Abstract

We propose simple experiments in cavity quantum electrodynamics leading, for the first time, to the measurement of negative values of the Wigner function of an electromagnetic field. We also show that the realization of a controlled-not gate within the framework of cavity QED is a special case of our proposal, and is equivalent to the measurement of the Wigner function of a one-photon field at the origin of phase space.

© Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. R. J. Glauber, Phys. Rev. 131, 2766 (1963).
    [CrossRef]
  2. R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963).
    [CrossRef]
  3. E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
    [CrossRef]
  4. D. Stoler, Phys. Rev. D 1, 3217 (1970).
    [CrossRef]
  5. D. Stoler, Phys. Rev. D 4, 1925 (1971).
    [CrossRef]
  6. H. P. Yuen, Phys. Lett. A 51, 1 (1976).
    [CrossRef]
  7. H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
    [CrossRef]
  8. C. M. Caves, Phys. Rev. D 23, 1693 (1981).
    [CrossRef]
  9. E. Wigner, Phys. Rev. 40, 749 (1932).
    [CrossRef]
  10. K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969).
    [CrossRef]
  11. K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969).
    [CrossRef]
  12. T. W. Marshall and E. Santos, quant-ph/9711046 (1997).
  13. K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).
    [CrossRef] [PubMed]
  14. U. Leonhardt and H. Paul, Phys. Rev. Lett. 72, 4086 (1994).
    [CrossRef] [PubMed]
  15. G. M. D'Ariano, U. Leonhardt and H. Paul, Phys. Rev. A 52, R1801 (1995).
    [CrossRef]
  16. D. T. Smithey, M. Beck, M. G. Raymer and A. Faridani, Phys. Rev. Lett. 70,1244 (1993).
    [CrossRef] [PubMed]
  17. G. Breitenbach, T. M" uller, S. F. Pereira, J.-Ph. Poizat, S. Schiller and J. Mlynek, J. Opt. Soc. Am. B 12, 2304 (1995).
    [CrossRef]
  18. S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller and J. Mlynek, Phys. Rev. Lett. 77, 2933 (1996).
    [CrossRef] [PubMed]
  19. D. Leibfried D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano and D. J. Wineland, Phys. Rev. Lett. 77, 4281 (1996).
    [CrossRef] [PubMed]
  20. L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997).
    [CrossRef]
  21. L. Davidovich, A. Maali, M. Brune, J. M. Raimond and S. Haroche, Phys. Rev. Lett. 71, 2360 (1993).
    [CrossRef] [PubMed]
  22. L. Davidovich, M. Brune, J. M. Raimond and S. Haroche, Phys. Rev. A 53, 1295 (1996).
    [CrossRef] [PubMed]
  23. M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, S. Haroche, Phys. Rev. Lett. 77, 4887 (1996).
    [CrossRef] [PubMed]
  24. X. Maitre, E. Hagley, G. Nogues and C. Wunderlich, Phys. Rev. Lett. 79, 769 (1997).
    [CrossRef]
  25. L. Davidovich, N. Zagury, M. Brune, J. M. Raimond and S. Haroche, Phys. Rev. A 50, R895 (1994).
    [CrossRef]

Other (25)

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963).
[CrossRef]

E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
[CrossRef]

D. Stoler, Phys. Rev. D 1, 3217 (1970).
[CrossRef]

D. Stoler, Phys. Rev. D 4, 1925 (1971).
[CrossRef]

H. P. Yuen, Phys. Lett. A 51, 1 (1976).
[CrossRef]

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

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

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969).
[CrossRef]

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

T. W. Marshall and E. Santos, quant-ph/9711046 (1997).

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

U. Leonhardt and H. Paul, Phys. Rev. Lett. 72, 4086 (1994).
[CrossRef] [PubMed]

G. M. D'Ariano, U. Leonhardt and H. Paul, Phys. Rev. A 52, R1801 (1995).
[CrossRef]

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

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

S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller and J. Mlynek, Phys. Rev. Lett. 77, 2933 (1996).
[CrossRef] [PubMed]

D. Leibfried D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano and D. J. Wineland, Phys. Rev. Lett. 77, 4281 (1996).
[CrossRef] [PubMed]

L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997).
[CrossRef]

L. Davidovich, A. Maali, M. Brune, J. M. Raimond and S. Haroche, Phys. Rev. Lett. 71, 2360 (1993).
[CrossRef] [PubMed]

L. Davidovich, M. Brune, J. M. Raimond and S. Haroche, Phys. Rev. A 53, 1295 (1996).
[CrossRef] [PubMed]

M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, S. Haroche, Phys. Rev. Lett. 77, 4887 (1996).
[CrossRef] [PubMed]

X. Maitre, E. Hagley, G. Nogues and C. Wunderlich, Phys. Rev. Lett. 79, 769 (1997).
[CrossRef]

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond and S. Haroche, Phys. Rev. A 50, R895 (1994).
[CrossRef]

Supplementary Material (2)

» Media 1: MOV (245 KB)     
» Media 2: MOV (701 KB)     

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 (4)

Figure 1.
Figure 1.

Experimental scheme

Figure 2.
Figure 2.

Atomic level scheme. The transition ie is detuned by δ from the frequency ω of a mode of cavity C, while the transition eg is resonant with the fields in R1 and R2 . State |g〉 is not affected by the field in C.

Figure 3.
Figure 3.

Animation showing the real part of the Wigner function for the “odd cat” (|α〉 - | - α〉)/N -, as ϕ changes from 0 to π. The frame above corresponds to ϕ = Pi/2. [Media 1]

Figure 4.
Figure 4.

Evolution of the Bloch vector (green) of an atom that crosses the proposed setup, interacting with the eletromagnetic field (red) in the two Ramsey zones, and having a dispersive interaction with the field in the superconducting cavity C, when the number of photons of the field in C has a well defined parity. The atom is initially in the state e. As the atom crosses the first Ramsey zone, its Bloch vector is rotated by π/2 around the vector representing the electromagnetic field along the real polarization axis, as shown in (a) and (d). As the atom crosses the cavity C, the Bloch vector rotates around the population axis. If the number of photons in the cavity is odd (b), the Bloch vector ends up pointing towards the opposite direction, and the rotation in the second zone leads the atom back to state |e〉 (c). On the other hand, if the number of photons in C is even, the Bloch vector turns by an integer multiple of 2π, so its direction does not change (d). The second Ramsey zone then brings the atom to |g〉 (f).

Equations (7)

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

W ( x , p ) = 1 2 π ħ e ipy / ħ x y 2 ρ ̂ x + y 2 dxdp .
W ( z , z * ) = 2 Tr [ ρ ̂ D ̂ ( z , z * ) e i π a ̂ a ̂ D ̂ 1 ( z , z * ) ] ,
Δ P = P e P g = e { e Tr [ D ̂ ( z , z * ) ρ ̂ D ̂ 1 ( z , z * ) e a a ] } .
Δ P = P e P g = W ( z , z * ) / 2 .
W ( z , z * , s ) = e z ξ * z * ξ e s ξ 2 / 2 Tr [ ρ ̂ D ̂ ( ξ , ξ * ) ] π 1 d 2 ξ .
W ( z , z * ϕ ) = 2 i e / 2 sin ϕ 2 Tr [ D ̂ ( z , z * ) ρ ̂ D ̂ ( z , z * ) e a ̂ a ̂ ] .
i W ( x , y , τ ) τ = 1 8 ( 2 x 2 + 2 y 2 ) W ( x , y , τ ) ,

Metrics