## Mode counting in high-dimensional orbital angular momentum entanglement

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

*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. 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]

*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. 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]

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]

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]

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]

*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]

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]

*two-photon*interference in an interferometer with built-in rotation.

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]

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]

_{01}versus TEM

_{00}) 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. 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]

**412**, 313–316 (2001). [CrossRef] [PubMed]

**93**, 053601 (2004). [CrossRef] [PubMed]

**95**, 240501 (2005). [CrossRef] [PubMed]

## 2. Experimental results

_{P}= 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]

_{00}pump profile.

*R*(

*θ*) consists of a fixed mirror M

_{1}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]

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]

_{2}cover layer) instead of dielectric mirrors. The measured phase difference

*ϕ*= 0.81

_{s-p}*π*between the (three-fold) reflected s- and p-polarized light, is in fact close enough to the ideal value of

*ϕ*=

_{s-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 SiO

_{2}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]

*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

*K*(see below). All curves are symmetric under the operation

_{az}*θ*↔

*θ*(

*θ*= 0° corresponds to no image rotation) and periodic in

*θ*↔

*θ*+ 180°. This last observation, that a rotation over 180° instead of 360° already produces identical physics, reflects the two-photon character of the interference.

*K*= 1) we obtained visibilities of at least 98%, independent of

_{az}*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

*K*) 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

_{az}*θ*= 90° for

*K*= 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.

_{az}*a*/

*w*)

_{d}^{2}sin

^{2}

*θ*and

*a*is the aperture radius. The diffraction waist

*w*= 2

_{d}*Lθ*, 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

_{p}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]

*w*= 1.4 mm, in agreement with the mentioned values of

_{d}*L*and

*θ*. 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

_{p}*K*(

_{az}*K*= 1.13 in Fig. 4), where the two-photon visibility should remain high over a large angular range. At large

_{az}*K*the visibility drop upon rotation is fast enough to dominate spurious misalignment effects.

_{az}## 3. Mode counting

*V*(

*θ*). More specifically, we find

*P*(with -∞ <

_{l}*l*< ∞ and ∑

_{l}

*P*= 1) is the probability to detected a photon pair with orbital angular momenta (

_{l}*l*, -

*l*) [14

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

*V*(

*θ*) as a weighted sum over contributions from groups of

*l*-modes, each contribution oscillating between

*V*= 1 (HOM dip) and

_{l}*V*= - 1 (HOM peak), with its own angular dependence. It shows the power of our experiment, where a simple Fourier transformation of the measured

_{l}*V*(

*θ*) yields the complete OAM distribution {

*P*}.

_{l}*V*(

*θ*)/

*V*(0) to obtain the probability distribution

*P*for each theoretical curve. The azimuthal Schmidt numbers that we calculated from these distributions are

_{l}*K*= 1.13 for the 1 mm aperture,

_{az}*K*= 2.9 for the 4 mm aperture, and

_{az}*K*= 8 for the 10 mm aperture. The aperture clearly allows us to tune the effective number of entangled modes.

_{az}*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.

*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]

*K*

_{2D}= 1 +(

*ã*/

*w*)

_{d}^{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

*P*as a power series of the form

_{l}*P*∝

_{l}*α*(with

^{l}*α*< 1).

## 4. Summary

*V*(θ) profile yields the full probability distribution over the OAM modes involved in the entanglement. Finally, we have calculated the azimuthal Schmidt number

*K*corresponding to the effective number of entangled OAM modes. At

_{az}*K*= 1 (fiber-coupled detection) the detected two-photon field is a direct product state that contains no spatial entanglement; at

_{az}*K*= 2 the two-photon field acts as a pair of entangled qubits; at

_{az}*K*= 8 we deal with entangled quNits (with

_{az}*N*= 8) with a much richer internal structure.

## Appendix

*V*(θ) and the OAM spectrum {

*P*}, 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:

_{l}*u*〉 and ȣ

_{i}*v*〉 are two sets of orthonormal transverse modes. The Schmidt number

_{i}*K*= 1/(Σ

_{i}λ

_{i}

^{2}), with Σλ

_{i}= 1, quantifies the effective number of participating modes.

*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 √γ

_{l,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

*P*= Σ

_{l}_{p}λ

_{l,p}.

*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}= λ

_{l,p}and φ

_{-i,p}= φ

_{l,p}) and the orthogonality of the (

*l,p*) states, finally yields

*R*(θ, ∆τ) ∝ 〈Ψ(θ)∣Ψ(θ)〉

_{cc}_{det}with the definition of the two-photon visibility, we obtain Eq. (2) as final result.

## Acknowledgements

## 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. |

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. |

3. | C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A |

4. | R.S. Bennink, S.J. Bentley, R.W. Boyd, and J.C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. |

5. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entangelement of the orbital angular momentum states of photons,” Nature |

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. |

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. |

8. | D.P. Caetano and P.H. Souto Ribeiro, “Generation of spatial antibunching with free-propagating twin beams,” Phys. Rev. A |

9. | S.P. Walborn, A.N. de Oliveira, S. Pédua, and C.H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. |

10. | W.A.T. Nogueira, S.P. Walborn, S. Pédua, and C.H. Monken, “Generation of a two-photon singlet beam,” Phys. Rev. Lett. |

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. |

12. | H. Bechmann-Pasquinucci and A. Peres, “Quantum cryptography with 3-state systems,” Phys. Rev. Lett. |

13. | J.P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A |

14. | L. Torner, J.P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express |

15. | C.K. Law and J.H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. |

16. | M. Segev, R. Solomon, and A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. |

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. |

18. | H. Sasada and M. Okamoto, “Tranverse-mode beam splitter of a light beam and its application to cryptography,” Phys. Rev. A |

19. | C.K. Hong, Z.Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. |

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 |

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. |

22. | R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. |

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 |

25. | R. Grobe, K. Rzazewski, and J.H. Eberly, “Measure of electron-electron correlation in atomic physics,” J. Phys. B |

26. | A. Ekert and P.L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. |

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 |

**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

- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- D.P. Caetano and P.H. Souto Ribeiro, "Generation of spatial antibunching with free-propagating twin beams," Phys. Rev. A 68, 043806 (2003). [CrossRef]
- S.P. Walborn, A.N. de Oliveira, S. P’adua, and C.H. Monken, "Multimode Hong-Ou-Mandel interference," Phys. Rev. Lett. 90, 143601 (2003). [CrossRef] [PubMed]
- W.A.T. Nogueira, S.P. Walborn, S. P’adua, and C.H. Monken, "Generation of a two-photon singlet beam," Phys. Rev. Lett. 92, 043602 (2004). [CrossRef] [PubMed]
- L. Neves, G. Lima, J.G. Aguirre Gomez, C.H. Monken, C. Saavedra, and S. P’adua, "Generation of entangled states of qudits using twin photons," Phys. Rev. Lett. 94, 100501 (2005). [CrossRef] [PubMed]
- H. Bechmann-Pasquinucci and A. Peres, "Quantum cryptography with 3-state systems," Phys. Rev. Lett. 85, 3313-3316 (2000). [CrossRef] [PubMed]
- J.P. Torres, A. Alexandrescu, and L. Torner, "Quantum spiral bandwidth of entangled two-photon states," Phys. Rev. A 68, 050301(R) (2003). [CrossRef]
- L. Torner, J.P. Torres, and S. Carrasco, "Digital spiral imaging," Opt. Express 13, 873-881 (2005). [CrossRef] [PubMed]
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