Abstract

We have extended Fourier transform of quantum light to a fractional Fourier processing, and demonstrated that a classical optical fractional Fourier processor can be used for the shaping of quantum correlations between two or more photons. Comparing the present method with that of Fourier processing, we find that fractional Fourier processing for quantum light possesses many advantages. Based on such a method, not only quantum correlations can be shaped more rich, but also the initial states can be easily identified. Moreover, the twisted phase information can be recovered and quantum states are easily controlled in performing quantum information experiments. Our findings open up new avenues for the manipulation of correlations between photons in optical quantum information processing.

©2014 Optical Society of America

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References

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  1. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980).
    [Crossref]
  2. A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39(2), 159–175 (1987).
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  6. D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, and H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12(11), 2424–2431 (1995).
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  7. B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226(1-6), 61–80 (2003).
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  8. Y. Zhang, B. Dong, B. Gu, and G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15(5), 1114–1120 (1998).
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  9. H. E. Hwang and P. Han, “Fractional Fourier transform optimization approach for analyzing optical beam propagation between two spherical surfaces,” Opt. Commun. 245(1-6), 11–19 (2005).
    [Crossref]
  10. Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005).
    [Crossref]
  11. Y. Cai and S. Y. Zhu, “Coincidence fractional Fourier transform implemented with partially coherent light radiation,” J. Opt. Soc. Am. A 22(9), 1798–1804 (2005).
    [Crossref] [PubMed]
  12. F. Wang, Y. J. Cai, and S. L. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006).
    [Crossref] [PubMed]
  13. J. Liu, A. Tan, and Z. Hong, “Experimental observation of coincidence fractional Fourier transform with entanglement photon pairs,” Opt. Commun. 282(17), 3524–3526 (2009).
    [Crossref]
  14. D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
    [Crossref]
  15. D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
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    [Crossref]
  20. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
    [Crossref]
  21. M. Atatüre, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Multiparameter entanglement in quantum interferometry,” Phys. Rev. A 66(2), 023822 (2002).
    [Crossref]
  22. B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94(22), 223601 (2005).
    [Crossref] [PubMed]
  23. R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74(1), 013801 (2006).
    [Crossref]
  24. A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
    [Crossref]
  25. A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010).
    [Crossref] [PubMed]
  26. E. Poem, Y. Gilead, Y. Lahini, and Y. Silberberg, “Fourier processing of quantum light,” Phys. Rev. A 86(2), 023836 (2012).
    [Crossref]
  27. H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
    [Crossref] [PubMed]

2012 (2)

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

E. Poem, Y. Gilead, Y. Lahini, and Y. Silberberg, “Fourier processing of quantum light,” Phys. Rev. A 86(2), 023836 (2012).
[Crossref]

2010 (1)

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010).
[Crossref] [PubMed]

2009 (2)

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[Crossref]

J. Liu, A. Tan, and Z. Hong, “Experimental observation of coincidence fractional Fourier transform with entanglement photon pairs,” Opt. Commun. 282(17), 3524–3526 (2009).
[Crossref]

2008 (3)

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[Crossref]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

2006 (2)

R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74(1), 013801 (2006).
[Crossref]

F. Wang, Y. J. Cai, and S. L. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006).
[Crossref] [PubMed]

2005 (4)

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94(22), 223601 (2005).
[Crossref] [PubMed]

H. E. Hwang and P. Han, “Fractional Fourier transform optimization approach for analyzing optical beam propagation between two spherical surfaces,” Opt. Commun. 245(1-6), 11–19 (2005).
[Crossref]

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005).
[Crossref]

Y. Cai and S. Y. Zhu, “Coincidence fractional Fourier transform implemented with partially coherent light radiation,” J. Opt. Soc. Am. A 22(9), 1798–1804 (2005).
[Crossref] [PubMed]

2003 (1)

B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226(1-6), 61–80 (2003).
[Crossref]

2002 (2)

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[Crossref]

M. Atatüre, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Multiparameter entanglement in quantum interferometry,” Phys. Rev. A 66(2), 023822 (2002).
[Crossref]

1998 (1)

1995 (1)

1993 (3)

1987 (1)

A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39(2), 159–175 (1987).
[Crossref]

1980 (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980).
[Crossref]

Abouraddy, A. F.

Atatüre, M.

M. Atatüre, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Multiparameter entanglement in quantum interferometry,” Phys. Rev. A 66(2), 023822 (2002).
[Crossref]

Barnett, S. M.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

Bitran, Y.

Boyd, R. W.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

Buller, G. S.

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

Cai, Y.

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005).
[Crossref]

Y. Cai and S. Y. Zhu, “Coincidence fractional Fourier transform implemented with partially coherent light radiation,” J. Opt. Soc. Am. A 22(9), 1798–1804 (2005).
[Crossref] [PubMed]

Cai, Y. J.

Chen, Z.-B.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Di Giuseppe, G.

M. Atatüre, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Multiparameter entanglement in quantum interferometry,” Phys. Rev. A 66(2), 023822 (2002).
[Crossref]

Dong, B.

Dorsch, R. G.

Edamatsu, K.

R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74(1), 013801 (2006).
[Crossref]

Franke-Arnold, S.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

Gilead, Y.

E. Poem, Y. Gilead, Y. Lahini, and Y. Silberberg, “Fourier processing of quantum light,” Phys. Rev. A 86(2), 023836 (2012).
[Crossref]

Gu, B.

Han, P.

H. E. Hwang and P. Han, “Fractional Fourier transform optimization approach for analyzing optical beam propagation between two spherical surfaces,” Opt. Commun. 245(1-6), 11–19 (2005).
[Crossref]

He, S. L.

Hennelly, B.

B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226(1-6), 61–80 (2003).
[Crossref]

Hong, Z.

J. Liu, A. Tan, and Z. Hong, “Experimental observation of coincidence fractional Fourier transform with entanglement photon pairs,” Opt. Commun. 282(17), 3524–3526 (2009).
[Crossref]

Hwang, H. E.

H. E. Hwang and P. Han, “Fractional Fourier transform optimization approach for analyzing optical beam propagation between two spherical surfaces,” Opt. Commun. 245(1-6), 11–19 (2005).
[Crossref]

Itoh, T.

R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74(1), 013801 (2006).
[Crossref]

Jack, B.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

Jha, A. K.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

Kerr, F. H.

A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39(2), 159–175 (1987).
[Crossref]

Lahini, Y.

E. Poem, Y. Gilead, Y. Lahini, and Y. Silberberg, “Fourier processing of quantum light,” Phys. Rev. A 86(2), 023836 (2012).
[Crossref]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref] [PubMed]

Leach, J.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

Lin, Q.

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005).
[Crossref]

Liu, J.

J. Liu, A. Tan, and Z. Hong, “Experimental observation of coincidence fractional Fourier transform with entanglement photon pairs,” Opt. Commun. 282(17), 3524–3526 (2009).
[Crossref]

Lohmann, A. W.

Lu, C.-Y.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

McBride, A. C.

A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39(2), 159–175 (1987).
[Crossref]

Mendlovic, D.

Morandotti, R.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref] [PubMed]

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980).
[Crossref]

Ozaktas, H.

Ozaktas, H. M.

Padgett, M. J.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

Pan, J.-W.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Pellat-Finet, P.

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[Crossref]

Perets, H. B.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref] [PubMed]

Poem, E.

E. Poem, Y. Gilead, Y. Lahini, and Y. Silberberg, “Fourier processing of quantum light,” Phys. Rev. A 86(2), 023836 (2012).
[Crossref]

Pozzi, F.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref] [PubMed]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94(22), 223601 (2005).
[Crossref] [PubMed]

M. Atatüre, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Multiparameter entanglement in quantum interferometry,” Phys. Rev. A 66(2), 023822 (2002).
[Crossref]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[Crossref]

Sergienko, A. V.

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94(22), 223601 (2005).
[Crossref] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[Crossref]

M. Atatüre, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Multiparameter entanglement in quantum interferometry,” Phys. Rev. A 66(2), 023822 (2002).
[Crossref]

Shaw, M. D.

M. Atatüre, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Multiparameter entanglement in quantum interferometry,” Phys. Rev. A 66(2), 023822 (2002).
[Crossref]

Sheridan, J. T.

B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226(1-6), 61–80 (2003).
[Crossref]

Shimizu, R.

R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74(1), 013801 (2006).
[Crossref]

Silberberg, Y.

E. Poem, Y. Gilead, Y. Lahini, and Y. Silberberg, “Fourier processing of quantum light,” Phys. Rev. A 86(2), 023836 (2012).
[Crossref]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref] [PubMed]

Sorel, M.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref] [PubMed]

Souto Ribeiro, P. H.

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[Crossref]

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[Crossref]

Tan, A.

J. Liu, A. Tan, and Z. Hong, “Experimental observation of coincidence fractional Fourier transform with entanglement photon pairs,” Opt. Commun. 282(17), 3524–3526 (2009).
[Crossref]

Tasca, D. S.

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[Crossref]

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[Crossref]

Teich, M. C.

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94(22), 223601 (2005).
[Crossref] [PubMed]

M. Atatüre, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Multiparameter entanglement in quantum interferometry,” Phys. Rev. A 66(2), 023822 (2002).
[Crossref]

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D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[Crossref]

Walborn, S. P.

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
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J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
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Zeilinger, A.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
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[Crossref]

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[Crossref]

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Rev. Mod. Phys. (1)

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
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D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, Berlin, 2000).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2005).

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Figures (6)
Fig. 1
Fig. 1 Schematic picture depicting Fractional Fourier processing of quantum light. The path-entangled photon-pairs are taken as source, and an initial phase factor can be added to one of the paths by phase control part. The photon-pairs go through a 4-f filter equipped with a spatial light modulator (SLM), in which a classic FrFT process is applied, and finally detected by two fiber-coupled detectors (D1, D2). Coincidence detection is manifested by a coincidence counting unit (CCU). The classic FrFT process, which could be realized by lenses shown in the line-box. Here f represents the focal length of the lens and f1 = Qf is called standard focal length. R and Q are related FrFT parameters, R = tan( ϕ /2) and Q = sin( ϕ ) for upper case, and R = sin( ϕ ) and Q = tan( ϕ /2) for lower case. Here ϕ = pπ/2, p represents fractional degree.

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Fig. 2
Fig. 2 (a) Schematic of the beam shaping for FrFT process with a sinusoidal phase mask. For FrFT process, the transform part will be replaced by the lens system in the line-box of Fig. 1. (b) Output intensity patterns for different values of the phase amplitude Ap for two input beams when fractional degree p is 1 (FT case). The output intensity patterns as a function of Ap and position for p = 1-6.66 × 10−9 (c) and p = 0.8 (d).

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Fig. 3
Fig. 3 Output intensity patterns as a function of fractional degree p and position with two input modes for Ap = 0.86π (a) and Ap = 1.67π (b). The other parameters are taken identical with those in Fig. 2.

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Fig. 4
Fig. 4 Normalized spatial correlation functions for the incident two-photon path-entangled state with different p. (a) Spatial correlation function of the input state without mask. (b) A section of the phase pattern created by the applied sinusoidal phase mask in the two-photon fractional Fourier plane. (c)–(f) correspond to the spatial correlation patterns at Ap = 1.67π for p = 0.95, 0.85, 0.75 and 0.65, respectively. The initial phase factor is taken as φ = 0.

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Fig. 5
Fig. 5 Spatial correlation functions for the incident two-photon path-entangled state at Ap = 1.67π and p = 0.65. (a)–(d) correspond to the phase difference between two photons φ = 0, φ = π, φ = π/2 and φ = -π/2, respectively. The other parameters are taken identical with those in Fig. 4.

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Fig. 6
Fig. 6 State information recovery under small perturbation by FrFT. (a) Sinusoidal phase mask pattern, spatial correlation function and output intensity patterns for the FT process. (b) 2D bend phase mask pattern, deformed spatial correlation function and output intensity patterns in the FT process. (c) State recovery by FrFT process. Here the fractional degree p = 0.992 is taken. The other parameters are taken identical with those in Fig. 4

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Equations (6)

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|Ψ〉= ∫ d x 1 d x 2 A S ( x 1 , x 2 ) a ^ † ( x 2 ) a ^ † ( x 1 )|0〉,
| Ψ out 〉= ∫ d x o1 d x o2 B S ( x o1 , x o2 ) b ^ † ( x o2 ) b ^ † ( x o1 )|0〉
B S ( x o1 , x o2 )= ∫ d x i1 d x i2 A S ( x i1 , x i2 )U( x o2 , x i2 )U( x o1 , x i1 ),
U F r ( x o , x i ) = exp [ i π λ f 1 ( x o 2 cot ϕ − 2 x o x i csc ϕ + x i 2 cot ϕ ) ] .
B S ( x o1 , x o2 )= ∫ d x i1 d x i2 A S ( x i1 , x i2 ) U T ( x o2 , x i2 ) U T ( x o1 , x i1 ) ∝ 1 2 [ U T ( x o1 , x a ) U T ( x o2 , x a )+ e iφ U T ( x o1 , x b ) U T ( x o2 , x b ) ],
U T ( x o , x i )= ∫ d x f U Fr ( x o , x f )M( x f ) U F ( x f , x i ) =exp[ iπ λ f 1 x o 2 cotϕ ]×( 1+i )× f 1 λtanϕ × ∑ n=−∞ n=∞ i n J n ( Ap )exp[ −iπ ( x o λ f 1 cscϕ+ x i λf −nυ ) 2 λ f 1 tanϕ ] .

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