Abstract

The different orbital modes that appear from the misalignment of the laser beams in the xy plane of a low-density magneto-optical trap are studied. Using the circular motion approximation, one can obtain a qualitative description of the phenomenon and can calculate the ring radii, the atom’s velocity in the ring, and the trap parameters in which the rings appear. A careful analysis of some theories previously employed in researching this topic shows that they have extrapolation problems that led to unreasonable conclusions. In particular, we compare the results of the circular motion approximation itself with those obtained from a direct numerical integration of the atomic trajectory. Within this approximation the influence of the increase in the number of trapped atoms over the ring is also investigated, and we observe an instability in a ring-shaped structure when the interaction between the atoms is included.

© 2000 Optical Society of America

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References

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  1. E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
    [CrossRef] [PubMed]
  2. T. Walker, D. Sesko, and C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64, 408–411 (1990); D. Sesko, T. Walker, and C. Wieman, “Behavior of neutral atoms in a spontaneous force trap,” J. Opt. Soc. Am. B 8, 946–958 (1991).
    [CrossRef] [PubMed]
  3. L. G. Marcassa, D. Milori, M. Ori, G. I. Surdutovich, S. C. Zilio, and V. S. Bagnato, “Magneto-optical trap for sodium atoms from a vapor cell and observation of spatial modes,” Braz. J. Phys. 22, 3–6 (1992); V. S. Bagnato, L. G. Marcassa, M. Oriá, G. I. Surdutovich, and S. C. Zilio, “Spatial distributions of optically trapped cooled neutral atoms,” Laser Phys. 2, 172–177 (1992).
  4. V. S. Bagnato, L. G. Marcassa, M. Ori, G. I. Surdutovich, R. Vitlina, and S. C. Zilio, “Spatial distribution of atoms in a magneto-optical trap,” Phys. Rev. A 48, 3771–3775 (1993).
    [CrossRef] [PubMed]
  5. M. T. de Araujo, L. G. Marcassa, S. C. Zilio, and V. S. Bagnato, “Double-ring structure: another variant in the spatial distribution of cold sodium atoms,” Phys. Rev. A 51, 4286–4288 (1995).
    [CrossRef] [PubMed]
  6. I. Guedes, M. T. de Araujo, D. M. B. P. Milori, G. I. Surdutovich, V. S. Bagnato, and S. C. Zilio, “Forces acting on magneto-optically trapped atoms,” J. Opt. Soc. Am. B 11, 1935–1940 (1994).
    [CrossRef]
  7. I. Guedes, H. F. Silva Filho, and F. D. Nunes, “Theoretical analysis of the spatial structures of atoms in magneto-optical traps,” Phys. Rev. A 55, 561–567 (1997).
    [CrossRef]
  8. V. G. Minogin and V. S. Letokov, Laser Light Pressure on Atoms (Gordon & Breach, New York, 1987); S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. 58, 699–739 (1986).
    [CrossRef]
  9. K. Lindquist, M. Stephens, and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 4082–4090 (1992).
    [CrossRef] [PubMed]
  10. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford U. Press, Oxford, UK, 1994).
  11. G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, Calif., 1985).
  12. A. M. Steane, M. Chowdhury, and C. J. Foot, “Radiation force in the magneto-optical trap,” J. Opt. Soc. Am. B 9, 2142–2158 (1992).
    [CrossRef]
  13. D. Felinto, L. G. Marcassa, V. S. Bagnato, and S. S. Vianna, “Influence of the number of atoms in a ring-shaped magneto-optical trap: observation of bifurcation,” Phys. Rev. A 60, 2591–2594 (1999).
    [CrossRef]

1999 (1)

D. Felinto, L. G. Marcassa, V. S. Bagnato, and S. S. Vianna, “Influence of the number of atoms in a ring-shaped magneto-optical trap: observation of bifurcation,” Phys. Rev. A 60, 2591–2594 (1999).
[CrossRef]

1997 (1)

I. Guedes, H. F. Silva Filho, and F. D. Nunes, “Theoretical analysis of the spatial structures of atoms in magneto-optical traps,” Phys. Rev. A 55, 561–567 (1997).
[CrossRef]

1995 (1)

M. T. de Araujo, L. G. Marcassa, S. C. Zilio, and V. S. Bagnato, “Double-ring structure: another variant in the spatial distribution of cold sodium atoms,” Phys. Rev. A 51, 4286–4288 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

V. S. Bagnato, L. G. Marcassa, M. Ori, G. I. Surdutovich, R. Vitlina, and S. C. Zilio, “Spatial distribution of atoms in a magneto-optical trap,” Phys. Rev. A 48, 3771–3775 (1993).
[CrossRef] [PubMed]

1992 (2)

K. Lindquist, M. Stephens, and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 4082–4090 (1992).
[CrossRef] [PubMed]

A. M. Steane, M. Chowdhury, and C. J. Foot, “Radiation force in the magneto-optical trap,” J. Opt. Soc. Am. B 9, 2142–2158 (1992).
[CrossRef]

1987 (1)

E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[CrossRef] [PubMed]

Bagnato, V. S.

D. Felinto, L. G. Marcassa, V. S. Bagnato, and S. S. Vianna, “Influence of the number of atoms in a ring-shaped magneto-optical trap: observation of bifurcation,” Phys. Rev. A 60, 2591–2594 (1999).
[CrossRef]

M. T. de Araujo, L. G. Marcassa, S. C. Zilio, and V. S. Bagnato, “Double-ring structure: another variant in the spatial distribution of cold sodium atoms,” Phys. Rev. A 51, 4286–4288 (1995).
[CrossRef] [PubMed]

I. Guedes, M. T. de Araujo, D. M. B. P. Milori, G. I. Surdutovich, V. S. Bagnato, and S. C. Zilio, “Forces acting on magneto-optically trapped atoms,” J. Opt. Soc. Am. B 11, 1935–1940 (1994).
[CrossRef]

V. S. Bagnato, L. G. Marcassa, M. Ori, G. I. Surdutovich, R. Vitlina, and S. C. Zilio, “Spatial distribution of atoms in a magneto-optical trap,” Phys. Rev. A 48, 3771–3775 (1993).
[CrossRef] [PubMed]

Cable, A.

E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[CrossRef] [PubMed]

Chowdhury, M.

Chu, S.

E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[CrossRef] [PubMed]

de Araujo, M. T.

M. T. de Araujo, L. G. Marcassa, S. C. Zilio, and V. S. Bagnato, “Double-ring structure: another variant in the spatial distribution of cold sodium atoms,” Phys. Rev. A 51, 4286–4288 (1995).
[CrossRef] [PubMed]

I. Guedes, M. T. de Araujo, D. M. B. P. Milori, G. I. Surdutovich, V. S. Bagnato, and S. C. Zilio, “Forces acting on magneto-optically trapped atoms,” J. Opt. Soc. Am. B 11, 1935–1940 (1994).
[CrossRef]

Felinto, D.

D. Felinto, L. G. Marcassa, V. S. Bagnato, and S. S. Vianna, “Influence of the number of atoms in a ring-shaped magneto-optical trap: observation of bifurcation,” Phys. Rev. A 60, 2591–2594 (1999).
[CrossRef]

Foot, C. J.

Guedes, I.

I. Guedes, H. F. Silva Filho, and F. D. Nunes, “Theoretical analysis of the spatial structures of atoms in magneto-optical traps,” Phys. Rev. A 55, 561–567 (1997).
[CrossRef]

I. Guedes, M. T. de Araujo, D. M. B. P. Milori, G. I. Surdutovich, V. S. Bagnato, and S. C. Zilio, “Forces acting on magneto-optically trapped atoms,” J. Opt. Soc. Am. B 11, 1935–1940 (1994).
[CrossRef]

Lindquist, K.

K. Lindquist, M. Stephens, and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 4082–4090 (1992).
[CrossRef] [PubMed]

Marcassa, L. G.

D. Felinto, L. G. Marcassa, V. S. Bagnato, and S. S. Vianna, “Influence of the number of atoms in a ring-shaped magneto-optical trap: observation of bifurcation,” Phys. Rev. A 60, 2591–2594 (1999).
[CrossRef]

M. T. de Araujo, L. G. Marcassa, S. C. Zilio, and V. S. Bagnato, “Double-ring structure: another variant in the spatial distribution of cold sodium atoms,” Phys. Rev. A 51, 4286–4288 (1995).
[CrossRef] [PubMed]

V. S. Bagnato, L. G. Marcassa, M. Ori, G. I. Surdutovich, R. Vitlina, and S. C. Zilio, “Spatial distribution of atoms in a magneto-optical trap,” Phys. Rev. A 48, 3771–3775 (1993).
[CrossRef] [PubMed]

Milori, D. M. B. P.

Nunes, F. D.

I. Guedes, H. F. Silva Filho, and F. D. Nunes, “Theoretical analysis of the spatial structures of atoms in magneto-optical traps,” Phys. Rev. A 55, 561–567 (1997).
[CrossRef]

Ori, M.

V. S. Bagnato, L. G. Marcassa, M. Ori, G. I. Surdutovich, R. Vitlina, and S. C. Zilio, “Spatial distribution of atoms in a magneto-optical trap,” Phys. Rev. A 48, 3771–3775 (1993).
[CrossRef] [PubMed]

Prentiss, M.

E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[CrossRef] [PubMed]

Pritchard, D.

E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[CrossRef] [PubMed]

Raab, E.

E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[CrossRef] [PubMed]

Silva Filho, H. F.

I. Guedes, H. F. Silva Filho, and F. D. Nunes, “Theoretical analysis of the spatial structures of atoms in magneto-optical traps,” Phys. Rev. A 55, 561–567 (1997).
[CrossRef]

Steane, A. M.

Stephens, M.

K. Lindquist, M. Stephens, and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 4082–4090 (1992).
[CrossRef] [PubMed]

Surdutovich, G. I.

I. Guedes, M. T. de Araujo, D. M. B. P. Milori, G. I. Surdutovich, V. S. Bagnato, and S. C. Zilio, “Forces acting on magneto-optically trapped atoms,” J. Opt. Soc. Am. B 11, 1935–1940 (1994).
[CrossRef]

V. S. Bagnato, L. G. Marcassa, M. Ori, G. I. Surdutovich, R. Vitlina, and S. C. Zilio, “Spatial distribution of atoms in a magneto-optical trap,” Phys. Rev. A 48, 3771–3775 (1993).
[CrossRef] [PubMed]

Vianna, S. S.

D. Felinto, L. G. Marcassa, V. S. Bagnato, and S. S. Vianna, “Influence of the number of atoms in a ring-shaped magneto-optical trap: observation of bifurcation,” Phys. Rev. A 60, 2591–2594 (1999).
[CrossRef]

Vitlina, R.

V. S. Bagnato, L. G. Marcassa, M. Ori, G. I. Surdutovich, R. Vitlina, and S. C. Zilio, “Spatial distribution of atoms in a magneto-optical trap,” Phys. Rev. A 48, 3771–3775 (1993).
[CrossRef] [PubMed]

Wieman, C.

K. Lindquist, M. Stephens, and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 4082–4090 (1992).
[CrossRef] [PubMed]

Zilio, S. C.

M. T. de Araujo, L. G. Marcassa, S. C. Zilio, and V. S. Bagnato, “Double-ring structure: another variant in the spatial distribution of cold sodium atoms,” Phys. Rev. A 51, 4286–4288 (1995).
[CrossRef] [PubMed]

I. Guedes, M. T. de Araujo, D. M. B. P. Milori, G. I. Surdutovich, V. S. Bagnato, and S. C. Zilio, “Forces acting on magneto-optically trapped atoms,” J. Opt. Soc. Am. B 11, 1935–1940 (1994).
[CrossRef]

V. S. Bagnato, L. G. Marcassa, M. Ori, G. I. Surdutovich, R. Vitlina, and S. C. Zilio, “Spatial distribution of atoms in a magneto-optical trap,” Phys. Rev. A 48, 3771–3775 (1993).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (2)

Phys. Rev. A (5)

D. Felinto, L. G. Marcassa, V. S. Bagnato, and S. S. Vianna, “Influence of the number of atoms in a ring-shaped magneto-optical trap: observation of bifurcation,” Phys. Rev. A 60, 2591–2594 (1999).
[CrossRef]

V. S. Bagnato, L. G. Marcassa, M. Ori, G. I. Surdutovich, R. Vitlina, and S. C. Zilio, “Spatial distribution of atoms in a magneto-optical trap,” Phys. Rev. A 48, 3771–3775 (1993).
[CrossRef] [PubMed]

M. T. de Araujo, L. G. Marcassa, S. C. Zilio, and V. S. Bagnato, “Double-ring structure: another variant in the spatial distribution of cold sodium atoms,” Phys. Rev. A 51, 4286–4288 (1995).
[CrossRef] [PubMed]

I. Guedes, H. F. Silva Filho, and F. D. Nunes, “Theoretical analysis of the spatial structures of atoms in magneto-optical traps,” Phys. Rev. A 55, 561–567 (1997).
[CrossRef]

K. Lindquist, M. Stephens, and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 4082–4090 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[CrossRef] [PubMed]

Other (5)

T. Walker, D. Sesko, and C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64, 408–411 (1990); D. Sesko, T. Walker, and C. Wieman, “Behavior of neutral atoms in a spontaneous force trap,” J. Opt. Soc. Am. B 8, 946–958 (1991).
[CrossRef] [PubMed]

L. G. Marcassa, D. Milori, M. Ori, G. I. Surdutovich, S. C. Zilio, and V. S. Bagnato, “Magneto-optical trap for sodium atoms from a vapor cell and observation of spatial modes,” Braz. J. Phys. 22, 3–6 (1992); V. S. Bagnato, L. G. Marcassa, M. Oriá, G. I. Surdutovich, and S. C. Zilio, “Spatial distributions of optically trapped cooled neutral atoms,” Laser Phys. 2, 172–177 (1992).

M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford U. Press, Oxford, UK, 1994).

G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, Calif., 1985).

V. G. Minogin and V. S. Letokov, Laser Light Pressure on Atoms (Gordon & Breach, New York, 1987); S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. 58, 699–739 (1986).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Laser beam configuration in the MOT with misalignment in the xy plane. The boldface lines indicate the directions of the six k wave vectors at the centers of their respective Gaussian spatial profiles. The parameter s is the misalignment from the coordinate axis.

Fig. 2
Fig. 2

Sign of Fy. In the hatched region we show Fy<0; in the plain region, Fy>0. The parameters used to generate this graph were δ=-Γ, s=0.8w, and w=5 mm.

Fig. 3
Fig. 3

Ring radius versus magnetic field gradient, for δ=-Γ, s=0.8w, and Ω0=13Γ. For values of b below bcut, no stable structure is observed. For values of b between bcut and bt, only one ring is observed. For values above bt, only a ball is observed.

Fig. 4
Fig. 4

Maximum ring radius versus misalignment for Ω0=0.1Γ (squares), Ω0=Γ (filled circles), Ω0=5Γ [inverted (filled) triangles], Ω0=15Γ (open triangles), and Ω0=21Γ (open circles). The solid curve is a plot of xc/2 versus misalignment. We also used δ=-Γ and w=5 mm.

Fig. 5
Fig. 5

Curve that separates the regions in which γbtxc/2 is greater or smaller than 0.1|δ| for different values of δ.

Fig. 6
Fig. 6

Ring radius versus magnetic field gradient, for the same parameters as in Fig. 3. The filled circles indicate the predictions of the numerical calculation with the LVA. The theory predicts regions with one ring, with two rings, with a ring plus a ball, and with only a ball. The open circles indicate the maximum radius at which the LVA is valid, as defined by condition (22), with D=0.5.

Fig. 7
Fig. 7

Ring radius versus magnetic field gradient, for the same parameters as in Fig. 3. The curve was generated with the use of numerically integrated trajectories.

Fig. 8
Fig. 8

Maximum ring radius versus misalignment. The squares indicate the results based on the UCMA, and the circles indicate the results obtained from the integrated trajectories. The other parameters are δ=-Γ and Ω0=21Γ. The dashed lines are just to guide the eye.

Fig. 9
Fig. 9

Variation of the ring shape with b, for (a) s=2.8w and (b) s=0.2w. The outer ring indicates the biggest observed ring (with radius equal to Rm). The ring radius becomes smaller as b is increased. The other parameters are δ=-Γ and Ω0=21Γ.

Fig. 10
Fig. 10

Form of fring(t) in the limit of a high number of atoms. We show the value of fring, at t=1, for N=104 (plus sign), N=106 (diamond), and N=108 (square).

Fig. 11
Fig. 11

Form of Fy(x) for N=103 [solid curve (see center of figure)], N=104 (open circles), and N=105 (filled circles). The other parameters are w=3 mm, s=w, Ω0=4Γ, δ=-2Γ, and b=5 G/cm.

Equations (37)

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

Fx±
=kΓΩ02 exp-(ys)2w2Γ2+2Ω02 exp-(ys)2w2+4(δ±kvx±γbx/2)2ıˆ,
Fy±
=kΓΩ02 exp-(x±s)2w2Γ2+2Ω02 exp-(x±s)2w2+4(δ±kvy±γby/2)2Jˆ,
Fx=kΓAΓ2+2A+4(δ-γbx/2)2-kΓAΓ2+2A+4(δ+γbx/2)2,
Fy=kΓA exp-(x2-2sx)w2Γ2+2A exp-(x2-2sx)w2+4(δ-kvy)2-kΓA exp-(x2+2sx)w2Γ2+2A exp-(x2+2sx)w2+4(δ+kvy)2,
A=Ω02 exp[-(s2/w2)].
vx=0,
Fy=0,
Fx=mvy2x.
va=Kmx(1+C1x2+C2x4)1/2,
K=8γb|δ|kΓA(Γ2+2A+4δ2)2,
C1=2(γb)2(Γ2+2A-4δ2)(Γ2+2A+4δ2)2,
C2=(γb)4(Γ2+2A+4δ2)2.
Fy(x, vy)=I(x, vy)[fr(x)-fa(vy)],
fr(x)=tanh(2xs/w2),
fa(vy)=8k|δ|vyΓ2+4δ2+4k2vy2,
[v0(x)]±=2|δ|±4δ2-(Γ2+4δ2)tanh22xsw21/22k tanh2xsw2
xc=w22sarctanh2|δ|Γ2+4δ2,
dfadx(x=0)>dfrdx(x=0),
bt=2sw2Γ2+4δ28k|δ|2m(Γ2+2A+4δ2)28kΓγA|δ|.
γbx/2|δ|,
kvy|δ|.
va=(K/m)1/2x.
γbtxc/2|δ|.
fa(x)=8k|δ|Km xΓ2+4δ2+4k2Kmx2.
Fy=FV-FA,
FV=kΓCexp-(x2-2sx)w21+2C exp-(x2-2sx)w2-exp-(x2+2sx)w21+2C exp-(x2+2sx)w2,
FA=8k2ΓC|δ|Γ2+4δ2exp-(x2-2sx)w21+2C exp-(x2-2sx)w22+exp-(x2+2sx)w21+2C exp-(x2+2sx)w22vy,
C=AΓ2+4δ2.
vy<D(|δ|/k),
Fring(r)=i=1Nαr-ri|r-ri|3,
Fring(r)=αNR2fring(r/R)rˆ,
fring(t)=1Ni=1Nt-cos θi(t2+1-2t cos θi)3/2,
fring(t)
=1π1t+t2K2tt+1-1t-t2E2tt+1t112ln2Nπt=1,
Fx(x)-Fring(x)=mvy2x

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