Abstract

We demonstrate a method to reconstruct the joint photon statistics of two or more modes of radiation by using on/off photodetection performed at different quantum efficiencies. The two-mode case is discussed in detail, and experimental results are presented for the bipartite states obtained after a beam splitter fed by a single photon state or a thermal state.

© 2006 Optical Society of America

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  1. See, e.g., M. Genovese, Phys. Rep. 413, 319 (2005).
  2. K. Banaszek and I. A. Walmsley, Opt. Lett. 28, 52 (2003).
    [CrossRef] [PubMed]
  3. J. Rehácek, Z. Hradil, O. Haderka, J. Perina Jr., and M. Hamar, Phys. Rev. A 67, 061801(R) (2003).
    [CrossRef]
  4. G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
    [CrossRef] [PubMed]
  5. M. Raymer and M. Beck, in Quantum States Estimation, M.G. A.Paris and J.Rehácek, eds., Lect. Notes Phys. 649 (Springer, 2004).
  6. D. Mogilevtsev, Acta Phys. Slov. 49, 743 (1999).
  7. A. R. Rossi, S. Olivares, and M. G. A. Paris, Phys. Rev. A 70, 055801 (2004).
    [CrossRef]
  8. A. P. Dempster, N. M. Laird, and D. B. Rubin, J. R. Stat. Soc. Ser. B (Methodol.) 39, 1 (1977).
  9. G. Brida, M. Genovese, and C. Novero, J. Mod. Opt. 47, 2099 (2000).
    [CrossRef]
  10. G. Brida, M. Genovese, and M. Gramegna, Laser Phys. Lett. 3, 115 (2006).
    [CrossRef]

2006 (1)

G. Brida, M. Genovese, and M. Gramegna, Laser Phys. Lett. 3, 115 (2006).
[CrossRef]

2005 (2)

See, e.g., M. Genovese, Phys. Rep. 413, 319 (2005).

G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
[CrossRef] [PubMed]

2004 (1)

A. R. Rossi, S. Olivares, and M. G. A. Paris, Phys. Rev. A 70, 055801 (2004).
[CrossRef]

2003 (2)

K. Banaszek and I. A. Walmsley, Opt. Lett. 28, 52 (2003).
[CrossRef] [PubMed]

J. Rehácek, Z. Hradil, O. Haderka, J. Perina Jr., and M. Hamar, Phys. Rev. A 67, 061801(R) (2003).
[CrossRef]

2000 (1)

G. Brida, M. Genovese, and C. Novero, J. Mod. Opt. 47, 2099 (2000).
[CrossRef]

1999 (1)

D. Mogilevtsev, Acta Phys. Slov. 49, 743 (1999).

1977 (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, J. R. Stat. Soc. Ser. B (Methodol.) 39, 1 (1977).

Andreoni, A.

G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
[CrossRef] [PubMed]

Banaszek, K.

Beck, M.

M. Raymer and M. Beck, in Quantum States Estimation, M.G. A.Paris and J.Rehácek, eds., Lect. Notes Phys. 649 (Springer, 2004).

Bondani, M.

G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
[CrossRef] [PubMed]

Brida, G.

G. Brida, M. Genovese, and M. Gramegna, Laser Phys. Lett. 3, 115 (2006).
[CrossRef]

G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
[CrossRef] [PubMed]

G. Brida, M. Genovese, and C. Novero, J. Mod. Opt. 47, 2099 (2000).
[CrossRef]

Dempster, A. P.

A. P. Dempster, N. M. Laird, and D. B. Rubin, J. R. Stat. Soc. Ser. B (Methodol.) 39, 1 (1977).

Genovese, M.

G. Brida, M. Genovese, and M. Gramegna, Laser Phys. Lett. 3, 115 (2006).
[CrossRef]

See, e.g., M. Genovese, Phys. Rep. 413, 319 (2005).

G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
[CrossRef] [PubMed]

G. Brida, M. Genovese, and C. Novero, J. Mod. Opt. 47, 2099 (2000).
[CrossRef]

Gramegna, M.

G. Brida, M. Genovese, and M. Gramegna, Laser Phys. Lett. 3, 115 (2006).
[CrossRef]

G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
[CrossRef] [PubMed]

Haderka, O.

J. Rehácek, Z. Hradil, O. Haderka, J. Perina Jr., and M. Hamar, Phys. Rev. A 67, 061801(R) (2003).
[CrossRef]

Hamar, M.

J. Rehácek, Z. Hradil, O. Haderka, J. Perina Jr., and M. Hamar, Phys. Rev. A 67, 061801(R) (2003).
[CrossRef]

Hradil, Z.

J. Rehácek, Z. Hradil, O. Haderka, J. Perina Jr., and M. Hamar, Phys. Rev. A 67, 061801(R) (2003).
[CrossRef]

Laird, N. M.

A. P. Dempster, N. M. Laird, and D. B. Rubin, J. R. Stat. Soc. Ser. B (Methodol.) 39, 1 (1977).

Mogilevtsev, D.

D. Mogilevtsev, Acta Phys. Slov. 49, 743 (1999).

Novero, C.

G. Brida, M. Genovese, and C. Novero, J. Mod. Opt. 47, 2099 (2000).
[CrossRef]

Olivares, S.

A. R. Rossi, S. Olivares, and M. G. A. Paris, Phys. Rev. A 70, 055801 (2004).
[CrossRef]

Paris, M. G. A.

G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
[CrossRef] [PubMed]

A. R. Rossi, S. Olivares, and M. G. A. Paris, Phys. Rev. A 70, 055801 (2004).
[CrossRef]

Perina, J.

J. Rehácek, Z. Hradil, O. Haderka, J. Perina Jr., and M. Hamar, Phys. Rev. A 67, 061801(R) (2003).
[CrossRef]

Raymer, M.

M. Raymer and M. Beck, in Quantum States Estimation, M.G. A.Paris and J.Rehácek, eds., Lect. Notes Phys. 649 (Springer, 2004).

Rehácek, J.

J. Rehácek, Z. Hradil, O. Haderka, J. Perina Jr., and M. Hamar, Phys. Rev. A 67, 061801(R) (2003).
[CrossRef]

Rossi, A.

G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
[CrossRef] [PubMed]

Rossi, A. R.

A. R. Rossi, S. Olivares, and M. G. A. Paris, Phys. Rev. A 70, 055801 (2004).
[CrossRef]

Rubin, D. B.

A. P. Dempster, N. M. Laird, and D. B. Rubin, J. R. Stat. Soc. Ser. B (Methodol.) 39, 1 (1977).

Walmsley, I. A.

Zambra, G.

G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
[CrossRef] [PubMed]

Acta Phys. Slov. (1)

D. Mogilevtsev, Acta Phys. Slov. 49, 743 (1999).

J. Mod. Opt. (1)

G. Brida, M. Genovese, and C. Novero, J. Mod. Opt. 47, 2099 (2000).
[CrossRef]

J. R. Stat. Soc. Ser. B (Methodol.) (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, J. R. Stat. Soc. Ser. B (Methodol.) 39, 1 (1977).

Laser Phys. Lett. (1)

G. Brida, M. Genovese, and M. Gramegna, Laser Phys. Lett. 3, 115 (2006).
[CrossRef]

Opt. Lett. (1)

Phys. Rep. (1)

See, e.g., M. Genovese, Phys. Rep. 413, 319 (2005).

Phys. Rev. A (2)

A. R. Rossi, S. Olivares, and M. G. A. Paris, Phys. Rev. A 70, 055801 (2004).
[CrossRef]

J. Rehácek, Z. Hradil, O. Haderka, J. Perina Jr., and M. Hamar, Phys. Rev. A 67, 061801(R) (2003).
[CrossRef]

Phys. Rev. Lett. (1)

G. Zambra, M. Bondani, A. Andreoni, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, Phys. Rev. Lett. 95, 063602 (2005).
[CrossRef] [PubMed]

Other (1)

M. Raymer and M. Beck, in Quantum States Estimation, M.G. A.Paris and J.Rehácek, eds., Lect. Notes Phys. 649 (Springer, 2004).

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Figures (3)

Fig. 1
Fig. 1

Schematic diagrams of the experimental setups. Left, setup to reconstruct the joint photon statistics of a bipartite state obtained by splitting a multithermal state by a balanced beam splitter. Right, setup to reconstruct the joint photon statistics of the bipartite states obtained by splitting a PDC-II heralded single-photon state by balanced and unbalanced beam-splitters. SCA, single-channel analyzer. Other abbreviations are defined in text.

Fig. 2
Fig. 2

Reconstruction of the joint photon distribution of a bipartite state: PDC heralded photon state split by a BS (balanced and unbalanced). Dashed lines represent one standard deviation.

Fig. 3
Fig. 3

Reconstruction of the joint photon distribution of a bipartite state: attenuated PDC multithermal distribution split by a balanced BS. The two histograms correspond to reconstructed distributions (dark gray) compared with multithermal ones. Bottom left, fidelity F of the reconstructed marginal distributions to multithermal ones as a function of the number of the iterations of the algorithm. Inset, total error ϵ. Bottom right, measured frequencies f 00 , f 01 , f 10 as function of the quantum efficiency compared with the expected ones (multithermal, solid curves).

Equations (3)

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g μ = p B μ p q p , μ = 1 , , 3 K , p = 1 , , ( 1 + N ) 2 ,
B μ p = { A μ n A μ k μ = 1 , , K A μ n ( 1 A μ k ) μ = K + 1 , , 2 K ( 1 A μ n ) A μ k μ = 2 K + 1 , , 3 K } .
q p ( i + 1 ) = q p ( i ) ( μ = 1 3 K B μ p ) 1 μ = 1 3 K B μ p h μ g μ [ { q p ( i ) } ] .

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