Abstract

A theoretical investigation into density, pressure, and temperature distributions in magneto-optical traps is presented. After a brief overview of the forces that arise from reradiation and absorption, a condition that the absorptive force be conservative is used to show that, if the temperature is uniform throughout the trap, any density solutions to the force equations will not be physical. Further, consistent density solutions are unlikely to exist at all. In contrast, with a varying temperature reasonable solutions are demonstrated, with some restrictions. Doppler forces involved in ring-shaped trap structures are used to calculate orbit radii in racetrack geometry traps, and corrections to the present discrepancy between theoretical and experimental studies are discussed in the context of reradiation and diffusion.

© 2000 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. G. Hillenbrand, C. J. Foot, and K. Burnett, “Heating due to long-range photon exchange interactions between cold atoms,” Phys. Rev. A 50, 1479–1489 (1994).
    [CrossRef] [PubMed]
  30. M. T. de Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, “Time of flight technique to determine temperature of atoms in a ring-shaped trap,” Opt. Commun. 119, 85–89 (1995).
    [CrossRef]
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1997 (1)

C. Gabbanini, A. Evangelista, S. Gozzini, A. Lucchesini, A. Fioretti, J. Müller, M. Colla, and E. Arimondo, “Scaling laws in magneto-optical traps,” Europhys. Lett. 37, 251–256 (1997).
[CrossRef]

1996 (1)

S. Grego, M. Colla, A. Fioretti, J. H. Müller, P. Verkerk, and E. Arimondo, “A cesium magneto-optical trap for cold-collision studies,” Opt. Commun. 132, 519–526 (1996).
[CrossRef]

1995 (5)

M. T. de Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, “Time of flight technique to determine temperature of atoms in a ring-shaped trap,” Opt. Commun. 119, 85–89 (1995).
[CrossRef]

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[CrossRef] [PubMed]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[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]

G. Hillenbrand, K. Burnett, and C. J. Foot, “Effect of scattered radiation on sub-Doppler cooling,” Phys. Rev. A 52, 4763–4786 (1995).
[CrossRef] [PubMed]

1994 (7)

K. Ellinger, J. Cooper, and P. Zoller, “Light-pressure force in N-atom systems,” Phys. Rev. A 49, 3909–3933 (1994).
[CrossRef] [PubMed]

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[CrossRef]

M. Gajda and J. Mostowski, “Three-dimensional theory of the magneto-optical trap: Doppler cooling in the low-intensity limit,” Phys. Rev. A 49, 4864–4875 (1994).
[CrossRef] [PubMed]

M. Drewsen, P. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in a cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[CrossRef]

G. Hillenbrand, C. J. Foot, and K. Burnett, “Heating due to long-range photon exchange interactions between cold atoms,” Phys. Rev. A 50, 1479–1489 (1994).
[CrossRef] [PubMed]

D. Hoffman, P. Feng, and T. Walker, “Measurement of Rb trap-loss collision spectra,” J. Opt. Soc. Am. B 11, 712–720 (1994).
[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]

1993 (2)

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

V. S. Bagnato, L. G. Marcassa, M. Oria, 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 (3)

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).

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]

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]

1991 (2)

D. W. Sesko, T. G. Walker, and C. E. Wieman, “Behavior of neutral atoms in a spontaneous force trap,” J. Opt. Soc. Am. B 8, 946–958 (1991).
[CrossRef]

A. M. Steane and C. J. Foot, “Laser cooling below the Doppler limit in a magneto-optical trap,” Europhys. Lett. 14, 231–236 (1991).
[CrossRef]

1990 (2)

C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571–1574 (1990).
[CrossRef] [PubMed]

T. Walker, D. Sesko, and C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64, 408–411 (1990).
[CrossRef] [PubMed]

1989 (4)

1988 (2)

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[CrossRef] [PubMed]

J. Dalibard, “Laser cooling of an optically thick gas: the simplest radiation pressure trap?,” Opt. Commun. 68, 203–208 (1988).
[CrossRef]

1987 (1)

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

Anderson, M. H.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef] [PubMed]

Arimondo, E.

C. Gabbanini, A. Evangelista, S. Gozzini, A. Lucchesini, A. Fioretti, J. Müller, M. Colla, and E. Arimondo, “Scaling laws in magneto-optical traps,” Europhys. Lett. 37, 251–256 (1997).
[CrossRef]

S. Grego, M. Colla, A. Fioretti, J. H. Müller, P. Verkerk, and E. Arimondo, “A cesium magneto-optical trap for cold-collision studies,” Opt. Commun. 132, 519–526 (1996).
[CrossRef]

Bagnato, V. S.

M. T. de Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, “Time of flight technique to determine temperature of atoms in a ring-shaped trap,” Opt. Commun. 119, 85–89 (1995).
[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. Oria, 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]

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).

Burnett, K.

G. Hillenbrand, K. Burnett, and C. J. Foot, “Effect of scattered radiation on sub-Doppler cooling,” Phys. Rev. A 52, 4763–4786 (1995).
[CrossRef] [PubMed]

G. Hillenbrand, C. J. Foot, and K. Burnett, “Heating due to long-range photon exchange interactions between cold atoms,” Phys. Rev. A 50, 1479–1489 (1994).
[CrossRef] [PubMed]

Cable, A.

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

Castin, Y.

M. Drewsen, P. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in a cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[CrossRef]

Chowdhury, M.

Chu, S.

P. J. Ungar, D. S. Weiss, E. Riis, and S. Chu, “Optical molasses and multilevel atoms: theory,” J. Opt. Soc. Am. B 6, 2058–2071 (1989).
[CrossRef]

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

Chudesnikov, D. O.

Clairon, A.

M. Drewsen, P. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in a cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[CrossRef]

Cohen-Tannoudji, C.

Colla, M.

C. Gabbanini, A. Evangelista, S. Gozzini, A. Lucchesini, A. Fioretti, J. Müller, M. Colla, and E. Arimondo, “Scaling laws in magneto-optical traps,” Europhys. Lett. 37, 251–256 (1997).
[CrossRef]

S. Grego, M. Colla, A. Fioretti, J. H. Müller, P. Verkerk, and E. Arimondo, “A cesium magneto-optical trap for cold-collision studies,” Opt. Commun. 132, 519–526 (1996).
[CrossRef]

Cooper, C. J.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[CrossRef] [PubMed]

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[CrossRef]

Cooper, J.

K. Ellinger, J. Cooper, and P. Zoller, “Light-pressure force in N-atom systems,” Phys. Rev. A 49, 3909–3933 (1994).
[CrossRef] [PubMed]

Cornell, E. A.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef] [PubMed]

Dalibard, J.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[CrossRef] [PubMed]

J. Dalibard and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,” J. Opt. Soc. Am. B 6, 2023–2045 (1989).
[CrossRef]

J. Dalibard, “Laser cooling of an optically thick gas: the simplest radiation pressure trap?,” Opt. Commun. 68, 203–208 (1988).
[CrossRef]

Davis, K. B.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

de Araujo, M. T.

M. T. de Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, “Time of flight technique to determine temperature of atoms in a ring-shaped trap,” Opt. Commun. 119, 85–89 (1995).
[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]

Drewsen, M.

M. Drewsen, P. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in a cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[CrossRef]

Edwards, N. H.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[CrossRef] [PubMed]

Ellinger, K.

K. Ellinger, J. Cooper, and P. Zoller, “Light-pressure force in N-atom systems,” Phys. Rev. A 49, 3909–3933 (1994).
[CrossRef] [PubMed]

Ensher, J. R.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef] [PubMed]

Evangelista, A.

C. Gabbanini, A. Evangelista, S. Gozzini, A. Lucchesini, A. Fioretti, J. Müller, M. Colla, and E. Arimondo, “Scaling laws in magneto-optical traps,” Europhys. Lett. 37, 251–256 (1997).
[CrossRef]

Feng, P.

Fioretti, A.

C. Gabbanini, A. Evangelista, S. Gozzini, A. Lucchesini, A. Fioretti, J. Müller, M. Colla, and E. Arimondo, “Scaling laws in magneto-optical traps,” Europhys. Lett. 37, 251–256 (1997).
[CrossRef]

S. Grego, M. Colla, A. Fioretti, J. H. Müller, P. Verkerk, and E. Arimondo, “A cesium magneto-optical trap for cold-collision studies,” Opt. Commun. 132, 519–526 (1996).
[CrossRef]

Foot, C. J.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[CrossRef] [PubMed]

G. Hillenbrand, K. Burnett, and C. J. Foot, “Effect of scattered radiation on sub-Doppler cooling,” Phys. Rev. A 52, 4763–4786 (1995).
[CrossRef] [PubMed]

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[CrossRef]

G. Hillenbrand, C. J. Foot, and K. Burnett, “Heating due to long-range photon exchange interactions between cold atoms,” Phys. Rev. A 50, 1479–1489 (1994).
[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]

A. M. Steane and C. J. Foot, “Laser cooling below the Doppler limit in a magneto-optical trap,” Europhys. Lett. 14, 231–236 (1991).
[CrossRef]

Gabbanini, C.

C. Gabbanini, A. Evangelista, S. Gozzini, A. Lucchesini, A. Fioretti, J. Müller, M. Colla, and E. Arimondo, “Scaling laws in magneto-optical traps,” Europhys. Lett. 37, 251–256 (1997).
[CrossRef]

Gajda, M.

M. Gajda and J. Mostowski, “Three-dimensional theory of the magneto-optical trap: Doppler cooling in the low-intensity limit,” Phys. Rev. A 49, 4864–4875 (1994).
[CrossRef] [PubMed]

Gould, P. L.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[CrossRef] [PubMed]

Gozzini, S.

C. Gabbanini, A. Evangelista, S. Gozzini, A. Lucchesini, A. Fioretti, J. Müller, M. Colla, and E. Arimondo, “Scaling laws in magneto-optical traps,” Europhys. Lett. 37, 251–256 (1997).
[CrossRef]

Grego, S.

S. Grego, M. Colla, A. Fioretti, J. H. Müller, P. Verkerk, and E. Arimondo, “A cesium magneto-optical trap for cold-collision studies,” Opt. Commun. 132, 519–526 (1996).
[CrossRef]

Grison, D.

M. Drewsen, P. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in a cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[CrossRef]

Guedes, I.

Hillenbrand, G.

G. Hillenbrand, K. Burnett, and C. J. Foot, “Effect of scattered radiation on sub-Doppler cooling,” Phys. Rev. A 52, 4763–4786 (1995).
[CrossRef] [PubMed]

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[CrossRef]

G. Hillenbrand, C. J. Foot, and K. Burnett, “Heating due to long-range photon exchange interactions between cold atoms,” Phys. Rev. A 50, 1479–1489 (1994).
[CrossRef] [PubMed]

Hoffman, D.

Horowicz, R. J.

M. T. de Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, “Time of flight technique to determine temperature of atoms in a ring-shaped trap,” Opt. Commun. 119, 85–89 (1995).
[CrossRef]

Joffe, M. A.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

Kaiser, R.

M. T. de Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, “Time of flight technique to determine temperature of atoms in a ring-shaped trap,” Opt. Commun. 119, 85–89 (1995).
[CrossRef]

Kazantsev, A. P.

Ketterle, W.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

Laurent, P.

M. Drewsen, P. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in a cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[CrossRef]

Lett, P. D.

P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, “Optical molasses,” J. Opt. Soc. Am. B 6, 2084–2107 (1989).
[CrossRef]

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[CrossRef] [PubMed]

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]

Lucchesini, A.

C. Gabbanini, A. Evangelista, S. Gozzini, A. Lucchesini, A. Fioretti, J. Müller, M. Colla, and E. Arimondo, “Scaling laws in magneto-optical traps,” Europhys. Lett. 37, 251–256 (1997).
[CrossRef]

Marcassa, L. G.

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. Oria, 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]

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).

Martin, A.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

Matthews, M. R.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef] [PubMed]

Metcalf, H. J.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[CrossRef] [PubMed]

Milori, D.

M. T. de Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, “Time of flight technique to determine temperature of atoms in a ring-shaped trap,” Opt. Commun. 119, 85–89 (1995).
[CrossRef]

Milori, D. M. B. P.

Monroe, C.

C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571–1574 (1990).
[CrossRef] [PubMed]

Mostowski, J.

M. Gajda and J. Mostowski, “Three-dimensional theory of the magneto-optical trap: Doppler cooling in the low-intensity limit,” Phys. Rev. A 49, 4864–4875 (1994).
[CrossRef] [PubMed]

Müller, J.

C. Gabbanini, A. Evangelista, S. Gozzini, A. Lucchesini, A. Fioretti, J. Müller, M. Colla, and E. Arimondo, “Scaling laws in magneto-optical traps,” Europhys. Lett. 37, 251–256 (1997).
[CrossRef]

Müller, J. H.

S. Grego, M. Colla, A. Fioretti, J. H. Müller, P. Verkerk, and E. Arimondo, “A cesium magneto-optical trap for cold-collision studies,” Opt. Commun. 132, 519–526 (1996).
[CrossRef]

Nadir, A.

M. Drewsen, P. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in a cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[CrossRef]

Oria, M.

V. S. Bagnato, L. G. Marcassa, M. Oria, 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]

Oriá, M.

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).

Perrin, H.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[CrossRef] [PubMed]

Phillips, W. D.

P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, “Optical molasses,” J. Opt. Soc. Am. B 6, 2084–2107 (1989).
[CrossRef]

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[CrossRef] [PubMed]

Prentiss, M.

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

Pritchard, D. E.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

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

Raab, E. L.

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

Riis, E.

Rink, J.

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[CrossRef]

Robinson, H.

C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571–1574 (1990).
[CrossRef] [PubMed]

Rolston, S. L.

Salomon, C.

M. Drewsen, P. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in a cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[CrossRef]

Santarelli, G.

M. Drewsen, P. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in a cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[CrossRef]

Sesko, D.

T. Walker, D. Sesko, and C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64, 408–411 (1990).
[CrossRef] [PubMed]

Sesko, D. W.

Steane, A. M.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[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]

A. M. Steane and C. J. Foot, “Laser cooling below the Doppler limit in a magneto-optical trap,” Europhys. Lett. 14, 231–236 (1991).
[CrossRef]

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. Oria, 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]

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).

A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, and V. P. Yakovlev, “Scattering, velocity bunching, and self-localization of atoms in a light field,” J. Opt. Soc. Am. B 6, 2130–2139 (1989).
[CrossRef]

Swann, W.

C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571–1574 (1990).
[CrossRef] [PubMed]

Szriftgiser, P.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[CrossRef] [PubMed]

Tanner, C. E.

Townsend, C. G.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[CrossRef] [PubMed]

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[CrossRef]

Tuboy, A.

M. T. de Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, “Time of flight technique to determine temperature of atoms in a ring-shaped trap,” Opt. Commun. 119, 85–89 (1995).
[CrossRef]

Ungar, P. J.

Verkerk, P.

S. Grego, M. Colla, A. Fioretti, J. H. Müller, P. Verkerk, and E. Arimondo, “A cesium magneto-optical trap for cold-collision studies,” Opt. Commun. 132, 519–526 (1996).
[CrossRef]

Vitlina, R.

V. S. Bagnato, L. G. Marcassa, M. Oria, 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]

Walker, T.

D. Hoffman, P. Feng, and T. Walker, “Measurement of Rb trap-loss collision spectra,” J. Opt. Soc. Am. B 11, 712–720 (1994).
[CrossRef]

T. Walker, D. Sesko, and C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64, 408–411 (1990).
[CrossRef] [PubMed]

Walker, T. G.

Watts, R. N.

P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, “Optical molasses,” J. Opt. Soc. Am. B 6, 2084–2107 (1989).
[CrossRef]

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[CrossRef] [PubMed]

Weiss, D. S.

Westbrook, C. I.

P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, “Optical molasses,” J. Opt. Soc. Am. B 6, 2084–2107 (1989).
[CrossRef]

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[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]

T. Walker, D. Sesko, and C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64, 408–411 (1990).
[CrossRef] [PubMed]

C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571–1574 (1990).
[CrossRef] [PubMed]

Wieman, C. E.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef] [PubMed]

D. W. Sesko, T. G. Walker, and C. E. Wieman, “Behavior of neutral atoms in a spontaneous force trap,” J. Opt. Soc. Am. B 8, 946–958 (1991).
[CrossRef]

Yakovlev, V. P.

Zetie, K.

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[CrossRef]

Zetie, K. P.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[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]

M. T. de Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, “Time of flight technique to determine temperature of atoms in a ring-shaped trap,” Opt. Commun. 119, 85–89 (1995).
[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]

V. S. Bagnato, L. G. Marcassa, M. Oria, 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]

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).

Zoller, P.

K. Ellinger, J. Cooper, and P. Zoller, “Light-pressure force in N-atom systems,” Phys. Rev. A 49, 3909–3933 (1994).
[CrossRef] [PubMed]

Appl. Phys. B (1)

M. Drewsen, P. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in a cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[CrossRef]

Europhys. Lett. (3)

A. M. Steane and C. J. Foot, “Laser cooling below the Doppler limit in a magneto-optical trap,” Europhys. Lett. 14, 231–236 (1991).
[CrossRef]

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[CrossRef]

C. Gabbanini, A. Evangelista, S. Gozzini, A. Lucchesini, A. Fioretti, J. Müller, M. Colla, and E. Arimondo, “Scaling laws in magneto-optical traps,” Europhys. Lett. 37, 251–256 (1997).
[CrossRef]

J. Opt. Soc. Am. B (8)

Laser Phys. (1)

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).

Opt. Commun. (3)

J. Dalibard, “Laser cooling of an optically thick gas: the simplest radiation pressure trap?,” Opt. Commun. 68, 203–208 (1988).
[CrossRef]

S. Grego, M. Colla, A. Fioretti, J. H. Müller, P. Verkerk, and E. Arimondo, “A cesium magneto-optical trap for cold-collision studies,” Opt. Commun. 132, 519–526 (1996).
[CrossRef]

M. T. de Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, “Time of flight technique to determine temperature of atoms in a ring-shaped trap,” Opt. Commun. 119, 85–89 (1995).
[CrossRef]

Phys. Rev. A (8)

G. Hillenbrand, C. J. Foot, and K. Burnett, “Heating due to long-range photon exchange interactions between cold atoms,” Phys. Rev. A 50, 1479–1489 (1994).
[CrossRef] [PubMed]

K. Ellinger, J. Cooper, and P. Zoller, “Light-pressure force in N-atom systems,” Phys. Rev. A 49, 3909–3933 (1994).
[CrossRef] [PubMed]

M. Gajda and J. Mostowski, “Three-dimensional theory of the magneto-optical trap: Doppler cooling in the low-intensity limit,” Phys. Rev. A 49, 4864–4875 (1994).
[CrossRef] [PubMed]

G. Hillenbrand, K. Burnett, and C. J. Foot, “Effect of scattered radiation on sub-Doppler cooling,” Phys. Rev. A 52, 4763–4786 (1995).
[CrossRef] [PubMed]

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]

V. S. Bagnato, L. G. Marcassa, M. Oria, 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]

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

T. Walker, D. Sesko, and C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64, 408–411 (1990).
[CrossRef] [PubMed]

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[CrossRef] [PubMed]

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

C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571–1574 (1990).
[CrossRef] [PubMed]

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

Science (1)

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef] [PubMed]

Other (1)

P. L. Gould, P. D. Lett, and W. D. Phillips, “New measurements with optical molasses,” in Laser Spectroscopy VIII, W. Persson and S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), pp. 64–67.

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Figures (8)

Fig. 1
Fig. 1

Typical 2D density distribution, n(x, y), derived from the condition that Fabs is conservative, with n(x, 0)=exp(-x2) and I=1. Along the |x|=|y| arms the density tends to the value n(0, 0)/2=0.5.

Fig. 2
Fig. 2

Typical (a) density and (b) temperature distributions in the variable-temperature regime. The pressure distribution has a similar shape to the density. The distributions were generated from the 1D distribution n(x, 0)=0.3×1011 exp(-kspringx2/2kBT0) cm-3 by use of kspring=5×10-20 Nm-1 and 3kspring/-kabsItot=-1011 cm-3.

Fig. 3
Fig. 3

Spherically symmetric density distributions (solid curves) at constant temperature T0=30 µK with nmax=3kspring/(krerad-kabs)Itot=1011 cm-3, kspring=5×10-20Nm-1 for n(0)=0.3×1011, n(0)=0.6×1011, and n(0)=0.951011 cm-3. The corresponding atom numbers are N=0.48106, N=1.5×106, and N=9.7×106, respectively. Also shown are corresponding distributions in the TL regime (dashed curves).

Fig. 4
Fig. 4

Typical spherically symmetric (a) pressure and (b) temperature for distributions corresponding to a Gaussian density profile given by n(r)=n(0)exp(-kspringr2/2kBT0) with kspring, T0, and nmax as in Fig. 3. The density maxima were n(0)=0.3×1011 (solid curves), n(0)=0.8×1011 (dashed curves), n(0)=1.3×1011 cm-3 (dashed–dotted curves).

Fig. 5
Fig. 5

Racetrack beam alignment typically used to create orbital modes in a MOT. The bold arrows indicate the axes and propagation directions of the four Gaussian laser beams in the z=0 plane. The z beams necessary for confinement in the z direction are not shown.

Fig. 6
Fig. 6

Difference, Fdiff(x)=Ftrap(x)-Fcent(x), between the trapping and the centripetal forces predicted by the nonlinear (solid curve) and the linear (dashed curve) theories for sodium atoms; b=15 G/cm, Δ=-2Γ, w=1 cm, s=1.2 cm, Ω0=0.15Γ. The linear theory predicts that a ring with a radius of 11 mm will be produced, whereas the more-exact theory predicts a radius of 4.6 mm.

Fig. 7
Fig. 7

Results of a MC simulation of a MOT under the same conditions as for Fig. 6. The simulation models 100 atoms released at the origin of a MOT and then allowed to reach equilibrium. The image is a conglomerate of snapshots from various times, where each dot represents an atom. The ring radius is r3 mm.

Fig. 8
Fig. 8

Comparison of the nonlinear (solid curves) and the linear (dashed curves) calculations of the trapping force Ftrap(x) and the required centripetal force Fcent(x). The linear and the nonlinear models for these forces agree at small x values. The same trap parameters are used as in Figs. 6 and 7.

Equations (22)

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

Ix±(r)=Ix expσL xn(u, y, z)du,
Fabsx=-2kabsIx 0x n(u, y, z)du,
Frerad(r)=krerad4πn(r)I(r) r-r|r-r|3d3r,
Frerad(r)=kreradItot 0r n(u)u2dur2.
P(r)=kB[T(r)n(r)+n(r)T(r)]=Ftotn(r).
kBT[ln n(r)]=Fspring+Fabs+Frerad=Ftot.
kBT2[ln n(r)]=·Ftot
=(krerad-kabs)Itotn(r)-3kspring,
Ix y0x n(u, y, z)du=Iy x0y n(x, u, z)du.
N=0j=0N cN-j,j jN-j+1xN-j+1yj-1=I N=0j=0N cN-j,j N-jj+1xN-j-1yj+1,
c2(l-k),2k=Ik2l2kc2l,0.
n(x, y)=l=0k=0lIk2l2kc2l,0x2(l-k)y2k.
n(x, x/I)=c0,0+12l=1c2l,0(2x)2l=½n(0, 0)+½n(2x, 0).
×[(Fspring+Fabs+Frerad)n(r)]=0.
Fabs-kabskreradFrerad=-σLσRFrerad.
Ftot(r)=-kspringr+(krerad-kabs)Itot 0r n(u)u2dur2.
dP(r)dr=kB d[n(r)T(r)]dr=-kspringr+(krerad-kabs)×Itot 0r n(u)u2dur2n(r).
kBT03kspringn(r)n(r)-1n2(r)[n(r)]2+2rn(r)n(r)
=-1+n(r)nmax,
RV=|Ftot/kB-T0[ln n(r)]|Max{|Ftot|/kB, |T0[ln n(r)]|}.
Ftrap=kLΩ02Γes2/w211+Itot(x)IS+2Γ2(Δ+ωBbx)2-11+Itot(x)IS+2Γ2(Δ-ωBbx)2,
Fy=kLΩ02Γexp[-(x+s)2/w2]1+Itot(x)IS+2Γ2(Δ-kLvy)2-exp[-(x-s)2/w2]1+Itot(x)IS+2Γ2(Δ+kLvy)2.

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