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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 3753–3772

Quantifying the non-Gaussianity of the state of spatially correlated down-converted photons

E. S. Gómez, W. A. T. Nogueira, C. H. Monken, and G. Lima  »View Author Affiliations

Optics Express, Vol. 20, Issue 4, pp. 3753-3772 (2012)

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The state of spatially correlated down-converted photons is usually treated as a two-mode Gaussian entangled state. While intuitively this seems to be reasonable, it is known that new structures in the spatial distributions of these photons can be observed when the phase-matching conditions are properly taken into account. Here, we study how the variances of the near- and far-field conditional probabilities are affected by the phase-matching functions, and we analyze the role of the EPR-criterion regarding the non-Gaussianity and entanglement detection of the spatial two-photon state of spontaneous parametric down-conversion (SPDC). Then we introduce a statistical measure, based on the negentropy of the joint distributions at the near- and far-field planes, which allows for the quantification of the non-Gaussianity of this state. This measure of non-Gaussianity requires only the measurement of the diagonal covariance sub-matrices, and will be relevant for new applications of the spatial correlation of SPDC in CV quantum information processing.

© 2012 OSA

OCIS Codes
(270.0270) Quantum optics : Quantum optics
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

Original Manuscript: December 12, 2011
Revised Manuscript: January 22, 2012
Manuscript Accepted: January 23, 2012
Published: January 31, 2012

E. S. Gómez, W. A. T. Nogueira, C. H. Monken, and G. Lima, "Quantifying the non-Gaussianity of the state of spatially correlated down-converted photons," Opt. Express 20, 3753-3772 (2012)

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  1. 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]
  2. M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett.92, 233601 (2004). [CrossRef]
  3. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev.47, 777 (1935). [CrossRef]
  4. E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett.82, 2868 (1999). [CrossRef]
  5. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett.99, 170408 (2007). [CrossRef] [PubMed]
  6. S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep.495, 87 (2010). [CrossRef]
  7. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A52, R3429 (1995). [CrossRef] [PubMed]
  8. A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett.90, 133603 (2003). [CrossRef] [PubMed]
  9. L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005). [CrossRef] [PubMed]
  10. M. N. O’Sullivan-Hale, I. Ali Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett.94, 220501 (2005). [CrossRef]
  11. A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett.104, 010501 (2010). [CrossRef] [PubMed]
  12. C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A31, 2409 (1985). [CrossRef] [PubMed]
  13. M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A505122 (1994). [CrossRef] [PubMed]
  14. C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A57, 3123 (1998). [CrossRef]
  15. 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]
  16. H. Di Lorenzo Pires, C. H. Monken, and M. P. van Exter, “Direct measurement of transverse-mode entanglement in two-photon states,” Phys. Rev. A80, 022307 (2009). [CrossRef]
  17. K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A75, 050101 (2007). [CrossRef]
  18. S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, “Angular Schmidt modes in spontaneous parametric down-conversion,” Phys. Rev. A83, 060302 (2011). [CrossRef]
  19. S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A76, 033801 (2007). [CrossRef]
  20. D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A78, 010304 (2008). [CrossRef]
  21. D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A79, 033801 (2009). [CrossRef]
  22. L. J. Zhang, L. Neves, J. S. Lundeen, and I. A. Walmsley, “A characterization of the single-photon sensitivity of an electron multiplying charge-coupled device,” J. Phys. B42, 114011 (2009). [CrossRef]
  23. H. Di Lorenzo Pires and M. P. van Exter, “Observation of near-field correlations in spontaneous parametric down-conversion,” Phys. Rev. A79, 041801 (2009). [CrossRef]
  24. M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys.81, 1727–1751 (2009). [CrossRef]
  25. R. M. Gomes, A. Salles, F. Toscano, P. H. S. Ribeiro, and S. P. Walborn, “Quantum entanglement beyond Gaussian criteria,” Proc. Natl. Acad. Sci. U.S.A.106, 21517 (2009). [CrossRef] [PubMed]
  26. A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001). [CrossRef]
  27. M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A78, 060303 (2008). [CrossRef]
  28. M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A82, 052341 (2010). [CrossRef]
  29. M. Ostermeyer, D. Korn, D. Puhlmann, C. Henkel, and J. Eisert, “Two-dimensional characterization of spatially entangled photon pairs,” J. Mod. Opt.56, 1829–1837 (2009). [CrossRef]
  30. H. Di Lorenzo Pires and M. P. van Exter, “Near-field correlations in the two-photon field,” Phys. Rev. A80, 053820 (2009). [CrossRef]
  31. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett.88, 120401 (2002). [CrossRef] [PubMed]
  32. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991). [CrossRef]
  33. P. Comon, “Independent component analysis, A new concept?,” Sig. Process.36, 287–314 (1994). [CrossRef]
  34. P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high dimensional entangled states,” arXiv:1107.5245v1[quant-ph].
  35. M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett.96, 080502 (2006). [CrossRef] [PubMed]

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