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.

[Optical Society of America ]

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  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. PRLTAO 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. OPLEDP 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. PRLTAO 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 JOBPDE 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. PRLTAO 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. PRLTAO 69 , 49 ( 1992
    [CrossRef] [PubMed]
  7. A. Hemmerich and T. W. Hansch , 2-dimensional atomic crystal bound by light , Phys. Rev. Lett. PRLTAO 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. PRLTAO 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. PRLTAO 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. PRLTAO 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. PRLTAO 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. PRLTAO 76 , 4512 ( 1996
    [CrossRef] [PubMed]
  13. J. P. Gordon and A. Ashkin , Motion of atoms in a radiation trap , Phys. Rev. A PLRAAN 21 , 1606 ( 1980
    [CrossRef]
  14. 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. JAPIAU 63 , 5191 ( 1988
    [CrossRef]
  15. L. S. Letokhov and V. G. Minogin , Trapping and storage of atoms in a laser field , Appl. Phys. APHYCC 17 , 99 ( 1978
    [CrossRef]
  16. 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. PRLTAO 74 , 1311 ( 1995
    [CrossRef] [PubMed]
  17. T. Pfau , Ch. Kurtsiefer , C. S. Adams , M. Sigel , and J. Mlynek , Magneto-optical beam splitter for atoms , Phys. Rev. Lett. PRLTAO 71 , 3427 ( 1993
    [CrossRef] [PubMed]
  18. 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. OPLEDP 20 , 1310 ( 1995
    [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. PRLTAO 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. JAPIAU 63 , 5191 ( 1988
[CrossRef]

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. PRLTAO 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. PRLTAO 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. PRLTAO 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. PRLTAO 71 , 3083 ( 1993
[CrossRef] [PubMed]

Chu, A

Courtois, J. Y

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. PRLTAO 68 , 3861 ( 1992
[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. PRLTAO 71 , 3083 ( 1993
[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. PRLTAO 75 , 2823 ( 1995
[CrossRef] [PubMed]

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. PRLTAO 75 , 4583 ( 1995
[CrossRef] [PubMed]

Johnson, K. S

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. JAPIAU 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. PRLTAO 71 , 3427 ( 1993
[CrossRef] [PubMed]

Letokhov, L. S

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

Lounis, B

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. PRLTAO 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. PRLTAO 76 , 4512 ( 1996
[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. PRLTAO 76 , 4512 ( 1996
[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. PRLTAO 76 , 4508 ( 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. PRLTAO 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. JAPIAU 63 , 5191 ( 1988
[CrossRef]

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. PRLTAO 75 , 4583 ( 1995
[CrossRef] [PubMed]

Other

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. PRLTAO 60 , 2137 ( 1988
[CrossRef] [PubMed]

M. A. Kasevich , D. S. Weiss , and S. Chu , Normal-incidence reflection of slow atoms from an optical evanescent wave , Opt. Lett. OPLEDP 15 , 607 ( 1990
[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. PRLTAO 71 , 3083 ( 1993
[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 JOBPDE 13 , 233 ( 1996
[CrossRef]

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. PRLTAO 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. PRLTAO 69 , 49 ( 1992
[CrossRef] [PubMed]

A. Hemmerich and T. W. Hansch , 2-dimensional atomic crystal bound by light , Phys. Rev. Lett. PRLTAO 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. PRLTAO 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. PRLTAO 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. PRLTAO 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. PRLTAO 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. PRLTAO 76 , 4512 ( 1996
[CrossRef] [PubMed]

J. P. Gordon and A. Ashkin , Motion of atoms in a radiation trap , Phys. Rev. A PLRAAN 21 , 1606 ( 1980
[CrossRef]

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. JAPIAU 63 , 5191 ( 1988
[CrossRef]

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

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. PRLTAO 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. PRLTAO 71 , 3427 ( 1993
[CrossRef] [PubMed]

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. OPLEDP 20 , 1310 ( 1995
[CrossRef] [PubMed]

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 (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)

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

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.

Metrics