Abstract

Quantum reflection of neutral atoms from a periodic far-detuned dipole potential is proposed and analyzed. This periodic atom mirror relies on constructive interference of small reflections from each cell to yield a high reflection coefficient even for very weak potentials. The reflected energy spectrum is calculated as a function of the potential height and the number of cells for both positive and negative potentials, and its relation to the reflection from one potential cell is derived. Two ways of increasing the reflection bandwidth, one based on changing the envelope of the potential and the other on changing its period gradually (chirp), are investigated. The phase of the reflected atoms and its dependence on experimental parameters are calculated, as well as the interaction time of the atoms with the potential and the spontaneous-emission rate during the reflection. Finally, it is shown that atoms with velocities of a few tens mm/s can be coherently reflected from a negative periodic potential with readily available laser diodes.

© 1998 Optical Society of America

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  1. V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnikov, and A. I. Sidorov, “Quantum-state-selective mirror of atoms by laser light,” Phys. Rev. Lett. 60, 2137 (1988).
    [Crossref] [PubMed]
  2. M. A. Kasevich, D. S. Weiss, and S. Chu, “Normal-incidence reflection of slow atoms from an optical evanescent wave,” Opt. Lett. 15, 607 (1990).
    [Crossref] [PubMed]
  3. C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
    [Crossref] [PubMed]
  4. C. Henkel, C. I. Westbrook, and A. Aspect, “Quantum reflection: atomic matter–wave optics in an exponential potential,” J. Opt. Soc. Am. B 13, 233 (1996).
    [Crossref]
  5. P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J. Y. Courtois, and G. Grynberg, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861 (1992).
    [Crossref] [PubMed]
  6. P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49 (1992).
    [Crossref] [PubMed]
  7. A. Hemmerich and T. W. Hansch, “2-dimensional atomic crystal bound by light,” Phys. Rev. Lett. 70, 410 (1993).
    [Crossref] [PubMed]
  8. G. Grynberg, B. Lounis, P. Verkerk, J. Y. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in 2-dimensional and 3-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249 (1993).
    [Crossref] [PubMed]
  9. G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
    [Crossref] [PubMed]
  10. M. Weidemuller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, “Bragg diffraction in an atomic lattice bound by light,” Phys. Rev. Lett. 75, 4583 (1995).
    [Crossref] [PubMed]
  11. M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508 (1996).
    [Crossref] [PubMed]
  12. S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, “Observation of atomic Wannier–Stark ladders in an accelerating optical potential,” Phys. Rev. Lett. 76, 4512 (1996).
    [Crossref] [PubMed]
  13. J. P. Gordon and A. Ashkin, “Motion of atoms in a radiation trap,” Phys. Rev. A 21, 1606 (1980).
    [Crossref]
  14. The mirror can also be formed in the z direction, such that the atoms move vertically, in a trampoline configuration. However, in this case the gravitation potential must be included in the analysis, which is not treated here.
  15. N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University, London, 1947), Chap. 2.
  16. A. N. Khondker, M. Rezwan Khan, and A. F. M. Anwar, “Transmission line analogy of resonance tunneling phenomena: the generalized impedance concept,” J. Appl. Phys. 63, 5191 (1988).
    [Crossref]
  17. L. S. Letokhov and V. G. Minogin, “Trapping and storage of atoms in a laser field,” Appl. Phys. 17, 99 (1978).
    [Crossref]
  18. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976), Chap. 9.
  19. [(∊-〈V〉)]1/2 was used in the argument of the sine (instead of ∊), which leads to a better accuracy in the phase.
  20. A truncated half-Gaussian envelope of the shape exp(-ξ2/2σ2) for 0≤ξ≤NΛ, and 0 elsewhere, was used. For a full-Gaussian envelope the reflected amplitudes from regions of equal height in both sides of the potential interfere, which adds narrow fringes to the reflected energy spectrum.
  21. Note that for spherical wave fronts the interference fringes are not perfectly parallel. With the parameters given in the text, the angle of the fringes will change by ±3° over a 1-mm interaction range, which might call for a two-dimensional treatment, depending on the size of the atomic cloud.
  22. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, 2nd ed. (Wiley, New York, 1977), Vol. 1, Chap. 1.
  23. In the case of total reflection the reflection coefficient can be written as r=exp(iϕ), and the probability distribution to the left of the potential is then |ψ(ξ)|2=a cos2(kξ-ϕ/2). The normalization used is a=1.
  24. N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, “Long atomic coherence times in an optical dipole trap,” Phys. Rev. Lett. 74, 1311 (1995).
    [Crossref] [PubMed]
  25. T. Pfau, Ch. Kurtsiefer, C. S. Adams, M. Sigel, and J. Mlynek, “Magneto-optical beam splitter for atoms,” Phys. Rev. Lett. 71, 3427 (1993).
    [Crossref] [PubMed]
  26. K. S. Johnson, A. Chu, T. W. Lynn, K. K. Berggren, M. S. Shahriar, and M. Prentiss, “Demonstration of a nonmagnetic blazed-grating atomic beam splitter,” Opt. Lett. 20, 1310 (1995).
    [Crossref] [PubMed]

1996 (3)

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508 (1996).
[Crossref] [PubMed]

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, “Observation of atomic Wannier–Stark ladders in an accelerating optical potential,” Phys. Rev. Lett. 76, 4512 (1996).
[Crossref] [PubMed]

C. Henkel, C. I. Westbrook, and A. Aspect, “Quantum reflection: atomic matter–wave optics in an exponential potential,” J. Opt. Soc. Am. B 13, 233 (1996).
[Crossref]

1995 (4)

K. S. Johnson, A. Chu, T. W. Lynn, K. K. Berggren, M. S. Shahriar, and M. Prentiss, “Demonstration of a nonmagnetic blazed-grating atomic beam splitter,” Opt. Lett. 20, 1310 (1995).
[Crossref] [PubMed]

N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, “Long atomic coherence times in an optical dipole trap,” Phys. Rev. Lett. 74, 1311 (1995).
[Crossref] [PubMed]

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

M. Weidemuller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, “Bragg diffraction in an atomic lattice bound by light,” Phys. Rev. Lett. 75, 4583 (1995).
[Crossref] [PubMed]

1993 (4)

A. Hemmerich and T. W. Hansch, “2-dimensional atomic crystal bound by light,” Phys. Rev. Lett. 70, 410 (1993).
[Crossref] [PubMed]

G. Grynberg, B. Lounis, P. Verkerk, J. Y. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in 2-dimensional and 3-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249 (1993).
[Crossref] [PubMed]

T. Pfau, Ch. Kurtsiefer, C. S. Adams, M. Sigel, and J. Mlynek, “Magneto-optical beam splitter for atoms,” Phys. Rev. Lett. 71, 3427 (1993).
[Crossref] [PubMed]

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[Crossref] [PubMed]

1992 (2)

P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J. Y. Courtois, and G. Grynberg, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861 (1992).
[Crossref] [PubMed]

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49 (1992).
[Crossref] [PubMed]

1990 (1)

1988 (2)

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnikov, and A. I. Sidorov, “Quantum-state-selective mirror of atoms by laser light,” Phys. Rev. Lett. 60, 2137 (1988).
[Crossref] [PubMed]

A. N. Khondker, M. Rezwan Khan, and A. F. M. Anwar, “Transmission line analogy of resonance tunneling phenomena: the generalized impedance concept,” J. Appl. Phys. 63, 5191 (1988).
[Crossref]

1980 (1)

J. P. Gordon and A. Ashkin, “Motion of atoms in a radiation trap,” Phys. Rev. A 21, 1606 (1980).
[Crossref]

1978 (1)

L. S. Letokhov and V. G. Minogin, “Trapping and storage of atoms in a laser field,” Appl. Phys. 17, 99 (1978).
[Crossref]

Adams, C. S.

N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, “Long atomic coherence times in an optical dipole trap,” Phys. Rev. Lett. 74, 1311 (1995).
[Crossref] [PubMed]

T. Pfau, Ch. Kurtsiefer, C. S. Adams, M. Sigel, and J. Mlynek, “Magneto-optical beam splitter for atoms,” Phys. Rev. Lett. 71, 3427 (1993).
[Crossref] [PubMed]

Aminoff, C. G.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[Crossref] [PubMed]

Anwar, A. F. M.

A. N. Khondker, M. Rezwan Khan, and A. F. M. Anwar, “Transmission line analogy of resonance tunneling phenomena: the generalized impedance concept,” J. Appl. Phys. 63, 5191 (1988).
[Crossref]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976), Chap. 9.

Ashkin, A.

J. P. Gordon and A. Ashkin, “Motion of atoms in a radiation trap,” Phys. Rev. A 21, 1606 (1980).
[Crossref]

Aspect, A.

Balykin, V. I.

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnikov, and A. I. Sidorov, “Quantum-state-selective mirror of atoms by laser light,” Phys. Rev. Lett. 60, 2137 (1988).
[Crossref] [PubMed]

Ben Dahan, M.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508 (1996).
[Crossref] [PubMed]

Berggren, K. K.

Bharucha, C. F.

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, “Observation of atomic Wannier–Stark ladders in an accelerating optical potential,” Phys. Rev. Lett. 76, 4512 (1996).
[Crossref] [PubMed]

Birkl, G.

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

Bouyer, P.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[Crossref] [PubMed]

Castin, Y.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508 (1996).
[Crossref] [PubMed]

Chu, A.

Chu, S.

N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, “Long atomic coherence times in an optical dipole trap,” Phys. Rev. Lett. 74, 1311 (1995).
[Crossref] [PubMed]

M. A. Kasevich, D. S. Weiss, and S. Chu, “Normal-incidence reflection of slow atoms from an optical evanescent wave,” Opt. Lett. 15, 607 (1990).
[Crossref] [PubMed]

Cohen-Tannoudji, C.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[Crossref] [PubMed]

P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J. Y. Courtois, and G. Grynberg, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861 (1992).
[Crossref] [PubMed]

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, 2nd ed. (Wiley, New York, 1977), Vol. 1, Chap. 1.

Courtois, J. Y.

G. Grynberg, B. Lounis, P. Verkerk, J. Y. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in 2-dimensional and 3-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249 (1993).
[Crossref] [PubMed]

P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J. Y. Courtois, and G. Grynberg, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861 (1992).
[Crossref] [PubMed]

Dalibard, J.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[Crossref] [PubMed]

Davidson, N.

N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, “Long atomic coherence times in an optical dipole trap,” Phys. Rev. Lett. 74, 1311 (1995).
[Crossref] [PubMed]

Desbiolles, P.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[Crossref] [PubMed]

Deutsch, I. H.

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, 2nd ed. (Wiley, New York, 1977), Vol. 1, Chap. 1.

Esslinger, T.

M. Weidemuller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, “Bragg diffraction in an atomic lattice bound by light,” Phys. Rev. Lett. 75, 4583 (1995).
[Crossref] [PubMed]

Gatzke, M.

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

Gerz, C.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49 (1992).
[Crossref] [PubMed]

Gordon, J. P.

J. P. Gordon and A. Ashkin, “Motion of atoms in a radiation trap,” Phys. Rev. A 21, 1606 (1980).
[Crossref]

Gorlitz, A.

M. Weidemuller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, “Bragg diffraction in an atomic lattice bound by light,” Phys. Rev. Lett. 75, 4583 (1995).
[Crossref] [PubMed]

Grynberg, G.

G. Grynberg, B. Lounis, P. Verkerk, J. Y. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in 2-dimensional and 3-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249 (1993).
[Crossref] [PubMed]

P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J. Y. Courtois, and G. Grynberg, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861 (1992).
[Crossref] [PubMed]

Hansch, T. W.

M. Weidemuller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, “Bragg diffraction in an atomic lattice bound by light,” Phys. Rev. Lett. 75, 4583 (1995).
[Crossref] [PubMed]

A. Hemmerich and T. W. Hansch, “2-dimensional atomic crystal bound by light,” Phys. Rev. Lett. 70, 410 (1993).
[Crossref] [PubMed]

Hemmerich, A.

M. Weidemuller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, “Bragg diffraction in an atomic lattice bound by light,” Phys. Rev. Lett. 75, 4583 (1995).
[Crossref] [PubMed]

A. Hemmerich and T. W. Hansch, “2-dimensional atomic crystal bound by light,” Phys. Rev. Lett. 70, 410 (1993).
[Crossref] [PubMed]

Henkel, C.

Jessen, P. S.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49 (1992).
[Crossref] [PubMed]

Johnson, K. S.

Kasevich, M.

N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, “Long atomic coherence times in an optical dipole trap,” Phys. Rev. Lett. 74, 1311 (1995).
[Crossref] [PubMed]

Kasevich, M. A.

Khondker, A. N.

A. N. Khondker, M. Rezwan Khan, and A. F. M. Anwar, “Transmission line analogy of resonance tunneling phenomena: the generalized impedance concept,” J. Appl. Phys. 63, 5191 (1988).
[Crossref]

Kurtsiefer, Ch.

T. Pfau, Ch. Kurtsiefer, C. S. Adams, M. Sigel, and J. Mlynek, “Magneto-optical beam splitter for atoms,” Phys. Rev. Lett. 71, 3427 (1993).
[Crossref] [PubMed]

Laloe, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, 2nd ed. (Wiley, New York, 1977), Vol. 1, Chap. 1.

Lee, H. J.

N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, “Long atomic coherence times in an optical dipole trap,” Phys. Rev. Lett. 74, 1311 (1995).
[Crossref] [PubMed]

Letokhov, L. S.

L. S. Letokhov and V. G. Minogin, “Trapping and storage of atoms in a laser field,” Appl. Phys. 17, 99 (1978).
[Crossref]

Letokhov, V. S.

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnikov, and A. I. Sidorov, “Quantum-state-selective mirror of atoms by laser light,” Phys. Rev. Lett. 60, 2137 (1988).
[Crossref] [PubMed]

Lett, P. D.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49 (1992).
[Crossref] [PubMed]

Lounis, B.

G. Grynberg, B. Lounis, P. Verkerk, J. Y. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in 2-dimensional and 3-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249 (1993).
[Crossref] [PubMed]

P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J. Y. Courtois, and G. Grynberg, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861 (1992).
[Crossref] [PubMed]

Lynn, T. W.

Madison, K. W.

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, “Observation of atomic Wannier–Stark ladders in an accelerating optical potential,” Phys. Rev. Lett. 76, 4512 (1996).
[Crossref] [PubMed]

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University, London, 1947), Chap. 2.

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976), Chap. 9.

Minogin, V. G.

L. S. Letokhov and V. G. Minogin, “Trapping and storage of atoms in a laser field,” Appl. Phys. 17, 99 (1978).
[Crossref]

Mlynek, J.

T. Pfau, Ch. Kurtsiefer, C. S. Adams, M. Sigel, and J. Mlynek, “Magneto-optical beam splitter for atoms,” Phys. Rev. Lett. 71, 3427 (1993).
[Crossref] [PubMed]

Niu, Q.

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, “Observation of atomic Wannier–Stark ladders in an accelerating optical potential,” Phys. Rev. Lett. 76, 4512 (1996).
[Crossref] [PubMed]

Ovchinnikov, Yu. B.

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnikov, and A. I. Sidorov, “Quantum-state-selective mirror of atoms by laser light,” Phys. Rev. Lett. 60, 2137 (1988).
[Crossref] [PubMed]

Peik, E.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508 (1996).
[Crossref] [PubMed]

Pfau, T.

T. Pfau, Ch. Kurtsiefer, C. S. Adams, M. Sigel, and J. Mlynek, “Magneto-optical beam splitter for atoms,” Phys. Rev. Lett. 71, 3427 (1993).
[Crossref] [PubMed]

Phillips, W. D.

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49 (1992).
[Crossref] [PubMed]

Prentiss, M.

Raizen, M. G.

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, “Observation of atomic Wannier–Stark ladders in an accelerating optical potential,” Phys. Rev. Lett. 76, 4512 (1996).
[Crossref] [PubMed]

Reichel, J.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508 (1996).
[Crossref] [PubMed]

Rezwan Khan, M.

A. N. Khondker, M. Rezwan Khan, and A. F. M. Anwar, “Transmission line analogy of resonance tunneling phenomena: the generalized impedance concept,” J. Appl. Phys. 63, 5191 (1988).
[Crossref]

Rolston, S. L.

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49 (1992).
[Crossref] [PubMed]

Salomon, C.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508 (1996).
[Crossref] [PubMed]

G. Grynberg, B. Lounis, P. Verkerk, J. Y. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in 2-dimensional and 3-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249 (1993).
[Crossref] [PubMed]

P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J. Y. Courtois, and G. Grynberg, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861 (1992).
[Crossref] [PubMed]

Shahriar, M. S.

Sidorov, A. I.

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnikov, and A. I. Sidorov, “Quantum-state-selective mirror of atoms by laser light,” Phys. Rev. Lett. 60, 2137 (1988).
[Crossref] [PubMed]

Sigel, M.

T. Pfau, Ch. Kurtsiefer, C. S. Adams, M. Sigel, and J. Mlynek, “Magneto-optical beam splitter for atoms,” Phys. Rev. Lett. 71, 3427 (1993).
[Crossref] [PubMed]

Spreeuw, R. J. C.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49 (1992).
[Crossref] [PubMed]

Steane, A. M.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[Crossref] [PubMed]

Verkerk, P.

G. Grynberg, B. Lounis, P. Verkerk, J. Y. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in 2-dimensional and 3-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249 (1993).
[Crossref] [PubMed]

P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J. Y. Courtois, and G. Grynberg, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861 (1992).
[Crossref] [PubMed]

Weidemuller, M.

M. Weidemuller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, “Bragg diffraction in an atomic lattice bound by light,” Phys. Rev. Lett. 75, 4583 (1995).
[Crossref] [PubMed]

Weiss, D. S.

Westbrook, C. I.

C. Henkel, C. I. Westbrook, and A. Aspect, “Quantum reflection: atomic matter–wave optics in an exponential potential,” J. Opt. Soc. Am. B 13, 233 (1996).
[Crossref]

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49 (1992).
[Crossref] [PubMed]

Wilkinson, S. R.

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, “Observation of atomic Wannier–Stark ladders in an accelerating optical potential,” Phys. Rev. Lett. 76, 4512 (1996).
[Crossref] [PubMed]

Appl. Phys. (1)

L. S. Letokhov and V. G. Minogin, “Trapping and storage of atoms in a laser field,” Appl. Phys. 17, 99 (1978).
[Crossref]

J. Appl. Phys. (1)

A. N. Khondker, M. Rezwan Khan, and A. F. M. Anwar, “Transmission line analogy of resonance tunneling phenomena: the generalized impedance concept,” J. Appl. Phys. 63, 5191 (1988).
[Crossref]

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

Opt. Lett. (2)

Phys. Rev. A (1)

J. P. Gordon and A. Ashkin, “Motion of atoms in a radiation trap,” Phys. Rev. A 21, 1606 (1980).
[Crossref]

Phys. Rev. Lett. (12)

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[Crossref] [PubMed]

P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J. Y. Courtois, and G. Grynberg, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861 (1992).
[Crossref] [PubMed]

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49 (1992).
[Crossref] [PubMed]

A. Hemmerich and T. W. Hansch, “2-dimensional atomic crystal bound by light,” Phys. Rev. Lett. 70, 410 (1993).
[Crossref] [PubMed]

G. Grynberg, B. Lounis, P. Verkerk, J. Y. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in 2-dimensional and 3-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249 (1993).
[Crossref] [PubMed]

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

M. Weidemuller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, “Bragg diffraction in an atomic lattice bound by light,” Phys. Rev. Lett. 75, 4583 (1995).
[Crossref] [PubMed]

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508 (1996).
[Crossref] [PubMed]

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, “Observation of atomic Wannier–Stark ladders in an accelerating optical potential,” Phys. Rev. Lett. 76, 4512 (1996).
[Crossref] [PubMed]

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnikov, and A. I. Sidorov, “Quantum-state-selective mirror of atoms by laser light,” Phys. Rev. Lett. 60, 2137 (1988).
[Crossref] [PubMed]

N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, “Long atomic coherence times in an optical dipole trap,” Phys. Rev. Lett. 74, 1311 (1995).
[Crossref] [PubMed]

T. Pfau, Ch. Kurtsiefer, C. S. Adams, M. Sigel, and J. Mlynek, “Magneto-optical beam splitter for atoms,” Phys. Rev. Lett. 71, 3427 (1993).
[Crossref] [PubMed]

Other (8)

The mirror can also be formed in the z direction, such that the atoms move vertically, in a trampoline configuration. However, in this case the gravitation potential must be included in the analysis, which is not treated here.

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University, London, 1947), Chap. 2.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976), Chap. 9.

[(∊-〈V〉)]1/2 was used in the argument of the sine (instead of ∊), which leads to a better accuracy in the phase.

A truncated half-Gaussian envelope of the shape exp(-ξ2/2σ2) for 0≤ξ≤NΛ, and 0 elsewhere, was used. For a full-Gaussian envelope the reflected amplitudes from regions of equal height in both sides of the potential interfere, which adds narrow fringes to the reflected energy spectrum.

Note that for spherical wave fronts the interference fringes are not perfectly parallel. With the parameters given in the text, the angle of the fringes will change by ±3° over a 1-mm interaction range, which might call for a two-dimensional treatment, depending on the size of the atomic cloud.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, 2nd ed. (Wiley, New York, 1977), Vol. 1, Chap. 1.

In the case of total reflection the reflection coefficient can be written as r=exp(iϕ), and the probability distribution to the left of the potential is then |ψ(ξ)|2=a cos2(kξ-ϕ/2). The normalization used is a=1.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram showing the two laser beams that interfere to form the periodic potential. (a) The orientation of the two laser beams (L1,2) and the cold atomic cloud to be reflected. v0 is the mean velocity of the atoms, and g indicates the gravitational acceleration. (b) The resulting 1-D potential, shown here with a Gaussian envelope. The propagation directions of the incident, reflected, and transmitted parts of the atomic wave function are also indicated.

Fig. 2
Fig. 2

Reflection coefficient from a periodic cosine potential with a rectangle envelope, for different numbers of potential cells, N=10, N=100, and N=1000 in dashed, solid, and dash-dotted curves, respectively. The normalized potential amplitude is V0=40, either blue or red detuned. The horizontal axis is the normalized atomic kinetic energy above the potential maximum, -Vmax.

Fig. 3
Fig. 3

Energy spectrum of the Mathieu equation. The vertical axis is the normalized energy of the atom above the potential maximum, and the horizontal axis is the normalized potential amplitude V0. The gray area indicates the allowed bands of the potential, and the white areas are the forbidden bands, in which reflection occurs. The arrows indicate the first three forbidden energy gaps of the V0=40 potential.

Fig. 4
Fig. 4

Reflection coefficient from a single V0=40 cosine potential cell versus the normalized kinetic energy above the potential maximum, for red (dotted curve) and blue (solid curve) detuning of the laser. The Born approximation of Eq. (8) is given as a dashed curve. The energies of the first three reflection bands of the periodic potential are also marked.

Fig. 5
Fig. 5

Width of the first five reflection bands of the V0=40 potential as a function of the band mean energy. Both energy mean and bandwidth are shown in normalized units. The values obtained from a numerical calculation of the energy bands of the Mathieu equation are marked with ‘○’, and the values obtained from approximation (10) are marked with ‘×’.

Fig. 6
Fig. 6

Second reflection band from a potential with a truncated half-Gaussian envelope (see text), solid curve, and from a chirped potential, dashed curve. Parameters of the half-Gaussian are σ=1770Λ and V0=42 (maximal potential height). Parameters for the chirped potential: period changes linearly from 2.01ξ to 1.985ξ and V0=40. The number of cells in both cases is N=700. The reflection from a uniform envelope and period potential with N=N0=63 is also shown (dotted curve) as a reference.

Fig. 7
Fig. 7

(a) Relative phase of the reflection from a V0=40 potential, in the first and second reflection bands. (b) Interaction time of the atom with the potential in the same energy bands. Time values are given both in normalized units and in seconds, for 85Rb atoms and parameters as in text. The solid curve is calculated from the phase of the reflection, with Eq. (11), and the asterisks are values calculated from the wave function by use of Eq. (13). Note the different scale of the energy axis in the two bands.

Tables (2)

Tables Icon

Table 1 Numerical Values of the Reflection Parameters for the First Three Reflection Bands from a V0=40 Potential

Tables Icon

Table 2 Numerical Values of the Velocities, Interaction Time, Gravitational Fall, and Spontaneous-Scattering Rates, for the First Three Reflection Bands of a V0=40 Periodic Potentiala

Equations (17)

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U(x)=Ωr2(x)/4δ,
-22m d2ψ(x)dx2+U(x)ψ(x)=Eψ(x),
U(x)=A(x)U02 [1+cos(2kΛx)].
-d2ψdξ2+A(ξ)V02 [1+cos(2ξ)]ψ=ψ.
ψ(ξ)=exp(ikξ)+r exp(-ikξ),forξ<0,
ψ(ξ)=t exp(ikξ),forξ>NΛ.
ϕ1=20πk(ξ)dξ,
reflectn2+V02+V0232n2.
R1()V02kΛ4 sin2[Λ(-V)]162(-kΛ2)2,
ΔϕN(0+Δ)=Nϕ1(0+Δ)Nϕ1(0)+Ndϕ1d=0Δ,
Δ403/2N0(40+V0).
τ=dϕd=0,
0|ψ(x)|2dx-X|ψ(x)|2dx=0|ψ(x)|2dx-X0|ψ(x)|2dx+0|ψ(x)|2dx
X0|ψ(x)|2dxX/2.
τ=20|ψ(ξ)|2dξk0,
γs(ξ)=γ38δ2×I(ξ)Is=γωΛV(ξ)δ.
S=20|ψ(ξ)|2γs(ξ)dξk0ωΛ.

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