Multiparameter entanglement in quantum interferometry

Transcription

1 PHYSICAL REVIEW A, 66, Multiarameter entanglement in quantum interferometry Mete Atatüre, 1 Giovanni Di Giusee, 2 Matthew D. Shaw, 2 Alexander V. Sergienko, 1,2 Bahaa E. A. Saleh, 2 and Malvin C. Teich 1,2 1 Quantum Imaging Laboratory, Deartment of Physics, Boston University, 8 Saint Mary s Street, Boston, Massachusetts Quantum Imaging Laboratory, Deartment of Electrical and Comuter Engineering, Boston University, 8 Saint Mary s Street, Boston, Massachusetts Received 28 June 2001; revised manuscrit received 26 October 2001; ublished 30 August 200 The role of multiarameter entanglement in quantum interference from collinear Tye-II sontaneous arametric down-conversion is exlored using a variety of aerture shaes and sizes, in regimes of both ultrafast and continuous-wave uming. We have develoed and exerimentally verified a theory of down-conversion which considers a quantum state that can be concurrently entangled in frequency, wavevector, and olarization. In articular, we demonstrate deviations from the familiar triangular interference di, such as asymmetry and eaking. These findings imrove our caacity to control the quantum state roduced by sontaneous arametric down-conversion, and should rove useful to those ursuing the many roosed alications of downconverted light. DOI: /PhysRevA PACS number s : Dv, Re, Ky, a I. INTRODUCTION In the nonlinear-otical rocess of sontaneous arametric down-conversion SPDC 1, in which a laser beam illuminates a nonlinear-otical crystal, airs of hotons are generated in a state that can be entangled concurrently in frequency, momentum, and olarization. A significant number of exerimental efforts designed to verify the entangled nature of such states have been carried out on states entangled in a single arameter, such as in energy 3, momentum 4, or olarization 5. In general, the quantum state roduced by SPDC is not factorizable into indeendently entangled single-arameter functions. Consequently any attemt to access one arameter is affected by the resence of the others. A common aroach to quantum interferometry to date has been to choose a single entangled arameter of interest and eliminate the deendence of the quantum state on all other arameters. For examle, when investigating olarization entanglement, sectral and satial filtering are tyically imosed in an attemt to restrict attention to olarization alone. A more general aroach to this roblem is to consider and exloit the concurrent entanglement from the outset. In this aroach, the observed quantum-interference attern in one arameter, such as olarization, can be modified at will by controlling the deendence of the state on the other arameters, such as frequency and transverse wave vector. This strong interdeendence has its origin in the nonfactorizability of the quantum state into roduct functions of the searate arameters. In this aer we theoretically and exerimentally study how the olarization quantum-interference attern, resented as a function of relative temoral delay between the hotons of an entangled air, is modified by controlling the otical system through different kinds of satial aertures. The effect of the sectral rofile of the um field is investigated by using both a continuous wave and a ulsed laser to generate SPDC. The role of the satial rofile of the um field is also studied exerimentally by restricting the um-beam diameter at the face of the nonlinear crystal. Satial effects in Tye-I SPDC have reviously been investigated, tyically in the context of imaging with satially resolving detection systems 6. The theoretical formalism resented here for Tye-II SPDC is suitable for extension to Tye-I in the resence of an arbitrary otical system and detection aaratus. Our study leads to a deeer hysical understanding of multiarameter entangled two-hoton states and concomitantly rovides a route for engineering these states for secific alications, including quantum information rocessing. II. MULTIPARAMETER ENTANGLED-STATE FORMALISM In this section we resent a multidimensional analysis of the entangled-hoton state generated via Tye-II SPDC. To admit a broad range of ossible exerimental schemes we consider, in turn, the three distinct stages of any exerimental aaratus: the generation, roagation, and detection of the quantum state 7. A. Generation By virtue of the relatively weak interaction in the nonlinear crystal, we consider the two-hoton state generated within the confines of first-order time-deendent erturbation theory, (2) i t dt Ĥint t 0. 1 t0 Here Ĥ int (t ) is the interaction Hamiltonian, t 0,t is the duration of the interaction, and 0 is the initial vacuum state. The interaction Hamiltonian governing this henomenon is 8 Ĥ int t (2) V dr Ê ( ) r,t Ê o ( ) r,t Ê e ( ) r,t H.c., /2002/66 / /$ The American Physical Society

2 METE ATATÜRE et al. where (2) is the second-order suscetibility and V is the volume of the nonlinear medium in which the interaction takes lace. The oerator Ê j ( ) (r,t ) reresents the ositive- negative- frequency ortion of the jth electric-field oerator, with the subscrit j reresenting the um ( ), ordinary (o), and extraordinary e waves at osition r and time t, and H.c. stands for Hermitian conjugate. Due to the high intensity of the um field it can be reresented by a classical c-number, rather than as an oerator, with an arbitrary satiotemoral rofile given by E r,t dk Ẽ k e ik r e i (k )t, 3 where Ẽ (k ) is the comlex-amlitude rofile of the field as a function of the wave vector k. We decomose the three-dimensional wave vector k into a two-dimensional transverse wave vector q and frequency, so that Eq. 3 takes the form FIG. 1. Decomosition of a three-dimensional wave vector (k) into longitudinal ( ) and transverse (q) comonents. The angle between the otical axis OA of the nonlinear crystal and the wave vector k is. The angle between the otical axis and the longitudinal axis (e 3 ) is denoted OA. The satial walk-off of the extraordinary olarization comonent of a field traveling through the nonlinear crystal is characterized by the quantity M. E r,t dq d Ẽ q ; e i z e iq x e i t, 4 q o,q e ; o, e Ẽ q o q e ; o e where x sans the transverse lane erendicular to the roagation direction z. In a similar way the ordinary and extraordinary fields can be exressed in terms of the quantum-mechanical creation oerators â (q, ) for the (q, ) modes as Ê j ( ) r,t dq j d j e i j z e iq j x e i j t â j q j, j, where the subscrit j o,e. The longitudinal comonent of k, denoted, can be written in terms of the (q, ) air as 7,9 n, c 2 q 2, where c is the seed of light in vacuum, is the angle between k and the otical axis of the nonlinear crystal see Fig. 1, and n(, ) is the index of refraction in the nonlinear medium. Note that the symbol n(, ) in Eq. 6 reresents the extraordinary refractive index n e (, ) when calculating for extraordinary waves, and the ordinary refractive index n o ( ) for ordinary waves. Substituting Eqs. 4 and 5 into Eqs. 1 and yields the quantum state at the outut of the nonlinear crystal, 5 6 L sinc L 2 e il /2, where L is the thickness of the crystal and o e where j ( j,o,e) is related to the indices (q j, j ) via relations similar to Eq. 6. The nonsearability of the function (q o,q e ; o, e ) in Eqs. 7 and 8, recalling Eq. 6, is the hallmark of concurrent multiarameter entanglement. B. Proagation Proagation of the down-converted light between the lanes of generation and detection is characterized by the transfer function of the otical system. The bihoton robability amlitude 8 at the sace-time coordinates (x A,t A ) and (x B,t B ) where detection takes lace is defined by A x A,x B ;t A,t B 0 Ê ( ) A x A,t A Ê ( ) B x B,t B (2). 9 The exlicit forms of the quantum oerators at the detection locations are reresented by 7 Ê A ( ) x A,t A dq d e i t A H Ae x A,q; â e q, H Ao x A,q; â o q,, 8 (2) dq o dq e d o d e q o,q e ; o, e Ê B ( ) x B,t B dq d e i t B H Be x B,q; â e q, with â o q o, o â e q e, e 0, 7 H Bo x B,q; â o q,, 10 where the transfer function H ij (i A,B and j e,o) describes the roagation of a (q, ) mode from the nonlinear

3 MULTIPARAMETER ENTANGLEMENT IN QUANTUM... FIG. 2. a Illustration of the idealized setu for observation of quantum interference using SPDC. BBO reresents a beta-barium borate nonlinear otical crystal, H ij (x i,q; ) is the transfer function of the system, and the detection lane is reresented by x i. b For most exerimental configurations the transfer function can be factorized into diffraction-deendent H(x i,q; ) and diffractionindeendent (T ij ) comonents. crystal outut lane to the detection lane. Substituting Eqs. 7 and 10 into Eq. 9 yields a general form for the bihoton robability amlitude, A x A,x B ;t A,t B dq o dq e d o d e q o,q e ; o, e H Ae x A,q e ; e H Bo x B,q o ; o e i( e t A o t B ) H Ao x A,q o ; o H Be x B,q e ; e e i( o t A e t B ). 11 By choosing otical systems with exlicit forms of the functions H Ae, H Ao, H Be, and H Bo, the overall bihoton robability amlitude can be constructed as desired. C. Detection The formulation of the detection rocess requires some knowledge of the detection aaratus. Slow detectors, for examle, imart temoral integration while detectors of finite area imart satial integration. One extreme case is realized when the temoral resonse of a oint detector is sread negligibly with resect to the characteristic time scale of SPDC, namely, the inverse of down-conversion bandwidth. In this limit the coincidence rate reduces to R A x A,x B ;t A,t B 2. 1 On the other hand, quantum-interference exeriments tyically make use of slow bucket detectors. Under these conditions, the coincidence count rate R is readily exressed in terms of the bihoton robability amlitude as R dx A dx B dt A dt B A x A,x B ;t A,t B FIG. 3. a Schematic of the exerimental setu for observation of quantum interference using Tye-II collinear SPDC see text for details. b Detail of the ath from the crystal outut lane to the detector inut lane. F( ) reresents an otional filter transmission function, (x) reresents an aerture function, and f is the focal length of the lens. III. MULTIPARAMETER ENTANGLED-STATE MANIPULATION In this section we aly the mathematical descrition resented above to secific configurations of a quantum interferometer. Since the evolution of the state is ultimately described by the transfer function H ij, an exlicit form of this function is needed for each configuration of interest. Almost all quantum-interference exeriments erformed to date have a common feature, namely, that the transfer function H ij in Eq. 11, with i A,B and j o,e, can be searated into diffraction-deendent and -indeendent terms as H ij x i,q; T ij H x i,q;, 14 where the diffraction-deendent terms are groued in H and the diffraction-indeendent terms are groued in T ij see Fig.. Free sace, aertures, and lenses, for examle, can be treated as diffraction-deendent elements while beam slitters, temoral delays, and wave lates can be considered as diffraction-indeendent elements. For collinear SPDC configurations, for examle, in the resence of a relative oticalath delay between the ordinary and the extraordinary olarized hotons, as illustrated in Fig. 3 a, T ij is simly T ij e i e j e i ej, 15 where the symbol ej is the Kronecker delta with ee 1 and eo 0. The unit vector e i describes the orientation of each olarization analyzer in the exerimental aaratus, while e j is the unit vector that describes the olarization of each down-converted hoton. Using the exression for H ij given in Eq. 14 in the general bihoton robability amlitude given in Eq. 11, we construct a comact exression for all systems that can be searated into diffraction-deendent and -indeendent elements as

4 METE ATATÜRE et al. A x A,x B ;t A,t B dq o dq e d o d e q o,q e ; o, e T Ae H x A,q e ; e T Bo H x B,q o ; o e i( e t A o t B ) T Ao H x A,q o ; o T Be H x B,q e ; e e i( o t A e t B ). 16 Given the general form of the bihoton robability amlitude for a searable system, we now roceed to investigate several secific exerimental arrangements. For the exerimental work resented in this aer the angle between e i and e j is 45, so T ij can be simlified by using (e i e j ) 1/ 2 5. Substituting this into Eq. 16, the bihoton robability amlitude becomes A x A,x B ;t A,t B dq o dq e d o d e Ẽ q o q e ; o e L sinc L 2 e il /2 e i e H x A,q e ; e H x B,q o ; o e i( e t A o t B ) H x A,q o ; o H x B,q e ; e e i( o t A e t B ) 17 where sinc(x) sin(x)/x. Substitution of Eq. 17 into Eq. 13 gives an exression for the coincidence-count rate given an arbitrary um rofile and otical system. However, this exression is unwieldy for uroses of redicting interference atterns excet in certain cases, where integration can be swiftly erformed. In articular, we consider three cases of the satial and sectral rofiles of the um field in Eq. 17 : 1 a olychromatic lane wave, a monochromatic lane wave, and 3 a monochromatic beam with arbitrary satial rofile. These various cases will be used subsequently for cw and ulse-umed SPDC studies. First, a nonmonochromatic lane-wave um field is described mathematically by Ẽ q ; q E 0, 18 where E ( 0 ) is the sectral rofile of the um field. Equation 17 then takes the form A x A,x B ;t A,t B dq d o d e E o e 0 L sinc L 2 e il /2 e i e H x A,q; e H x B, q; o e i( e t A o t B ) H x A, q; o H x B,q; e e i( o t A e t B ). 19 Second, a monochromatic lane-wave um field is described by Ẽ q ; q 0, whereuon Eq. 19 becomes A x A,x B ;t A,t B dq d L sinc L 2 20 e il /2 e i e i 0 (ta t B ) H x A,q; H x B, q; 0 e i (t A t B ) H x A, q; 0 H x B,q; e i (t A t B ). 21 In this case the nonfactorizability of the state is due solely to the hase matching. Third, we examine the effect of the satial distribution of the um field by considering a monochromatic field with an arbitrary satial rofile described by Ẽ q ; q 0, 2 where (q ) characterizes the satial rofile of the um field through transverse wave vectors. In this case Eq. 17 simlifies to A x A,x B ;t A,t B dq o dq e d q e q o L sinc L 2 e il /2 e i e i 0 (ta t B ) H x A,q e ; H x B,q o ; 0 e i (t A t B ) H x A,q o ; 0 H x B,q e ; e i (t A t B ). 23 Using Eq. 21 as the bihoton robability amlitude for cw-umed SPDC and Eq. 19 for ulse-umed SPDC in Eq. 13, we can now investigate the behavior of the quantum-interference attern for otical systems described by secific transfer functions H. Equation 23 will be considered in Sec. III C to investigate the limit where the um satial rofile has a considerable effect on the quantuminterference attern. The diffraction-deendent elements in most of these exerimental arrangements are illustrated in Fig. 3 b. To describe this system mathematically via the function H, we need to derive the imulse resonse function, also known as the oint-sread function for otical systems. A tyical aerture diameter of b 1 cm at a distance d 1 m from the crystal outut lane yields b 4 /4 d using 0.5 m, guaranteeing the validity of the Fresnel aroximation. We therefore roceed with the calculation of the imulse resonse function in this aroximation. Without loss of generality, we now resent a two-dimensional one longi

5 MULTIPARAMETER ENTANGLEMENT IN QUANTUM... tudinal and one transverse analysis of the imulse resonse function, extension to three dimensions being straightforward. Referring to Fig. 3 b, the overall roagation through this system is broken into free-sace roagation from the nonlinear-crystal outut surface (x,0) to the lane of the aerture (x,d 1 ), free-sace roagation from the aerture lane to the thin lens (x,d 1 d 2 ), and finally free-sace roagation from the lens to the lane of detection (x i,d 1 d 2 f ), with i A,B. Free-sace roagation of a monochromatic sherical wave with frequency from (x,0) to (x,d 1 ) over a distance r is e i /c r e i /c d 2 1 (x x ) 2 e i /c d 1 e i /2cd 1 (x x )2. 24 The sectral filter is reresented mathematically by a function F( ) and the aerture is reresented by the function (x). In the (x,d 1 ) lane, at the location of the aerture, the imulse resonse function of the otical system between lanes x and x takes the form h x,x; F x e i /c d 1e i /2cd 1 (x x )2. 25 Also, the imulse resonse function for the single-lens system from the lane (x,d 1 ) to the lane (x i,d 1 d 2 f ), as shown in Fig. 3 b, is h x i,x ; e i /c (d 2 f ) ex i x i 2 2cf d 2 f 1 e i x i x /cf. 26 Combining this with Eq. 25 rovides h x i,x; F e i /c (d 1 d 2 ex f ) i x i 2 2cf d 2 f 1 e i x2 /2cd 1 x e i x 2 /2cd 1 ex i c x x x i d 1 f dx, 27 which is the imulse resonse function of the entire otical system from the crystal outut lane to the detector inut lane. We use this imulse resonse function to determine the transfer function of the system in terms of transverse wave vectors via H x i,q; dx h x i,x; e iq x, so that the transfer function exlicitly takes the form H x i,q; e i /c (d 1 d 2 ex f ) i x i 2 2cf d 2 f 1 e i cd 1 / q 2 P cf x i q F, where the function P ( /cf)xi q is defined by P cf x i q dx x e i x x i /cf e iq x. 30 Using Eq. 29 we can now describe the roagation of the down-converted light from the crystal lane to the detection lanes. Since no birefringence is assumed for any material in the system considered to this oint, this transfer function is identical for both olarization modes (o,e). In some of the exerimental arrangements discussed in this aer, a rism is used to searate the um field from the SPDC. The alteration of the transfer function H by the resence of this rism is found mathematically to be negligible see Aendix A and the effect of the rism is neglected. A arallel set of exeriments conducted without the use of a rism further justifies this conclusion see Sec. III C. Continuing the analysis in the Fresnel aroximation and the aroximation that the SPDC fields are quasimonochromatic, we can derive an analytical form for the coincidencecount rate defined in Eq. 13, R R 0 1 V, 31 where R 0 is the coincidence rate outside the region of quantum interference. In the absence of sectral filtering V LD sinc 1 0 L 2 M 2 LD LD 1 P A 0 LM LD e P B 0 LM LD e, 3 where D 1/u o 1/u e with u j denoting the grou velocity for the j-olarized hoton ( j o,e), M ln n e ( 0 /2, OA )/ e 10, and (x) 1 x for 1 x 1, and zero otherwise. A derivation of Eq. 3, along with the definitions of all quantities in this exression, is resented in Aendix B. The function P i with i A,B) is the normalized Fourier transform of the squared magnitude of the aerture function i (x); it is given by dy i y i * y e iy q P i q. 33 dy i y i * y The rofile of the function P i within Eq. 3 lays a key role in the results resented in this aer. The common exerimental ractice is to use extremely small aertures to reach the one-dimensional lane-wave limit. As shown in Eq. 3, this gives P i functions that are broad in comarison with so that determines the shae of the quantum interference attern, resulting in a symmetric triangular di. The sinc function in Eq. 3 is aroximately equal to unity for all ractical exerimental configurations and therefore lays an insignificant role. On the other hand, this sinc function reresents the difference between the familiar onedimensional model which redicts R( ) R

6 METE ATATÜRE et al. (/LD 1), a erfectly triangular interference di and a three-dimensional model in the resence of a very small onaxis aerture. Note that for symmetric aertures i (x) i ( x), so from Eq. 33 the functions P i are symmetric as well. However, within Eq. 3 the P i functions, which are centered at 0, are shifted with resect to the function, which is symmetric about LD/2. Since Eq. 3 is the roduct of functions with different centers of symmetry, it redicts asymmetric quantum-interference atterns, as have been observed in recent exeriments 11,1. When the aertures are wide, the i (x) are broad functions that result in narrow P i, so that the interference attern is strongly influenced by the shae of the functions P i. If, in addition, the aertures are satially shifted in the transverse lane, the P i become oscillatory functions that result in sinusoidal modulation of the interference attern. This can result in a artial inversion of the di into a eak for certain ranges of the delay, as will be discussed subsequently. In short, it is clear from Eq. 3 that V( ) can be altered dramatically by carefully selecting the aerture rofile. When SPDC is generated using a finite-bandwidth-ulsed um field, Eq. 3 becomes V LD 1 V P A 0 LM LD e P B 0 LM LD e, 34 where the sinc function in Eq. 3 is simly relaced by V, which is given by d V E 0 2 sinc D L 0 0 L 2 M 2 LD LD 1 35 d E 0 2 with D 1/u 1 2 (1/u o 1/u e ), where u j denotes the grou velocity for the j-olarized hoton ( j, o, and e) and all other arameters are identical to those in Eq. 3. The visibility of the di in this case is governed by the bandwidth of the um field. A. Quantum interference with circular aertures Of ractical interest are the effects of the aerture shae and size, via the function P (q), on olarization quantuminterference atterns. To this end, we consider the exerimental arrangement illustrated in Fig. 3 a in the resence of a circular aerture with diameter b. The mathematical reresentation of this aerture is given in terms of the Bessel function J 1, J 1 b q P q b q For the exeriments conducted with cw-umed SPDC the um was a single-mode cw argon-ion laser with a wavelength of nm and a ower of 200 mw. The um light was delivered to a -BaB 2 O 4 BBO crystal with a thickness of 1.5 mm. The crystal was aligned to roduce collinear and degenerate Tye-II sontaneous arametric downconversion. The residual um light was removed from the signal and idler beams with a fused-silica disersion rism. The collinear beams were then sent through a delay line comrised of a z-cut crystalline quartz element fast axis orthogonal to the fast axis of the BBO crystal whose thickness was varied to control the relative otical-ath delay between the hotons of a down-converted air. The characteristic thickness of the quartz element was much less than the distance between the crystal and the detection lanes, yielding negligible effects on the satial roerties of the SPDC. The hoton airs were then sent to a nonolarizing beam slitter. Each arm of the olarization intensity interferometer following this beam slitter contained a Glan-Thomson olarization analyzer set to 45, a convex lens to focus the incoming beam, and an actively quenched Peltier-cooled single-hoton-counting avalanche hotodiode detector denoted D i with i A,B in Fig. 3 a. No sectral filtering was used in the selection of the signal and idler hotons for detection. The counts from the detectors were conveyed to a coincidence counting circuit with a 3-ns-coincidence-time window. Corrections for accidental coincidences were not necessary. The exeriments with ulse-umed SPDC were carried out using the same interferometer as that used in the cwumed SPDC exeriments, but with ultrafast laser ulses in lace of cw laser light. The um field was obtained by frequency doubling the radiation from an actively modelocked Ti:sahire laser, which emitted ulses of light at 830 nm. After doubling, 80-fs ulses full width at half maximum were roduced at 415 nm, with a reetition rate of 80 MHz and an average ower of 15 mw. For the cw case, the observed normalized coincidence rates quantum-interference atterns from a cw-umed 1.5-mm BBO crystal symbols, along with the exected theoretical curves solid, are dislayed in Fig. 4 as a function of relative otical-ath delay for various values of aerture diameter b laced 1 m from the crystal. Clearly the observed

7 MULTIPARAMETER ENTANGLEMENT IN QUANTUM... FIG. 4. Normalized coincidence-count rate R( )/R 0, as a function of the relative otical-ath delay, for different diameters of a circular aerture laced 1 m from the crystal. The symbols are the exerimental results and the solid curves are the theoretical lots for each aerture diameter. The data were obtained using a 351-nm cw um and no sectral filters. No fitting arameters are used. The dashed curve reresents the one-dimensional 1D lane-wave theory, which is clearly inadequate for large aerture diameters. FIG. 6. Normalized coincidence-count rate, as a function of the relative otical-ath delay, for 0.5-mm-, 1.5-mm-, and 3-mmthick BBO crystals as indicated in the lot. A 5-mm circular aerture was laced 1 m from the crystal. The symbols are the exerimental results reorted in Refs. 11,1, while the solid curves are the corresonding theoretical lots associated with the multiarameter formalism resented here and in Ref. 11. The data were obtained using an 80-fs-ulsed um centered at 415 nm and no sectral filters. No fitting arameters are used. Adated from Refs. 11,1. interference attern is more strongly asymmetric for larger values of b. As the aerture becomes wider, the hasematching condition between the um and the generated down-conversion allows a greater range of (q, ) modes to be admitted. The (q, ) modes that overla less introduce more distinguishability. This inherent distinguishability, in turn, reduces the visibility of the quantum-interference attern and introduces an asymmetric shae. The theoretical lots of the visibility of the quantuminterference attern at the full-comensation delay LD/2, as a function of the crystal thickness, is lotted in Fig. 5 for various aerture diameters laced 1 m from the crystal. Full visibility is exected only in the limit of extremely thin crystals, or with the use of an extremely small aerture where the one-dimensional limit is alicable. As the crystal thickness increases, the visibility deends more dramatically on the aerture diameter. Similarly, as the aerture diameter increases, the visibility deends more dramatically on the crystal thickness. This is clear from the exlicit form of Eq. 3, V 4 sinc 0 L 2 M 2 8cd 1 1 J 0 M Lb 0 M Lb 2 37 FIG. 5. Solid curves reresent theoretical coincidence visibility of the quantum-interference attern for cw SPDC as a function of crystal thickness, for various circular-aerture diameters b laced 1 m from the crystal. The dashed line reresents the one-dimensional 1D lane-wave limit of the multiarameter formalism. Visibility is calculated for a relative otical-ath delay LD/2. The symbols reresent exerimental data collected using a 1.5-mm-thick BBO crystal, and selected aerture diameters b of2mm circle, 3mm triangle, and5mm square, as indicated on the lot. for LD/2 and a circular aerture rofile for which we used Eq. 36. It is aarent that Eq. 37 deends on the roduct Lb. The exerimentally observed visibility for various aerture diameters, using the 1.5-mm-thick BBO crystal emloyed in our exeriments symbols in Fig. 5, is consistent with the theory. If the um field in Fig. 3 is ulsed, then there are additional limitations on the visibility that emerge as a result of the broad sectral bandwidth of the um field The validity of our analysis in the case of ulse-umed SPDC has been confirmed in our earlier work 11, the results of which are reroduced in Fig. 6. This figure is resented here exressly to rovide a comarison with Fig. 12, showing that asymmetry in the quantum-interference attern is obtained whether a rism or a dichroic mirror is used to remove the femtosecond-ulsed um. The asymmetry of the interference attern for increasing crystal thickness is more visible in the ulsed than in the cw regime. With resect to visibilities, Fig. 7 shows a lot for the ulsed-um case similar to Fig. 5 for the cw case

8 METE ATATÜRE et al. FIG. 7. Solid curves reresent theoretical coincidence visibility of the quantum-interference attern for ulsed SPDC as a function of crystal thickness, for various circular-aerture diameters b laced 1 m from the crystal. Visibility is calculated for a relative oticalath delay LD/2. The dashed curve reresents the onedimensional 1D lane-wave limit of the multiarameter formalism. The symbols reresent exerimental data collected using a 3.0-mm aerture and BBO crystals of thickness 0.5 mm hexagon, 1.5 mm triangle, and 3.0 mm circle. Comarison should be made with Fig. 5 for the cw SPDC. B. Quantum interference with slit aertures For the majority of quantum-interference exeriments involving relative otical-ath delay, circular aertures are the norm. In this section we consider the use of a vertical slit aerture to investigate the transverse symmetry of the generated hoton airs. Since the exerimental arrangement of Fig. 3 a remains identical excet for the aerture, Eq. 21 still holds and (x) takes the exlicit form P q sin b e 1 q b e 1 q sin a e 2 q. 38 a e 2 q FIG. 8. Normalized coincidence-count rate as a function of the relative otical-ath delay for a 1-mm 7-mm horizontal slit triangles. The data were obtained using a 351-nm cw um and no sectral filters. Exerimental results for a vertical slit are indicated by squares. Solid curves are the theoretical lots for the two orientations. The data shown by squares in Fig. 8 are the observed normalized coincidence rates for a cw-umed 1.5-mm BBO in the resence of a vertical slit aerture with a 7 mm and b 1 mm. The quantum-interference attern is highly asymmetric and has low visibility, and indeed is similar to that obtained using a wide circular aerture see Fig. 4. The solid curve is the theoretical quantum-interference attern exected for the vertical slit aerture used. In order to investigate the transverse symmetry, the comlementary exeriment has also been erformed using a horizontal slit aerture. For the horizontal slit, the arameters a and b in Eq. 38 are interchanged so that a 1 mm and b 7 mm. The data shown by triangles in Fig. 8 are the observed normalized coincidence rates for a cw-umed 1.5-mm BBO in the resence of this aerture. The most dramatic effect observed is the symmetrization of the quantuminterference attern and the recovery of the high visibility, desite the wide aerture along the horizontal axis. A ractical benefit of such a slit aerture is that the count rate is increased considerably, which is achieved by limiting the range of transverse wave vectors along the otical axis of the crystal to induce indistinguishability and allowing a wider range along the orthogonal axis to increase the collection efficiency of the SPDC hoton airs. This finding is of significant value, since a high count rate is required for many alications of entangled hoton airs and, indeed, many researchers have suggested more comlex means of generating high-flux hoton airs 15. Noting that the otical axis of the crystal falls along the vertical axis, these results verify that the dominating ortion of distinguishability lies, as exected, along the otical axis. The orthogonal axis horizontal in this case rovides a negligible contribution to distinguishability, so that almost full visibility can be achieved desite the wide aerture along the horizontal axis. The otical axis of the crystal in the exerimental arrangements discussed above is vertical with resect to the lab frame. This coincides with the olarization basis of the down-converted hotons. To stress the indeendence of these two axes, we wish to make the symmetry axis for the satial distribution of SPDC distinct from the olarization axes of the down-converted hotons. One way of achieving this exerimentally is illustrated in Fig. 9 a. Rather than using a simle BBO crystal we incororate two half-wave lates, one laced before the crystal and aligned at 22.5 with resect to the vertical axis of the laboratory frame, and the other after the crystal and aligned at The BBO crystal is rotated by 45 with resect to the vertical axis of the laboratory frame. Consequently, SPDC is generated in a secial distribution with a 45 -rotated axis of symmetry, while keeing the olarization of the hotons aligned with the horizontal and the vertical axes. Rotated-slit-aerture exeriments with cw-umed SPDC, similar to those resented in Fig. 8, were carried out using this arrangement. The results are shown in Fig. 9 b. The highest visibility in the quantuminterference attern occurred when the vertical slit was rotated 45 triangles, while the lowest and the most asymmetric attern occurred when the vertical slit aerture was rotated 45 squares. This verifies that the effect of axis selection is solely due to the satiotemoral distribution

9 MULTIPARAMETER ENTANGLEMENT IN QUANTUM... FIG. 9. a A half-wave late at 22.5 rotation, and another half-wave late at 22.5 rotation are laced before and after the BBO crystal in Fig. 3 a, resectively. This arrangement results in SPDC olarized along the horizontal-vertical axes and the axis of symmetry rotated 45 with resect to the vertical axis. b Normalized coincidence-count rate from cw-umed SPDC as a function of the relative otical-ath delay for a 1-mm 7-mm vertical slit rotated 45 triangles. Exerimental results for a vertical slit rotated 45 are indicated by squares. Solid curves are the theoretical lots for the two orientations. of SPDC, and not related to any birefringence or disersion effects associated with the linear elements in the exerimental arrangement. C. Quantum interference with increased accetance angle A otential obstacle for accessing a wider range of transverse wave vectors is the resence of disersive elements in the otical system. One or more disersion risms, for examle, are often used to searate the intense um field from the down-converted hotons 16. As discussed in Aendix A, the finite angular resolution of the system aerture and collection otics can, in certain limits, cause the rism to act as a sectral filter. To increase the limited accetance angle of the detection system and more fully robe the multiarameter interference features of the entangled-hoton airs, we carry out exeriments using the alternate setu shown in Fig. 10. Note that a dichroic mirror is used in lace of a rism. Moreover, the effective accetance angle is increased by reducing the distance between the crystal and the aerture lane. This allows us to access a greater range of transverse wave vectors with our interferometer, facilitating the observation of the effects discussed in Secs. III A and III B without the use of a rism. Using this exerimental arrangement, we reeated the circular-aerture exeriments, the results of which were resented in Fig. 4. Figure 11 dislays the observed quantuminterference atterns normalized coincidence rates from a cw-umed 1.5-mm BBO crystal symbols along with the FIG. 10. Schematic of alternate exerimental setu for observation of quantum interference using cw-umed Tye-II collinear SPDC. The configuration illustrated here makes use of a dichroic mirror in lace of the rism used in Fig. 3 a, thereby admitting greater accetance of the transverse-wave comonents. The dichroic mirror reflects the um wavelength while transmitting a broad wavelength range that includes the bandwidth of the SPDC. The single aerture shown in Fig. 3 a is relaced by searate aertures laced equal distances from the beam slitter in each arm of the interferometer. exected theoretical curves solid as a function of relative otical-ath delay for various values of the aerture diameter b laced 750 mm from the crystal. For the data on the curve with the lowest visibility squares, the limiting aertures in the system were determined not by the irises as shown in Fig. 11, but by the dimensions of the Glan-Thomson olarization analyzers, which measure 7 mm across. Similar exeriments were conducted with ulse-umed SPDC in the absence of risms. The resulting quantuminterference atterns are shown as symbols in Fig. 12. The observed quantum-interference atterns normalized coincidence rates from a 1.5-mm BBO crystal symbols along with the exected theoretical curves solid as a function of relative otical-ath delay for two values of the aerture di- FIG. 11. Normalized coincidence-count rate as a function of the relative otical-ath delay, for different diameters of an aerture that is circular in the configuration of Fig. 10. The symbols are the exerimental results and the solid curves are the theoretical lots for each aerture diameter. The data were obtained using a 351-nm cw um and no sectral filters. No fitting arameters are used. The behavior of the interference attern is similar to that observed in Fig. 4; the deendence on the diameter of the aerture is slightly stronger in this case

10 METE ATATÜRE et al. FIG. 12. Normalized coincidence-count rate as a function of the relative otical-ath delay, for different diameters of an aerture that is circular in the configuration of Fig. 10. The symbols are the exerimental results and the solid curves are the theoretical lots for each aerture diameter. The data were obtained using an 80-fs ulsed um centered at 415 nm and no sectral filters. In contrast to the results resented in Fig. 6, a dichroic mirror was used to remove the femtosecond um. No fitting arameters are used. Analogous results for cw-umed down-conversion are shown in Figs. 4 and 11. ameter b laced 750 mm from the crystal. The solid curves are the interference atterns calculated by using the model given in Sec. II, again assuming a Gaussian sectral rofile for the um. Comaring Fig. 12 with Fig. 6, we note that the asymmetry in the quantum-interference atterns is maintained irresective of whether a rism or a dichroic mirror is used to remove the femtosecond um. This is consistent with the multiarameter entangled nature of SPDC. An exerimental study directed secifically toward examining the role of the rism, which is used to remove the residual um light, in a similar quantum-interference exeriment has recently been resented in Ref. 17. In that work, the authors carried out a set of exeriments both with and without risms. They reorted that the interference atterns observed with a rism in the aaratus are asymmetric, while those obtained in the absence of such a rism are symmetric. The authors claim that the asymmetry of the quantum-interference attern is an artifact of the resence of the rism. In contrast to the conclusions of that study, we show that asymmetrical atterns are, in fact, observed in the absence of a rism see Fig. 1. Indeed, our theory and exeriments show that interference atterns become symmetric when narrow aertures are used, either in the absence or in the resence of a rism. This indicates conclusively that transverse effects are resonsible for asymmetry in the interference attern in the resent exeriment, which may not have been the case in revious exeriments. FIG. 13. Normalized coincidence-count rate as a function of the relative otical-ath delay, for different diameters of the um beam in the configuration of Fig. 10. The symbols are the exerimental results and the solid curve is the theoretical lot of the quantum-interference attern for an infinite lane-wave um. The data were obtained using a 351-nm cw um and no sectral filters. The circular aerture in the otical system for the down-converted light was 2.5 mm at a distance of 750 mm. No fitting arameters are used. D. Pum-field diameter effects The examles of the um field we have considered are all lane waves. In this section, and in the latter art of Aendix B, we demonstrate the validity of this assumtion under our exerimental conditions and find a limit where this assumtion is no longer valid. To demonstrate the indeendence of the interference attern on the size of the um, we laced a variety of aertures directly at the front surface of the crystal. Figure 13 shows the observed normalized coincidence rates from a cw-umed 1.5-mm BBO crystal symbols as a function of the relative otical-ath delay for various values of um beam diameter. The accetance angle of the otical system for the down-converted light is determined by a 2.5-mm aerture at a distance of 750 mm from the crystal. The theoretical curve solid corresonds to the quantum-interference attern for an infinite lane-wave um. Figure 14 shows similar lots as in Fig. 13 in the resence of a 5-mm aerture in the otical system for the down-converted light. The tyical value of the um beam diameter in quantum-interference exeriments is 5 mm. The exerimental results from a 5-mm-, 1.5-mm-, and 0.2-mmdiameter um all lie within exerimental uncertainty, and are ractically identical excet for the extreme reduction in count rate due to the reduced um intensity. FIG. 14. Plots similar to those in Fig. 13 in the resence of a 5.0-mm circular aerture in the otical system for the downconverted light

11 MULTIPARAMETER ENTANGLEMENT IN QUANTUM... This behavior of the interference attern suggests that the deendence of the quantum-interference attern on the diameter of the um beam is negligible within the limits considered in this work. Indeed, if the um diameter is comarable to the satial walk-off of the um beam within the nonlinear crystal, then the lane-wave aroximation is not valid and the roer satial rofile of the um beam must be considered in Eq. 23. For the 1.5-mm BBO used in our exeriments this limit is 70 m, which is smaller than any aerture we could use without facing rohibitively low count rates. E. Shifted-aerture effects In the work resented thus far, the otical elements in the system are laced concentrically about the longitudinal z axis. In this condition, the sole aerture before the beam slitter, as shown in Fig. 3 a, yields the same transfer function as two identical aertures laced in each arm after the beam slitter, as shown in Fig. 10. In this section we show that the observed quantum-interference attern is also sensitive to a relative shift of the aertures in the transverse lane. To account for this, we must include an additional factor in Eq. 3, cos LM LD e 2 s A s B, 39 where s i with i A,B) is the dislacement of each aerture from the longitudinal z axis. This extra factor rovides yet another degree of control on the quantum-interference attern for a given aerture form. 1. Quantum interference with shifted-slit aertures First, we revisit the case of slit aertures, discussed above in Sec. III B. Using the setu shown in Fig. 10, we laced identically oriented slit aertures in each arm of the interferometer, which can be hysically shifted u and down in the transverse lane. A satially shifted aerture introduces an extra hase into the P (q) functions, which, in turn, results in the sinusoidal modulation of the quantum-interference attern as shown in Eq. 39 above. The two sets of data shown in Fig. 15 reresent the observed normalized coincidence rates for a cw-umed 1.5-mm BBO crystal in the resence of identical aertures laced without shift in each arm as shown in Fig. 10. The triangular oints corresond to the use of 1-mm 7-mm horizontal slits. The square oints corresond to the same aertures rotated 90 to form vertical slits. Since this configuration, as shown in Fig. 10, accesses a wider range of accetance angles, the dimensions of the other otical elements become relevant as effective aertures in the system. Although the aertures themselves are aligned symmetrically, an effective vertical shift of s A s B 1.6 mm is induced by the relative dislacement of the two olarization analyzers. The solid curves in Fig. 15 are the theoretical lots for the two aerture orientations. Note that as in the exeriments described in Sec. III B, the horizontal slits give a high visibility interference attern, and the vertical slits give an FIG. 15. Normalized coincidence-count rate as a function of the relative otical-ath delay for identical 1-mm 7-mm horizontal slits laced in each arm in the configuration of Fig. 10 triangles. The data were obtained using a 351-nm cw um and no sectral filters. Exerimental results are also shown for two identical 1-mm 7-mm vertical slits, but shifted with resect to each other by 1.6 mm along the long axis of the slit squares. Solid curves are the theoretical lots for the two orientations. asymmetric attern with low visibility, even in the absence of a rism. Note further that the cosine modulation of Eq. 39 results in eaking of the interference attern when the vertical slits are used. 2. Quantum interference with shifted-ring aertures Given the exerimental setu shown in Fig. 10 with an annular aerture in arm A and a 7-mm circular aerture in arm B, we obtained the quantum-interference atterns shown in Fig. 16. The annular aerture used had an outer diameter of b 4 mm and an inner diameter of a 2 mm, yielding an aerture function FIG. 16. Normalized coincidence-count rate as a function of the relative otical-ath delay, for an annular aerture internal and external diameters of 2 and 4 mm, resectively in one of the arms of the interferometer in the configuration of Fig. 10. A 7-mm circular aerture is laced in the other arm. The data were obtained using a 351-nm cw um and no sectral filters. The symbols are exerimental results for different relative shifts of the annulus along the direction of the otical axis of the crystal vertical. The solid curves are the theoretical lots without any fitting arameters

12 METE ATATÜRE et al. P 2 A q b a J 1 b q q J 1 a q q. 40 The symbols give the exerimental results for various values of the relative shift s A s B, as denoted in the legend. Note that as in the case of the shifted slit, V( ) becomes negative for certain values of the relative otical-ath delay ( ), and the interference attern dislays a eak rather than the familiar triangular di usually exected in this tye of exeriment. IV. CONCLUSION In summary, we observe that the multiarameter entangled nature of the two-hoton state generated by SPDC allows transverse satial effects to lay a role in olarizationbased quantum-interference exeriments. The interference atterns generated in these exeriments are, as a result, governed, in art, by the rofiles of the aertures in the otical system, which admit wave vectors in secified directions. Including a finite bandwidth for the um field strengthens this deendence on the aerture rofiles, clarifying why the asymmetry was first observed in the ultrafast regime 11. The henomenological analysis rovided in our earlier aer 1 was confined to a 1D theoretical construct and, as such, needed to invoke distinguishability to characterize the data. It is gratifying that the multiarameter theory resented here is caable of fitting both cw and ultrafast SPDC quantuminterference data, taking into account the full roerties of the down-conversion rocess and obviating the need for a henomenological construct. The effect of the um-beam diameter on the quantum-interference attern has also been shown to be negligible for a tyical range of um diameter values used in similar exerimental arrangements. Moreover, the multiarameter formalism resented here is suitable for characterizing SPDC from a nonlinear medium with inhomogeneous nonlinearity; this has been exlicitly demonstrated for a cascade of two nonlinear crystals searated by an air ga 18. In contrast to the usual single-direction olarizationentangled state, the wide-angle olarization-entangled state offers a richness that can be exloited in a variety of alications involving quantum-information rocessing. ACKNOWLEDGMENTS This work was suorted by the National Science Foundation, the David and Lucile Packard Foundation, and the Defense Advanced Research Projects Agency. The authors thank A. F. Abouraddy and M. C. Booth for valuable suggestions. APPENDIX A: EFFECT OF PRISM ON SYSTEM TRANSFER FUNCTION A comlete mathematical analysis was carried out to analyze the effects of the rism on the system transfer function under the standard araxial aroximation 16. This aendix sets forth the hysical understanding of this analysis, which is rather lengthy and is therefore not rovided here. FIG. 17. A lot of Eq. A, the relation between the magnitudes of the transverse wave vectors entering (q) and exiting (q ) a fused-silica rism with an aex angle of 60 see inset at lower right. The range of transverse wave vectors allowed by the otical system in our exeriments lies comletely within the area included in the tiny box at the center of the lot. The uer left inset shows details of this central region. The dashed curve is a line with unity sloe reresenting the q q ma. The dotted lines denote the deviation from this maing arising from the sectral bandwidth ( ) of the incident light beam. As long as the width of the shaded area, which denotes the angular disersion of the rism, is smaller than the angular resolution (/b) of the aerture-lens combination, the effect of the rism is negligible. We resent a mathematical analysis of the effect of the rism used in some of the exeriments resented above on the satiotemoral distribution of SPDC. We begin by assuming that the central wavelength for down-conversion is aligned at the minimum deviation angle 0, so that the inut and the outut angles of the rism are equal. The otical axis z used in the calculation of the system transfer function follows these angles as shown in the inset of Fig. 17. Within the araxial and quasimonochromatic field aroximations the rism is reresented by a maing of each (q, ) mode to a(q, ) mode. Using Snell s Law, the relation between q and q at a given frequency is dictated by S q,q S q,q n 2 sin, A1 where is the aex angle of the rism. The function S is given by S q,q sin 0 cos 0 c q e 1 sin 0 c 2 2 n 2 sin 0 2 sin 0 c q e 1 cos 0 c 2 2 q e 1 1/2 q e 1. A Figure 17 shows a lot of q as a function of q. Plotting this curve for various frequencies roduces no deviation visible within the resolution of the rinted grah. The small box at the center of the lot highlights the range of transverse wave vectors limited by the accetance angle of the otical

13 MULTIPARAMETER ENTANGLEMENT IN QUANTUM... system used in the above exeriments. The dashed line is the q q curve with unity sloe. We now consider a linear exansion in frequencies and transverse wave vectors of the above-mentioned maing of a(q, ) mode to a (q, ) mode in the form q e 1 q e 1 0 0, c 2 q e 2 q e 2, A3 where the negative sign multilying q e 1 indicates that a ray of light at the inut face of the rism with a small deviation in one direction is maed to a ray of light at the outut face with a corresonding deviation in the oosite direction see Eq. 16 in Sec of Ref. 19. The arameter corresonds to the angular disersion arameter of the rism 19 with the exlicit form given by on the quantum-interference attern is negligible in comarison with other satial and sectral effects. APPENDIX B: DERIVATION OF VISIBILITY IN EQ. 32 The urose of this aendix is to derive Eq. 3 using Eqs. 13, 21, and 29. To obtain an analytical solution within the Fresnel aroximation we assume quasimonochromatic fields and erform an exansion in terms of a small angular frequency sread ( ) around the central angular frequency ( 0 /2) associated with degenerate downconversion, and small transverse comonents q with resect to the total wave vector k j for collinear downconversion. In short, we use the fact that 0 /2, with 0 /2, and q 2 k 2. In these limits we obtain o,q K o 0 /2 q 2 u o 2K o B1 sin dn cos 0 cos / d. A4 e,q K e 0 /2 q 2 Me u e 2K 2 q, e B To find the range of aerture diameters b, where the effect of the rism can be considered negligible, we need to comare the angular resolution of the aerture-lens combination at the detection lane to the angular disersion of the rism. A system with an infinite aerture and an infinite lens mas each wave vector into a distinct oint at the detection lane. In ractice, of course, a finite aerture limits the angular resolution of the system at the detection lane to the order of /b, where is the central wavelength of the downconverted light and b is the diameter of the aerture. Now, the rism mas each frequency into a distinct wave vector. The angular disersion of the rism is on the order of where is the bandwidth of the incident light beam. If the angular disersion of the rism is less than the angular resolution of the combined aerture and lens, the disersive roerties of the rism have negligible effect on the quantum-interference attern. For an SPDC bandwidth of 10 nm around a central wavelength of 702 nm and a value of given by s calculated from the material roerties of fused silica, the effect of the angular disersion introduced by the rism on the exeriments resented in this aer can be safely neglected for aerture diameters less than 20 mm. For aerture diameters in the vicinity of this value and higher, the effect on the quantum-interference attern is a sectral-filter-like smoothing of the edges. The chromatic disersion exerienced by the down-converted light when roagating through such disersive elements has already been examined 14 ; the disersiveness of the material must be an order of magnitude higher than the values used in the exeriments considered here to have a significant effect on the downconverted light. We conclude that the transfer function of the system, rovided in Eq. 29, is not affected by the resence of the rism. Moreover, the results discussed in Secs. III A, III B, and III C exerimentally confirm that the effect of the rism where the exlicit forms for K j, u j j o,e, and Me 2 are 10 K j k j,q 0 /2,q 0, Me 2 k e q k e e 0 /2,q 0, 1 u j j,q 0 /2,q 0, M ln n e, e e 0 /2, e OA. B3 Using the results in Eqs. B1, B, and B3 we can now rovide an aroximate form for, which is the argument of the sinc function in Eq. 21, as D 2c q 2 Me 2 q, 0 B4 where D 1/u o 1/u e. Using this aroximate form for in the integral reresentation of sinc(x) sinc L 2 e il /2 0 dz e iz, B5 L with the assumtion that L d 1, we obtain Eq. 31 with R 0 d L 0 dz e id z L 0 dz e id z J 0 z,z, B