Abstract

A classical collective behavior is observed in the spatial distributions of a cloud of optically trapped neutral atoms. They include extended uniform-density ellipsoids, rings of atoms around a small central ball, and clumps of atoms orbiting a central core. The distributions depend sensitively on the number of atoms and the alignment of the laser beams. Abrupt bistable transitions between different distributions are seen. This system is studied in detail, and much of this behavior can be explained by the incorporation of long-range interactions between the atoms in the equation of equilibrium. It is shown how attenuation and multiple scattering of the incident photons lead to these interactions.

© 1991 Optical Society of America

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References

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  1. D. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986).
    [Crossref] [PubMed]
  2. E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987).
    [Crossref] [PubMed]
  3. T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64, 408 (1990).
    [Crossref] [PubMed]
  4. A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63, 957 (1989).
    [Crossref] [PubMed]
  5. D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989); M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Roab, and D. Pritchard, Opt. Lett. 13, 452 (1988).
    [Crossref] [PubMed]
  6. J. Dalibard, Opt. Commun. 68, 203 (1988).
    [Crossref]
  7. A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, and V. P. Yakovlev, J. Opt. Soc. Am. B 6, 2130 (1989).
    [Crossref]
  8. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1989), p. 6.
  9. B. R. Mollow, Phys. Rev. 188, 1969 (1969); Phys. Rev. A 5, 2217 (1972); D. A. Holm, M. Sargent, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
    [Crossref]
  10. S. Gilbert, Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1984).
  11. B. Dahamani, L. Hollberg, and R. Drullenger, Opt. Lett. 12, 876 (1988).
    [Crossref]
  12. P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
    [Crossref] [PubMed]
  13. R. Sinclair (Division of Physics, National Science Foundation, Washington, D.C. 20550, personal communication) has pointed out that the clumping is similar in character to the negative-mass instability that is observed in electron plasmas. We are now investigating this analogy.

1990 (1)

T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64, 408 (1990).
[Crossref] [PubMed]

1989 (3)

A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63, 957 (1989).
[Crossref] [PubMed]

D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989); M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Roab, and D. Pritchard, Opt. Lett. 13, 452 (1988).
[Crossref] [PubMed]

A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, and V. P. Yakovlev, J. Opt. Soc. Am. B 6, 2130 (1989).
[Crossref]

1988 (3)

J. Dalibard, Opt. Commun. 68, 203 (1988).
[Crossref]

B. Dahamani, L. Hollberg, and R. Drullenger, Opt. Lett. 12, 876 (1988).
[Crossref]

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
[Crossref] [PubMed]

1987 (1)

E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987).
[Crossref] [PubMed]

1986 (1)

D. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986).
[Crossref] [PubMed]

1969 (1)

B. R. Mollow, Phys. Rev. 188, 1969 (1969); Phys. Rev. A 5, 2217 (1972); D. A. Holm, M. Sargent, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
[Crossref]

Bagnato, V.

D. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986).
[Crossref] [PubMed]

Cable, A. E.

E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987).
[Crossref] [PubMed]

Chu, S.

E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987).
[Crossref] [PubMed]

Chudesnikov, D. O.

Dahamani, B.

Dalibard, J.

J. Dalibard, Opt. Commun. 68, 203 (1988).
[Crossref]

Drullenger, R.

Gallagher, A.

D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989); M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Roab, and D. Pritchard, Opt. Lett. 13, 452 (1988).
[Crossref] [PubMed]

A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63, 957 (1989).
[Crossref] [PubMed]

Gilbert, S.

S. Gilbert, Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1984).

Gould, P. L.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
[Crossref] [PubMed]

Hollberg, L.

Kazantsev, A. P.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1989), p. 6.

Lett, P. D.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
[Crossref] [PubMed]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1989), p. 6.

Metcalf, H. J.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
[Crossref] [PubMed]

Mollow, B. R.

B. R. Mollow, Phys. Rev. 188, 1969 (1969); Phys. Rev. A 5, 2217 (1972); D. A. Holm, M. Sargent, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
[Crossref]

Monroe, C.

D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989); M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Roab, and D. Pritchard, Opt. Lett. 13, 452 (1988).
[Crossref] [PubMed]

Phillips, W. D.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
[Crossref] [PubMed]

Prentiss, M. G.

E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987).
[Crossref] [PubMed]

Pritchard, D.

A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63, 957 (1989).
[Crossref] [PubMed]

D. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986).
[Crossref] [PubMed]

Pritchard, D. E.

E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987).
[Crossref] [PubMed]

Raab, E.

D. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986).
[Crossref] [PubMed]

Raab, E. L.

E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987).
[Crossref] [PubMed]

Sesko, D.

T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64, 408 (1990).
[Crossref] [PubMed]

D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989); M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Roab, and D. Pritchard, Opt. Lett. 13, 452 (1988).
[Crossref] [PubMed]

Sinclair, R.

R. Sinclair (Division of Physics, National Science Foundation, Washington, D.C. 20550, personal communication) has pointed out that the clumping is similar in character to the negative-mass instability that is observed in electron plasmas. We are now investigating this analogy.

Surdutovich, G. I.

Walker, T.

T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64, 408 (1990).
[Crossref] [PubMed]

D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989); M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Roab, and D. Pritchard, Opt. Lett. 13, 452 (1988).
[Crossref] [PubMed]

Watts, R.

D. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986).
[Crossref] [PubMed]

Watts, R. N.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
[Crossref] [PubMed]

Westbrook, C. I.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
[Crossref] [PubMed]

Wieman, C.

T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64, 408 (1990).
[Crossref] [PubMed]

D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989); M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Roab, and D. Pritchard, Opt. Lett. 13, 452 (1988).
[Crossref] [PubMed]

D. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986).
[Crossref] [PubMed]

Yakovlev, V. P.

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

Opt. Commun. (1)

J. Dalibard, Opt. Commun. 68, 203 (1988).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (1)

B. R. Mollow, Phys. Rev. 188, 1969 (1969); Phys. Rev. A 5, 2217 (1972); D. A. Holm, M. Sargent, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
[Crossref]

Phys. Rev. Lett. (6)

D. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986).
[Crossref] [PubMed]

E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987).
[Crossref] [PubMed]

T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64, 408 (1990).
[Crossref] [PubMed]

A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63, 957 (1989).
[Crossref] [PubMed]

D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989); M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Roab, and D. Pritchard, Opt. Lett. 13, 452 (1988).
[Crossref] [PubMed]

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
[Crossref] [PubMed]

Other (3)

R. Sinclair (Division of Physics, National Science Foundation, Washington, D.C. 20550, personal communication) has pointed out that the clumping is similar in character to the negative-mass instability that is observed in electron plasmas. We are now investigating this analogy.

S. Gilbert, Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1984).

L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1989), p. 6.

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Figures (14)

Fig. 1
Fig. 1

Spatial distributions of trapped atoms. (a) With fewer than 108 atoms, the cloud forms a uniform-density sphere. (b) Top view of rotating clump without strobing. (c) View of (b) with camera strobed at 110 Hz. The clump is rotating around the nucleus in a counterclockwise direction. (d) Top view of rotating clump without strobing but with a smaller radius than in (b) (the clump is one quarter of the area of the total fluorescence). (e) Top view of continuous ring. (f) Side view of (e). The horizontal full scale for (a), (d), (e), and (f) is 1.0 cm. For (b) and (c) it is 0.8 cm.

Fig. 2
Fig. 2

Forces that arise within an optically thick cloud of atoms. The diameter of the cloud is 0.2 cm. (a) The change in the intensity of the trapping light due to absorption across the cloud of atoms produces the attenuation force FAX ∝〈σL〉(I+I). (b) The spontaneous emission of two atoms separated by a distance d produces the repulsive radiation trapping force between the atoms, FRX ∝ 〈σFd−2. (c) The three forces in units of kelvins per centimeter. Note that the total force is FT = −kx + FAX + FRX = 0 within the cloud.

Fig. 3
Fig. 3

Absorption (solid curve) and emission (dotted-dashed curve) profiles calculated for a two-level atom in a one-dimensional standing-wave field. The curves are for an average intensity of 12 mW/cm2 and a laser detuning of −1.5ΔN. The emission profile is in arbitrary units. The zero-width elastic emission peak (which constitutes 50% of the total) is represented by the spike at −1.5ΔN.

Fig. 4
Fig. 4

Geometry of the misalignment in the horizontal plane that gives rise to the orbital modes. The bold boundary and the central ball show the positions of the atoms. The arrow indicates the direction of the orbit. The radiation pressure from the central core is also illustrated.

Fig. 5
Fig. 5

Trajectories of an atom numerically calculated from our model. The trajectories are shown for ten initial positions and velocities of the atom. The trap parameters were chosen to be the same as for the conditions in Fig. 1(e) (Δ = −1.5ΔN, I/IS = 12, B′ = 15 G/cm, w = 6 mm, and s = 1.5 mm). The frequency of the orbit is 82 Hz.

Fig. 6
Fig. 6

Schematic of the experimental apparatus. The third trapping beam, perpendicular to the page, is not shown.

Fig. 7
Fig. 7

Displacement of a small ball of atoms versus intensity of the pushing beam (circles). The solid-line] fit shows that the force is harmonic and gives a spring constant of 6 K/cm2 The laser detuning was −1.5ΔN, the total intensity was I/IS = 12, and the magnetic field gradient was 15 G/cm. The pushing beam was at the same frequency as the trap laser.

Fig. 8
Fig. 8

The dots show the TOF spectrum, after the trap light is turned off and the fluorescence is measured, of a probe beam 4 mm to the side of the cloud. The solid curve is a fit to the data for a Maxwell–Boltzmann distribution (T = 265 μK) that heavily weights the rising edge of the TOF spectrum.

Fig. 9
Fig. 9

Measured temperature of the atoms versus the diameter (FWHM) of the cloud for detunings of −1.5ΔN (filled circles) and −2.5ΔN (open circles). (b) Measured temperature of the atoms versus detuning for a 1.5-mm-diameter cloud. The total laser intensity is 12 mW/cm2, and the magnetic field gradient is 15 G/cm.

Fig. 10
Fig. 10

Top view of fluorescence profile of the cloud. (a) Data for the ideal-gas mode (circles) with a fit (solid curve) for a Gaussian distribution. (b) Data for the static mode (circles) with a fit (solid curve) assuming that we are looking along the minor axis of a ellipsoid of constant density. The fit gives a ratio of 1.5 for the ellipsoidal axes.

Fig. 11
Fig. 11

Plot of the diameter (FWHM) of the cloud of atoms as a function of the number of atoms in the cloud (circles). For the main figure the magnetic field gradient is 9 G/cm, and for the inset it is 16.5 G/cm. The laser detuning is −1.5ΔN, and the total laser intensity is 12 mW/cm2. The solid curves show the predictions of the mode that are described in the text.

Fig. 12
Fig. 12

Fluorescence from a cloud of trapped atoms as a function of time after the loading is terminated. The main figure shows the decay of atoms from the static mode into the ideal-gas mode. The inset shows the decay from an orbiting mode through an abrupt transition (indicated by the arrow) to the static mode.

Fig. 13
Fig. 13

(a) Time dependence of total fluorescence for the rotating clump without modulation of the magnetic field. (b) Beat signal of total fluorescence produced after the transition is induced to the rotating clump mode. The orbital frequency was 112 Hz, and the magnetic field was modulated at 119 Hz. The peak-to-peak modulation was approximately 40% of the total fluorescence for both (a) and (b).

Fig. 14
Fig. 14

Dependence of the rotational frequency of the clump on (a) the laser detuning (B′ = 12.5 G/cm, I = 12 mW/cm2), (b) the magnetic field gradient (Δ = −1.5ΔN, I = 12 mW/cm2), and (c) the total average intensity (Δ = −2.0ΔN, B′ = 15 G/cm). The circles are the experimental data, and the pluses give the results of our numerical model.

Equations (23)

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A X ( r ) = σ L - x x n ( r ) d x ,
I X ± ( r ) = I { 1 - [ A T ( y , z ) ± A X ( r ) ] / 2 } ,
F A X ( F ) = - ϕ L I A X ( F ) / c ,
· F A = - 6 σ L 2 I n / c .
I rad = σ L I / 4 π d 2 .
F R = σ F σ L I / 4 π c d 2 .
F R ( r ) = σ F σ L I 4 π c n ( r ) r - r r - r 3 d 3 r ,
· F R = 6 σ F σ L I n ( r ) / c .
F R = σ F I rad / c ,
I rad = σ L I N / 4 π R 2 .
n max = c k / 2 σ L ( σ F - σ L ) I ,
T n ( r ) = ( F A + F R - k r ) n ( r ) .
T 2 ln [ n ( r ) ] = 3 k ( n / n max - 1 ) .
I ( x , y , z ) = 6 I cos 2 ( 2 π x λ + 2 π y λ + 2 π z λ + ϕ ) ,
σ L = σ 0 { [ 1 + 4 ( Δ Δ N ) 2 ] [ 1 + 6 I I S + 4 ( Δ Δ N ) 2 ] } - 1 / 2 .
m d 2 r d t 2 = - k r - γ d r d t - k z ^ × r + α N r 2 r ^ .
N = N [ η c + η r ln ( 2 η r N / π ) 2 π ] .
R = [ α N / ( k - m ω 2 ) ] 1 / 3 ,
F X ± = π h Δ N λ I ± ( y ) I S 1 + I ( r ) I S + 4 ( Δ ± ω B x ± v X / λ ) 2 Δ N 2 ,
I ± ( y ) = I exp { - [ 2 ( y s / 2 ) w ] 2 ln 2 } I g ± ,
F R X = α N x / ( x 2 + y 2 ) 3 / 2 ,
F X = F R X + π h Δ N I λ I S × ( g + - g - ) { 1 + I ( r ) I S + 4 [ Δ 2 + ( ω B x + v X / λ ) 2 ] Δ N 2 } + ( g + + g - ) Δ ( ω B x + v X / λ ) Δ N 2 { 1 + I ( r ) I s + 4 [ Δ 2 + ( ω B x + v X / λ ) 2 ] Δ N 2 - 64 Δ 2 ( ω B x + v X / λ ) 2 Δ N 4 } 2 .
F = F X x ^ + F Y y ^ ,

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