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.

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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. Muller, 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, E. Giacobino, Laser Spectroscopy XIII international conference, 305 (World Scientific, 1997).
  8. M. J. Collett, 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).

Other (12)

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

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

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]

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

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

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

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

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

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

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

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).

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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)

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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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