Abstract

The design and implementation of a novel source of degenerate polarization entangled photon pairs in the telecom band, based on a cavity enhanced parametric downconversion process, is presented. Two of the four maximally entangled Bell states are produced; the remaining two are obtainable by the addition of a half wave plate into the setup. The coincident photon detection rate in the A/D basis between two detectors at the output of the device revealed the production of highly entangled states, resulting in quantum interference visibilities of 0.971 ± 0.041 (ϕ = 0 state) and 0.932 ± 0.036 (ϕ = π state) respectively. The entangled states were found to break the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality by around 6 standard deviations. From the measured coincidence counting rates and the optical system losses, an entangled photon pair production rate of 8.9 × 104 s−1 mW−1 pump was estimated.

© 2010 OSA

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2010 (4)

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancment cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4(3), 170–173 (2010).
[CrossRef]

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair souce based on a type-II PPLN waveguide emitting at a telecom wavelength,” N. J. Phys. 12(10), 103005 (2010).
[CrossRef]

M. Medic, J. B. Altepeter, M. A. Hall, M. Patel, and P. Kumar, “Fiber-based telecommuncation-band source of degenerate entangled photons,” Opt. Lett. 35, 802–804 (2010).
[CrossRef] [PubMed]

P.J. Thomas, M.H. Dunn, D.J.M. Stothard, D.A. Walsh, and C.J. Chunnilall, “A pump enhanced source of telecom-band correlated photon pairs,” submitted to J. Mod. Opt. (2010).
[PubMed]

2009 (2)

T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled KTiOPO4 waveguide,” Opt. Express 17(14), 12019–12030 (2009).
[CrossRef] [PubMed]

M. Scholz, L. Koch, and O. Benson, “Statistics of narrow-band single photons for quantum memories generated by ultrabright cavity-enhanced parametric down-conversion,” Phys. Rev. Lett. 102(6), 063603 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (4)

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express 15(23), 15377–15386 (2007).
[CrossRef] [PubMed]

S. Odate, A. Yoshizawa, and H. Tsuchida, “Polarisation-entangled photon-pair source at 1550 nm using 1 mm-long PPLN waveguide in fibre-loop configuration,” Electron. Lett. 43(24), 1376–1377 (2007).
[CrossRef]

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

H. Hübel, M. R. Vanner, T. Lederer, B. Blauensteiner, T. Lorünser, A. Poppe, and A. Zeilinger, “High-fidelity transmission of polarization encoded qubits from an entangled source over 100 km of fiber,” Opt. Express 15(12), 7853–7862 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (3)

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94(5), 053601 (2005).
[CrossRef] [PubMed]

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94(8), 083601-1-4 (2005).

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Erratum: Two-Photon Coincident-Frequency Entanglement via Extended Phase Matching [Phys. Rev. Lett. 94, 083601 (2005)],” Phys. Rev. Lett. 94(16), 169903–1 (2005).
[CrossRef]

2004 (2)

C. E. Kuklewicz, M. Fiorentino, G. Messin, and J. H. Shapiro, “High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter,” Phys. Rev. A 69(1), 013807 (2004).
[CrossRef]

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69(4), 041801 (2004).
[CrossRef]

2003 (1)

A. Yoshizawa, R. Kaji, and H. Tsuchida, “Generation of polarisation-entangled photon pairs at 1550 nm using two PPLN waveguides,” Electron. Lett. 39(7), 621–622 (2003).
[CrossRef]

2002 (1)

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Extended phase-matching conditions for improved entanglement generation,” Phys. Rev. A 66(4), 043813 (2002).
[CrossRef]

2001 (5)

W. J. Alford and A. V. Smith, “Wavelength variation of the second-order nonlinear coefficients of KNbO3, KTiOPO4, KTiOAsO4, LiNbO3, LiIO3, β-BaB2O4, KH2PO4, and LiB3O5 crystals: a test of Miller wavelength scaling,” J. Opt. Soc. Am. B 18(4), 524–533 (2001).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[CrossRef]

J. Volz, C. Kurtsiefer, and H. Weinfurter, “Compact all-solid-state source of polarization-entangled photon pairs,” Appl. Phys. Lett. 79(6), 869–871 (2001).
[CrossRef]

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65(1), 012314 (2001).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64(6), 062311 (2001).
[CrossRef]

2000 (1)

M. Oberparleiter and H. Weinfurter, “Cavity-enhanced generation of polarisation-entangled photon pairs,” Opt. Commun. 183(1-4), 133–137 (2000).
[CrossRef]

1999 (1)

D. Gottesman and I. L. Chuang, “Demonstrating the viabilty of universal quantum computation using teleportation and single-qubit operations,” Nature 402(6760), 390–393 (1999).
[CrossRef]

1995 (1)

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(24), 4337–4341 (1995).
[CrossRef] [PubMed]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[CrossRef] [PubMed]

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories,” Phys. Rev. Lett. 23(15), 880–884 (1969).
[CrossRef]

Albota, M. A.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94(8), 083601-1-4 (2005).

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Erratum: Two-Photon Coincident-Frequency Entanglement via Extended Phase Matching [Phys. Rev. Lett. 94, 083601 (2005)],” Phys. Rev. Lett. 94(16), 169903–1 (2005).
[CrossRef]

Alford, W. J.

Alibart, O.

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair souce based on a type-II PPLN waveguide emitting at a telecom wavelength,” N. J. Phys. 12(10), 103005 (2010).
[CrossRef]

Altepeter, J. B.

Barbieri, C.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Battle, P.

Benson, O.

M. Scholz, L. Koch, and O. Benson, “Statistics of narrow-band single photons for quantum memories generated by ultrabright cavity-enhanced parametric down-conversion,” Phys. Rev. Lett. 102(6), 063603 (2009).
[CrossRef] [PubMed]

Blauensteiner, B.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

H. Hübel, M. R. Vanner, T. Lederer, B. Blauensteiner, T. Lorünser, A. Poppe, and A. Zeilinger, “High-fidelity transmission of polarization encoded qubits from an entangled source over 100 km of fiber,” Opt. Express 15(12), 7853–7862 (2007).
[CrossRef] [PubMed]

Chen, J.

Chuang, I. L.

D. Gottesman and I. L. Chuang, “Demonstrating the viabilty of universal quantum computation using teleportation and single-qubit operations,” Nature 402(6760), 390–393 (1999).
[CrossRef]

Chunnilall, C.J.

P.J. Thomas, M.H. Dunn, D.J.M. Stothard, D.A. Walsh, and C.J. Chunnilall, “A pump enhanced source of telecom-band correlated photon pairs,” submitted to J. Mod. Opt. (2010).
[PubMed]

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories,” Phys. Rev. Lett. 23(15), 880–884 (1969).
[CrossRef]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Dunn, M.H.

P.J. Thomas, M.H. Dunn, D.J.M. Stothard, D.A. Walsh, and C.J. Chunnilall, “A pump enhanced source of telecom-band correlated photon pairs,” submitted to J. Mod. Opt. (2010).
[PubMed]

Fedrizzi, A.

Fiorentino, M.

T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A 73(1), 012316 (2006).
[CrossRef]

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94(8), 083601-1-4 (2005).

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Erratum: Two-Photon Coincident-Frequency Entanglement via Extended Phase Matching [Phys. Rev. Lett. 94, 083601 (2005)],” Phys. Rev. Lett. 94(16), 169903–1 (2005).
[CrossRef]

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69(4), 041801 (2004).
[CrossRef]

C. E. Kuklewicz, M. Fiorentino, G. Messin, and J. H. Shapiro, “High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter,” Phys. Rev. A 69(1), 013807 (2004).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Franson, J. D.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64(6), 062311 (2001).
[CrossRef]

Fürst, M.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Giovannetti, V.

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Extended phase-matching conditions for improved entanglement generation,” Phys. Rev. A 66(4), 043813 (2002).
[CrossRef]

Gottesman, D.

D. Gottesman and I. L. Chuang, “Demonstrating the viabilty of universal quantum computation using teleportation and single-qubit operations,” Nature 402(6760), 390–393 (1999).
[CrossRef]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Hall, M. A.

Herbst, T.

Herrmann, H.

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair souce based on a type-II PPLN waveguide emitting at a telecom wavelength,” N. J. Phys. 12(10), 103005 (2010).
[CrossRef]

Holt, R. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories,” Phys. Rev. Lett. 23(15), 880–884 (1969).
[CrossRef]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[CrossRef] [PubMed]

Horne, M. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories,” Phys. Rev. Lett. 23(15), 880–884 (1969).
[CrossRef]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Hübel, H.

Issautier, A.

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair souce based on a type-II PPLN waveguide emitting at a telecom wavelength,” N. J. Phys. 12(10), 103005 (2010).
[CrossRef]

Jacobs, B. C.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64(6), 062311 (2001).
[CrossRef]

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[CrossRef]

Jennewein, T.

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express 15(23), 15377–15386 (2007).
[CrossRef] [PubMed]

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Kaji, R.

A. Yoshizawa, R. Kaji, and H. Tsuchida, “Generation of polarisation-entangled photon pairs at 1550 nm using two PPLN waveguides,” Electron. Lett. 39(7), 621–622 (2003).
[CrossRef]

Kärtner, F. X.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Erratum: Two-Photon Coincident-Frequency Entanglement via Extended Phase Matching [Phys. Rev. Lett. 94, 083601 (2005)],” Phys. Rev. Lett. 94(16), 169903–1 (2005).
[CrossRef]

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94(8), 083601-1-4 (2005).

Kiesel, N.

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancment cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4(3), 170–173 (2010).
[CrossRef]

Kikuchi, K.

Kim, T.

T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A 73(1), 012316 (2006).
[CrossRef]

Koch, L.

M. Scholz, L. Koch, and O. Benson, “Statistics of narrow-band single photons for quantum memories generated by ultrabright cavity-enhanced parametric down-conversion,” Phys. Rev. Lett. 102(6), 063603 (2009).
[CrossRef] [PubMed]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Krischek, R.

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancment cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4(3), 170–173 (2010).
[CrossRef]

Kuklewicz, C. E.

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69(4), 041801 (2004).
[CrossRef]

C. E. Kuklewicz, M. Fiorentino, G. Messin, and J. H. Shapiro, “High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter,” Phys. Rev. A 69(1), 013807 (2004).
[CrossRef]

Kumar, P.

Kurtsiefer, C.

J. Volz, C. Kurtsiefer, and H. Weinfurter, “Compact all-solid-state source of polarization-entangled photon pairs,” Appl. Phys. Lett. 79(6), 869–871 (2001).
[CrossRef]

Kuzucu, O.

O. Kuzucu and F. N. C. Wong, “Pulsed Sagnac source of narrow-band polarization-entangled photons,” Phys. Rev. A 77(3), 032314 (2008).
[CrossRef]

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94(8), 083601-1-4 (2005).

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Erratum: Two-Photon Coincident-Frequency Entanglement via Extended Phase Matching [Phys. Rev. Lett. 94, 083601 (2005)],” Phys. Rev. Lett. 94(16), 169903–1 (2005).
[CrossRef]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[CrossRef]

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(24), 4337–4341 (1995).
[CrossRef] [PubMed]

Lederer, T.

Lee, K. F.

Levin, T.

Li, X.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94(5), 053601 (2005).
[CrossRef] [PubMed]

Liang, C.

Lim, H. C.

Lindenthal, M.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Lorünser, T.

Maccone, L.

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Extended phase-matching conditions for improved entanglement generation,” Phys. Rev. A 66(4), 043813 (2002).
[CrossRef]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[CrossRef] [PubMed]

Martin, A.

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair souce based on a type-II PPLN waveguide emitting at a telecom wavelength,” N. J. Phys. 12(10), 103005 (2010).
[CrossRef]

Mattle, K.

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(24), 4337–4341 (1995).
[CrossRef] [PubMed]

Medic, M.

Messin, G.

C. E. Kuklewicz, M. Fiorentino, G. Messin, and J. H. Shapiro, “High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter,” Phys. Rev. A 69(1), 013807 (2004).
[CrossRef]

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69(4), 041801 (2004).
[CrossRef]

Meyenburg, M.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Michelberger, P.

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancment cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4(3), 170–173 (2010).
[CrossRef]

Milburn, G. J.

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65(1), 012314 (2001).
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Munro, W. J.

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65(1), 012314 (2001).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[CrossRef]

Oberparleiter, M.

M. Oberparleiter and H. Weinfurter, “Cavity-enhanced generation of polarisation-entangled photon pairs,” Opt. Commun. 183(1-4), 133–137 (2000).
[CrossRef]

Odate, S.

S. Odate, A. Yoshizawa, and H. Tsuchida, “Polarisation-entangled photon-pair source at 1550 nm using 1 mm-long PPLN waveguide in fibre-loop configuration,” Electron. Lett. 43(24), 1376–1377 (2007).
[CrossRef]

Ömer, B.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Ostrowsky, D. B.

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair souce based on a type-II PPLN waveguide emitting at a telecom wavelength,” N. J. Phys. 12(10), 103005 (2010).
[CrossRef]

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[CrossRef] [PubMed]

Ozawa, A.

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancment cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4(3), 170–173 (2010).
[CrossRef]

Patel, M.

Perdigues, J.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Pittman, T. B.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64(6), 062311 (2001).
[CrossRef]

Poppe, A.

Ralph, T. C.

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65(1), 012314 (2001).
[CrossRef]

Rarity, J.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Roberts, T. D.

Scheidl, T.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Schmitt-Manderbach, T.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Scholz, M.

M. Scholz, L. Koch, and O. Benson, “Statistics of narrow-band single photons for quantum memories generated by ultrabright cavity-enhanced parametric down-conversion,” Phys. Rev. Lett. 102(6), 063603 (2009).
[CrossRef] [PubMed]

Sergienko, A. V.

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(24), 4337–4341 (1995).
[CrossRef] [PubMed]

Shapiro, J. H.

C. E. Kuklewicz, M. Fiorentino, G. Messin, and J. H. Shapiro, “High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter,” Phys. Rev. A 69(1), 013807 (2004).
[CrossRef]

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69(4), 041801 (2004).
[CrossRef]

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Extended phase-matching conditions for improved entanglement generation,” Phys. Rev. A 66(4), 043813 (2002).
[CrossRef]

Sharping, J. E.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94(5), 053601 (2005).
[CrossRef] [PubMed]

Shih, Y.

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(24), 4337–4341 (1995).
[CrossRef] [PubMed]

Shimony, A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories,” Phys. Rev. Lett. 23(15), 880–884 (1969).
[CrossRef]

Smith, A. V.

Sodnik, Z.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Sohler, W.

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair souce based on a type-II PPLN waveguide emitting at a telecom wavelength,” N. J. Phys. 12(10), 103005 (2010).
[CrossRef]

Stothard, D.J.M.

P.J. Thomas, M.H. Dunn, D.J.M. Stothard, D.A. Walsh, and C.J. Chunnilall, “A pump enhanced source of telecom-band correlated photon pairs,” submitted to J. Mod. Opt. (2010).
[PubMed]

Tanzilli, S.

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair souce based on a type-II PPLN waveguide emitting at a telecom wavelength,” N. J. Phys. 12(10), 103005 (2010).
[CrossRef]

Thomas, P.J.

P.J. Thomas, M.H. Dunn, D.J.M. Stothard, D.A. Walsh, and C.J. Chunnilall, “A pump enhanced source of telecom-band correlated photon pairs,” submitted to J. Mod. Opt. (2010).
[PubMed]

Tiefenbacher, F.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Trojek, P.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Tsuchida, H.

H. C. Lim, A. Yoshizawa, H. Tsuchida, and K. Kikuchi, “Stable source of high quality telecom-band polarization-entangled photon-pairs based on a single, pulse-pumped, short PPLN waveguide,” Opt. Express 16(17), 12460–12468 (2008).
[CrossRef] [PubMed]

S. Odate, A. Yoshizawa, and H. Tsuchida, “Polarisation-entangled photon-pair source at 1550 nm using 1 mm-long PPLN waveguide in fibre-loop configuration,” Electron. Lett. 43(24), 1376–1377 (2007).
[CrossRef]

A. Yoshizawa, R. Kaji, and H. Tsuchida, “Generation of polarisation-entangled photon pairs at 1550 nm using two PPLN waveguides,” Electron. Lett. 39(7), 621–622 (2003).
[CrossRef]

Udem, T.

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancment cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4(3), 170–173 (2010).
[CrossRef]

Ursin, R.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Vanner, M. R.

Volz, J.

J. Volz, C. Kurtsiefer, and H. Weinfurter, “Compact all-solid-state source of polarization-entangled photon pairs,” Appl. Phys. Lett. 79(6), 869–871 (2001).
[CrossRef]

Voss, P. L.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94(5), 053601 (2005).
[CrossRef] [PubMed]

Walsh, D.A.

P.J. Thomas, M.H. Dunn, D.J.M. Stothard, D.A. Walsh, and C.J. Chunnilall, “A pump enhanced source of telecom-band correlated photon pairs,” submitted to J. Mod. Opt. (2010).
[PubMed]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Weier, H.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

Weinfurter, H.

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancment cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4(3), 170–173 (2010).
[CrossRef]

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
[CrossRef]

J. Volz, C. Kurtsiefer, and H. Weinfurter, “Compact all-solid-state source of polarization-entangled photon pairs,” Appl. Phys. Lett. 79(6), 869–871 (2001).
[CrossRef]

M. Oberparleiter and H. Weinfurter, “Cavity-enhanced generation of polarisation-entangled photon pairs,” Opt. Commun. 183(1-4), 133–137 (2000).
[CrossRef]

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(24), 4337–4341 (1995).
[CrossRef] [PubMed]

White, A. G.

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65(1), 012314 (2001).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[CrossRef]

Wieczorek, W.

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancment cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4(3), 170–173 (2010).
[CrossRef]

Wong, F. N. C.

T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled KTiOPO4 waveguide,” Opt. Express 17(14), 12019–12030 (2009).
[CrossRef] [PubMed]

O. Kuzucu and F. N. C. Wong, “Pulsed Sagnac source of narrow-band polarization-entangled photons,” Phys. Rev. A 77(3), 032314 (2008).
[CrossRef]

T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A 73(1), 012316 (2006).
[CrossRef]

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94(8), 083601-1-4 (2005).

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Erratum: Two-Photon Coincident-Frequency Entanglement via Extended Phase Matching [Phys. Rev. Lett. 94, 083601 (2005)],” Phys. Rev. Lett. 94(16), 169903–1 (2005).
[CrossRef]

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69(4), 041801 (2004).
[CrossRef]

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Extended phase-matching conditions for improved entanglement generation,” Phys. Rev. A 66(4), 043813 (2002).
[CrossRef]

Yoshizawa, A.

H. C. Lim, A. Yoshizawa, H. Tsuchida, and K. Kikuchi, “Stable source of high quality telecom-band polarization-entangled photon-pairs based on a single, pulse-pumped, short PPLN waveguide,” Opt. Express 16(17), 12460–12468 (2008).
[CrossRef] [PubMed]

S. Odate, A. Yoshizawa, and H. Tsuchida, “Polarisation-entangled photon-pair source at 1550 nm using 1 mm-long PPLN waveguide in fibre-loop configuration,” Electron. Lett. 43(24), 1376–1377 (2007).
[CrossRef]

A. Yoshizawa, R. Kaji, and H. Tsuchida, “Generation of polarisation-entangled photon pairs at 1550 nm using two PPLN waveguides,” Electron. Lett. 39(7), 621–622 (2003).
[CrossRef]

Zeilinger, A.

H. Hübel, M. R. Vanner, T. Lederer, B. Blauensteiner, T. Lorünser, A. Poppe, and A. Zeilinger, “High-fidelity transmission of polarization encoded qubits from an entangled source over 100 km of fiber,” Opt. Express 15(12), 7853–7862 (2007).
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Figures (3)

Fig. 1
Fig. 1

Scheme for producing the two superposition components of the entangled state in Eq. (3). The ket terms are either associated with downconversion photons that are produced while the pump beam is travelling from (a) right to left (forwards) or (b) left to right (backwards).

Fig. 2
Fig. 2

Schematic for source of polarization entangled photon pairs in the telecom band. The blue and red lines denote 792 nm pump light and downconversion respectively, while the black lines are electrical connections.

Fig. 3
Fig. 3

(a) Measurements of the number of coincident photon detections at D1 and D2 (see Fig. 2) as a function of θ 2 for θ 1 = 0° (triangles) and θ 1 = 90° (circles). (b) Measurements of the number of coincidences as a function of θ2 with fixed θ 1 = 45°. Data sets correspond to θsilica = 30.0° (ϕ = 0 state, hollow circles), θsilica = 32.1 (ϕ = π/2 state, filled circles) and θsilica = 34.4° (ϕ = π state, triangles). Measurements at each setting of θ2 were made over a 300 s time period which corresponds to a detector ‘on time’ of ~1.85 s. For both (a) and (b), the lines are least squares fit to the experimental data, the square data points are the measured singles counts at D2 over 300 s as a function of θ 2 . The pump power into the enhancement cavity was 50 mW. The error bars shown were used as weightings for the curve fitting and are based upon Poissonian N uncertainties.

Equations (20)

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| ψ ± = [ | V 1 H 2 ± | H 1 V 2 ] / 2 ,
| ϕ ± = [ | V 1 V 2 ± | H 1 H 2 ] / 2 .
| ψ = | V 1 H 2 + e i ϕ | H 1 V 2 .
| ψ = η F W e ϕ F W | V 1 H 2 + η B W e ϕ B W | H 1 V 2 ,
ϕ F W E C = k p [ L D + n y L C ] + [ k s + k i ] L B + N e v e n N [ { L B + n y L C + L D } k p + π ] ,
ϕ B W E C = k p [ L B + n y L C ] + [ k s + k i ] L D + N o d d N [ { L B + n y L C + L D } k p + π ] ,
[ L B + n y L C + L D ] k p = q k p λ p 2 = q π .
ϕ F W E C = k p L D + [ k s + k i ] L B + N e v e n N [ q π + π ] ,
ϕ B W E C = k p L B + [ k s + k i ] L D + N o d d N [ q π + π ] .
ϕ F W E X T = [ k s + k i ] L A + ϕ M 2 ( s , i ) + ϕ H W P ( s , i ) + 3 π ,
ϕ B W E X T = [ k s + k i ] L E + ϕ M 1 ( s , i ) + 3 π .
η F W = r I p ( R 2 R 1 ) r = I p 1 R 2 R 1 ,
η B W = r I p R 2 ( R 2 R 1 ) r = I p R 2 1 R 2 R 1 .
| ψ = I p 1 R 2 R 1 | V 1 H 2 + I p R 2 1 R 2 R 1 e [ k s + k i ] [ L D + L E L A L B ] + k p [ L B L D ] ϕ M 2 ( s , i ) + ϕ M 1 ( s , i ) ϕ H W P ( s , i ) | H 1 V 2 .
N c l a s s c i a l , θ 1 = 45 ° ( θ 2 ) = 1 2 [ sin 2 θ 2 + cos 2 θ 2 ] .
N e n t a n g l e d , θ 1 = 45 ( θ 2 ) = N c l a s s c i a l , θ 1 = 45 ° ( θ 2 ) + cos θ 2 sin θ 2 cos ϕ .
S = | E ( θ 1 , θ 2 ) + E ( θ 1 , θ ' 2 ) E ( θ ' 1 , θ 2 ) + E ( θ ' 1 , θ ' 2 ) | ,
E ( θ 1 , θ 2 ) = N + + N + N + + N N + + + N + + N + + N .
E ( π / 4 , 0 )   = 0. 869  ±   0.0 68 , E ( π / 4 ,  7 π / 8 )   = 0. 546  ±   0.0 56 ,
E ( 5 π / 8 ,   0 )   =   0. 5 0 ±   0.0 55 , E ( 5 π / 8 ,  7 π / 8 )   = 0. 813  ±   0.0 64 .

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