Abstract

We explore spatial correlations created by stimulated pair emission in frequency degenerate parametric down-conversion from a periodically poled KTP crystal pumped by ∼2 ps duration laser pulses. The ratio of stimulated pairs over spontaneous pairs reaches as high 0.8 in the experiment. This ratio is a direct measure of the total number of modes relevant to the down-conversion process. We identify a universal curve for this ratio that accounts for the effect of the focused pump, introducing a coherence diameter r0 related to the diffraction limited size of the pump beam in the far-field. Measurements of the spatial correlations of the PDC light for longer crystals and tight focusing conditions show that the description given in terms of a universal curve is surprisingly robust and breaks down only for a laser beam focussed to a waist smaller than 40 μm in a 2 mm long PPKTP crystal.

© 2014 Optical Society of America

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    [Crossref]
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    [Crossref]
  36. A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “X entanglement: The nonfactorable spatiotemporal structure of biphoton correlation,” Phys. Rev. Lett. 102, 223601 (2009).
    [Crossref] [PubMed]
  37. L. Caspani, E. Brambilla, and A. Gatti, “Tailoring the spatiotemporal structure of biphoton entanglement intype-i parametric down-conversion,” Phys. Rev. A 81, 033808 (2010).
    [Crossref]
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    [Crossref]

2013 (1)

T. H. Yang and M. Nevascues, “Robust self-testing of unknown quantum systems into any entangled two-qubit states,” Phys. Rev. A 87, 050102(R) (2013).
[Crossref]

2012 (2)

A. J. H. van der Torren, S. C. Yorulmaz, J. J. Renema, M. P. van Exter, and M. J. A. de Dood, “Spatially entangled four-photon states from a periodically poled potassium-titanyl-phosphate crystal,” Phys. Rev. A 85, 043837 (2012).
[Crossref]

S. C. Yorulmaz and M. J. A. de Dood, “Characterization of pulsed parametric down-conversion in ppktp crystals,” Proc. SPIE 8440, 84400G (2012).
[Crossref]

2010 (3)

L. Caspani, E. Brambilla, and A. Gatti, “Tailoring the spatiotemporal structure of biphoton entanglement intype-i parametric down-conversion,” Phys. Rev. A 81, 033808 (2010).
[Crossref]

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref] [PubMed]

S. Walborn, C. Monken, S. Pdua, and P. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports 495, 87 (2010).
[Crossref]

2009 (2)

C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “X entanglement: The nonfactorable spatiotemporal structure of biphoton correlation,” Phys. Rev. Lett. 102, 223601 (2009).
[Crossref] [PubMed]

2008 (5)

W. H. Peeters and M. P. van Exter, “Optical characterization of periodically-poled ktiopo4,” Optics Express 16, 7344 (2008).
[Crossref] [PubMed]

S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. S. Ribeiro, “Schemes for quantum key distribution with higher-order alphabets using single-photon fractional fourier optics,” Phys. Rev. A 77, 062323 (2008).
[Crossref]

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[Crossref] [PubMed]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled fock states for robust quantum optical metrology, imaging and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Y.-X. Gong, X.-B. Zou, X.-L. Niu, J. Li, Y.-F. Huang, and G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[Crossref]

2006 (4)

S. P. Walborn, D. S. Lemelle, M. P. Almedia, and P. H. S. Ribeiro, “Quantum key distribution with higher-order alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[Crossref] [PubMed]

D. S. Lemelle, M. P. Almeida, P. H. S. Ribeiro, and S. P. Walborn, “A simple optical demonstration of quantum cryptography using transverse position and momentum variables,” Am. J.of Phys. 74, 542 (2006).
[Crossref]

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]

J.-L. Chen, C. Wu, L. C. Kwek, C. H. Oh, and M.-L. Ge, “Violating bell inequalities maximally for two d-dimensional systems,” Phys. Rev. A 74, 032106 (2006).
[Crossref]

2005 (2)

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett. 95, 201502 (2005).
[Crossref]

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “Experimental quantum coin tossing,” Phys. Rev. Lett. 94, 040501 (2005).
[Crossref] [PubMed]

2004 (4)

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the einstein-podolsky-rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

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]

H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, and N. Gisin, “Two independent photon pairs versus four-photon entangled states in parametric down conversion,” J. Mod. Opt. 51, 1637 (2004).
[Crossref]

2002 (1)

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176 (2002).
[Crossref]

2000 (1)

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

1999 (2)

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[Crossref]

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[Crossref]

1998 (2)

P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595 (1998).
[Crossref]

C. H. Monken, P. H. S. Ribereiro, and S. Padua, “Optimizing the photon pair collection efficiency: A step toward a loophole-free bell’s inequalities experiment,” Phys. Rev. A 57, R2267 (1998).
[Crossref]

1997 (1)

T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A 56, 1534 (1997).
[Crossref]

1995 (3)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600 (1995).
[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]

1989 (1)

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205–2208 (1989).
[Crossref] [PubMed]

1985 (1)

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54, 2011–2013 (1985).
[Crossref] [PubMed]

1982 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental realization of einstein-podolsky-rosen-bohm Gedankenexperiment: A new violation of bell’s inequalities,” Phys. Rev. Lett. 49, 91–94 (1982).
[Crossref]

1981 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via bell’s theorem,” Phys. Rev. Lett. 47, 460–463 (1981).
[Crossref]

Acín, A.

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

Acn, A.

H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, and N. Gisin, “Two independent photon pairs versus four-photon entangled states in parametric down conversion,” J. Mod. Opt. 51, 1637 (2004).
[Crossref]

Aiello, A.

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]

Almedia, M. P.

S. P. Walborn, D. S. Lemelle, M. P. Almedia, and P. H. S. Ribeiro, “Quantum key distribution with higher-order alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[Crossref] [PubMed]

Almeida, M. P.

D. S. Lemelle, M. P. Almeida, P. H. S. Ribeiro, and S. P. Walborn, “A simple optical demonstration of quantum cryptography using transverse position and momentum variables,” Am. J.of Phys. 74, 542 (2006).
[Crossref]

Aspect, A.

A. Aspect, P. Grangier, and G. Roger, “Experimental realization of einstein-podolsky-rosen-bohm Gedankenexperiment: A new violation of bell’s inequalities,” Phys. Rev. Lett. 49, 91–94 (1982).
[Crossref]

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via bell’s theorem,” Phys. Rev. Lett. 47, 460–463 (1981).
[Crossref]

Beenakker, C. W. J.

C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
[Crossref] [PubMed]

Bennink, R. S.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the einstein-podolsky-rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

Bentley, S. J.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the einstein-podolsky-rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

Boyd, R. W.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the einstein-podolsky-rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

Brambilla, E.

L. Caspani, E. Brambilla, and A. Gatti, “Tailoring the spatiotemporal structure of biphoton entanglement intype-i parametric down-conversion,” Phys. Rev. A 81, 033808 (2010).
[Crossref]

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “X entanglement: The nonfactorable spatiotemporal structure of biphoton correlation,” Phys. Rev. Lett. 102, 223601 (2009).
[Crossref] [PubMed]

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176 (2002).
[Crossref]

Caspani, L.

L. Caspani, E. Brambilla, and A. Gatti, “Tailoring the spatiotemporal structure of biphoton entanglement intype-i parametric down-conversion,” Phys. Rev. A 81, 033808 (2010).
[Crossref]

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “X entanglement: The nonfactorable spatiotemporal structure of biphoton correlation,” Phys. Rev. Lett. 102, 223601 (2009).
[Crossref] [PubMed]

Chen, J.-L.

J.-L. Chen, C. Wu, L. C. Kwek, C. H. Oh, and M.-L. Ge, “Violating bell inequalities maximally for two d-dimensional systems,” Phys. Rev. A 74, 032106 (2006).
[Crossref]

Cirac, J. I.

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

de Dood, M. J. A.

A. J. H. van der Torren, S. C. Yorulmaz, J. J. Renema, M. P. van Exter, and M. J. A. de Dood, “Spatially entangled four-photon states from a periodically poled potassium-titanyl-phosphate crystal,” Phys. Rev. A 85, 043837 (2012).
[Crossref]

S. C. Yorulmaz and M. J. A. de Dood, “Characterization of pulsed parametric down-conversion in ppktp crystals,” Proc. SPIE 8440, 84400G (2012).
[Crossref]

Dowling, J. P.

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled fock states for robust quantum optical metrology, imaging and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Dür, W.

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

Eberly, J. H.

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]

Franson, J. D.

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205–2208 (1989).
[Crossref] [PubMed]

Friberg, S.

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54, 2011–2013 (1985).
[Crossref] [PubMed]

Gatti, A.

L. Caspani, E. Brambilla, and A. Gatti, “Tailoring the spatiotemporal structure of biphoton entanglement intype-i parametric down-conversion,” Phys. Rev. A 81, 033808 (2010).
[Crossref]

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “X entanglement: The nonfactorable spatiotemporal structure of biphoton correlation,” Phys. Rev. Lett. 102, 223601 (2009).
[Crossref] [PubMed]

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176 (2002).
[Crossref]

Ge, M.-L.

J.-L. Chen, C. Wu, L. C. Kwek, C. H. Oh, and M.-L. Ge, “Violating bell inequalities maximally for two d-dimensional systems,” Phys. Rev. A 74, 032106 (2006).
[Crossref]

Gisin, N.

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, and N. Gisin, “Two independent photon pairs versus four-photon entangled states in parametric down conversion,” J. Mod. Opt. 51, 1637 (2004).
[Crossref]

Gong, Y.-X.

Y.-X. Gong, X.-B. Zou, X.-L. Niu, J. Li, Y.-F. Huang, and G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[Crossref]

Grangier, P.

A. Aspect, P. Grangier, and G. Roger, “Experimental realization of einstein-podolsky-rosen-bohm Gedankenexperiment: A new violation of bell’s inequalities,” Phys. Rev. Lett. 49, 91–94 (1982).
[Crossref]

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via bell’s theorem,” Phys. Rev. Lett. 47, 460–463 (1981).
[Crossref]

Gühne, O.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett. 95, 201502 (2005).
[Crossref]

Guo, G.-C.

Y.-X. Gong, X.-B. Zou, X.-L. Niu, J. Li, Y.-F. Huang, and G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[Crossref]

Hong, C. K.

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54, 2011–2013 (1985).
[Crossref] [PubMed]

Howell, J. C.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the einstein-podolsky-rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

Huang, Y.-F.

Y.-X. Gong, X.-B. Zou, X.-L. Niu, J. Li, Y.-F. Huang, and G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[Crossref]

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S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled fock states for robust quantum optical metrology, imaging and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Jedrkiewicz, O.

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “X entanglement: The nonfactorable spatiotemporal structure of biphoton correlation,” Phys. Rev. Lett. 102, 223601 (2009).
[Crossref] [PubMed]

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M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[Crossref]

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T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A 56, 1534 (1997).
[Crossref]

Kiesel, N.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett. 95, 201502 (2005).
[Crossref]

Klyshko, D. N.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600 (1995).
[Crossref] [PubMed]

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M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
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J.-L. Chen, C. Wu, L. C. Kwek, C. H. Oh, and M.-L. Ge, “Violating bell inequalities maximally for two d-dimensional systems,” Phys. Rev. A 74, 032106 (2006).
[Crossref]

Kwiat, P. G.

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).
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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).
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S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. S. Ribeiro, “Schemes for quantum key distribution with higher-order alphabets using single-photon fractional fourier optics,” Phys. Rev. A 77, 062323 (2008).
[Crossref]

S. P. Walborn, D. S. Lemelle, M. P. Almedia, and P. H. S. Ribeiro, “Quantum key distribution with higher-order alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[Crossref] [PubMed]

D. S. Lemelle, M. P. Almeida, P. H. S. Ribeiro, and S. P. Walborn, “A simple optical demonstration of quantum cryptography using transverse position and momentum variables,” Am. J.of Phys. 74, 542 (2006).
[Crossref]

Li, J.

Y.-X. Gong, X.-B. Zou, X.-L. Niu, J. Li, Y.-F. Huang, and G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[Crossref]

Lugiato, L. A.

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “X entanglement: The nonfactorable spatiotemporal structure of biphoton correlation,” Phys. Rev. Lett. 102, 223601 (2009).
[Crossref] [PubMed]

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176 (2002).
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S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54, 2011–2013 (1985).
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H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, and N. Gisin, “Two independent photon pairs versus four-photon entangled states in parametric down conversion,” J. Mod. Opt. 51, 1637 (2004).
[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, 4337–4341 (1995).
[Crossref] [PubMed]

Moerman, J. J. D.

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “Experimental quantum coin tossing,” Phys. Rev. Lett. 94, 040501 (2005).
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Monken, C.

S. Walborn, C. Monken, S. Pdua, and P. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports 495, 87 (2010).
[Crossref]

Monken, C. H.

C. H. Monken, P. H. S. Ribereiro, and S. Padua, “Optimizing the photon pair collection efficiency: A step toward a loophole-free bell’s inequalities experiment,” Phys. Rev. A 57, R2267 (1998).
[Crossref]

Murao, M.

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[Crossref]

Nevascues, M.

T. H. Yang and M. Nevascues, “Robust self-testing of unknown quantum systems into any entangled two-qubit states,” Phys. Rev. A 87, 050102(R) (2013).
[Crossref]

Nienhuis, G.

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]

Niu, X.-L.

Y.-X. Gong, X.-B. Zou, X.-L. Niu, J. Li, Y.-F. Huang, and G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[Crossref]

Oemrawsingh, S. S. R.

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]

Oh, C. H.

J.-L. Chen, C. Wu, L. C. Kwek, C. H. Oh, and M.-L. Ge, “Violating bell inequalities maximally for two d-dimensional systems,” Phys. Rev. A 74, 032106 (2006).
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Z. J. Ou, Multi-photon quantum interference (Springer, New York, 2007).

Padua, S.

C. H. Monken, P. H. S. Ribereiro, and S. Padua, “Optimizing the photon pair collection efficiency: A step toward a loophole-free bell’s inequalities experiment,” Phys. Rev. A 57, R2267 (1998).
[Crossref]

Pdua, S.

S. Walborn, C. Monken, S. Pdua, and P. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports 495, 87 (2010).
[Crossref]

Peeters, W. H.

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref] [PubMed]

W. H. Peeters and M. P. van Exter, “Optical characterization of periodically-poled ktiopo4,” Optics Express 16, 7344 (2008).
[Crossref] [PubMed]

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref] [PubMed]

Plenio, M. B.

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[Crossref]

Rarity, J. G.

P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595 (1998).
[Crossref]

Renema, J. J.

A. J. H. van der Torren, S. C. Yorulmaz, J. J. Renema, M. P. van Exter, and M. J. A. de Dood, “Spatially entangled four-photon states from a periodically poled potassium-titanyl-phosphate crystal,” Phys. Rev. A 85, 043837 (2012).
[Crossref]

Ribeiro, P. H. S.

S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. S. Ribeiro, “Schemes for quantum key distribution with higher-order alphabets using single-photon fractional fourier optics,” Phys. Rev. A 77, 062323 (2008).
[Crossref]

S. P. Walborn, D. S. Lemelle, M. P. Almedia, and P. H. S. Ribeiro, “Quantum key distribution with higher-order alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[Crossref] [PubMed]

D. S. Lemelle, M. P. Almeida, P. H. S. Ribeiro, and S. P. Walborn, “A simple optical demonstration of quantum cryptography using transverse position and momentum variables,” Am. J.of Phys. 74, 542 (2006).
[Crossref]

Ribeiro, P. S.

S. Walborn, C. Monken, S. Pdua, and P. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports 495, 87 (2010).
[Crossref]

Ribereiro, P. H. S.

C. H. Monken, P. H. S. Ribereiro, and S. Padua, “Optimizing the photon pair collection efficiency: A step toward a loophole-free bell’s inequalities experiment,” Phys. Rev. A 57, R2267 (1998).
[Crossref]

Riedmatten, H. D.

H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, and N. Gisin, “Two independent photon pairs versus four-photon entangled states in parametric down conversion,” J. Mod. Opt. 51, 1637 (2004).
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Roger, G.

A. Aspect, P. Grangier, and G. Roger, “Experimental realization of einstein-podolsky-rosen-bohm Gedankenexperiment: A new violation of bell’s inequalities,” Phys. Rev. Lett. 49, 91–94 (1982).
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A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via bell’s theorem,” Phys. Rev. Lett. 47, 460–463 (1981).
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Rubin, M. H.

T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A 56, 1534 (1997).
[Crossref]

Scarani, V.

H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, and N. Gisin, “Two independent photon pairs versus four-photon entangled states in parametric down conversion,” J. Mod. Opt. 51, 1637 (2004).
[Crossref]

Schmid, C.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett. 95, 201502 (2005).
[Crossref]

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, 4337–4341 (1995).
[Crossref] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600 (1995).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[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, 4337–4341 (1995).
[Crossref] [PubMed]

Shih, Y. H.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600 (1995).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref] [PubMed]

Silberhorn, C.

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[Crossref] [PubMed]

Strekalov, D. V.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600 (1995).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref] [PubMed]

Tapster, P. R.

P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595 (1998).
[Crossref]

Tasca, D. S.

S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. S. Ribeiro, “Schemes for quantum key distribution with higher-order alphabets using single-photon fractional fourier optics,” Phys. Rev. A 77, 062323 (2008).
[Crossref]

Thew, R. T.

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

Tittel, W.

H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, and N. Gisin, “Two independent photon pairs versus four-photon entangled states in parametric down conversion,” J. Mod. Opt. 51, 1637 (2004).
[Crossref]

Tóth, G.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett. 95, 201502 (2005).
[Crossref]

Ursin, R.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett. 95, 201502 (2005).
[Crossref]

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “Experimental quantum coin tossing,” Phys. Rev. Lett. 94, 040501 (2005).
[Crossref] [PubMed]

van der Torren, A. J. H.

A. J. H. van der Torren, S. C. Yorulmaz, J. J. Renema, M. P. van Exter, and M. J. A. de Dood, “Spatially entangled four-photon states from a periodically poled potassium-titanyl-phosphate crystal,” Phys. Rev. A 85, 043837 (2012).
[Crossref]

van Exter, M. P.

A. J. H. van der Torren, S. C. Yorulmaz, J. J. Renema, M. P. van Exter, and M. J. A. de Dood, “Spatially entangled four-photon states from a periodically poled potassium-titanyl-phosphate crystal,” Phys. Rev. A 85, 043837 (2012).
[Crossref]

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref] [PubMed]

C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
[Crossref] [PubMed]

W. H. Peeters and M. P. van Exter, “Optical characterization of periodically-poled ktiopo4,” Optics Express 16, 7344 (2008).
[Crossref] [PubMed]

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]

Vaziri, A.

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “Experimental quantum coin tossing,” Phys. Rev. Lett. 94, 040501 (2005).
[Crossref] [PubMed]

Vedral, V.

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[Crossref]

Venderbos, J. W. F.

C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
[Crossref] [PubMed]

Vidal, G.

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

Walborn, S.

S. Walborn, C. Monken, S. Pdua, and P. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports 495, 87 (2010).
[Crossref]

Walborn, S. P.

S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. S. Ribeiro, “Schemes for quantum key distribution with higher-order alphabets using single-photon fractional fourier optics,” Phys. Rev. A 77, 062323 (2008).
[Crossref]

S. P. Walborn, D. S. Lemelle, M. P. Almedia, and P. H. S. Ribeiro, “Quantum key distribution with higher-order alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[Crossref] [PubMed]

D. S. Lemelle, M. P. Almeida, P. H. S. Ribeiro, and S. P. Walborn, “A simple optical demonstration of quantum cryptography using transverse position and momentum variables,” Am. J.of Phys. 74, 542 (2006).
[Crossref]

Walmsley, I. A.

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[Crossref] [PubMed]

Weber, U.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett. 95, 201502 (2005).
[Crossref]

Weinfurter, H.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett. 95, 201502 (2005).
[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, 4337–4341 (1995).
[Crossref] [PubMed]

Wildfeuer, C. F.

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled fock states for robust quantum optical metrology, imaging and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Woerdman, J. P.

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]

Wu, C.

J.-L. Chen, C. Wu, L. C. Kwek, C. H. Oh, and M.-L. Ge, “Violating bell inequalities maximally for two d-dimensional systems,” Phys. Rev. A 74, 032106 (2006).
[Crossref]

Yang, T. H.

T. H. Yang and M. Nevascues, “Robust self-testing of unknown quantum systems into any entangled two-qubit states,” Phys. Rev. A 87, 050102(R) (2013).
[Crossref]

Yorulmaz, S. C.

A. J. H. van der Torren, S. C. Yorulmaz, J. J. Renema, M. P. van Exter, and M. J. A. de Dood, “Spatially entangled four-photon states from a periodically poled potassium-titanyl-phosphate crystal,” Phys. Rev. A 85, 043837 (2012).
[Crossref]

S. C. Yorulmaz and M. J. A. de Dood, “Characterization of pulsed parametric down-conversion in ppktp crystals,” Proc. SPIE 8440, 84400G (2012).
[Crossref]

Zbinden, H.

H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, and N. Gisin, “Two independent photon pairs versus four-photon entangled states in parametric down conversion,” J. Mod. Opt. 51, 1637 (2004).
[Crossref]

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

Zeilinger, A.

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “Experimental quantum coin tossing,” Phys. Rev. Lett. 94, 040501 (2005).
[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]

Zhang, L.

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[Crossref] [PubMed]

Zou, X.-B.

Y.-X. Gong, X.-B. Zou, X.-L. Niu, J. Li, Y.-F. Huang, and G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[Crossref]

Am. J.of Phys. (1)

D. S. Lemelle, M. P. Almeida, P. H. S. Ribeiro, and S. P. Walborn, “A simple optical demonstration of quantum cryptography using transverse position and momentum variables,” Am. J.of Phys. 74, 542 (2006).
[Crossref]

J. Mod. Opt. (2)

H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, and N. Gisin, “Two independent photon pairs versus four-photon entangled states in parametric down conversion,” J. Mod. Opt. 51, 1637 (2004).
[Crossref]

P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595 (1998).
[Crossref]

J. Opt. B (1)

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176 (2002).
[Crossref]

Optics Express (1)

W. H. Peeters and M. P. van Exter, “Optical characterization of periodically-poled ktiopo4,” Optics Express 16, 7344 (2008).
[Crossref] [PubMed]

Phys. Rev. A (13)

T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A 56, 1534 (1997).
[Crossref]

L. Caspani, E. Brambilla, and A. Gatti, “Tailoring the spatiotemporal structure of biphoton entanglement intype-i parametric down-conversion,” Phys. Rev. A 81, 033808 (2010).
[Crossref]

C. H. Monken, P. H. S. Ribereiro, and S. Padua, “Optimizing the photon pair collection efficiency: A step toward a loophole-free bell’s inequalities experiment,” Phys. Rev. A 57, R2267 (1998).
[Crossref]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref] [PubMed]

A. J. H. van der Torren, S. C. Yorulmaz, J. J. Renema, M. P. van Exter, and M. J. A. de Dood, “Spatially entangled four-photon states from a periodically poled potassium-titanyl-phosphate crystal,” Phys. Rev. A 85, 043837 (2012).
[Crossref]

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[Crossref]

Y.-X. Gong, X.-B. Zou, X.-L. Niu, J. Li, Y.-F. Huang, and G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[Crossref]

S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. S. Ribeiro, “Schemes for quantum key distribution with higher-order alphabets using single-photon fractional fourier optics,” Phys. Rev. A 77, 062323 (2008).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Experimental setup for generating and characterizing spatially entangled four-photon states. UV pump pulses from a frequency doubled Ti:Sapphire laser are focused by lens L1 in a PPKTP crystal. Down-converted photons are collected by the lens L2 (f2 = 270 mm) and filtered by a bandpass filters F at a wavelength of 826.4 nm with 1 nm or 0.4 nm FWHM bandwidths. Photons are split by a beam splitter (BS) and detected by fiber-coupled APDs D1 and D2 placed on computer controlled translation stages. Spatial-mode selection occurs by means of tunable detector apertures (A) and a lens L3 mounted on a translation stage. Photon counts and coincidences are recorded as a function of the angular positions q (horizontal direction) and p (vertical direction) of the detectors in the far-field.

Fig. 2
Fig. 2

Variation of the visibility χ of the four-photon state as a function of aperture size. The PDC light is generated in a 2 mm PPKTP crystal and detected using 1 nm FWHM bandpass filter (black open symbols) and 0.4 nm FWHM bandpass filter in combination with the 1 nm FWHM filter (red solid symbols) at 826.4 nm wavelength. The solid lines (with shaded areas) correspond to calculations based on Eq. (8) with pump beam diameter of 85±10 μm for each of the bandpass filters. The inset shows the measured transmission spectra of the filters.

Fig. 3
Fig. 3

(a) Visibility χ of four-photon states obtained using different pump beam diameters equal to wp = 55 ± 10 μm (black symbols), wp = 85 ± 10 μm (red symbols) and wp = 155 ± 10 μm (blue symbols) as a function of the diameter of the detector aperture. Photon pairs are created in a 2 mm long PPKTP crystal and filtered by a 1 nm FWHM bandpass filter. (b) Universal curve for visibility of four-photon states as a function of the normalized aperture diameter a/r0. The inset illustrates one side of the PDC ring with an area selected by an aperture of diameter a together with the characteristic diameter r0 = λf/(πwp) determined by the diameter of the pump beam wp.

Fig. 4
Fig. 4

Measured visibility χ of the four-photon state generated with pump beam diameters equal to wp = 55 μm (blue triangles), wp = 85 μm (red circles) and wp = 155 μm (green squares). The data are shown as a function of crystal length, using an aperture diameter of 1 mm and a 1 nm FWHM bandpass filter. The solid lines represent fits to the data (see text). The shaded areas indicate the confidence interval of the fit taking into account a ±10 μm uncertainty in the beam diameter wp. The inset shows the walk-off length obtained from the fit of Eq. (6) to the data.

Fig. 5
Fig. 5

Measured joint spatial distribution of genuine four-photon states as a function of the angular positions of q1 and q2 of the detectors D1 and D2 in the far-field using a 1 nm FWHM bandpass filter and a 1.5 mm aperture size at f = 270 mm. Down-converted photons are created by a pump beam diameter of (a) wp = 35 ± 5 μm (b) wp = 45 ± 5 μm (c) wp = 85 ± 5 μm.

Tables (1)

Tables Icon

Table 1 The experimental and calculated values of characteristic diameter r′1 and the visibility χt χs (see text).

Equations (11)

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C ( q s , q i , ω s , ω i ) = E p ( q s + q i ; ω s + ω i ) T ( ω s ) T ( ω i ) sinc ( 1 2 Δ k L ) ,
1 2 Δ k ( ω ) L = b 2 | q s q i | 2 + φ 0 + η ( δ ω s + δ ω i ) ,
E p ( q s + q i ; ω s + ω i ) exp ( | q s + q i | 2 / σ 2 ) exp ( τ 2 ( δ ω s + δ ω i ) 2 / 2 )
C ( q s , q i , ω s , ω i ) e ( η ) 2 ( δ ω s + δ ω i ) 2 × e ( τ 2 / 2 ) ( δ ω s + δ ω i ) 2 e ( 1 / ( 2 ) ) 2 ( δ ω s 2 + δ ω i 2 ) × e ( b 2 | q s q i | 2 + φ 0 ) 2 e ( 1 / σ ) 2 | q s + q i | 2 .
χ t = 1 1 + ( τ ) 2
χ w = 1 1 + ( L 2 L 0 ) 2 ,
χ s ( r 1 , r 2 ) = exp ( | r 1 r 2 | 2 / r 0 2 ) .
χ s ( a ) = 1 π 2 a 4 χ s ( r 1 , r 2 ) Θ ( a 2 | r 1 | ) Θ ( a 2 | r 2 | ) d r 1 d r 2 ,
χ s ( a ) = 1 π 0 1 ( arccos ( x ) x 1 x 2 ) exp ( ( x a / r 0 ) 2 ) d x .
R 12 n s = N P ( η 2 ) 2 P 2 2 2 ,
R 0 n s = N P ( η 2 ) 2 P 2 2 2 ( 1 + χ ) ,

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