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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  7. Y.-X. Gong, X.-B. Zou, X.-L. Niu, J. Li, Y.-F. Huang, G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  36. A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, 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, A. Gatti, “Tailoring the spatiotemporal structure of biphoton entanglement intype-i parametric down-conversion,” Phys. Rev. A 81, 033808 (2010).
    [CrossRef]
  38. C. H. Monken, P. H. S. Ribereiro, 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]

2013

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

2012

A. J. H. van der Torren, S. C. Yorulmaz, J. J. Renema, M. P. van Exter, 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, M. J. A. de Dood, “Characterization of pulsed parametric down-conversion in ppktp crystals,” Proc. SPIE 8440, 84400G (2012).
[CrossRef]

2010

L. Caspani, E. Brambilla, 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, M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[CrossRef] [PubMed]

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

2009

C. W. J. Beenakker, J. W. F. Venderbos, 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, L. A. Lugiato, “X entanglement: The nonfactorable spatiotemporal structure of biphoton correlation,” Phys. Rev. Lett. 102, 223601 (2009).
[CrossRef] [PubMed]

2008

W. H. Peeters, 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, 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, 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, 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, G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[CrossRef]

2006

S. P. Walborn, D. S. Lemelle, M. P. Almedia, 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, 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, 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, M.-L. Ge, “Violating bell inequalities maximally for two d-dimensional systems,” Phys. Rev. A 74, 032106 (2006).
[CrossRef]

2005

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, 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, A. Zeilinger, “Experimental quantum coin tossing,” Phys. Rev. Lett. 94, 040501 (2005).
[CrossRef] [PubMed]

2004

J. C. Howell, R. S. Bennink, S. J. Bentley, 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, N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[CrossRef]

C. K. Law, 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, N. Gisin, “Two independent photon pairs versus four-photon entangled states in parametric down conversion,” J. Mod. Opt. 51, 1637 (2004).
[CrossRef]

2002

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

2000

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

1999

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

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

1998

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

C. H. Monken, P. H. S. Ribereiro, 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

T. E. Keller, 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

T. B. Pittman, Y. H. Shih, D. V. Strekalov, 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, 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, Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

1989

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

1985

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

1982

A. Aspect, P. Grangier, 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

A. Aspect, P. Grangier, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176 (2002).
[CrossRef]

Caspani, L.

L. Caspani, E. Brambilla, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[CrossRef]

Huver, S. D.

S. D. Huver, C. F. Wildfeuer, 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, L. A. Lugiato, “X entanglement: The nonfactorable spatiotemporal structure of biphoton correlation,” Phys. Rev. Lett. 102, 223601 (2009).
[CrossRef] [PubMed]

Jonathan, D.

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

Keller, T. E.

T. E. Keller, 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]

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

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

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P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, 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, 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, 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, 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, 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, G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[CrossRef]

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A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, “X entanglement: The nonfactorable spatiotemporal structure of biphoton correlation,” Phys. Rev. Lett. 102, 223601 (2009).
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L. A. Lugiato, A. Gatti, E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176 (2002).
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H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, 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, Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

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

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G. Molina-Terriza, A. Vaziri, R. Ursin, A. Zeilinger, “Experimental quantum coin tossing,” Phys. Rev. Lett. 94, 040501 (2005).
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S. Walborn, C. Monken, S. Pdua, P. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports 495, 87 (2010).
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C. H. Monken, P. H. S. Ribereiro, S. Padua, “Optimizing the photon pair collection efficiency: A step toward a loophole-free bell’s inequalities experiment,” Phys. Rev. A 57, R2267 (1998).
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M. Murao, D. Jonathan, M. B. Plenio, V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[CrossRef]

Nevascues, M.

T. H. Yang, 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, J. P. Woerdman, “Effect of spatial filtering on the schmidt decomposition of entangled photons,” Phys. Rev. A 74, 012309 (2006).
[CrossRef]

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Y.-X. Gong, X.-B. Zou, X.-L. Niu, J. Li, Y.-F. Huang, 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, 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, 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).

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C. H. Monken, P. H. S. Ribereiro, 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, 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, M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[CrossRef] [PubMed]

W. H. Peeters, M. P. van Exter, “Optical characterization of periodically-poled ktiopo4,” Optics Express 16, 7344 (2008).
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T. B. Pittman, Y. H. Shih, D. V. Strekalov, 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, V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
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Rarity, J. G.

P. R. Tapster, J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595 (1998).
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A. J. H. van der Torren, S. C. Yorulmaz, J. J. Renema, M. P. van Exter, 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, 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, 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, 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, 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, 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, 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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A. Aspect, P. Grangier, 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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T. E. Keller, 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, 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, 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, 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, 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, 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, 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, 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, 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, I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[CrossRef] [PubMed]

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, 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, Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef] [PubMed]

Tapster, P. R.

P. R. Tapster, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, A. Zeilinger, “Experimental quantum coin tossing,” Phys. Rev. Lett. 94, 040501 (2005).
[CrossRef] [PubMed]

Vedral, V.

M. Murao, D. Jonathan, M. B. Plenio, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, G.-C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[CrossRef]

Am. J.of Phys.

D. S. Lemelle, M. P. Almeida, P. H. S. Ribeiro, 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.

H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acn, W. Tittel, H. Zbinden, 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, J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595 (1998).
[CrossRef]

J. Opt. B

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

Optics Express

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

Phys. Rev. A

T. E. Keller, 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, 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, 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, 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, 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, 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, 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, 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]

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

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

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

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

M. P. van Exter, A. Aiello, S. S. R. Oemrawsingh, G. Nienhuis, J. P. Woerdman, “Effect of spatial filtering on the schmidt decomposition of entangled photons,” Phys. Rev. A 74, 012309 (2006).
[CrossRef]

Phys. Rev. Lett.

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

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

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