Abstract

Photonic quantum information experiments demand bright and highly entangled photon pair sources. The combination of periodic poling and collinear excitation geometry allows the use of considerably longer crystals for parametric down-conversion. We demonstrate a picosecond-pulsed laser pumped source of high quality polarization entangled photon pairs. The phase of the output biphoton state is affected by the relative phase of the two-color interferometer and the phase of the nonlinearly interacting Gaussian beams. We measure the influence of these onto the phase of the output state. The presented source is a promising candidate for a compact, semiconductor laser driven source of entangled photon pairs.

© 2012 OSA

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  1. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003).
    [CrossRef] [PubMed]
  4. J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett.86, 4435–4438 (2001).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A77, 032314 (2008).
  13. O. Kuzucu and F. N. C. Wong “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express15, 15377–15386 (2007).
  14. M. Hentschel, H. Hübel, A. Poppe, and A. Zeilinger “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express15, 15377–15386 (2007).
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    [CrossRef]
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    [CrossRef]
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2010 (1)

R. S. Bennink“Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A81, 053805 (2010).
[CrossRef]

2009 (1)

2008 (3)

H. E. Major, C. B. E. Gawith, and P. G. R. Smith “Gouy phase compensation in quasi phase matching,” Opt. Commun.281, 5036–5040 (2008).
[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008).
[CrossRef] [PubMed]

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A77, 032314 (2008).

2007 (4)

2006 (1)

T. Kim, M. Fiorentino, and F. N. C. Wong “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A73, 012316 (2006).
[CrossRef]

2004 (3)

M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett.92, 077901 (2004).
[CrossRef] [PubMed]

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A70, 043818 (2004).
[CrossRef]

B.-S. Shi and A. Tomita “Generation of a pulsed polarization entangled photon pair using a Sagnac interferometer,” Phys. Rev. A69, 013803 (2004).
[CrossRef]

2003 (1)

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003).
[CrossRef] [PubMed]

2002 (1)

T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger “Experimental nonlocality proof of quantum teleportation and entanglement swapping,” Phys. Rev. Lett.88, 017903 (2002).
[CrossRef] [PubMed]

2001 (2)

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett.86, 4435–4438 (2001).
[CrossRef] [PubMed]

D. F. V. James, P. Kwiat, W. Munro, and A. White“Measurement of qubits,” Phys. Rev. A64, 052312 (2001).
[CrossRef]

1997 (1)

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997).
[CrossRef]

1996 (1)

1968 (1)

G. D. Boyd and D. A. Kleiman “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39, 3597–3640 (1968).
[CrossRef]

Bennink, R. S.

R. S. Bennink“Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A81, 053805 (2010).
[CrossRef]

Bourennane, M.

M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett.92, 077901 (2004).
[CrossRef] [PubMed]

Bouwmeester, D.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleiman “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39, 3597–3640 (1968).
[CrossRef]

Ciddor, P. E.

Daniell, M.

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett.86, 4435–4438 (2001).
[CrossRef] [PubMed]

Eibl, M.

M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett.92, 077901 (2004).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997).
[CrossRef]

Fedrizzi, A.

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A77, 032314 (2008).

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. A73, 012316 (2006).
[CrossRef]

Gasparoni, S.

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003).
[CrossRef] [PubMed]

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett.86, 4435–4438 (2001).
[CrossRef] [PubMed]

Gawith, C. B. E.

H. E. Major, C. B. E. Gawith, and P. G. R. Smith “Gouy phase compensation in quasi phase matching,” Opt. Commun.281, 5036–5040 (2008).
[CrossRef]

Goggin, M.

Hamel, D. R.

D. R. Hamel “Realization of novel entangled photon sources using periodically poled materials,” Master’s thesis, University of Waterloo (2010).

Hendrych, M.

Hentschel, M.

Herbst, T.

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A77, 032314 (2008).

Hübel, H.

James, D. F. V.

D. F. V. James, P. Kwiat, W. Munro, and A. White“Measurement of qubits,” Phys. Rev. A64, 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,” Phys. Rev. A77, 032314 (2008).

T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger “Experimental nonlocality proof of quantum teleportation and entanglement swapping,” Phys. Rev. Lett.88, 017903 (2002).
[CrossRef] [PubMed]

Kiesel, N.

M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett.92, 077901 (2004).
[CrossRef] [PubMed]

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. A73, 012316 (2006).
[CrossRef]

Kleiman, D. A.

G. D. Boyd and D. A. Kleiman “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39, 3597–3640 (1968).
[CrossRef]

Kurtsiefer, C.

M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett.92, 077901 (2004).
[CrossRef] [PubMed]

Kuzucu, O.

Kwiat, P.

R. Rangarajan, M. Goggin, and P. Kwiat “Optimizing type-I polarization-entangleg photons,” Opt. Express17, 18920–18933 (2009).
[CrossRef]

D. F. V. James, P. Kwiat, W. Munro, and A. White“Measurement of qubits,” Phys. Rev. A64, 052312 (2001).
[CrossRef]

Lastzka, N.

Lee, P. S. K.

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A70, 043818 (2004).
[CrossRef]

Lundeen, J. S.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008).
[CrossRef] [PubMed]

Major, H. E.

H. E. Major, C. B. E. Gawith, and P. G. R. Smith “Gouy phase compensation in quasi phase matching,” Opt. Commun.281, 5036–5040 (2008).
[CrossRef]

Mattle, K.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997).
[CrossRef]

Micuda, M.

Mosley, P. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008).
[CrossRef] [PubMed]

Munro, W.

D. F. V. James, P. Kwiat, W. Munro, and A. White“Measurement of qubits,” Phys. Rev. A64, 052312 (2001).
[CrossRef]

Pan, J.-W.

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003).
[CrossRef] [PubMed]

T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger “Experimental nonlocality proof of quantum teleportation and entanglement swapping,” Phys. Rev. Lett.88, 017903 (2002).
[CrossRef] [PubMed]

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett.86, 4435–4438 (2001).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997).
[CrossRef]

Poppe, A.

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A77, 032314 (2008).

M. Hentschel, H. Hübel, A. Poppe, and A. Zeilinger “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express15, 15377–15386 (2007).

Rangarajan, R.

Schnabel, R.

Shi, B.-S.

B.-S. Shi and A. Tomita “Generation of a pulsed polarization entangled photon pair using a Sagnac interferometer,” Phys. Rev. A69, 013803 (2004).
[CrossRef]

Silberhorn, C.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008).
[CrossRef] [PubMed]

Smith, B. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008).
[CrossRef] [PubMed]

Smith, P. G. R.

H. E. Major, C. B. E. Gawith, and P. G. R. Smith “Gouy phase compensation in quasi phase matching,” Opt. Commun.281, 5036–5040 (2008).
[CrossRef]

Tomita, A.

B.-S. Shi and A. Tomita “Generation of a pulsed polarization entangled photon pair using a Sagnac interferometer,” Phys. Rev. A69, 013803 (2004).
[CrossRef]

Torres, J. P.

U’Ren, A. B.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008).
[CrossRef] [PubMed]

Ursin, R.

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003).
[CrossRef] [PubMed]

van Exter, M. P.

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A70, 043818 (2004).
[CrossRef]

Walmsley, I. A.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008).
[CrossRef] [PubMed]

Wasylczyk, P.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008).
[CrossRef] [PubMed]

Weihs, G.

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003).
[CrossRef] [PubMed]

T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger “Experimental nonlocality proof of quantum teleportation and entanglement swapping,” Phys. Rev. Lett.88, 017903 (2002).
[CrossRef] [PubMed]

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett.86, 4435–4438 (2001).
[CrossRef] [PubMed]

Weinfurter, H.

M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett.92, 077901 (2004).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997).
[CrossRef]

White, A.

D. F. V. James, P. Kwiat, W. Munro, and A. White“Measurement of qubits,” Phys. Rev. A64, 052312 (2001).
[CrossRef]

Woerdman, J. P.

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A70, 043818 (2004).
[CrossRef]

Wong, F. N. C.

O. Kuzucu and F. N. C. Wong “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express15, 15377–15386 (2007).

T. Kim, M. Fiorentino, and F. N. C. Wong “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A73, 012316 (2006).
[CrossRef]

Zeilinger, A.

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A77, 032314 (2008).

M. Hentschel, H. Hübel, A. Poppe, and A. Zeilinger “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express15, 15377–15386 (2007).

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003).
[CrossRef] [PubMed]

T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger “Experimental nonlocality proof of quantum teleportation and entanglement swapping,” Phys. Rev. Lett.88, 017903 (2002).
[CrossRef] [PubMed]

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett.86, 4435–4438 (2001).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (1)

G. D. Boyd and D. A. Kleiman “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39, 3597–3640 (1968).
[CrossRef]

Nature (2)

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997).
[CrossRef]

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

H. E. Major, C. B. E. Gawith, and P. G. R. Smith “Gouy phase compensation in quasi phase matching,” Opt. Commun.281, 5036–5040 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (6)

R. S. Bennink“Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A81, 053805 (2010).
[CrossRef]

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A70, 043818 (2004).
[CrossRef]

B.-S. Shi and A. Tomita “Generation of a pulsed polarization entangled photon pair using a Sagnac interferometer,” Phys. Rev. A69, 013803 (2004).
[CrossRef]

T. Kim, M. Fiorentino, and F. N. C. Wong “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A73, 012316 (2006).
[CrossRef]

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A77, 032314 (2008).

D. F. V. James, P. Kwiat, W. Munro, and A. White“Measurement of qubits,” Phys. Rev. A64, 052312 (2001).
[CrossRef]

Phys. Rev. Lett. (4)

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett.86, 4435–4438 (2001).
[CrossRef] [PubMed]

M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett.92, 077901 (2004).
[CrossRef] [PubMed]

T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger “Experimental nonlocality proof of quantum teleportation and entanglement swapping,” Phys. Rev. Lett.88, 017903 (2002).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008).
[CrossRef] [PubMed]

Other (2)

Engineering metrology toolbox, http://emtoolbox.nist.gov/Wavelength/Ciddor.asp .

D. R. Hamel “Realization of novel entangled photon sources using periodically poled materials,” Master’s thesis, University of Waterloo (2010).

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Figures (4)

Fig. 1
Fig. 1

Experimental set-up. The frequency doubled Ti:Sapphire picosecond laser (blue) is directed via a single mode fiber (SMF) onto the Sagnac loop. The pump light is split at the polarizing beamsplitter (PBS). A half-waveplate (HWP) placed inside the loop rotates the polarization of one of the pump directions in front of the crystal (PPKTP). The photon pairs produced in the counter-clockwise direction pass through the same waveplate on their way out of the loop. The photons leave the loop through the polarizing beamsplitter (PBS) and are coupled into single mode optical fibers (output 1 and output 2). The remaining pump light is eliminated by two color glass filters (CG). The phase φp of the pump beam is used to adjust the phase between the two conversion processes and thus to set the phase of the output entangled state.

Fig. 2
Fig. 2

Coincidence counts from the picosecond pulsed Sagnac source as a function of the polarizer angle in one output port. For the circles / squares the polarizer in the other output port was set to horizontal (H) / −45° (A), respectively. The solid lines are sinusoidal fits to the data. The Poissonian error bars are smaller than the symbols. The lower count rate of this measurement compared with the reported brightness originates from linear polarizers used in the measurement which exhibit low transmission (≈62%).

Fig. 3
Fig. 3

The error bars are smaller than the symbols. (a) The fidelity of the generated entangled state to the states Ψ+ and Ψ changes anti-symmetrically as the crystal is displaced from the initial position (near 4th data point). (b) The real and the imaginary parts of the off-diagonal density matrix element indicate the changing phase among the |HV〉 and |VH〉 components. (c) The tangle for all positions of the crystal.

Fig. 4
Fig. 4

(a) The phase of the entangled state as a function of the crystal displacement from the Sagnac loop center shows a nearly linear dependence on the crystal displacement, where zero corresponds to the nominal loop center. The symbols represent the phases reconstructed from the tomography measurements (the statistical errors are smaller than the symbols). The solid line is a fit of the nonlinear Gouy phase model including the air dispersive shift. The contribution of the air at the mean pressure and humidity on the afternoon of the measurement (see the main text) is drawn as the blue dashed straight line. (b) In order to highlight the influence of the Gouy phase here we show data, where φair was subtracted. The fit to this data is based on Boyd and Kleinman’s theory and results in ξ = 0.706 ± 0.076 for the focusing parameter and 6.0±1.6mm for the initial displacement of the beam waists from the loop center. In addition we plot, for identical parameters, φGouy obtained from Boyd and Kleinman’s up-conversion theory, but for coincident clock- and counter-clockwise beam waist positions (black dashed line).

Equations (3)

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| Ψ = | H V + e i φ | V H
H ( σ , ξ , f ) = f / z 0 ξ f / z 0 exp ( i σ τ ) 1 + i τ d τ .
φ ( x ) = φ p + φ air ( x ) + φ Gouy ( σ opt ( f ( x ) ) , f ( x ) ) ,

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