Azimuthal distinguishability of entangled photons generated in spontaneous parametric down-conversion
Clara I. Osorio, Gabriel Molina-Terriza, Blanca G. Font, Juan P. Torres
aa r X i v : . [ qu a n t - ph ] S e p Azimuthal distinguishability of entangled photons generated in spontaneousparametric down-conversion
Clara I. Osorio, Gabriel Molina-Terriza,
1, 2
Blanca G. Font,
1, 3 and Juan P. Torres
1, 3 ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain ICREA-Institucio Catalana de Recerca i Estudis Avancats, 08010 Barcelona, Spain Dept. Signal Theory and Communications, Universitat Politecnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona Spain
We experimentally demonstrate that paired photons generated in different sections of a down-conversion cone, when some of the interacting waves show Poynting vector walk-off, carry differentspatial correlations, and therefore a different degree of spatial entanglement. This is shown to be inagreement with theoretical results. We also discuss how this azimuthal distinguishing informationof the down-conversion cone is relevant for the implementation of quantum sources aimed at thegeneration of entanglement in other degrees of freedom, such as polarization.
INTRODUCTION
Paired photons entangled in the spatial degree of freedom are represented by an infinite dimensional Hilbert space.This offers the possibility to implement quantum algorithms that either inherently use dimensions higher than twoor exhibit enhanced efficiency in increasingly higher dimensions (see [1] and references inside). These include thedemonstration of the violation of bipartite, three dimensional Bell inequalities [2], the implementation of the quantumcoin tossing protocol with qutrits [3], and the generation of quantum states in ultra-high dimensional spaces [4].Actually, the amount of spatial bandwidth, and the degree of spatial entanglement, can be tailored [5, 6], beingpossible to control the effective dimensionality where spatial entanglement resides.The most widely used source for generating paired photons with entangled spatial properties is spontaneous para-metric down-conversion (SPDC) [7, 8]. In this process, photons are known to be emitted in cones whose shape dependsof the phase matching conditions inside the nonlinear crystal. All relevant experiments reported to date make useof a small section of the full down-conversion cone. But the spatial properties of different sections of the cone havebeen unexplored experimentally up to now. This could be done, for example, by relocating the single photon countingmodules. Then, one question naturally arises:
Are the entangled spatial properties of the photons modified dependingof the location in the down-conversion cone where they are detected?
The answer to this question is of great relevance for the implementation of many quantum information schemes.When considering entanglement in the spatial degree of freedom, one should determine whether pairs of photons withdifferent azimuthal angle of emission might show different spatial quantum correlations, since all quantum informationapplications are based on the availability and use of specific quantum states.Additionally, the spatial properties of entangled two-photon states have to be taken into account even when en-tanglement takes place in other degrees of freedom, such as polarization. In general, it is required to suppress anyspatial “which-path” information that otherwise degrades the degree of entanglement. This is especially true forconfigurations that make use of a large spatial bandwidth [9] and in certain SPDC configurations where horizontallyand vertically polarized photons are generated in different sections of the down-conversion cone [10, 11]. Finally, thegeneration of heralded single photons with well defined spatial properties, i.e. a gaussian shape for optimum couplinginto monomode optical fibers, depends on the angle of emission [12].Here we experimentally demonstrate that the presence of Poynting vector walk-off, which is unavoidable in mostSPDC configurations currently being used, introduces azimuthal distinguishing information in the down-conversioncone . Paired photons generated with different azimuthal angles show correspondingly different spatial quantumcorrelations and amount of entanglement. We also show that this spatial distinguishing information can severelydegrade the quality of polarization entanglement, since the full quantum state that describes the entangled photonsis a nonseparable mixture of polarization and spatial variables.
EXPERIMENTAL SET-UP AND RESULTS
In Fig. 1 we present a scheme of our experimental set-up. The output beam of a CW diode laser emitting at λ p = 405 nm , is appropriately spatially filtered to obtain a beam with a gaussian profile, while a half wave plate(HWP) is used to control the polarization. The pump beam is focalized to w = 136 µm beam waist on the input faceof a L = 5mm thick lithium iodate crystal, cut at 42 ◦ for Type I degenerate collinear phase matching. The generated [t]FIG. 1: (a) Diagram of the experimental set up and (b) The down-conversion cone. Single photon detectors are located inopposite sides of the cone, forming an angle α with the Y Z plane. photons, signal and idler, are ordinary polarized, in opposition to the extraordinary polarized pump beam. The crystalis tilted to generate paired photons which propagate inside the nonlinear crystal with a non-collinear angle of ϕ = 4 ◦ .Due to the crystal birefringence, the pump beam exhibits Pointing vector walk-off with angle ρ = 4 . ◦ , while thegenerated photons do not exhibit spatial walk-off. Fig. 1(b) represents the transversal section of the down-conversioncone. The directions of propagation of the signal and the idler photons over this ring are determined by the azimuthalangle α , which is the angle between the plane of propagation of the down-converted photons and the Y Z plane. Todetermine experimentally the position of the crystal optics axis, and the origin of α , we measure the relative positionof the pump beam in the plane XY at the input and output faces of the nonlinear crystal using a CCD camera.Right after the crystal, each of the generated photons traverse a 2 − f system with focal length f=50 cm . Low-pass filters are used to remove the remaining pump beam radiation. After the filters, the photons are coupled intomultimode fibers. In order to increase our spatial resolution, we use small pinholes of 300 µ m of diameter. We keepthe idler pinhole fixed and measure the coincidence rate while scanning the signal photon transverse spatial shapewith a motorized XY translation stage. Finally, as we are interested in the different spatial correlations at differentazimuthal positions of the downconversion ring, instead of rotating the whole detection system, the nonlinear crystaland the polarization of the pump beam are rotated around the propagation direction. Due to slight misalignments ofthe rotation axis of the crystal, after every rotation it is necessary to adjust the tilt of the crystal to achieve generationof photons at the same non-collinear angle in all the cases.Images for different azimuthal sections of the cone were taken. We present a sample of them in the upper row ofFig. 2, which summarizes our main experimental results. Each column shows the coincidence rate for α = 0 ◦ , 90 ◦ ,180 ◦ and 270 ◦ . The movie shows the experimental and theoretical spatial shape of the signal photon correspondingto other values of the angle α . Each point of these images corresponds to the recording of a 10 s measurement. Thetypical number of coincidences at the maximum is around 10 photons per second. The resolution of the experimentalimages is 50 ×
50 pixels. The different spatial shapes measured of the mode function of the signal photons clearlyshow that the down-conversion cone does not posses azimuthal symmetry. This agrees with the theoretical predictionspresented in the lower row of Fig. 2. Note that no fitting parameter has been used whatsoever. Slight discrepanciesbetween experimental data and theoretical predictions might be due to the small, but not negligible, bandwidth ofthe pump beam and to the fact that the resolution of our system is limited by the detection pinholes size.An interesting feature that can be observed in these images is that the mode function in Fig. 2(b), correspondingto the case α = 90 ◦ presents a nearly gaussian shape. We will show below that this effect happens whenever ϕ ≃ ρ ,which corresponds to our experimental conditions. On the other hand the mode function shown in Figs. 2 (a) and(c) are highly elliptical. y coordinate (mm) x c oo r d i na t e ( mm ) −5 0 5 −50 5 (a) (b) (c) (d) FIG. 2: Images showing the spatial shape of the mode function of the signal photon when measuring coincidences rates. Upperrow corresponds to theoretical predictions, and the lower row corresponds to experimental data. (a) α = 0 ◦ ; (b) α = 90 ◦ ; (c) α = 180 ◦ and (d) α = 270 ◦ . See also the corresponding movie. AZIMUTHAL DISTINGUISHABILITY OF PAIRED PHOTONS GENERATED IN DIFFERENTSECTIONS OF THE THE DOWN-CONVERSION CONE
To gain further insight, we turn to the theoretical description of this problem. The signal photon propagates alongthe direction z (see Fig. 1) with longitudinal wavevector k s = [( ω s n s /c ) − | p | ] / , and transverse wavevector p = ( p x , p y ). Similarly, the idler photon propagates along the z direction with longitudinal wavevector k i , andtransverse wavevector q . Here we consider the signal and idler photons as purely monochromatic, due to the use ofa narrow pinhole in the idler side, which selects a very small bandwidth of frequencies of the down-converted ring.Although photons detected in different parts of the down-conversion cone might present slightly different polarizations[13], this is a small effect, and therefore we neglect it.The quantum two-photon state at the output face of the nonlinear crystal, within the first order perturbationtheory, can be written as | Ψ i = R d p d q Φ( p , q ) a † s ( p ) a † i ( q ) | , i , where the mode function writes [12, 14]Φ ( p , q ) = N exp ( − (Γ L ) k + i ∆ k L ) × exp ( − ( p x + q x ) w + ( p y + q y ) w cos ϕ ) × exp (cid:26) − | p | w s − | q | w s (cid:27) (1)where ∆ k = tan ρ [( p x + q x ) cos α + ( p y + q y ) cos ϕ sin α ] − ( p y − q y ) sin ϕ comes from the phase matching conditionin the z direction, N is a normalization constant, and we assume that the pump beam shows a gaussian beam profilewith beam width w at the input face of the nonlinear crystal. We neglect the transverse momentum dependence ofall longitudinal wavevectors. The phase matching function, sinc (∆ k L/
2) has been approximated by an exponentialfunction that has the same width at the 1 /e of the intensity: sinc ( bx ) ≃ exp[ − (Γ b ) x ], with Γ = 0 . w s describes the effect of the unavoidable spatial filtering produced by the specific optical detection system used.In our experimental set-up, the probability to detect a signal photon at x in coincidence with an idler photon at thefixed pinhole position x = 0 is given by R c ( x , x = 0) = | Φ (2 π x / ( λ s f ) , x = 0) | .Eq. 1 shows that the spatial mode function shape shows ellipticity. The amount of ellipticity depends on thenon-collinear configuration [15], and on the azimuthal angle of emission ( α ) due to the presence of spatial walk off. The latter is the cause of the azimuthal symmetry breaking of the down-conversion cone . Both effects turn out to beimportant when the length of the crystal L is larger than the non-collinear length L nc = w / sin ϕ and the walk-offlength L w = w / tan ρ . Our experimental configuration is fully in this regime. We should notice that in a collinearSPDC configuration, Poynting vector walk-off also introduces ellipticity of the mode function [16].The theory also predicts the orientation of the spatial mode function of the signal photon, as shown in 2. Thisis given by the slope tan β in the ( p x , p y ) plane of the loci of perfect phase matching transverse momentum, which
0 90 180 270 3600.51 angle (degrees) w e i gh t (a)
0 90 180 270 360 angle (degrees)(b) −5 0 500.51 mode number w e i gh t (c) −5 0 5 mode number (e) −5 0 5 mode number (d) −5 0 5 mode number (f)
0º 90º 0º 90º
FIG. 3: Weight of the OAM modes l s = 0 (solid line), and all other modes (dashed lines) as a function of the angle α . (a),(c) and (d) w = 100 µm ; (b), (e) and (f) w = 600 µm . (c) and (e) show the OAM distribution for α = 0 ◦ , and (d) and (f)corresponds to α = 90 ◦ . We assume negligible spatial filtering ( w s ≃ writes tan β = (sin ϕ − tan ρ cos ϕ sin α ) / (tan ρ cos α ). If ϕ ≃ ρ and α = 90 ◦ , the spatial mode function of thesignal photons shows a nearly gaussian shape, due to the compensation of the non-collinear and walk-off effects. Allthese results are in agreement with experimental data in Fig. 2.This azimuthal variation of the spatial correlations can be made clearer if we express the mode function ofthe signal photon, Φ s ( p ) = Φ ( p , q = ) in terms of orbital angular momentum modes. The mode function canbe described by superposition of spiral harmonics [17] Φ s ( ρ, ϕ ) = (2 π ) − / P m a m ( ρ ) exp ( imϕ ), where a m ( ρ ) =1 / (2 π ) / R dϕ Φ s ( ρ, ϕ ) exp ( − imϕ ), and ρ and ϕ are cylindrical coordinates in the transverse wave-number space.The weight of the m -harmonic is given by C m = R ρdρ | a m ( ρ ) | .The gaussian pump beam corresponds to a mode with l p = 0, while the idler photon is projected into q = 0, whichcorresponds to projection into a large area gaussian mode ( l i = 0). Fig. 3(a) and (b) show the weight of the mode l s = 0, and the weight of all other OAM modes, as a function of the angle α for two different pump beam widths.We observe that the OAM correlations of the two-photon state change along the down-conversion cone due to theazimuthal symmetry breaking induced by the spatial walk-off. This implies that the correlations between OAM modesdo not follow the relationship l p = l s + l i . From Fig. 3 it is clearly observed that for larger pump beams the azimuthalchanges are smoothed out, since in this case the non-collinear and walk-off lengths are much larger than the crystallength.Figure 3(c) and 3(d) plots the OAM decomposition for w = 100 µ m, and Figs. 3(e) and 3(f) for w = 600 µ m, for α = 0 , ◦ . Notice that the weight of the l s = 0 mode is maximum for α = 90 ◦ , which therefore is the optimum anglefor the generation of heralded single photons with a gaussian-like shape. This effect can be clearly observed in Figs.2(b), 3(d) and 3(f). On the contrary, for α = 270 ◦ , the combined effects of the noncollinear and walk off effects makethe weight of the l s = 0 mode to obtain its minimum value. This is of relevance in any quantum information protocolwhere the generated photons, no matter the degree of freedom where the quantum information is encoded, are to becoupled into single mode fibers.Importantly, the degree of spatial entanglement of the two-photon state also shows azimuthal variations, dependingon the direction of emission of the down-converted photons. Fig. 4 shows the Schmidt number K = 1 /T rρ s , where ρ s = T r i | Ψ ih Ψ | , is the density matrix that describe the quantum state of the signal photon, after tracing out thespatial variables corresponding to the idler photon. The Schmidt number [6] is a measure of the degree of entanglementof the spatial two photon state, K = 1 corresponding to a product state, while larger values of K corresponds toincreasingly larger values of the degree of entanglement. The degree of entanglement is maximum for α = 0, and
50 60 70 80 90 100 collection mode ( µ m) angle (degree) S c h m i d t N u m be r α =0º α =45º α =90º FIG. 4: (a) Schmidt number (K) as a function of the angle α . The width of the collection mode is w s = 50 µ m. (b) Schmidtnumber as a function of the width of the collection mode for different values of α . In all cases, the pump beam width is w = 100 µ m. The Schmidt number for the case with negligible walk-off (dashed lines) is shown for comparison. minimum for α = 90 ◦ , as shown in Fig. 4(a). The degree of entanglement is known to decrease with increasingfiltering [18], i.e., larger values of w s , as shown in Fig. 4(b), and to increase for larger values of the pump beam width( w ). EFFECTS ON THE GENERATION OF POLARIZATION ENTANGLEMENT
The azimuthal distinguishing information introduced by walking SPDC affect the quantum properties ofpolarization-entangled states, when photons generated in different sections of the down-conversion cone are used.This is the case when using two type I SPDC crystal whose optical axis are rotated 90 ◦ . This configuration, originallydemonstrated for the generation of polarization-entangled photons [10], has been used as well for the generation ofhyperentangled quantum states [4]. The quantum state of the two-photon state writes | Ψ i = 1 √ Z d p d q [Φ ( p , q ) | H, p i s | H, q i i +Φ ( p , q ) | V, p i s | V, q i i ] (2)Φ ( p , q ) = Φ ( α = 0 , p , q ) exp ( ip y tan ρ s L + iq y tan ρ i L ) describes the spatial shape of the photons generated in thefirst nonlinear crystal, ρ s,i are the spatial walk-off angles of the down-converted photons traversing the second nonlinearcrystal, and Φ ( p , q ) = Φ ( α = 90 ◦ , p , q ) corresponds to the photons generated in the second nonlinear crystal. Thequantum state in the polarization space is obtained tracing out the spatial variables, i.e., ρ p = T r s | Ψ ih Ψ | , which gives ρ p = 12 {| H i s | H i i h H | s h H | i + | V i s | V i i h V | s h V | i + ξ [ | H i s | H i i h V | s h V | i + | V i s | V i i h H | s h H | i ] } (3)where ξ = R d p d q Φ ( p , q ) Φ ∗ ( p , q ).The degree of mixture of polarization and spatial variables is determined by the purity ( P ) of the quantum stategiven by Eq. (3), which writes P = 1 / (cid:0) | ξ | (cid:1) . The concurrence of the polarization-entangled state, which writeswrites C = | ξ | , quantifies the quality of the polarization entangled state. Fig. 5 shows the concurrence of the quantumstate for two different crystal lengths. If spatial walk-off effects are negligible, | ξ | = 1 and spatial and polarizationvariables can be separated. Therefore, both the purity and the concurrence are equal to 1. This is the case shown inFig. 5 for a crystal length of L = 0 .
50 250 50000.51 beam waist ( µ m) c on c u rr en c e L=2mmL=0.5mm
FIG. 5: Concurrence (C) of the polarization entangled bi-photon state generated in a two crystal configuration, as a functionof the pump beam, for two different values of the crystal length. The non-collinear angle is ϕ = 2 ◦ , and the pump beam waistis w = 100 µ m. CONCLUSIONS
We have shown theoretically and experimentally, that the presence of Poynting vector walk-off in SPDC configura-tions introduces azimuthal distinguishing information of paired photons emitted in different directions of propagation.The quantum correlations of the spatial two-photon state and, consequently, the degree of entanglement show az-imuthal variations that are enhanced when using highly focused pump beams and broadband spatial filters. Thisbreaking of the azimuthal symmetry of the down-conversion cone has important consequences when designing andimplementing sources of paired photons with entangled properties.
ACKNOWLEDGEMENTS
We want to thank X. Vidal and M. Navascues for helpful discussions. This work was supported by projects FIS2004-03556 and Consolider-Ingenio 2010 QOIT from Spain, by the European Commission under the Integrated ProjectQubit Applications (Contract No. 015848) and by the Generalitat de Catalunya. [1] G. Molina-Terriza, J. P. Torres and L. Torner, “Twisted photons”, Nature Phys. , 305 (2007).[2] A. Vaziri, J. Pan, T. Jennewein, G. Weihs and A. Zeilinger, “Concentration of Higher Dimensional Entanglement: Qutritsof Photon Orbital Angular Momentum”, Phys. Rev. Lett. , 227902 (2003).[3] G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “ Experimental Quantum Coin Tossing”, Phys. Rev. Lett. ,040501 (2005).[4] J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat , “Generation of Hyperentangled Photon Pairs”, Phys. Rev.Lett. , 260501 (2005).[5] J. P. Torres, A. Alexandrescu and L. Torner, “Quantum spiral bandwidth of entangled two-photon states”, Phys. Rev. A , 127903 (2004).[7] H. H. Arnaut and G. A. Barbosa, “Orbital and Intrinsic Angular Momentum of Single Photons and Entangled Pairs ofPhotons Generated by Parametric Down-Conversion ”, Phys. Rev. Lett. , 286 (2000).[8] A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons”, Nature , 313 (2001). [9] P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametricdown-conversion”, Phys. Rev. A , 033803 (2005).[10] P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum and P. H. Eberhard , “Ultrabright source of polarization-entangledphotons”, Phys. Rev. A , R773 (1999).[11] J. Altepeter, E. Jeffrey, and P. Kwiat, “Phase-compensated ultra-bright source of entangled photons”, Opt. Express, ,8951 (2005).[12] J. P. Torres, G. Molina-Terriza and L. Torner, “The spatial shape of entangled photon states generated in non-collinear,walking parametric downconversion”, J. Opt. B: Quantum Semiclass. Opt. , 235 (2005).[13] A. Migdall, “Polarization directions of noncollinear phase-matched optical parametric downconversion output”, J. Opt.Soc. Am. B, , 5349 (1996).[15] G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych and J. P. Torres, “Control of the shape of thespatial mode function of photons generated in noncollinear spontaneous parametric down-conversion”, Phys. Rev. A ,065802 (2005).[16] M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe and S. P. Kulik, “Anisotropically and HighEntanglement of Biphoton States Generated in Spontaneous Parametric Down-Conversion”, Phys. Rev. Lett. , 063901(2007).[17] G. Molina-Terriza, J. P. Torres and L. Torner, “Management of the Angular Momentum of Light: Preparation of Photonsin Multidimensional Vector States of Angular Momentum”, Phys. Rev. Lett. , 013601 (2002).[18] M. P. van Exter, A. Aiello, S. S. R. Oemrawsingh, G. Nienhuis, and J. P. Woerdman, “Effect of spatial filtering on theSchmidt decomposition of entangled photons”, Phys. Rev. A74