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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 10 — May. 14, 2007
  • pp: 6431–6438
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Mode counting in high-dimensional orbital angular momentum entanglement

M.P. van Exter, P.S.K. Lee, S. Doesburg, and J.P. Woerdman  »View Author Affiliations


Optics Express, Vol. 15, Issue 10, pp. 6431-6438 (2007)
http://dx.doi.org/10.1364/OE.15.006431


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Abstract

We study the high-dimensional orbital angular momentum (OAM) entanglement contained in the spatial profiles of two quantum-correlated photons. For this purpose, we use a multi-mode two-photon interferometer with an image rotator in one of the interferometer arms. By measuring the two-photon visibility as a function of the image rotation angle we measure the azimuthal Schmidt number, i.e., we count the number of OAM modes involved in the entanglement; in our setup this number is tunable from 1 to 8.

© 2007 Optical Society of America

1. Introduction

The most popular variety of quantum entanglement involves the polarization degree of freedom of two photons; in this case we deal obviously with two (polarization) modes per photon [1

1. Z.Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988). [CrossRef] [PubMed]

, 2

2. P.G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A.V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995). [CrossRef] [PubMed]

, 3

3. C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A 64, 023802 (2001). [CrossRef]

]. Recently, there has been a lot of interest in spatial entanglement of two photons; in this case the number of modes per photon can be much larger than two so that entanglement is correspondingly richer [4

4. R.S. Bennink, S.J. Bentley, R.W. Boyd, and J.C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004). [CrossRef] [PubMed]

, 5

5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entangelement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef] [PubMed]

, 6

6. N.K. Langford, R.B. Dalton, M.D. Harvey, J.L. O’Brien, G.J. Pryde, A. Gilchrist, S.D. Bartlett, and A.G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004). [CrossRef] [PubMed]

, 7

7. S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005). [CrossRef] [PubMed]

, 8

8. D.P. Caetano and P.H. Souto Ribeiro, “Generation of spatial antibunching with free-propagating twin beams,” Phys. Rev. A 68, 043806 (2003). [CrossRef]

, 9

9. S.P. Walborn, A.N. de Oliveira, S. Pédua, and C.H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003). [CrossRef] [PubMed]

, 10

10. W.A.T. Nogueira, S.P. Walborn, S. Pédua, and C.H. Monken, “Generation of a two-photon singlet beam,” Phys. Rev. Lett. 92, 043602 (2004). [CrossRef] [PubMed]

, 11

11. L. Neves, G. Lima, J.G. Aguirre Gomez, C.H. Monken, C. Saavedra, and S. Pédua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005). [CrossRef] [PubMed]

]. This interest is motivated, fundamentally, by the desire to understand the nature of quantum entanglement in a high-dimensional Hilbert space. From the point of view of applications the high-dimensional case is important since it holds promise for implementing high-dimensional alphabets for quantum information, e.g. for quantum key distribution [12

12. H. Bechmann-Pasquinucci and A. Peres, “Quantum cryptography with 3-state systems,” Phys. Rev. Lett. 85, 3313–3316 (2000). [CrossRef] [PubMed]

]. A popular basis for the spatial modes is the basis in which the modes are distinguished on account of their orbital angular momentum (OAM) [5

5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entangelement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef] [PubMed]

, 6

6. N.K. Langford, R.B. Dalton, M.D. Harvey, J.L. O’Brien, G.J. Pryde, A. Gilchrist, S.D. Bartlett, and A.G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004). [CrossRef] [PubMed]

, 7

7. S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005). [CrossRef] [PubMed]

]. An issue of much discussion in high-dimensional entanglement, OAM or otherwise, is how many modes are involved, beyond the statement that this number can be much larger than 2 [4

4. R.S. Bennink, S.J. Bentley, R.W. Boyd, and J.C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004). [CrossRef] [PubMed]

, 5

5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entangelement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef] [PubMed]

, 6

6. N.K. Langford, R.B. Dalton, M.D. Harvey, J.L. O’Brien, G.J. Pryde, A. Gilchrist, S.D. Bartlett, and A.G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004). [CrossRef] [PubMed]

, 7

7. S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005). [CrossRef] [PubMed]

, 13

13. J.P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301(R) (2003). [CrossRef]

, 14

14. L. Torner, J.P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13, 873–881 (2005). [CrossRef] [PubMed]

, 15

15. C.K. Law and J.H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004). [CrossRef] [PubMed]

]. In this article we demonstrate a practical method to quantify the number of OAM spatial modes involved in bi-photon entanglement; in our experiment this number has been varied in a controlled way from 1 to 8. This result has been achieved by using a special two-photon interferometer.

Our two-photon interferometer contains an image rotator in one of its arms (see Fig. 1). Similar interferometers with built-in rotation have only been tested at the one-photon level, where the rotation has been linked to a topological (Berry) phase [16

16. M. Segev, R. Solomon, and A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992). [CrossRef] [PubMed]

]. A one-photon interferometer with an image reversal has been shown to act as a sorter between even and odd spatial modes [17

17. H. Wei, X. Xue, J. Leach, M.J. Padgett, S.M. Barnett, S. Franke-Arnold, E. Yao, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003). [CrossRef]

, 18

18. H. Sasada and M. Okamoto, “Tranverse-mode beam splitter of a light beam and its application to cryptography,” Phys. Rev. A 68, 012323 (2003). [CrossRef]

]. We will instead consider two-photon interference in an interferometer with built-in rotation.

In two-photon interference experiments, two photons are combined on a beam splitter, before being detected. These experiments, which have been pioneered by Hong, Ou and Mandel (HOM) [19

19. C.K. Hong, Z.Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987). [CrossRef] [PubMed]

], demonstrate an effective bunching between the photons in each pair, but only if the optical beams have good spatial and temporal overlap. More recent versions of these “HOM” experiments study the generation of spatial anti-bunching [8

8. D.P. Caetano and P.H. Souto Ribeiro, “Generation of spatial antibunching with free-propagating twin beams,” Phys. Rev. A 68, 043806 (2003). [CrossRef]

], and the effect of a modified pump profile (TEM01 versus TEM00) on the interference pattern (bunching versus anti-bunching) [9

9. S.P. Walborn, A.N. de Oliveira, S. Pédua, and C.H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003). [CrossRef] [PubMed]

, 10

10. W.A.T. Nogueira, S.P. Walborn, S. Pédua, and C.H. Monken, “Generation of a two-photon singlet beam,” Phys. Rev. Lett. 92, 043602 (2004). [CrossRef] [PubMed]

].

The key question that we will address is what the two-photon interference in our interferometer-with-built-in-rotation tells us about the spatial entanglement between the two multi-mode beams. As our geometry leads to an effective separation of the radial and azimuthal degrees of freedom, where only the latter are manipulated, the experiment only provides information on the entanglement between the orbital angular momenta (OAM) of the two photons [5

5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entangelement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef] [PubMed]

, 6

6. N.K. Langford, R.B. Dalton, M.D. Harvey, J.L. O’Brien, G.J. Pryde, A. Gilchrist, S.D. Bartlett, and A.G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004). [CrossRef] [PubMed]

, 7

7. S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005). [CrossRef] [PubMed]

]. It allows to measure the azimuthal Schmidt number, i.e., to count the number of entangled OAM modes.

2. Experimental results

Figure 1 is a schematic view of our two-photon interferometer. We mildly focus light from a krypton ion laser (γ =407 nm, θP = 0.50 mrad divergence) onto a 1-mm-thick β-barium borate (BBO) crystal to generate quantum-entangled photon pairs at 814 nm via (type-I) spontaneous parametric down-conversion. These twin photons travel along individual interferometer arms, one of them through an image rotator R(θ), before they are combined at a beam splitter. The experimental results shown in Figs. 2–4 are obtained with an interferometer that contains an odd number of mirrors (as in Fig. 1). Two-photon interference is observed by recording the number of coincidences as a function of the delay ∆t between the two arms with single-photon counters (SPC). An adjustable circular aperture, positioned in one detection arm at L = 1.5 m from the crystal (≈ far field), allows us to control the detected number of entangled spatial modes. The limited detection bandwidth (5 nm) and detection angle (< 7 mrad) assure operation in the so-called thin-crystal limit [20

20. B.E.A. Saleh, A.F. Abouraddy, A.V. Sergienko, and M.C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000). [CrossRef]

], where phase matching is automatically fulfilled. In this limit, the spatial properties of the detected two-photon field are solely determined by the TEM00 pump profile.

Fig. 1. Schematic view of the experimental setup, representing a two-photon interferometer with an image rotator R(θ), comprising four out-of-plane mirrors, in one arm.

The image rotator R(θ) consists of a fixed mirror M1 and a rotatable “open Dove prism”, comprising three separate mirrors, arranged in an equilateral triangle, i.e., operating at reflection angles of 60°,30° and 60°. This construction has several advantages over alternative arrangements. By working with a unit of 3 mirrors instead of a glass Dove prism [21

21. J. Leach, M.J. Padgett, S.M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002). [CrossRef] [PubMed]

], we avoid any detrimental effects of wavelength dispersion, which could lead to temporal labelling and reduced interference. Furthermore, the adjustability of the mirrors allows us to reduce unwanted beam deflection to angles ≤ 0.1 mrad, whereas glass Dove prisms have typical wedge angles of ≈ 1 mrad. Finally, whereas image rotations are generally accompanied by polarization rotations [16

16. M. Segev, R. Solomon, and A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992). [CrossRef] [PubMed]

], our rotator hardly changes the polarization. This convenient property is obtained by using silver mirrors (protected by a thin SiO2 cover layer) instead of dielectric mirrors. The measured phase difference ϕs-p = 0.81π between the (three-fold) reflected s- and p-polarized light, is in fact close enough to the ideal value of ϕs-p = π needed for a polarization-insensitive rotator, to limit the measured power in the orthogonal polarization to at most 8% (a precise quantitative check of these values based upon the optical constants of silver is hampered by the fact that we do not know the precise thickness and porosity of the protective SiO2 coating). As both polarization components have the same spatial profile, we simply remove this small unwanted orthogonal component with a fixed polarizer (P 1 in Fig. 1). The topological phase [22

22. R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988). [CrossRef] [PubMed]

] that originates from the mentioned polarization changes goes unnoticed, as two-photon interferometers are insensitive to the optical phase.

Fig. 2. Two-photon coincidence rate versus the time delay Δt between the two interferometer arms, measured at a fixed rotation angle of θ = -30° behind a 1 mm aperture (dots) and a 10 mm aperture (squares). The coincidence rate measured for the 1 mm aperture has been multiplied by the area ratio (≈ 100x) for a direct comparison with the other geometry.

Fig. 3. Two-photon visibility versus the aperture diameter 2a, measured at a fixed rotation angle of θ = -30°. The solid curve represents a fit. The two encircled data points correspond to the interference patterns shown in Fig. (2).
Fig. 4. Two-photon visibility measured as a function of the rotation angle θ behind different aperture geometries (specified by the azimuthal Schmidt number Kaz) and behind single-mode fibers (Kaz = 1). The three dashed lines have been calculated from Eq. (1)

By repeating the measurements shown in Fig. 3 for a series of fixed rotation angles we obtain a two-dimensional table of visibilities V(a, θ) and from that the visibility V(θ) as a function of rotation angle θ for various fixed detection geometries. Figure 4 shows these results for four different geometries, which are specified by their azimuthal Schmidt number Kaz (see below). All curves are symmetric under the operation θθ(θ = 0° corresponds to no image rotation) and periodic in θθ + 180°. This last observation, that a rotation over 180° instead of 360° already produces identical physics, reflects the two-photon character of the interference.

For detection behind single-mode fibers (labeled as Kaz = 1) we obtained visibilities of at least 98%, independent of θ. As the fundamental fiber mode is rotationally symmetric, spatial labeling and loss of interference under image rotation will not occur. For free-space detection behind small apertures (small Kaz) the effect of image rotation on the two-photon interference is relatively mild. For larger apertures, this effect is much more drastic and leads to a visibility as low as 4% at θ = 90° for Kaz = 8. The reason for this reduction is that free-space detectors also monitor linear combinations of higher-order modes, which are no longer invariant under rotation and thus provide labeling information.

The theoretical curves in Figs. 3 and 4 are based on the following analytic expression [23

23. W.H. Peeters, E.J.K. Verstegen, and M.P. van Exter, “Orbital angular momentum analysis of high-dimensional entanglement,” submitted to Phys. Rev. A.

]

V(asinθ)=(1exp(ξ))ξ,
(1)

where ξ = 2(a/wd)2 sin2 θ and a is the aperture radius. The diffraction waist wd = 2p, or angular spread of one photon at a fixed position of the other, is twice the size of the pump in the (far-field) detection plane [24

24. C.H. Monken, P.H. Souto Ribeiro, and S. Padua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123–3126 (1998). [CrossRef]

]. The solid curve in Fig. 3 is a fit based on wd = 1.4 mm, in agreement with the mentioned values of L and θp. The three dashed curves in Fig. 4 are based on the same value and contain no adjustable parameter, apart from a small uniform scaling of the vertical axis. We attribute the (small) deviations between theory and experiment to imperfect beam alignment. These deviations show up most prominently at small Kaz (Kaz = 1.13 in Fig. 4), where the two-photon visibility should remain high over a large angular range. At large Kaz the visibility drop upon rotation is fast enough to dominate spurious misalignment effects.

3. Mode counting

V(θ)=lPlcos(2),
(2)

where the spiral weight Pl (with -∞ < l < ∞ and ∑l Pl = 1) is the probability to detected a photon pair with orbital angular momenta (l, -l) [14

14. L. Torner, J.P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13, 873–881 (2005). [CrossRef] [PubMed]

]. Equation (2) expresses the observed visibility V(θ) as a weighted sum over contributions from groups of l-modes, each contribution oscillating between Vl = 1 (HOM dip) and Vl = - 1 (HOM peak), with its own angular dependence. It shows the power of our experiment, where a simple Fourier transformation of the measured V(θ) yields the complete OAM distribution {Pl}.

Based on the above description, we count the number of entangled OAM modes in our experiment as follows: For the three lower curves in Fig. 4 we first performed a Fourier analysis of the normalized V(θ)/V(0) to obtain the probability distribution Pl for each theoretical curve. The azimuthal Schmidt numbers that we calculated from these distributions are Kaz = 1.13 for the 1 mm aperture, Kaz = 2.9 for the 4 mm aperture, and Kaz = 8 for the 10 mm aperture. The aperture clearly allows us to tune the effective number of entangled modes.

One might wonder whether, and if so, in what sense, our experiment proofs the existence of spatial entanglement and the conservation of OAM in the SPDC pair production. Suppose we would have based our analysis on a more general two-photon input state that is not restricted to l 1 + l 2 = 0. The calculated visibility V(θ) for an interferometer with an odd number of mirrors would then contain terms of the form P l1,l2 cos[(l 1 -l 2)θ], again using the mirror symmetry P l1,l2 = P l2,l1. We can not exclude this possibility a priori. For an interferometer with an even number of mirrors, however, V(θ) would then contain terms of the form P l1,l2 cos[(l 1 + l 2)θ] instead. Our observation that V(θ) ≈ 1 at any angle θ in the “even-mirror geometry”, can thus be interpreted as a real proof of the existence of OAM entanglement; any photon pair with l 1 ≠ -l 2 would make V(θ) angular dependent.

From a theoretical perspective, it is instructive to also consider a configuration with a Gaussian instead of a hard-edged transmission profile, having an aperture with an intensity transmission profile T(r) = exp(-r 2/ã 2). This combination allows for a complete (radial and azimuthal) analytic Schmidt decomposition of the detected field[27

27. M.P. van Exter, A. Aiello, S.S.R. Oemrawsingh, G. Nienhuis, and J.P. Woerdman, “Effect of spatial filtering on the Schmidt decomposition of entangled photons,” Phys. Rev. A 74, 012309 (2006). [CrossRef]

], and yields[23

23. W.H. Peeters, E.J.K. Verstegen, and M.P. van Exter, “Orbital angular momentum analysis of high-dimensional entanglement,” submitted to Phys. Rev. A.

]

V(θ)=11+(K2D1)sin2θ,
(3)

where K 2D = 1 +(ã/wd)2 is the 2D Schmidt number. The Airy profile of Eq. (3) has almost the same shape as the function described by Eq. (1). It again allows for a Fourier decomposition of the form (2), providing analytic expressions for Pl as a power series of the form Plαl (with α < 1).

4. Summary

In summary, we have demonstrated how the high-dimensional entanglement of orbital angular momentum (OAM) can be characterized with a two-photon interferometer that contains an odd number of mirrors and an image rotator in one of its interferometer arms. We have shown how a Fourier analysis of the observed angle-dependent visibility V(θ) profile yields the full probability distribution over the OAM modes involved in the entanglement. Finally, we have calculated the azimuthal Schmidt number Kaz corresponding to the effective number of entangled OAM modes. At Kaz = 1 (fiber-coupled detection) the detected two-photon field is a direct product state that contains no spatial entanglement; at Kaz = 2 the two-photon field acts as a pair of entangled qubits; at Kaz = 8 we deal with entangled quNits (with N = 8) with a much richer internal structure.

Appendix

The derivation of the Fourier relation between the visibility V(θ) and the OAM spectrum {Pl}, as presented in Eq. (2), starts with a (Schmidt) decomposition of the two-photon field in a sum over discrete spatial modes, instead of an integral over a plane-wave continuum. The two-photon field is thus represented by the pure state:

Ψ=iλi|ui|vi,
(4)

where ∣ui〉 and ȣvi〉 are two sets of orthonormal transverse modes. The Schmidt number K = 1/(Σiλi 2), with Σλi = 1, quantifies the effective number of participating modes.

As a second ingredient, we use the rotation symmetry of the pump beam and the detection apertures, in combination with the small-angle approximation, to single out the azimuthal dependence (OAM) of the two-photon field. Our Schmidt decomposition thus factorizes to

|Ψin=lpλl,pel,p|l,p|l,p,
(5)

where l and p are the azimuthal and radial quantum numbers and ∣l,p〉′ and ∣ - l,p〉″ are the Schmidt eigenmodes of the detected field. The rotation symmetry restricts these modes to “Laguerre-Gaussian-like” field profiles of which the precise radial distribution is co-determined by the detection apertures. As our amplitude coefficients √γl,p already contain the effects of aperture filtering, they will decrease rapidly both for high p and high l values (high l-states are quite extended even for p = 0). A summation over the radial mode number p yields the OAM probability Pl = Σpλl,p.

As a last step, we propagate the two-photon field of Eq. (5) through our interferometer and calculate the expected two-photon visibility V(θ). This propagation will modify the two-photon field in the following ways: every mirror reflection changes the handedness by inverting the OAM of each l-state from l to −l. The image rotation R(θ) adds a phase factor exp(ilθ) to each l-state. The relevant beam splitter operations are the double transmission, which leaves the l-states unaffected, and the double reflection, which swaps the labels and changes the handedness. Application of these operations to the state of Eq. (5), in combination with the “signal ↔ idler” symmetry (λ-l,p = λl,p and φ-i,p = φl,p) and the orthogonality of the (l,p) states, finally yields

|Ψ(θ)det=lpλl,pel,p(ei(+12Δωτ)ei(+12Δωτ))l,pl,p,
(6)

for the combined field behind a (50/50) beam splitter, where ∆π is the frequency difference between the two detected photons and τ is the time delay difference in the interferometer. By comparing the expected coincidence rate Rcc(θ, ∆τ) ∝ 〈Ψ(θ)∣Ψ(θ)〉det with the definition of the two-photon visibility, we obtain Eq. (2) as final result.

Acknowledgements

This work is supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM).

References and links

1.

Z.Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988). [CrossRef] [PubMed]

2.

P.G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A.V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995). [CrossRef] [PubMed]

3.

C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A 64, 023802 (2001). [CrossRef]

4.

R.S. Bennink, S.J. Bentley, R.W. Boyd, and J.C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004). [CrossRef] [PubMed]

5.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entangelement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef] [PubMed]

6.

N.K. Langford, R.B. Dalton, M.D. Harvey, J.L. O’Brien, G.J. Pryde, A. Gilchrist, S.D. Bartlett, and A.G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004). [CrossRef] [PubMed]

7.

S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005). [CrossRef] [PubMed]

8.

D.P. Caetano and P.H. Souto Ribeiro, “Generation of spatial antibunching with free-propagating twin beams,” Phys. Rev. A 68, 043806 (2003). [CrossRef]

9.

S.P. Walborn, A.N. de Oliveira, S. Pédua, and C.H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003). [CrossRef] [PubMed]

10.

W.A.T. Nogueira, S.P. Walborn, S. Pédua, and C.H. Monken, “Generation of a two-photon singlet beam,” Phys. Rev. Lett. 92, 043602 (2004). [CrossRef] [PubMed]

11.

L. Neves, G. Lima, J.G. Aguirre Gomez, C.H. Monken, C. Saavedra, and S. Pédua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005). [CrossRef] [PubMed]

12.

H. Bechmann-Pasquinucci and A. Peres, “Quantum cryptography with 3-state systems,” Phys. Rev. Lett. 85, 3313–3316 (2000). [CrossRef] [PubMed]

13.

J.P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301(R) (2003). [CrossRef]

14.

L. Torner, J.P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13, 873–881 (2005). [CrossRef] [PubMed]

15.

C.K. Law and J.H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004). [CrossRef] [PubMed]

16.

M. Segev, R. Solomon, and A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992). [CrossRef] [PubMed]

17.

H. Wei, X. Xue, J. Leach, M.J. Padgett, S.M. Barnett, S. Franke-Arnold, E. Yao, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003). [CrossRef]

18.

H. Sasada and M. Okamoto, “Tranverse-mode beam splitter of a light beam and its application to cryptography,” Phys. Rev. A 68, 012323 (2003). [CrossRef]

19.

C.K. Hong, Z.Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987). [CrossRef] [PubMed]

20.

B.E.A. Saleh, A.F. Abouraddy, A.V. Sergienko, and M.C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000). [CrossRef]

21.

J. Leach, M.J. Padgett, S.M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002). [CrossRef] [PubMed]

22.

R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988). [CrossRef] [PubMed]

23.

W.H. Peeters, E.J.K. Verstegen, and M.P. van Exter, “Orbital angular momentum analysis of high-dimensional entanglement,” submitted to Phys. Rev. A.

24.

C.H. Monken, P.H. Souto Ribeiro, and S. Padua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123–3126 (1998). [CrossRef]

25.

R. Grobe, K. Rzazewski, and J.H. Eberly, “Measure of electron-electron correlation in atomic physics,” J. Phys. B 27, L503–L508 (1994). [CrossRef]

26.

A. Ekert and P.L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995). [CrossRef]

27.

M.P. van Exter, A. Aiello, S.S.R. Oemrawsingh, G. Nienhuis, and J.P. Woerdman, “Effect of spatial filtering on the Schmidt decomposition of entangled photons,” Phys. Rev. A 74, 012309 (2006). [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(270.0270) Quantum optics : Quantum optics
(270.4180) Quantum optics : Multiphoton processes

ToC Category:
Quantum Optics

History
Original Manuscript: March 5, 2007
Revised Manuscript: April 13, 2007
Manuscript Accepted: April 14, 2007
Published: May 11, 2007

Citation
M. P. van Exter, P. S. K. Lee, S. Doesburg, and J. P. Woerdman, "Mode counting in high-dimensional orbital angular momentum entanglement," Opt. Express 15, 6431-6438 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6431


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References

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  19. C.K. Hong, Z.Y. Ou and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987). [CrossRef] [PubMed]
  20. B.E.A. Saleh, A.F. Abouraddy, A.V. Sergienko, and M.C. Teich, "Duality between partial coherence and partial entanglement," Phys. Rev. A 62, 043816 (2000). [CrossRef]
  21. J. Leach, M.J. Padgett, S.M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88, 257901 (2002). [CrossRef] [PubMed]
  22. R. Bhandari and J. Samuel, "Observation of topological phase by use of a laser interferometer," Phys. Rev. Lett. 60, 1211-1213 (1988). [CrossRef] [PubMed]
  23. W.H. Peeters, E.J.K. Verstegen, and M.P. van Exter, "Orbital angular momentum analysis of high-dimensional entanglement," submitted toPhys. Rev. A.
  24. C.H. Monken, P.H. Souto Ribeiro, and S. Padua, "Transfer of angular spectrum and image formation in spontaneous parametric down-conversion," Phys. Rev. A 57, 3123-3126 (1998). [CrossRef]
  25. R. Grobe, K. Rzazewski, and J.H. Eberly, "Measure of electron-electron correlation in atomic physics," J. Phys. B 27, L503-L508 (1994). [CrossRef]
  26. A. Ekert and P.L. Knight, "Entangled quantum systems and the Schmidt decomposition," Am. J. Phys. 63, 415- 423 (1995). [CrossRef]
  27. M.P. van Exter, A. Aiello, S.S.R. Oemrawsingh, G. Nienhuis, and J.P. Woerdman, "Effect of spatial filtering on the Schmidt decomposition of entangled photons," Phys. Rev. A 74, 012309 (2006). [CrossRef]

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