Abstract

We present the reconstruction of the Wigner function of some classical pulsed optical states obtained by direct measurement of the detected-photon probability distributions of the state displaced by a coherent field. We use a photodetector endowed with internal gain, which is operated in the non-photon-resolving regime. The measurements are performed up to mesoscopic intensities (up to more than 30 photons per pulse). The method can be applied to characterize nonclassical continuous-variable states.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. U. Leonhardt, H. Paul, and G. M. D'Ariano, Phys. Rev. A 52, 4899 (1995).
    [CrossRef] [PubMed]
  2. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
    [CrossRef] [PubMed]
  3. A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
    [CrossRef]
  4. M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
    [CrossRef]
  5. K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969).
    [CrossRef]
  6. S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528 (1996).
    [CrossRef] [PubMed]
  7. K. Banaszek and K. Wódkiewicz, Phys. Rev. Lett. 76, 4344 (1996).
    [CrossRef] [PubMed]
  8. K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, Phys. Rev. A 60, 674 (1999).
    [CrossRef]
  9. A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.
  10. T. Kiss, U. Herzog, and U. Leonhardt, Phys. Rev. A 52, 2433 (1995).
    [CrossRef] [PubMed]
  11. F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
    [CrossRef]
  12. M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, J. Mod. Opt. 56, 226 (2009).
    [CrossRef]
  13. M. Bondani, A. Allevi, and A. Andreoni, arXiv:quant-ph/0810.4026.
  14. L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. London 84, 435 (1964).
    [CrossRef]
  15. P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
    [CrossRef]
  16. G. S. Agarwal, J. Mod. Opt. 34, 909 (1987).
    [CrossRef]

2009 (2)

A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[CrossRef]

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, J. Mod. Opt. 56, 226 (2009).
[CrossRef]

1999 (1)

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, Phys. Rev. A 60, 674 (1999).
[CrossRef]

1996 (2)

S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528 (1996).
[CrossRef] [PubMed]

K. Banaszek and K. Wódkiewicz, Phys. Rev. Lett. 76, 4344 (1996).
[CrossRef] [PubMed]

1995 (2)

T. Kiss, U. Herzog, and U. Leonhardt, Phys. Rev. A 52, 2433 (1995).
[CrossRef] [PubMed]

U. Leonhardt, H. Paul, and G. M. D'Ariano, Phys. Rev. A 52, 4899 (1995).
[CrossRef] [PubMed]

1993 (1)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

1987 (1)

G. S. Agarwal, J. Mod. Opt. 34, 909 (1987).
[CrossRef]

1984 (1)

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

1969 (1)

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969).
[CrossRef]

1965 (1)

F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
[CrossRef]

1964 (2)

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. London 84, 435 (1964).
[CrossRef]

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, J. Mod. Opt. 34, 909 (1987).
[CrossRef]

Agliati, A.

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, J. Mod. Opt. 56, 226 (2009).
[CrossRef]

Allevi, A.

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, J. Mod. Opt. 56, 226 (2009).
[CrossRef]

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

M. Bondani, A. Allevi, and A. Andreoni, arXiv:quant-ph/0810.4026.

Andreoni, A.

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, J. Mod. Opt. 56, 226 (2009).
[CrossRef]

M. Bondani, A. Allevi, and A. Andreoni, arXiv:quant-ph/0810.4026.

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

Arecchi, F. T.

F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
[CrossRef]

Banaszek, K.

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, Phys. Rev. A 60, 674 (1999).
[CrossRef]

K. Banaszek and K. Wódkiewicz, Phys. Rev. Lett. 76, 4344 (1996).
[CrossRef] [PubMed]

Beck, M.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

Bondani, M.

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, J. Mod. Opt. 56, 226 (2009).
[CrossRef]

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

M. Bondani, A. Allevi, and A. Andreoni, arXiv:quant-ph/0810.4026.

Brida, G.

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

Cahill, K. E.

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969).
[CrossRef]

D'Ariano, G. M.

U. Leonhardt, H. Paul, and G. M. D'Ariano, Phys. Rev. A 52, 4899 (1995).
[CrossRef] [PubMed]

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

Genovese, M.

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

Glauber, R. J.

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969).
[CrossRef]

Gramegna, M.

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

Herzog, U.

T. Kiss, U. Herzog, and U. Leonhardt, Phys. Rev. A 52, 2433 (1995).
[CrossRef] [PubMed]

Hillery, M.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Kiss, T.

T. Kiss, U. Herzog, and U. Leonhardt, Phys. Rev. A 52, 2433 (1995).
[CrossRef] [PubMed]

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Krasinski, J. S.

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, Phys. Rev. A 60, 674 (1999).
[CrossRef]

Leonhardt, U.

U. Leonhardt, H. Paul, and G. M. D'Ariano, Phys. Rev. A 52, 4899 (1995).
[CrossRef] [PubMed]

T. Kiss, U. Herzog, and U. Leonhardt, Phys. Rev. A 52, 2433 (1995).
[CrossRef] [PubMed]

Lvovsky, A. I.

A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[CrossRef]

Mandel, L.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. London 84, 435 (1964).
[CrossRef]

O'Connell, R. F.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Olivares, S.

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

Paris, M. G. A.

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

Paul, H.

U. Leonhardt, H. Paul, and G. M. D'Ariano, Phys. Rev. A 52, 4899 (1995).
[CrossRef] [PubMed]

Radzewicz, C.

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, Phys. Rev. A 60, 674 (1999).
[CrossRef]

Raymer, M. G.

A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[CrossRef]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

Scully, M. O.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

Sudarshan, E. C. G.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. London 84, 435 (1964).
[CrossRef]

Traina, P.

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

Vogel, W.

S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528 (1996).
[CrossRef] [PubMed]

Wallentowitz, S.

S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528 (1996).
[CrossRef] [PubMed]

Wigner, E. P.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Wódkiewicz, K.

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, Phys. Rev. A 60, 674 (1999).
[CrossRef]

K. Banaszek and K. Wódkiewicz, Phys. Rev. Lett. 76, 4344 (1996).
[CrossRef] [PubMed]

Wolf, E.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. London 84, 435 (1964).
[CrossRef]

Zambra, G.

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

J. Mod. Opt. (2)

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, J. Mod. Opt. 56, 226 (2009).
[CrossRef]

G. S. Agarwal, J. Mod. Opt. 34, 909 (1987).
[CrossRef]

Phys. Rep. (1)

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Phys. Rev. (2)

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969).
[CrossRef]

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Phys. Rev. A (4)

T. Kiss, U. Herzog, and U. Leonhardt, Phys. Rev. A 52, 2433 (1995).
[CrossRef] [PubMed]

S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528 (1996).
[CrossRef] [PubMed]

U. Leonhardt, H. Paul, and G. M. D'Ariano, Phys. Rev. A 52, 4899 (1995).
[CrossRef] [PubMed]

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, Phys. Rev. A 60, 674 (1999).
[CrossRef]

Phys. Rev. Lett. (3)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

K. Banaszek and K. Wódkiewicz, Phys. Rev. Lett. 76, 4344 (1996).
[CrossRef] [PubMed]

F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
[CrossRef]

Proc. Phys. Soc. London (1)

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. London 84, 435 (1964).
[CrossRef]

Rev. Mod. Phys. (1)

A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[CrossRef]

Other (2)

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, arXiv:quant-ph/0903.0104.

M. Bondani, A. Allevi, and A. Andreoni, arXiv:quant-ph/0810.4026.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Scheme of the experimental setup: HPD, hybrid photodetector; BS, beam splitter; NF, neutral density filter; P, polarizer. Dotted boxes, elements to produce vacuum state (V), single-mode thermal state (T), and phase-averaged coherent state (AV).

Fig. 2
Fig. 2

Vacuum state. (a) Inset, Fano factor, F v , as a function of v ¯ . Main, reconstructed p m , β distributions at different β values (bars) and theoretical curves ( symbols + lines ) ; (b) section of the reconstructed Wigner function (dots) and theoretical surface.

Fig. 3
Fig. 3

(a) Phase-averaged coherent state. Main: reconstructed p m , β distributions for two values of β (bars) and theoretical curves corrected for the overlap parameter (black symbols + line ) or not (white symbols + line ). Inset, section of the reconstructed Wigner function (dots) superimposed to the theoretical surface; (b) same as (a) for the thermal state.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

W ( α ) = 2 π n = 0 ( 1 ) n p n , α ,
W ¯ ( β ) = 2 π ( 1 η ) d 2 β e 2 β β 2 / ( 1 η ) W ( β η )
W ¯ ( β ) = 2 π 1 2 m th + 1 exp ( 2 β β 0 2 2 m th + 1 ) ,
W ¯ ( β ) = 2 π I 0 ( 4 β β 0 ) exp [ 2 ( β 2 + β 0 2 ) ] ,

Metrics