Abstract

Optical homodyne tomography (OHT) is a tool that allows the reconstruction of Wigner functions for each detection frequency of a propagating optical beam. It can measure probability distribution functions (PDF’s) of the field amplitude for any given quadrature of interest. We demonstrate OHT for a range of classical optical states with constant and time varying modulations and show the advantage of OHT over conventional homodyne detection. The OHT simultaneously determines the signal to noise ratio in both amplitude and phase quadratures. We show that highly non-Gaussian Wigner functions can be obtained from incoherent superpositions of optical states.

© 1998 Optical Society of America

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References

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  1. E. P. Wigner, Phys. Rev. 40, 749 (1932).
    [Crossref]
  2. K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).
    [Crossref] [PubMed]
  3. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
    [Crossref] [PubMed]
  4. M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. 18, 1259 (1993).
    [Crossref] [PubMed]
  5. G. Breitenbach, T. Müller, S. F. Pereira, J. -Ph. Poizat, S. Schiller, and J. Mlynek, J. Opt. Soc. Am. B 12, 2304 (1995).
    [Crossref]
  6. G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997).
    [Crossref]
  7. T. Coudreau, A. Z. Khoury, and E. Giacobino, Laser Spectroscopy XIII international conference, 305 (World Scientific, 1997).
  8. M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
    [Crossref]
  9. H. P. Yuen and V. W. S. Chen, Opt. Lett. 18, 177 (1983).
    [Crossref]
  10. B. Schumaker, Opt. Lett. 19, 189 (1984).
    [Crossref]
  11. Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150 (1997).
    [Crossref]
  12. H.-A. Bachor, M. Taubman, A. G. White, T. Ralph, and D. E. McClalland, Proceedings of the Fourth International Conference on Squeezed States and Uncertainty Relations, 381(NASA, Goddard Space Flight Center, Greenbelt, Maryland:1996).
  13. T. Felbinger, S. Schiller, and J. Mlynek, Phys. Rev. Lett. 80, 492 (1998).
    [Crossref]
  14. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press:1997).

1998 (1)

T. Felbinger, S. Schiller, and J. Mlynek, Phys. Rev. Lett. 80, 492 (1998).
[Crossref]

1997 (2)

Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150 (1997).
[Crossref]

G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997).
[Crossref]

1995 (1)

1993 (2)

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

M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. 18, 1259 (1993).
[Crossref] [PubMed]

1989 (1)

K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).
[Crossref] [PubMed]

1984 (2)

B. Schumaker, Opt. Lett. 19, 189 (1984).
[Crossref]

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[Crossref]

1983 (1)

H. P. Yuen and V. W. S. Chen, Opt. Lett. 18, 177 (1983).
[Crossref]

1932 (1)

E. P. Wigner, Phys. Rev. 40, 749 (1932).
[Crossref]

Bachor, H.-A.

H.-A. Bachor, M. Taubman, A. G. White, T. Ralph, and D. E. McClalland, Proceedings of the Fourth International Conference on Squeezed States and Uncertainty Relations, 381(NASA, Goddard Space Flight Center, Greenbelt, Maryland:1996).

Beck, M.

M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. 18, 1259 (1993).
[Crossref] [PubMed]

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

Breitenbach, G.

Chen, V. W. S.

H. P. Yuen and V. W. S. Chen, Opt. Lett. 18, 177 (1983).
[Crossref]

Collett, M. J.

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[Crossref]

Cooper, J.

Coudreau, T.

T. Coudreau, A. Z. Khoury, and E. Giacobino, Laser Spectroscopy XIII international conference, 305 (World Scientific, 1997).

Faridani, A.

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

Felbinger, T.

T. Felbinger, S. Schiller, and J. Mlynek, Phys. Rev. Lett. 80, 492 (1998).
[Crossref]

Gardiner, C. W.

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[Crossref]

Giacobino, E.

T. Coudreau, A. Z. Khoury, and E. Giacobino, Laser Spectroscopy XIII international conference, 305 (World Scientific, 1997).

Khoury, A. Z.

T. Coudreau, A. Z. Khoury, and E. Giacobino, Laser Spectroscopy XIII international conference, 305 (World Scientific, 1997).

Kurtsiefer, Ch.

Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150 (1997).
[Crossref]

Leonhardt, U.

U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press:1997).

McClalland, D. E.

H.-A. Bachor, M. Taubman, A. G. White, T. Ralph, and D. E. McClalland, Proceedings of the Fourth International Conference on Squeezed States and Uncertainty Relations, 381(NASA, Goddard Space Flight Center, Greenbelt, Maryland:1996).

Mlynek, J.

T. Felbinger, S. Schiller, and J. Mlynek, Phys. Rev. Lett. 80, 492 (1998).
[Crossref]

Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150 (1997).
[Crossref]

G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997).
[Crossref]

G. Breitenbach, T. Müller, S. F. Pereira, J. -Ph. Poizat, S. Schiller, and J. Mlynek, J. Opt. Soc. Am. B 12, 2304 (1995).
[Crossref]

Müller, T.

Pereira, S. F.

Pfau, T.

Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150 (1997).
[Crossref]

Poizat, J. -Ph.

Ralph, T.

H.-A. Bachor, M. Taubman, A. G. White, T. Ralph, and D. E. McClalland, Proceedings of the Fourth International Conference on Squeezed States and Uncertainty Relations, 381(NASA, Goddard Space Flight Center, Greenbelt, Maryland:1996).

Raymer, M. G.

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

M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. 18, 1259 (1993).
[Crossref] [PubMed]

Risken, H.

K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).
[Crossref] [PubMed]

Schiller, S.

T. Felbinger, S. Schiller, and J. Mlynek, Phys. Rev. Lett. 80, 492 (1998).
[Crossref]

G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997).
[Crossref]

G. Breitenbach, T. Müller, S. F. Pereira, J. -Ph. Poizat, S. Schiller, and J. Mlynek, J. Opt. Soc. Am. B 12, 2304 (1995).
[Crossref]

Schumaker, B.

Smithey, D. T.

M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. 18, 1259 (1993).
[Crossref] [PubMed]

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

Taubman, M.

H.-A. Bachor, M. Taubman, A. G. White, T. Ralph, and D. E. McClalland, Proceedings of the Fourth International Conference on Squeezed States and Uncertainty Relations, 381(NASA, Goddard Space Flight Center, Greenbelt, Maryland:1996).

Vogel, K.

K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).
[Crossref] [PubMed]

White, A. G.

H.-A. Bachor, M. Taubman, A. G. White, T. Ralph, and D. E. McClalland, Proceedings of the Fourth International Conference on Squeezed States and Uncertainty Relations, 381(NASA, Goddard Space Flight Center, Greenbelt, Maryland:1996).

Wigner, E. P.

E. P. Wigner, Phys. Rev. 40, 749 (1932).
[Crossref]

Yuen, H. P.

H. P. Yuen and V. W. S. Chen, Opt. Lett. 18, 177 (1983).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature (2)

G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997).
[Crossref]

Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150 (1997).
[Crossref]

Opt. Lett. (3)

Phys. Rev. (1)

E. P. Wigner, Phys. Rev. 40, 749 (1932).
[Crossref]

Phys. Rev. A (2)

K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).
[Crossref] [PubMed]

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[Crossref]

Phys. Rev. Lett. (2)

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

T. Felbinger, S. Schiller, and J. Mlynek, Phys. Rev. Lett. 80, 492 (1998).
[Crossref]

Other (3)

U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press:1997).

H.-A. Bachor, M. Taubman, A. G. White, T. Ralph, and D. E. McClalland, Proceedings of the Fourth International Conference on Squeezed States and Uncertainty Relations, 381(NASA, Goddard Space Flight Center, Greenbelt, Maryland:1996).

T. Coudreau, A. Z. Khoury, and E. Giacobino, Laser Spectroscopy XIII international conference, 305 (World Scientific, 1997).

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Figures (5)

Figure 1.
Figure 1.

Experimental setup. EOM: Electro-optic modulator; ND: neu- tral-density filter; PZT: Piezo; PD: photodetector; LPF: low-pass filter; VCA: Voltage controlled attenuator; SA: Spectrum Analyser.

Figure 2.
Figure 2.

The data processing of a typical OHT experiment. (a) One segment of the measured time trace of quadrature amplitude value by scanning the PZT; (b)The PDF after binning the time trace; (c) The reconstructed Wigner function. a.u.: arbitrary unit.

Figure 3.
Figure 3.

Phase modulation with variable depth of modulation. The four Wigner function contours shown in one figure. The contours change from symmetric to non-symmetric when the modulation is increased.

Figure 4.
Figure 4.

The reconstructed Wigner function and contour plot of a classical mixture state. (a) Wigner function; (b) contour plot.

Figure 5.
Figure 5.

The reconstructed Wigner function and contour plot by using a phase-unlocked scheme. (a) Wigner function; (b) contour plot.

Equations (7)

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

P i ( Ω d , θ ) P LO V Ω d ( θ )
a ̂ ( t ) = [ E 0 + δ a ̂ ( t ) ] ( 1 + β e i Ω m t β e i Ω m t )
E 0 2 E 0 sin ( Ω m t ) + δ a ̂ ( t ) ,
i ( θ , t ) E 0 cos ( θ ) 2 β E 0 sin ( Ω m t ) sin ( θ ) + δ X ̂ ( θ , t ) .
i Ω m ( θ , ψ ; t ) = β E 0 cos ( ψ ) sin ( θ ) cos ( ψ ) δ X ci ( θ , Ω m ; t ) + sin ( ψ ) δ X cr ( θ , Ω m ; t )
i Ω m ( θ , 0 ; t ) = β E 0 sin ( θ ) δ X ci ( θ , Ω m ; t ) .
W Ω d ( x 1 , x 2 ) = 1 4 π 2 + + 0 π w Ω d ( x , θ ) exp [ ( x x 1 cos θ x 2 sin θ ) ] η dxdηdθ

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