Abstract

In this paper we report a polarization-entangled photon-pair source in a 1.5-μm band which can generate arbitrary entangled states including four maximum entangled states (Bell states) by using cascaded optical second nonlinearities (second-harmonic generation and the following spontaneous parametric down conversion) in a periodically poled LiNbO3 (PPLN) ridge-waveguide device. Exchange among the Bell states was achieved by using an optical phase bias compensator (OPBC) in a Sagnac loop interferometer and a half-wave plate outside the loop for polarization conversion. Quantitative evaluation was made on the performance of the photon-pair source through the experiments of two-photon interferences, quantum state tomography, and test of violation of Bell inequality. We observed high visibilities of 96%, fidelities of 97%, and 2.71 of the S parameter in inequality of Clauser, Horne, Shimony, and Holt (CHSH). The experimental values, including peak coincidence counts in the two-photon interference (approximately 170 counts per second), remained almost unchanged in despite of the exchange among the Bell states. They were also in good agreement with the theoretical assumption from the mean number of the photon-pairs under the test (0.04 per pulse). More detailed experimental studies on the dependence of the mean number of the photon-pairs revealed that the quantum states were well understood as the Werner state.

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  1. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 14(7), 983–985 (2002).
    [CrossRef]
  2. H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A 70(3), 031802–031805 (2004).
    [CrossRef]
  3. H. Takesue and K. Inoue, “1.5-µm band quantum-correlated photon pair generation in dispersion-shifted fiber: suppression of noise photons by cooling fiber,” Opt. Express 13(20), 7832–7839 (2005).
    [CrossRef] [PubMed]
  4. S. D. Dyer, M. J. Stevens, B. Baek, and S. W. Nam, “High-efficiency, ultra low-noise all-fiber photon-pair source,” Opt. Express 16(13), 9966–9977 (2008).
    [CrossRef] [PubMed]
  5. A. Yoshizawa, R. Kaji, and H. Tsuchida, “Generation of polarization-entangled photon pairs at 1550 nm using two PPLN waveguides,” Electron. Lett. 39(7), 621–622 (2003).
    [CrossRef]
  6. S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
    [CrossRef]
  7. 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]
  8. Q. Zhang, X. Xie, H. Takesue, S. W. Nam, C. Langrock, M. M. Fejer, and Y. Yamamoto, “Correlated photon-pair generation in reverse-proton-exchange PPLN waveguides with integrated mode demultiplexer at 10 GHz clock,” Opt. Express 15(16), 10288–10293 (2007).
    [CrossRef] [PubMed]
  9. H. Takesue, K. Inoue, O. Tadanaga, Y. Nishida, and M. Asobe, “Generation of pulsed polarization-entangled photon pairs in a 1.55-µm band with a periodically poled lithium niobate waveguide and an orthogonal polarization delay circuit,” Opt. Lett. 30(3), 293–295 (2005).
    [CrossRef] [PubMed]
  10. J. Chen, A. J. Pearlman, A. Ling, J. Fan, and A. L. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17(8), 6727–6740 (2009).
    [CrossRef] [PubMed]
  11. A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength,” New J. Phys. 12(10), 103005 (2010).
    [CrossRef]
  12. K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31(12), 1905–1907 (2006).
    [CrossRef] [PubMed]
  13. S. Arahira, N. Namekata, T. Kishimoto, H. Yaegashi, and S. Inoue, “Generation of polarization entangled photon pairs at telecommunication wavelength using cascaded χ(2) processes in a periodically poled LiNbO3 ridge waveguide,” Opt. Express 19(17), 16032–16043 (2011).
    [CrossRef] [PubMed]
  14. 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]
  15. T. Kishimoto and K. Nakamura, “Periodically poled MgO-doped stoichiometric LiNbO3 wavelength convertor with ridge-type annealed proton-exchanged waveguide,” IEEE Photon. Technol. Lett. 23(3), 161–163 (2011).
    [CrossRef]
  16. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312–052326 (2001).
    [CrossRef]
  17. H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S. Itabashi, “Generation of polarization entangled photon pairs using silicon wire waveguide,” Opt. Express 16(8), 5721–5727 (2008).
    [CrossRef] [PubMed]
  18. K. Edamatsu, “Entangled photons: generation, observation, and characterization,” Jpn. J. Appl. Phys. 46(11), 7175–7187 (2007).
    [CrossRef]

2011

T. Kishimoto and K. Nakamura, “Periodically poled MgO-doped stoichiometric LiNbO3 wavelength convertor with ridge-type annealed proton-exchanged waveguide,” IEEE Photon. Technol. Lett. 23(3), 161–163 (2011).
[CrossRef]

S. Arahira, N. Namekata, T. Kishimoto, H. Yaegashi, and S. Inoue, “Generation of polarization entangled photon pairs at telecommunication wavelength using cascaded χ(2) processes in a periodically poled LiNbO3 ridge waveguide,” Opt. Express 19(17), 16032–16043 (2011).
[CrossRef] [PubMed]

2010

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

2009

2008

2007

2006

2005

2004

H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A 70(3), 031802–031805 (2004).
[CrossRef]

2003

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

2002

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
[CrossRef]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 14(7), 983–985 (2002).
[CrossRef]

2001

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312–052326 (2001).
[CrossRef]

1969

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]

Alibart, O.

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

Arahira, S.

Asobe, M.

Baek, B.

Baldi, P.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
[CrossRef]

Chen, J.

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]

De Micheli, M.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
[CrossRef]

De Riedmatten, H.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
[CrossRef]

Dyer, S. D.

Edamatsu, K.

K. Edamatsu, “Entangled photons: generation, observation, and characterization,” Jpn. J. Appl. Phys. 46(11), 7175–7187 (2007).
[CrossRef]

Fan, J.

Fejer, M. M.

Fiorentino, M.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 14(7), 983–985 (2002).
[CrossRef]

Fukuda, H.

Gisin, N.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
[CrossRef]

Herrmann, H.

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength,” New 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]

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]

Inoue, K.

Inoue, S.

Issautier, A.

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

Itabashi, S.

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), 052312–052326 (2001).
[CrossRef]

Kaji, R.

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

Kikuchi, K.

Kishimoto, T.

S. Arahira, N. Namekata, T. Kishimoto, H. Yaegashi, and S. Inoue, “Generation of polarization entangled photon pairs at telecommunication wavelength using cascaded χ(2) processes in a periodically poled LiNbO3 ridge waveguide,” Opt. Express 19(17), 16032–16043 (2011).
[CrossRef] [PubMed]

T. Kishimoto and K. Nakamura, “Periodically poled MgO-doped stoichiometric LiNbO3 wavelength convertor with ridge-type annealed proton-exchanged waveguide,” IEEE Photon. Technol. Lett. 23(3), 161–163 (2011).
[CrossRef]

Kumar, P.

K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31(12), 1905–1907 (2006).
[CrossRef] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 14(7), 983–985 (2002).
[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), 052312–052326 (2001).
[CrossRef]

Langrock, C.

Lee, K. F.

Li, X.

Liang, C.

Lim, H. C.

Ling, A.

Martin, A.

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

Migdall, A. L.

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312–052326 (2001).
[CrossRef]

Nakamura, K.

T. Kishimoto and K. Nakamura, “Periodically poled MgO-doped stoichiometric LiNbO3 wavelength convertor with ridge-type annealed proton-exchanged waveguide,” IEEE Photon. Technol. Lett. 23(3), 161–163 (2011).
[CrossRef]

Nam, S. W.

Namekata, N.

Nishida, Y.

Ostrowsky, D. B.

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

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
[CrossRef]

Pearlman, A. J.

Sharping, J. E.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 14(7), 983–985 (2002).
[CrossRef]

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]

Sohler, W.

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

Stevens, M. J.

Tadanaga, O.

Takesue, H.

Tanzilli, S.

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

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
[CrossRef]

Tittel, W.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
[CrossRef]

Tokura, Y.

Tsuchida, H.

Tsuchizawa, T.

Voss, P. L.

K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31(12), 1905–1907 (2006).
[CrossRef] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 14(7), 983–985 (2002).
[CrossRef]

Watanabe, T.

White, A. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312–052326 (2001).
[CrossRef]

Xie, X.

Yaegashi, H.

Yamada, K.

Yamamoto, Y.

Yoshizawa, A.

Zbinden, H.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
[CrossRef]

Zhang, Q.

Electron. Lett.

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

Eur. Phys. J. D

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18(2), 155–160 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 14(7), 983–985 (2002).
[CrossRef]

T. Kishimoto and K. Nakamura, “Periodically poled MgO-doped stoichiometric LiNbO3 wavelength convertor with ridge-type annealed proton-exchanged waveguide,” IEEE Photon. Technol. Lett. 23(3), 161–163 (2011).
[CrossRef]

Jpn. J. Appl. Phys.

K. Edamatsu, “Entangled photons: generation, observation, and characterization,” Jpn. J. Appl. Phys. 46(11), 7175–7187 (2007).
[CrossRef]

New J. Phys.

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

Opt. Express

H. Takesue and K. Inoue, “1.5-µm band quantum-correlated photon pair generation in dispersion-shifted fiber: suppression of noise photons by cooling fiber,” Opt. Express 13(20), 7832–7839 (2005).
[CrossRef] [PubMed]

Q. Zhang, X. Xie, H. Takesue, S. W. Nam, C. Langrock, M. M. Fejer, and Y. Yamamoto, “Correlated photon-pair generation in reverse-proton-exchange PPLN waveguides with integrated mode demultiplexer at 10 GHz clock,” Opt. Express 15(16), 10288–10293 (2007).
[CrossRef] [PubMed]

H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S. Itabashi, “Generation of polarization entangled photon pairs using silicon wire waveguide,” Opt. Express 16(8), 5721–5727 (2008).
[CrossRef] [PubMed]

S. D. Dyer, M. J. Stevens, B. Baek, and S. W. Nam, “High-efficiency, ultra low-noise all-fiber photon-pair source,” Opt. Express 16(13), 9966–9977 (2008).
[CrossRef] [PubMed]

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]

J. Chen, A. J. Pearlman, A. Ling, J. Fan, and A. L. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17(8), 6727–6740 (2009).
[CrossRef] [PubMed]

S. Arahira, N. Namekata, T. Kishimoto, H. Yaegashi, and S. Inoue, “Generation of polarization entangled photon pairs at telecommunication wavelength using cascaded χ(2) processes in a periodically poled LiNbO3 ridge waveguide,” Opt. Express 19(17), 16032–16043 (2011).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. A

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312–052326 (2001).
[CrossRef]

H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A 70(3), 031802–031805 (2004).
[CrossRef]

Phys. Rev. Lett.

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]

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Figures (9)

Fig. 1
Fig. 1

Polarization-entangled photon-pair source with variable Bell states. PPLN: PPLN device (module). PBSC: polarization beam splitter/combiner. BSC: Babinet-Soleil compensator. FR: Faraday rotator. OPBC: optical phase-bias compensator. The position of the BSC could be precisely shifted by a micrometer.

Fig. 2
Fig. 2

Cascaded SHG/SPDC (c-SHG/SPDC) process in a PPLN device used in this study.

Fig. 3
Fig. 3

Experimental setup of polarization entanglement with variable Bell states.

Fig. 4
Fig. 4

Two-photon interference fringes of | Φ ( + ) state at the H/V basis (black) and the diagonal basis (red). Polarizer angle of the signal polarizer (θs) were 0 ° (black) and 45 ° (red), respectively. Closed circles are experimental data. The solid curves in the figure are fitting curves assuming cos 2 ( θ s θ i ) .

Fig. 5
Fig. 5

Reconstructed density matrix of the | Φ ( + ) state. (a) Real part. (b) Imaginary part.

Fig. 6
Fig. 6

Two-photon interference fringes of (a) | Φ ( ) , (b) | Ψ ( + ) , and (c) | Ψ ( ) state, respectively. Black: H/V basis. Red: diagonal basis. Polarizer angle of the signal polarizer (θs) were 0 ° (black) and 45 ° (red), respectively. Closed circles are experimental data. The solid curves in the figure are fitting curves assuming sinusoidal curves.

Fig. 7
Fig. 7

Two-photon interference fringes when the system was set to be ( |H s |H i +exp( iπ/2 ) |V s |V i ) / 2 state. Polarizer angle of the signal polarizer (θs) were 0 ° (black) and 45 ° (red), respectively.

Fig. 8
Fig. 8

Reconstructed density matrices of the | Φ ( + ) states as a function of the mean number of the photon-pair (μc). (a), (e) μc=0.00854/pulse. (b), (f) μc=0.1098/pulse. (c), (g) μc=0.2309/pulse. (d), (h) μc=0.4623/pulse. (a)-(d) Real parts. (e)-(h) Imaginary parts.

Fig. 9
Fig. 9

Dependences of the visibility (V), the purity (P), and the fidelity (F) on the mean number of the photon-pair (μc). Black: visibility. Red: purity. Blue: fidelity. Open circles: experimental values from reconstructed density matrices. Solid curves: calculations assuming Werner states. Gray closed squares are fidelities to the Werner state (FW).

Tables (1)

Tables Icon

Table 1 Summary of the Experimental Results

Equations (11)

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| Φ ( ± ) = 1 2 ( |H s |H i ± |V s |V i )
| Ψ ( ± ) = 1 2 ( |H s |V i ± |V s |H i )
V= μ c α s α i 2 μ c α s α i 2 +2( μ c α s 2 + d s )( μ c α i 2 + d i )
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 )= C( θ 1 , θ 2 )+C( θ 1 , θ 2 )C( θ 1 , θ 2 )C( θ 1 , θ 2 ) C( θ 1 , θ 2 )+C( θ 1 , θ 2 )+C( θ 1 , θ 2 )+C( θ 1 , θ 2 )
S=| E( θ 1 , θ 2 )E( θ 1 , θ 2 ) |+| E( θ 1 , θ 2 )+E( θ 1 , θ 2 ) |
S=2 2 V
ρ W =( 1a )| Φ ( + ) Φ ( + ) |+ 1 4 aI= 1 2 ( 1a/2 0 0 1a 0 a/2 0 0 0 0 a/2 0 1a 0 0 1a/2 )
V=1a
F= 1+3V 4
P= 1+3 V 2 4

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