Abstract

We demonstrate a unique atomic interferometer employing collective radiation from the atoms to measure the atomic phase. Atomic coherence is detected by interference of a continuous free-induction decay field and a continuous photon-echo field that originate in spatially separated regions of an atomic beam. Using this system, we measure the 180° phase shift of a J = 0 atomic state that undergoes a 2π-rotation of the Bloch vector or a two-level optical transition.

© 1995 Optical Society of America

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References

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  1. For a recent review of geometric phase measurements seeJ. W. Zwanziger, M. Koenig, A. Pines, Annu. Rev. Phys. Chem. 41, 601 (1990).
    [CrossRef]
  2. R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
    [CrossRef]
  3. R. Beach, S. R. Hartmann, R. Friedberg, Phys. Rev. A 25, 2658 (1982); R. Friedberg, S. R. Hartmann, Phys. Rev. A. 48, 1446 (1993).
    [CrossRef] [PubMed]
  4. C. Schnurr, K. D. Stokes, G. R. Welch, J. E. Thomas, Opt. Lett. 15, 1997 (1990).
    [CrossRef]
  5. V. P. Chebotayev, B. Ya. Dubetsky, A. P. Kasanantsev, V. P. Yaklovlev, J. Opt. Soc. Am. B 2, 1791 (1985).
    [CrossRef]
  6. Ch. J. Bordé, Phys. Lett. A 140, 10 (1989).
    [CrossRef]
  7. M. Kasevich, S. Chu, Phys. Rev. Let. 67, 181 (1991).
    [CrossRef]
  8. F. Riehle, A. Witte, Th. Kisters, J. Helmcke, Appl. Phys. B 54, 333 (1992).
    [CrossRef]
  9. K. D. Stokes, C. Schnurr, J. Gardner, M. Marable, S. Shaw, M. Goforth, D. E. Holmgren, J. E. Thomas, Opt. Lett. 14, 1324 (1989).
    [CrossRef] [PubMed]
  10. K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Chap. 6, p. 276.

1992 (1)

F. Riehle, A. Witte, Th. Kisters, J. Helmcke, Appl. Phys. B 54, 333 (1992).
[CrossRef]

1991 (1)

M. Kasevich, S. Chu, Phys. Rev. Let. 67, 181 (1991).
[CrossRef]

1990 (2)

For a recent review of geometric phase measurements seeJ. W. Zwanziger, M. Koenig, A. Pines, Annu. Rev. Phys. Chem. 41, 601 (1990).
[CrossRef]

C. Schnurr, K. D. Stokes, G. R. Welch, J. E. Thomas, Opt. Lett. 15, 1997 (1990).
[CrossRef]

1989 (2)

1985 (1)

1982 (1)

R. Beach, S. R. Hartmann, R. Friedberg, Phys. Rev. A 25, 2658 (1982); R. Friedberg, S. R. Hartmann, Phys. Rev. A. 48, 1446 (1993).
[CrossRef] [PubMed]

1957 (1)

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Beach, R.

R. Beach, S. R. Hartmann, R. Friedberg, Phys. Rev. A 25, 2658 (1982); R. Friedberg, S. R. Hartmann, Phys. Rev. A. 48, 1446 (1993).
[CrossRef] [PubMed]

Bordé, Ch. J.

Ch. J. Bordé, Phys. Lett. A 140, 10 (1989).
[CrossRef]

Chebotayev, V. P.

Chu, S.

M. Kasevich, S. Chu, Phys. Rev. Let. 67, 181 (1991).
[CrossRef]

Dubetsky, B. Ya.

Feynman, R. P.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Friedberg, R.

R. Beach, S. R. Hartmann, R. Friedberg, Phys. Rev. A 25, 2658 (1982); R. Friedberg, S. R. Hartmann, Phys. Rev. A. 48, 1446 (1993).
[CrossRef] [PubMed]

Gardner, J.

Goforth, M.

Gottfried, K.

K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Chap. 6, p. 276.

Hartmann, S. R.

R. Beach, S. R. Hartmann, R. Friedberg, Phys. Rev. A 25, 2658 (1982); R. Friedberg, S. R. Hartmann, Phys. Rev. A. 48, 1446 (1993).
[CrossRef] [PubMed]

Hellwarth, R. W.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Helmcke, J.

F. Riehle, A. Witte, Th. Kisters, J. Helmcke, Appl. Phys. B 54, 333 (1992).
[CrossRef]

Holmgren, D. E.

Kasanantsev, A. P.

Kasevich, M.

M. Kasevich, S. Chu, Phys. Rev. Let. 67, 181 (1991).
[CrossRef]

Kisters, Th.

F. Riehle, A. Witte, Th. Kisters, J. Helmcke, Appl. Phys. B 54, 333 (1992).
[CrossRef]

Koenig, M.

For a recent review of geometric phase measurements seeJ. W. Zwanziger, M. Koenig, A. Pines, Annu. Rev. Phys. Chem. 41, 601 (1990).
[CrossRef]

Marable, M.

Pines, A.

For a recent review of geometric phase measurements seeJ. W. Zwanziger, M. Koenig, A. Pines, Annu. Rev. Phys. Chem. 41, 601 (1990).
[CrossRef]

Riehle, F.

F. Riehle, A. Witte, Th. Kisters, J. Helmcke, Appl. Phys. B 54, 333 (1992).
[CrossRef]

Schnurr, C.

Shaw, S.

Stokes, K. D.

Thomas, J. E.

Vernon, F. L.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Welch, G. R.

C. Schnurr, K. D. Stokes, G. R. Welch, J. E. Thomas, Opt. Lett. 15, 1997 (1990).
[CrossRef]

Witte, A.

F. Riehle, A. Witte, Th. Kisters, J. Helmcke, Appl. Phys. B 54, 333 (1992).
[CrossRef]

Yaklovlev, V. P.

Zwanziger, J. W.

For a recent review of geometric phase measurements seeJ. W. Zwanziger, M. Koenig, A. Pines, Annu. Rev. Phys. Chem. 41, 601 (1990).
[CrossRef]

Annu. Rev. Phys. Chem. (1)

For a recent review of geometric phase measurements seeJ. W. Zwanziger, M. Koenig, A. Pines, Annu. Rev. Phys. Chem. 41, 601 (1990).
[CrossRef]

Appl. Phys. B (1)

F. Riehle, A. Witte, Th. Kisters, J. Helmcke, Appl. Phys. B 54, 333 (1992).
[CrossRef]

J. Appl. Phys. (1)

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

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

Opt. Lett. (2)

Phys. Lett. A (1)

Ch. J. Bordé, Phys. Lett. A 140, 10 (1989).
[CrossRef]

Phys. Rev. A (1)

R. Beach, S. R. Hartmann, R. Friedberg, Phys. Rev. A 25, 2658 (1982); R. Friedberg, S. R. Hartmann, Phys. Rev. A. 48, 1446 (1993).
[CrossRef] [PubMed]

Phys. Rev. Let. (1)

M. Kasevich, S. Chu, Phys. Rev. Let. 67, 181 (1991).
[CrossRef]

Other (1)

K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Chap. 6, p. 276.

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Figures (4)

Fig. 1
Fig. 1

Schematic of an atom interferometer using collective radiation. (a) Atoms traverse two continuous x-polarized laser fields (polarized with P1) of frequency ω, resonant with the |g〉 → |e〉 transition (see Fig. 2). These fields create a rephasing optical polarization that radiates a coherent echo field. The interference between the continuous FID field radiated from the π/2 region and the continuous echo field is measured in the Fourier plane of the lens system L1, L2 with diode array D to determine the phase of the atomic coherence. (b) Detail of the atom interferometer shown as the parallelogram in (a).

Fig. 2
Fig. 2

Energy-level diagram. Continuous photon echoes are excited on the J = 0 → J = 1, M = 1 transition by use of x-polarized fields of frequency ω. A 2π rotation of the J = 0, M = 0 state is accomplished with a z-polarized field of frequency ω′, resonant with the J = 0 → J = 1, M = 0 transition.

Fig. 3
Fig. 3

Fringes obtained in the Fourier plane for interference between continuous FID and echo fields. The filled circles are the experimental data plotted versus position on the diode array, and the solid curve is the fit obtained with Eq. (2).

Fig. 4
Fig. 4

Phase shift arising from a 2π rotation of the Bloch vector for a two-level optical transition coupled to the J = 0 ground state. Experimental data are shown as filled circles. The solid (dotted) curves are fits to the data with the z-polarized laser field off (on), producing a 0 (2π) pulse in the atom frame. The 180° phase shift between the solid and dotted curves is apparent.

Equations (2)

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E f ( p ) d y exp ( - i p y ) [ P FID ( y ) + P echo ( y - 2 y 21 ) ] .
I f ( p ) = I FID ( p ) + I echo ( p ) + 2 γ 21 [ I FID ( p ) I echo ( p ) ] 1 / 2 cos ( 2 p y 21 + ϕ ) ,

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