, H i () where, C C k W y k L x C ex inc ; =, (a) N 4 Here, N nhniv and N nv n ih. The outut tate will be maximally entangled tate for and will be generated with maximum efficiency for ky and kx i.e. C C. N Next, we derive an exreion for the ower of ignal generated in lanar waveguide through tye SPDC roce and for thi, we will follow the ame aroach a mentioned in Ref. [, ]. We firt calculate the tranition rate uing the Fermi Golden Rule. To calculate the tranition rate, we need to find the denity of tate. The number of H (V) - olarized ignal and V (H)- olarized idler tate in the element d k d k i H ( V ) iv ( H ) given a: A dn() d k 4 H ( V ) d k (a) iv ( H ) ( ) Therefore, the denity of tate in lanar waveguide i given a: A n H ( V ) n P iv ( H ) () dk 4 H ( V ) ydkiv ( H ) yd (b) ( ) c Thu, the tranition rate for the = and rocee i: ˆ ( P) T H( V ), V ( H) H, () The () i int i () H( V ), V( H) a a, tate correond to i H ( V ) iv ( H ) i final ate having one ignal and one idler hoton in horizontal (vertical) and vertical (horizontal) olarization reectively. The down converted ignal ower in a frequency interval, d or wavelength interval, d i, dp T Thu, we get the following exreion for ignal ower in = and SPDC rocee: cd P 3/ L W P dp ( ) dk 4 iydky n N i (3) kw y k xl ex ic n d The radiated ignal ower in bulk, dp B i given a [9]: dp cd P L W d (4) ( B) P B() () 4 nnh ( V ) ni V ( H ) i where, K y() Kz() W B() e x K x() L inc dk iy d kizdkydkz K, K and K i the hae mimatch along the x-, y- x y and z-direction reectively for = and rocee in bulk. n order to make coefficient C and C to be equal, eriodic oling of ubtrate i needed. For generation of olarization entangled hoton air through tye collinear (< ) and non-degenerate SPDC roce we require two linear QPM grating to atify the two longitudinal hae matching condition only but for non collinear emiion (> ) of olarization entangled hoton

4 air, two lanted eriodical oling of ubtrate i needed to atify all the four QPM condition imultaneouly. Alo, the tranvere hae matching condition dictate that if the emiion angle of ignal i in firt quadrant then idler emiion angle mut be in fourth quadrant and vice vera. t may alo be worth mentioning here that the two lant grating mut be aligned in two ooite direction with reect to roagation axi x in order to atify the two tranvere hae matching condition imultaneouly for non collinear emiion. n lanar waveguide, an angular degree of freedom i available with reect to the hae matching condition of the two rocee (ee Eq. (3a), (3b), (3c) and (3d)), o that tuning of the um wavelength lead to the generation of maximally entangled hoton air having ecific combination of ignal and idler wavelength emitted along a articular air of emiion angle. n contrat, channel waveguide in nonlinear material can confine light in both tranvere direction, imlying a retriction to only one tranlation degree of freedom, i.e. the roagation direction of the interacting hoton, o no angular degree of freedom i available and thu tunability of generated hoton air i not oible. n addition, in the cae of lanar waveguide ignal and idler hoton are hyically earated and o there i no need of any device to earate them unlike in the cae of channel waveguide. V. NUMERCAL SMULATONS n order to validate our analyi, we reent numerical reult for an ion exchanged otaium titanyl hohate (KTP) [3, 4] lanar waveguide of deth, d = 3. µm with a te index refractive rofile. For the numerical imulation, the value of the KTP ubtrate refractive indice (n ) for different wavelength and different olarization were calculated uing Sellmeier equation given in [5] and the refractive index difference (Δn) for a waveguide i taken to be. [3]. We have carried out the modal analyi [6] for lanar waveguide of deth, d (Fig. 3) to calculate the eigenmode and thu, obtained the roagation contant of H-olarized and V-olarized fundamental mode at um, ignal and idler wavelength by olving eigenvalue equation of TE mode and TM mode reectively. Figure 3: Cro ectional view of the lanar waveguide Uing the modal field ditribution of the H- and V- olarized fundamental mode of um, ignal and idler o evaluated from modal analyi of lanar waveguide, we have calculated the overla integral z and z defined by Eq. (8c) and Eq. (8d) reectively and it i found that the overla integral for both the rocee are almot equal. We firt demontrate that the roduction rate of the down converted hoton air generated through SPDC roce in a eriodically oled KTP (PPKTP) lanar waveguide i more than in eriodically oled bulk KTP nonlinear crytal. For thi, we conider a tye collinear and non-degenerate SPDC roce auming a 45 nm um of beam wait, W = µm with a um ower of mw in a PPKTP of length, L = mm. The variation of ignal ower denity with ignal wavelength i hown in Fig.4 (a) for eriodically oled lanar waveguide of grating eriod, Λ =.8 µm. Fig. 4(b) how the ower denity in cae of eriodically oled bulk ubtrate with QPM grating eriod, Λ = 3.97 µm (thi eriod i different from the waveguide cae a in thi cae the effective index i ut the bulk index of the ubtrate). /d [W/nm] dp (P) /d [W/nm] dp (B) (a) [nm].35.3 (b) [nm] Figure 4: Variation of ignal ower denity with wavelength in (a) lanar waveguide (b) bulk t can be een from Fig. 4(a) and Fig. 4(b) that ignal ower denity in the lanar waveguide i larger by a factor of about comared to that in bulk which in turn lead to higher rate of generation of SPDC air in lanar waveguide in comarion to bulk. Now, we how the generation of olarization entangled hoton air through tye non collinear and nondegenerate SPDC roce in KTP having two lant eriodic domain reveral grating. Simulation have been carried out for generation of maximally entangled hoton air around the air of angle (, - ) with reect to the x-axi. We have carried out numerical imulation and otimization of the waveguide arameter for a um wavelength of 45 nm and um beam wait, W = µm and obtained the otimized waveguide arameter for a

5 waveguide length of mm a: deth = 3. µm, QPM grating eriod of both the grating = 7. µm, lant grating angle, 5.73 and 8.5. For thee arameter the g g QPM condition for both the rocee can be imultaneouly atified and the olarization entangled hoton air aear with maximum robability along the, 9.95, 9.93 correonding to the air of angle i air of wavelength, i = ( nm, nm) (ee Fig. 5(a)).t can be een from Fig. 5 (a) that the two rocee have almot overlaing ectra with identical bandwidth of.3 nm. Thu, and thi lead to maximally entangled tate over an entire region of ectral overla. n context to the tunability of generated hoton air in lanar waveguide, we tudy generation of entangled hoton air at different wavelength along different air of angle uing different um wavelength. Table how the value of emiion angle and the correonding um wavelength required to generate different et of entangled air of hoton demontrating the tunability of the deign. For examle, we ee that by tuning the um wavelength by ± nm, ignal and idler wavelength change by aroximately ± nm to generate the maximally entangled hoton air. TABLE. Maximally entangled hoton air for different value of ignal and idler wavelength and their emiion angle with the correonding value of um wavelength. (nm), i (nm, nm), i (degree) 4 (84, 847.) 9.68, (84.987, ) 9.738, ( , ) 9.845, ( , ) 9.95, (848.94, 855.8).54,.34 for two roce till overla almot on the entire band and thu enuring generation of maximally entangled tate of different wavelength air at different angle. t may be mentioned here that in the cae of channel waveguide, in view of the hae matching condition any change in um wavelength by a much a. nm lead to reduction in SPDC roce and doe not lead to generation of maximally entangled hoton air. Normalized C Normalized C [nm] (b) = 44 nm.8 H H +V i H.6 V +H i.4. (a) = 45 nm Figure 5: Variation of Normalized C H H a a function of ignal wavelength for a um wavelength (a) 45nm (b) 44 nm H +V i V +H i [nm] t may be worth mentioning here that the entangled hoton air can alo be generated for maller emiion angle with higher efficiency by changing the grating eriod but thi i at a cot of tunability of generated hoton air. 47 (85.9, 857.).54,.38 Figure 5(b) how the normalized outut ectra at the ignal wavelength correonding to both of the SPDC rocee for um wavelength of 44 nm. t can be een that by changing the um wavelength to 44 nm entangled hoton air can be generated with maximum efficiency for the wavelength air of, i = ( nm, nm) at a lightly different angle air of, 9.845, A we can ee that even by i changing the um wavelength by nm the outut ectra Figure 6: To view of the waveguide deign for generating and collecting of olarization entangled hoton air. Here, region () i a lanar waveguide with two lant QPM grating for generating different olarization entangled tat by changing the um wavelength tate and region () how the collection of hoton air along the different channel waveguide. Figure 6 how a oible imlementation of the rooal in integrated otic form. At the end of the lanar waveguide ection channel waveguide can be oitioned at aroriate angle a er the deign to which the generated hoton air can get couled (much like in the cae of arrayed waveguide grating ued in otical fiber communication [7]. The outut of the channel waveguide

6 can be couled to otical fiber for further roceing. Deending on the um wavelength different wavelength air of entangled ignal and idler wavelength will exit from different air of channel waveguide. V. CONCLUSON We have hown that lanar waveguide rovide u an otimized configuration for generation of olarization entangled hoton air uing SPDC with efficiencie higher than for bulk with the oibility of eae of collection like. P. G. Kwiat, L. Mattle, H. Weinfurter, Z. Zeilinger, A. V. Sergienko, and Y. Shih, New High-intenity ource of olarization-entangled hoton air, Phy. Rev. Lett. 75, 4337 (995). P. G. Kwiat, E. Wak, A. G. White,. Aelbaum, and P. H. Eberhard, Ultrabright ource of olarization entangled hoton, Phy. Rev. A 6, R773 R776 (999). 3. S. Tanzilli, H. De Riedmatten, H. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Otrowky, N. Giin, Highly efficient hotonair ource uing eriodically oled lithium niobate waveguide, Electron. Lett. 37, 8(). 4. K. Sanaka, K. Kawahara, and T. Kuga, New High-Efficiency Source of Photon Pair for Engineering Quantum Entanglement, Phy. Rev. Lett., vol. 86, 56 (). 5. K. Banazek, A. B. U Ren, and. A. Walmley, Generation of correlated hoton in controlled atial mode by downconverion in nonlinear waveguide, Ot. Lett. 6, 367 (). 6. C. E. Kuklewicz, M. Fiorentino, G. Mein, F. N. C. Wong, and J. H. Shairo, High-flux ource of olarization-entangled hoton from a eriodically oled KTiOPO4 arametric downconverter, Phy. Rev. A69, 387 (4). 7. Y. Saaki, A. Yuri, K. Kawae, and H. to, Terahertz-wave urface-emitted difference frequency generation in lant-trie-tye eriodically oled LiNbO 3 crytal, Al. Phy. Lett. 8, 333 (). 8. W.Q. Zhang, Grou-velocity-matched otical arametric ocillator in tilted quai-haematched grating, Ot. Commun. 5 (5) 9. D. A. Kleinman, Theory of otical arametric noie Phy. Rev. 74, 7 4 (968). K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakian, and J. M. Liu, Hot Sot in Parametric Fluorecence with a Pum Beam of Finite Cro Section, EEE J. of Quantum Electron (995). in channel waveguide with the additional feature of tunability. Such a deign hould find alication in integrated quantum otic. ACKNOWLEDGMENTS Thi work ha been uorted by Minitry of Human Reource Deartment (MHRD), New Delhi, under the Senior Reearch Fellow (SRF) cheme.. Divya Bharadwa, K. Thyagaraan, Michal Karinki, and Konrad Banazek, Generation of higher-dimenional modal entanglement uing a three-waveguide directional couler, Phy. Rev. A, 9, 3384 (5).. M. Fiorentino, S. M. Sillane, R. G. Beauoleil, T. D. Robert, P. Battle, and M. W. Munro, Sontaneou arametric down-converion in eriodically oled KTP waveguide and bulk crytal, Ot. Exre 5, (7). 3. J. D. Bierlein, A. Ferretti, L. H. Brixner, W. Y. Hu, Fabrication and characterization of otical waveguide in KTiOPO4, Al. Phy. Lett. 5, 6 (987). 4. R. W. Rik, Fabrication and characterization of lanar ion-exchanged ktioo4 waveguide for frequency doubling, Al. Phy. Lett. 58, 9 (99). 5. K. Kato and E. Takaoka, Sellmeier and thermootic dierion formula for KTP, A. Ot. 4, () 6. A. K. Ghatak and K. Thyagaraan, Otical Electronic (Cambridge Univerity, 989). 7. P Kaminow, T. Li and A. Wilner, Otical Fiber Telecommunication, V A, Comonent and Subytem (Elevier, Amterdam, 8).