Abstract

An overview of the theory of the magneto-optical trap is presented, along with measurements of the effect of an imbalance in the intensities of the trapping beams. This investigation tests the theory of the spring constant of the trap and confirms that the confining force at the center of the trap results from an induced orientation of the atomic ground state. The experimental results give the magnitude of this force, which has not yet been calculated accurately. We calculate the radiation field in the three-dimensional molasses, finding that the relative time phase of the orthogonal standing waves is significant, and we give some insight into the phenomenon of interference fringes when the beams are misaligned. We also discuss the limitation of the trapped atomic density resulting from photon scattering within the cloud, predicting that densities above 1013 atoms/cm3 could be achieved in a trap operating at low saturation of the atomic transition. Finally, we briefly consider collisional loss at low densities, finding an especially large contribution from resonant dipole–dipole scattering.

© 1992 Optical Society of America

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  1. E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631 (1987).
    [CrossRef] [PubMed]
  2. S. Chu, C. Wieman, eds., feature on laser cooling and trapping of atoms, J. Opt. Soc. Am. B6, 2019–2278 (1989).
  3. A. Steane, C. Foot, “Laser Cooling below the Doppler limit in a magneto-optical trap,” Europhys. Lett. 14, 231 (1991).
    [CrossRef]
  4. D. Sesko, T. Walker, C. Wieman, “Behavior of neutral atoms in a spontaneous force trap,” J. Opt. Soc. Am. B 8, 946 (1991).
    [CrossRef]
  5. K. Molmer, “Friction and diffusion coefficients for cooling of atoms in laser fields with multidimensional periodicity,” Phys. Rev. A 44, 5820 (1991).
    [CrossRef] [PubMed]
  6. N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, New York, 1954), Eq. (297), p. 40.
  7. J. Nellessen, J. Werner, W. Ertmer, “Magneto-optical compression of a monoenergetic sodium atomic beam,” Opt. Commun. 78, 300 (1990).
    [CrossRef]
  8. D. Sesko, T. Walker, C. Monroe, A. Gallagher, C. Wieman, “Collisional losses from a light-force atom trap,” Phys. Rev. Lett. 63, 961 (1989).
    [CrossRef] [PubMed]
  9. C. Monroe, W. Swann, H. Robinson, C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571 (1990).
    [CrossRef] [PubMed]
  10. H. Metcalf, “Magneto-optical trapping and its application to helium metastables,” J. Opt. Soc. Am. B 6, 2206 (1989).
    [CrossRef]
  11. E. A. Power, Introductory Quantum Electronics (American Elsevier, New York, 1965).
  12. M. Prentiss, A. Cable, J. E. Bjorkholm, S. Chu, E. L. Raab, D. Pritchard, “Atomic-density-dependent losses in an optical trap,” Opt. Lett. 13, 452 (1988).
    [CrossRef] [PubMed]
  13. A. Cable, M. Prentiss, N. P. Bigelow, “Observations of sodium atoms in a magnetic molasses trap loaded by a continuous uncooled source,” Opt. Lett. 15, 507 (1990).
    [CrossRef] [PubMed]
  14. E. L. Raab, “Trapping sodium with light,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1988).
  15. P. Julienne, F. Mies, “Collisions of ultracold trapped atoms,” J. Opt. Soc. Am. B 6, 2257 (1989).
    [CrossRef]
  16. H. S. W. Massey, Electronic and Ionic Impact Phenomena (Oxford U. Press, London, 1971), Vol. 3.
  17. A. Corney, Atomic and Laser Spectroscopy (Oxford U. Press, London, 1977), Chap. 8.
  18. T. Walker, D. Sesko, C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64, 408 (1990).
    [CrossRef] [PubMed]
  19. J. Dalibard, “Laser cooling of an optically thick gas: the simplest radiation pressure trap?” Opt. Commun. 68, 203 (1988).
    [CrossRef]
  20. R. Loudon, The Quantum Theory of Light (Oxford U. Press, London, 1983), Chap. 8.
  21. B. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 188, 1969 (1969).
    [CrossRef]
  22. B. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217 (1972).
    [CrossRef]
  23. W. Phillips, H. Metcalf, “Laser deceleration of an atomic beam,” Phys. Rev. Lett. 48, 596 (1982).
    [CrossRef]
  24. P. Lett, W. Phillips, S. Rolston, C. Tanner, R. Watts, C. Westbrook, “Optical molasses,” J. Opt. Soc. Am. B 6, 2084 (1989).
    [CrossRef]
  25. E. Riis, D. Weiss, K. Moler, S. Chu, “Atom funnel for the production of a slow, high-density atomic beam,” Phys. Rev. Lett. 64, 1658 (1990).
    [CrossRef] [PubMed]
  26. J. Dalibard, C. Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,” J. Opt. Soc. Am. B 6, 2023 (1989).
    [CrossRef]
  27. B. Sheehy, S.-Q. Shang, P. van der Straten, S. Hatamian, H. Metcalf, “Magnetic-field induced laser cooling below the Doppler limit,” Phys. Rev. Lett. 64, 858 (1990).
    [CrossRef] [PubMed]
  28. S.-Q. Shang, B. Sheehy, P. van der Straten, H. Metcalf, “Velocity-selective magnetic resonance laser cooling,” Phys. Rev. Lett. 65, 317 (1990).
    [CrossRef] [PubMed]
  29. S.-Q. Shang, B. Sheehy, H. Metcalf, “Velocity-selective resonances and sub-Doppler laser cooling,” Phys. Rev. Lett. 67, 1094 (1991).
    [CrossRef] [PubMed]
  30. We use this expression following Refs. 28 and 29, although the name is slightly misleading, since there is no oscillating magnetic field: The mathematical description is merely analagous to a magnetic resonance.
  31. A. Hemmerich, D. Schropp, T. Hänsch, “Light forces in two crossed standing waves with controlled time phase difference,” Phys. Rev. A 44, 1910 (1991).
    [CrossRef] [PubMed]
  32. N. Bigelow, M. Prentiss, “Observation of channeling of atoms in the three-dimensional interference pattern of optical standing waves,” Phys. Rev. Lett. 65, 29 (1990).
    [CrossRef] [PubMed]
  33. N. Bigelow, M. Prentiss, “Decreased damping of ultracold atoms in optical molasses: predictions and a possible solution,” Opt. Lett. 15, 1479 (1990).
    [CrossRef] [PubMed]
  34. G. Nienhuis, P. van der Straten, S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A 44, 462 (1991);A. Steane, G. Hillenbrand, C. Foot, “Polarization gradient cooling in a one-dimensional σ+–σ− configuration for any atomic transition,” J. Phys. B (to be published).
    [CrossRef] [PubMed]
  35. T. Bergman, G. Erez, H. Metcalf, “Magnetostatic trapping fields for neutral atoms,” Phys. Rev. A 35, 1535 (1987).N. B.: There is an error in Table II. Bρ for n= 3 is missing a z and should read −3ρz2/2 + 3ρ2z/8.
    [CrossRef]
  36. C. Wieman, L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1 (1991).
    [CrossRef]
  37. “Pressure measurement and electron beam guns,” in Vacuum Generators Product Manual (Vacuum Generators Ltd., Hastings, UK, 1991), Sec. 07;S. Dushman, Scientific Foundations of Vacuum Technique (Wiley, New York, 1949).
  38. C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, S. Guellati, “Laser cooling of cesium atoms below 3 μK,” Europhys. Lett. 12, 683 (1990).
    [CrossRef]

1991 (7)

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

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

K. Molmer, “Friction and diffusion coefficients for cooling of atoms in laser fields with multidimensional periodicity,” Phys. Rev. A 44, 5820 (1991).
[CrossRef] [PubMed]

S.-Q. Shang, B. Sheehy, H. Metcalf, “Velocity-selective resonances and sub-Doppler laser cooling,” Phys. Rev. Lett. 67, 1094 (1991).
[CrossRef] [PubMed]

A. Hemmerich, D. Schropp, T. Hänsch, “Light forces in two crossed standing waves with controlled time phase difference,” Phys. Rev. A 44, 1910 (1991).
[CrossRef] [PubMed]

G. Nienhuis, P. van der Straten, S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A 44, 462 (1991);A. Steane, G. Hillenbrand, C. Foot, “Polarization gradient cooling in a one-dimensional σ+–σ− configuration for any atomic transition,” J. Phys. B (to be published).
[CrossRef] [PubMed]

C. Wieman, L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1 (1991).
[CrossRef]

1990 (10)

C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, S. Guellati, “Laser cooling of cesium atoms below 3 μK,” Europhys. Lett. 12, 683 (1990).
[CrossRef]

B. Sheehy, S.-Q. Shang, P. van der Straten, S. Hatamian, H. Metcalf, “Magnetic-field induced laser cooling below the Doppler limit,” Phys. Rev. Lett. 64, 858 (1990).
[CrossRef] [PubMed]

S.-Q. Shang, B. Sheehy, P. van der Straten, H. Metcalf, “Velocity-selective magnetic resonance laser cooling,” Phys. Rev. Lett. 65, 317 (1990).
[CrossRef] [PubMed]

N. Bigelow, M. Prentiss, “Observation of channeling of atoms in the three-dimensional interference pattern of optical standing waves,” Phys. Rev. Lett. 65, 29 (1990).
[CrossRef] [PubMed]

N. Bigelow, M. Prentiss, “Decreased damping of ultracold atoms in optical molasses: predictions and a possible solution,” Opt. Lett. 15, 1479 (1990).
[CrossRef] [PubMed]

J. Nellessen, J. Werner, W. Ertmer, “Magneto-optical compression of a monoenergetic sodium atomic beam,” Opt. Commun. 78, 300 (1990).
[CrossRef]

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

A. Cable, M. Prentiss, N. P. Bigelow, “Observations of sodium atoms in a magnetic molasses trap loaded by a continuous uncooled source,” Opt. Lett. 15, 507 (1990).
[CrossRef] [PubMed]

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

E. Riis, D. Weiss, K. Moler, S. Chu, “Atom funnel for the production of a slow, high-density atomic beam,” Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

1989 (5)

1988 (2)

M. Prentiss, A. Cable, J. E. Bjorkholm, S. Chu, E. L. Raab, D. Pritchard, “Atomic-density-dependent losses in an optical trap,” Opt. Lett. 13, 452 (1988).
[CrossRef] [PubMed]

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

1987 (2)

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

T. Bergman, G. Erez, H. Metcalf, “Magnetostatic trapping fields for neutral atoms,” Phys. Rev. A 35, 1535 (1987).N. B.: There is an error in Table II. Bρ for n= 3 is missing a z and should read −3ρz2/2 + 3ρ2z/8.
[CrossRef]

1982 (1)

W. Phillips, H. Metcalf, “Laser deceleration of an atomic beam,” Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

1972 (1)

B. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217 (1972).
[CrossRef]

1969 (1)

B. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 188, 1969 (1969).
[CrossRef]

Bergman, T.

T. Bergman, G. Erez, H. Metcalf, “Magnetostatic trapping fields for neutral atoms,” Phys. Rev. A 35, 1535 (1987).N. B.: There is an error in Table II. Bρ for n= 3 is missing a z and should read −3ρz2/2 + 3ρ2z/8.
[CrossRef]

Bigelow, N.

N. Bigelow, M. Prentiss, “Observation of channeling of atoms in the three-dimensional interference pattern of optical standing waves,” Phys. Rev. Lett. 65, 29 (1990).
[CrossRef] [PubMed]

N. Bigelow, M. Prentiss, “Decreased damping of ultracold atoms in optical molasses: predictions and a possible solution,” Opt. Lett. 15, 1479 (1990).
[CrossRef] [PubMed]

Bigelow, N. P.

Bjorkholm, J. E.

Cable, A.

Chu, S.

E. Riis, D. Weiss, K. Moler, S. Chu, “Atom funnel for the production of a slow, high-density atomic beam,” Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

M. Prentiss, A. Cable, J. E. Bjorkholm, S. Chu, E. L. Raab, D. Pritchard, “Atomic-density-dependent losses in an optical trap,” Opt. Lett. 13, 452 (1988).
[CrossRef] [PubMed]

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

Clairon, A.

C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, S. Guellati, “Laser cooling of cesium atoms below 3 μK,” Europhys. Lett. 12, 683 (1990).
[CrossRef]

Cohen-Tannoudji, C.

Corney, A.

A. Corney, Atomic and Laser Spectroscopy (Oxford U. Press, London, 1977), Chap. 8.

Dalibard, J.

C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, S. Guellati, “Laser cooling of cesium atoms below 3 μK,” Europhys. Lett. 12, 683 (1990).
[CrossRef]

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

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

Erez, G.

T. Bergman, G. Erez, H. Metcalf, “Magnetostatic trapping fields for neutral atoms,” Phys. Rev. A 35, 1535 (1987).N. B.: There is an error in Table II. Bρ for n= 3 is missing a z and should read −3ρz2/2 + 3ρ2z/8.
[CrossRef]

Ertmer, W.

J. Nellessen, J. Werner, W. Ertmer, “Magneto-optical compression of a monoenergetic sodium atomic beam,” Opt. Commun. 78, 300 (1990).
[CrossRef]

Foot, C.

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

Gallagher, A.

D. Sesko, T. Walker, C. Monroe, A. Gallagher, C. Wieman, “Collisional losses from a light-force atom trap,” Phys. Rev. Lett. 63, 961 (1989).
[CrossRef] [PubMed]

Guellati, S.

C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, S. Guellati, “Laser cooling of cesium atoms below 3 μK,” Europhys. Lett. 12, 683 (1990).
[CrossRef]

Hänsch, T.

A. Hemmerich, D. Schropp, T. Hänsch, “Light forces in two crossed standing waves with controlled time phase difference,” Phys. Rev. A 44, 1910 (1991).
[CrossRef] [PubMed]

Hatamian, S.

B. Sheehy, S.-Q. Shang, P. van der Straten, S. Hatamian, H. Metcalf, “Magnetic-field induced laser cooling below the Doppler limit,” Phys. Rev. Lett. 64, 858 (1990).
[CrossRef] [PubMed]

Hemmerich, A.

A. Hemmerich, D. Schropp, T. Hänsch, “Light forces in two crossed standing waves with controlled time phase difference,” Phys. Rev. A 44, 1910 (1991).
[CrossRef] [PubMed]

Hollberg, L.

C. Wieman, L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1 (1991).
[CrossRef]

Julienne, P.

Lett, P.

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, London, 1983), Chap. 8.

Massey, H. S. W.

H. S. W. Massey, Electronic and Ionic Impact Phenomena (Oxford U. Press, London, 1971), Vol. 3.

Metcalf, H.

S.-Q. Shang, B. Sheehy, H. Metcalf, “Velocity-selective resonances and sub-Doppler laser cooling,” Phys. Rev. Lett. 67, 1094 (1991).
[CrossRef] [PubMed]

B. Sheehy, S.-Q. Shang, P. van der Straten, S. Hatamian, H. Metcalf, “Magnetic-field induced laser cooling below the Doppler limit,” Phys. Rev. Lett. 64, 858 (1990).
[CrossRef] [PubMed]

S.-Q. Shang, B. Sheehy, P. van der Straten, H. Metcalf, “Velocity-selective magnetic resonance laser cooling,” Phys. Rev. Lett. 65, 317 (1990).
[CrossRef] [PubMed]

H. Metcalf, “Magneto-optical trapping and its application to helium metastables,” J. Opt. Soc. Am. B 6, 2206 (1989).
[CrossRef]

T. Bergman, G. Erez, H. Metcalf, “Magnetostatic trapping fields for neutral atoms,” Phys. Rev. A 35, 1535 (1987).N. B.: There is an error in Table II. Bρ for n= 3 is missing a z and should read −3ρz2/2 + 3ρ2z/8.
[CrossRef]

W. Phillips, H. Metcalf, “Laser deceleration of an atomic beam,” Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

Mies, F.

Moler, K.

E. Riis, D. Weiss, K. Moler, S. Chu, “Atom funnel for the production of a slow, high-density atomic beam,” Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

Mollow, B.

B. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217 (1972).
[CrossRef]

B. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 188, 1969 (1969).
[CrossRef]

Molmer, K.

K. Molmer, “Friction and diffusion coefficients for cooling of atoms in laser fields with multidimensional periodicity,” Phys. Rev. A 44, 5820 (1991).
[CrossRef] [PubMed]

Monroe, C.

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

D. Sesko, T. Walker, C. Monroe, A. Gallagher, C. Wieman, “Collisional losses from a light-force atom trap,” Phys. Rev. Lett. 63, 961 (1989).
[CrossRef] [PubMed]

Nellessen, J.

J. Nellessen, J. Werner, W. Ertmer, “Magneto-optical compression of a monoenergetic sodium atomic beam,” Opt. Commun. 78, 300 (1990).
[CrossRef]

Nienhuis, G.

G. Nienhuis, P. van der Straten, S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A 44, 462 (1991);A. Steane, G. Hillenbrand, C. Foot, “Polarization gradient cooling in a one-dimensional σ+–σ− configuration for any atomic transition,” J. Phys. B (to be published).
[CrossRef] [PubMed]

Phillips, W.

P. Lett, W. Phillips, S. Rolston, C. Tanner, R. Watts, C. Westbrook, “Optical molasses,” J. Opt. Soc. Am. B 6, 2084 (1989).
[CrossRef]

W. Phillips, H. Metcalf, “Laser deceleration of an atomic beam,” Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

Phillips, W. D.

C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, S. Guellati, “Laser cooling of cesium atoms below 3 μK,” Europhys. Lett. 12, 683 (1990).
[CrossRef]

Power, E. A.

E. A. Power, Introductory Quantum Electronics (American Elsevier, New York, 1965).

Prentiss, M.

Pritchard, D.

M. Prentiss, A. Cable, J. E. Bjorkholm, S. Chu, E. L. Raab, D. Pritchard, “Atomic-density-dependent losses in an optical trap,” Opt. Lett. 13, 452 (1988).
[CrossRef] [PubMed]

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

Raab, E. L.

M. Prentiss, A. Cable, J. E. Bjorkholm, S. Chu, E. L. Raab, D. Pritchard, “Atomic-density-dependent losses in an optical trap,” Opt. Lett. 13, 452 (1988).
[CrossRef] [PubMed]

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

E. L. Raab, “Trapping sodium with light,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1988).

Riis, E.

E. Riis, D. Weiss, K. Moler, S. Chu, “Atom funnel for the production of a slow, high-density atomic beam,” Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

Robinson, H.

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

Rolston, S.

Salomon, C.

C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, S. Guellati, “Laser cooling of cesium atoms below 3 μK,” Europhys. Lett. 12, 683 (1990).
[CrossRef]

Schropp, D.

A. Hemmerich, D. Schropp, T. Hänsch, “Light forces in two crossed standing waves with controlled time phase difference,” Phys. Rev. A 44, 1910 (1991).
[CrossRef] [PubMed]

Sesko, D.

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

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

D. Sesko, T. Walker, C. Monroe, A. Gallagher, C. Wieman, “Collisional losses from a light-force atom trap,” Phys. Rev. Lett. 63, 961 (1989).
[CrossRef] [PubMed]

Shang, S.-Q.

S.-Q. Shang, B. Sheehy, H. Metcalf, “Velocity-selective resonances and sub-Doppler laser cooling,” Phys. Rev. Lett. 67, 1094 (1991).
[CrossRef] [PubMed]

G. Nienhuis, P. van der Straten, S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A 44, 462 (1991);A. Steane, G. Hillenbrand, C. Foot, “Polarization gradient cooling in a one-dimensional σ+–σ− configuration for any atomic transition,” J. Phys. B (to be published).
[CrossRef] [PubMed]

B. Sheehy, S.-Q. Shang, P. van der Straten, S. Hatamian, H. Metcalf, “Magnetic-field induced laser cooling below the Doppler limit,” Phys. Rev. Lett. 64, 858 (1990).
[CrossRef] [PubMed]

S.-Q. Shang, B. Sheehy, P. van der Straten, H. Metcalf, “Velocity-selective magnetic resonance laser cooling,” Phys. Rev. Lett. 65, 317 (1990).
[CrossRef] [PubMed]

Sheehy, B.

S.-Q. Shang, B. Sheehy, H. Metcalf, “Velocity-selective resonances and sub-Doppler laser cooling,” Phys. Rev. Lett. 67, 1094 (1991).
[CrossRef] [PubMed]

S.-Q. Shang, B. Sheehy, P. van der Straten, H. Metcalf, “Velocity-selective magnetic resonance laser cooling,” Phys. Rev. Lett. 65, 317 (1990).
[CrossRef] [PubMed]

B. Sheehy, S.-Q. Shang, P. van der Straten, S. Hatamian, H. Metcalf, “Magnetic-field induced laser cooling below the Doppler limit,” Phys. Rev. Lett. 64, 858 (1990).
[CrossRef] [PubMed]

Steane, A.

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

Swann, W.

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

Tanner, C.

van der Straten, P.

G. Nienhuis, P. van der Straten, S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A 44, 462 (1991);A. Steane, G. Hillenbrand, C. Foot, “Polarization gradient cooling in a one-dimensional σ+–σ− configuration for any atomic transition,” J. Phys. B (to be published).
[CrossRef] [PubMed]

S.-Q. Shang, B. Sheehy, P. van der Straten, H. Metcalf, “Velocity-selective magnetic resonance laser cooling,” Phys. Rev. Lett. 65, 317 (1990).
[CrossRef] [PubMed]

B. Sheehy, S.-Q. Shang, P. van der Straten, S. Hatamian, H. Metcalf, “Magnetic-field induced laser cooling below the Doppler limit,” Phys. Rev. Lett. 64, 858 (1990).
[CrossRef] [PubMed]

Walker, T.

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

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

D. Sesko, T. Walker, C. Monroe, A. Gallagher, C. Wieman, “Collisional losses from a light-force atom trap,” Phys. Rev. Lett. 63, 961 (1989).
[CrossRef] [PubMed]

Watts, R.

Wax, N.

N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, New York, 1954), Eq. (297), p. 40.

Weiss, D.

E. Riis, D. Weiss, K. Moler, S. Chu, “Atom funnel for the production of a slow, high-density atomic beam,” Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

Werner, J.

J. Nellessen, J. Werner, W. Ertmer, “Magneto-optical compression of a monoenergetic sodium atomic beam,” Opt. Commun. 78, 300 (1990).
[CrossRef]

Westbrook, C.

Wieman, C.

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

C. Wieman, L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1 (1991).
[CrossRef]

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

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

D. Sesko, T. Walker, C. Monroe, A. Gallagher, C. Wieman, “Collisional losses from a light-force atom trap,” Phys. Rev. Lett. 63, 961 (1989).
[CrossRef] [PubMed]

Europhys. Lett. (2)

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

C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, S. Guellati, “Laser cooling of cesium atoms below 3 μK,” Europhys. Lett. 12, 683 (1990).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (2)

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

J. Nellessen, J. Werner, W. Ertmer, “Magneto-optical compression of a monoenergetic sodium atomic beam,” Opt. Commun. 78, 300 (1990).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. (1)

B. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 188, 1969 (1969).
[CrossRef]

Phys. Rev. A (5)

B. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217 (1972).
[CrossRef]

G. Nienhuis, P. van der Straten, S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A 44, 462 (1991);A. Steane, G. Hillenbrand, C. Foot, “Polarization gradient cooling in a one-dimensional σ+–σ− configuration for any atomic transition,” J. Phys. B (to be published).
[CrossRef] [PubMed]

T. Bergman, G. Erez, H. Metcalf, “Magnetostatic trapping fields for neutral atoms,” Phys. Rev. A 35, 1535 (1987).N. B.: There is an error in Table II. Bρ for n= 3 is missing a z and should read −3ρz2/2 + 3ρ2z/8.
[CrossRef]

A. Hemmerich, D. Schropp, T. Hänsch, “Light forces in two crossed standing waves with controlled time phase difference,” Phys. Rev. A 44, 1910 (1991).
[CrossRef] [PubMed]

K. Molmer, “Friction and diffusion coefficients for cooling of atoms in laser fields with multidimensional periodicity,” Phys. Rev. A 44, 5820 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (10)

D. Sesko, T. Walker, C. Monroe, A. Gallagher, C. Wieman, “Collisional losses from a light-force atom trap,” Phys. Rev. Lett. 63, 961 (1989).
[CrossRef] [PubMed]

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

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

N. Bigelow, M. Prentiss, “Observation of channeling of atoms in the three-dimensional interference pattern of optical standing waves,” Phys. Rev. Lett. 65, 29 (1990).
[CrossRef] [PubMed]

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

W. Phillips, H. Metcalf, “Laser deceleration of an atomic beam,” Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

E. Riis, D. Weiss, K. Moler, S. Chu, “Atom funnel for the production of a slow, high-density atomic beam,” Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

B. Sheehy, S.-Q. Shang, P. van der Straten, S. Hatamian, H. Metcalf, “Magnetic-field induced laser cooling below the Doppler limit,” Phys. Rev. Lett. 64, 858 (1990).
[CrossRef] [PubMed]

S.-Q. Shang, B. Sheehy, P. van der Straten, H. Metcalf, “Velocity-selective magnetic resonance laser cooling,” Phys. Rev. Lett. 65, 317 (1990).
[CrossRef] [PubMed]

S.-Q. Shang, B. Sheehy, H. Metcalf, “Velocity-selective resonances and sub-Doppler laser cooling,” Phys. Rev. Lett. 67, 1094 (1991).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

C. Wieman, L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1 (1991).
[CrossRef]

Other (9)

“Pressure measurement and electron beam guns,” in Vacuum Generators Product Manual (Vacuum Generators Ltd., Hastings, UK, 1991), Sec. 07;S. Dushman, Scientific Foundations of Vacuum Technique (Wiley, New York, 1949).

We use this expression following Refs. 28 and 29, although the name is slightly misleading, since there is no oscillating magnetic field: The mathematical description is merely analagous to a magnetic resonance.

S. Chu, C. Wieman, eds., feature on laser cooling and trapping of atoms, J. Opt. Soc. Am. B6, 2019–2278 (1989).

N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, New York, 1954), Eq. (297), p. 40.

E. L. Raab, “Trapping sodium with light,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1988).

E. A. Power, Introductory Quantum Electronics (American Elsevier, New York, 1965).

R. Loudon, The Quantum Theory of Light (Oxford U. Press, London, 1983), Chap. 8.

H. S. W. Massey, Electronic and Ionic Impact Phenomena (Oxford U. Press, London, 1971), Vol. 3.

A. Corney, Atomic and Laser Spectroscopy (Oxford U. Press, London, 1977), Chap. 8.

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Figures (10)

Fig. 1
Fig. 1

Geometry of a collision in which the fast atom is almost unperturbed.

Fig. 2
Fig. 2

Presence of Sisyphus cooling in the trap. (a)–(c) show the parameter p defined in the text. The calculation added six plane waves of unit amplitude, each pair having opposite circular polarizations, as for a trap. (a) Aligned beams, all at the same phase angle. (b) Aligned beams, with the x beam pair shifted in phase by π/2 rad with respect to the others. (c) As in (a) but with the x beams misaligned in the xy plane, one by +0.05 rad, the other by −0.05 rad. Each graph shows the variation of p along a set of four lines parallel to the y axis (which is also the quantization axis), the lines being separated in the x direction by a quarter of a wavelength. For some cases p = 0 is obtained for all y. (d) Intensity for the misaligned-beams case. The wavelength-scale intensity fluctuations occur along the same lines for which circular polarization is present but at different y positions. All the graphs are at z = 0. At z = +0.25 wavelengths, we still find variations in p similar to those shown in (c), but the intensity in (d) fluctuates between approximately 2 and 10 for all x and y. The wavelength-averaged optical potential has an almost uniform depth throughout the trap.

Fig. 3
Fig. 3

Magnetic-field lines and contours near the center of the trap. The field lines satisfy dz/dρ = Bz/Bp = −2z/ρ. The contours of constant field are ellipses.

Fig. 4
Fig. 4

Various significant radii in the trap. Top, the radius of the cloud, as given by relation (57) (3D) and by a calculation in which all the parameters are taken from the one-dimensional induced-orientation theory (1D). Bottom, radii at which the Larmor frequency is equal to the optical pumping rate and to the light shift. Note the different vertical scales. The example given is for Ω = Γ/2 and a field gradient of 10 G/cm.

Fig. 5
Fig. 5

Trapping experiments. Top: GL1, GL2, grating-stabilized lasers tuned to F = 4 → 5 and F = 3 → 4 transitions, respectively; IL, injection-locked laser; D, frequency diagnostics (saturated absorption cell); arrowed boxes, optical isolators. Bottom: BE, beam expander; H’s, half-wave plates; Q’s, quarter-wave plates; V, vertical beams; CCD, video camera. The diagram is not exactly to scale, but rough dimensions are indicated.

Fig. 6
Fig. 6

Inverse of the trap lifetime as a function of background pressure. The trapping light was at 6-MHz detuning, 7 mW/cm2 per beam. The experimental uncertainty in the lifetime measurements is indicated by the size of the symbols. The pressure given is that indicated by the ion gauge (calibrated for N2). If the background gas is cesium, the actual pressure is less than that indicated by the gauge.

Fig. 7
Fig. 7

Imbalance measurements. All the data are shown. The vertical axis shows the magnetic field, in gauss, to which the trapped cloud moves, per unit intensity imbalance w in the vertical beams. The straight line is a fit to the data points for which Ω2/δΓ < 0.4. The symbols indicate the intensity per beam as follows: I/IS = 2.3, ●; 2, ◊; 1.7, ○; 1.4, □; 1, ▲; 0.7, ∇; 0.55, Δ.

Fig. 8
Fig. 8

Imbalance measurements as in Fig. 7, rescaled by the expected detuning dependence in order to reveal the dependence on the average intensity per trapping beam. Only the data points for detunings larger than 10 MHz and Ω2/δΓ smaller than 0.4 are included. The line is a linear fit to the data.

Fig. 9
Fig. 9

Imbalance measurements. The data are as for Fig. 8, rescaled in order to reveal the dependence on detuning. The line is a linear fit.

Fig. 10
Fig. 10

Imbalance measurements, showing the data at small values of the light shift and detunings larger than 10 MHz.

Tables (1)

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Table 1 Estimates of Cross Section σ from Measurements of Lifetime τ

Equations (74)

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f i ( r , v ) = j κ i j r j j α i j υ j for small r , υ .
k B T = D 1 / α .
Number of atoms in the trap N , Number density of atoms n , Lifetime for an atom to remain in the trap τ , Maximum velocity for an atom to be captured by the trap υ c , Escape velocity for an atom in the center of the trap υ esc .
s = Ω 2 / 2 δ 2 + Γ 2 / 4 .
d N d t = R N τ β N 2 V .
N ( t ) = [ N ( 0 ) R τ ] exp ( t / τ ) + R τ .
N = R τ 1 5 ( υ c υ ¯ ) 4 A σ ,
C 3 R 3 = | d | 2 4 π 0 R 3
C 3 = 3 4 ( λ 2 π ) 3 Γ
= 6.56 × 10 48 Jm 3 for cesium ,
τ = ( π m k B T / 8 ) 1 / 2 σ P .
σ = f ( n ) ( C n / υ ) 2 / n 1 ,
Δ p = F d t ,
F = n C n R n + 1 , R = b cos θ , tan θ = υ t b .
Δ p = π / 2 π / 2 n C n υ b n ( cos n θ ) d θ
= f ( n ) C n υ b n [ f ( 3 ) = 4 , f ( 6 ) = 15 π 8 ]
σ = π ( f ( n ) C n m υ esc υ ) 2 / n .
½ k B T i = ½ κ i i r i 2 .
F R = k Γ 2 s ( s + 1 ) σ R 4 π r 2 r ̂
= I σ L c σ R 4 π r 2 r ̂ ,
σ L = ω L Γ 2 s I ( s + 1 )
= the cross section for absorption of photons from the laser field .
F A = I σ L 2 c 4 π r 2 r ̂ .
κ r = 4 3 π r 3 n I σ L 2 ( σ R / σ L 1 ) c 4 π r 2
n = 3 κ c I σ L 2 ( σ R / σ L 1 ) .
I el I el + I inel = 1 s + 1 .
δ 2 + Γ 2 / 4 + Ω 2 / 2 Γ 2 / 4 + Ω 2 / 2 σ L = s + 1 s + Γ 2 / ( 4 δ 2 + Γ 2 ) σ L .
σ R 1 s + 1 σ L + ( 1 1 s + 1 ) s + 1 s + Γ 2 / ( 4 δ 2 + Γ 2 ) σ L
σ R σ L 1 ( s s + 1 ) δ 2 s ( δ 2 + Γ 2 / 4 ) + Γ 2 / 4 .
z = exp ( α 2 m t ) [ A exp ( β 2 m t ) + B exp ( β 2 m t ) ] ,
β = ( α 2 4 m κ ) 1 / 2 ,
A B = β + α β α
β = 0 κ / α = α / 4 m .
2 m α β α κ ( 1 m κ α 2 + ) .
α υ κ z = α [ υ υ ( z ) ] ,
υ ( z ) = ( κ / α ) z .
g μ B B > Γ s ,
E ( r ) n = E n n exp ( i k n · r + ϕ n ) ,
ξ ± = E T · [ ( q ) ± ] * .
J z = M | 1 M 1 M | M + M | 2 M 2 M | M ,
μ z = M | 1 M 1 M | g J 1 μ B M M | 2 M 2 M | g J 2 μ B M ,
μ z = M | 1 M 1 M | g J 1 μ B M = g J 1 μ B J z .
υ B = g J 1 μ B B / k .
f = α ( υ υ B ) = α υ κ z ,
κ = g J 1 μ B k d B d z α ,
α = 120 17 δ Γ 5 Γ 2 + 4 δ 2 k 2
κ = g J 1 μ B k d B d z 120 17 δ Γ 5 Γ 2 + 4 δ 2 .
= 1 2 ( x ̂ b + i ŷ b ) .
= 1 2 ( x ̂ cos θ + i ŷ sin θ ) .
= 1 2 ( cos θ + 1 ) + 1 2 ( cos θ 1 ) + 1 2 ( sin θ ) 0 ,
B z = b 1 z + b 3 ( z 3 3 2 z ρ 2 ) + ,
B ρ = 1 2 b 1 ρ + b 3 ( 3 2 ρ z 2 + 3 8 ρ 2 z ) + ,
b 1 = μ 0 4 π 12 N I π R 2 A ( R 2 + A 2 ) 3 / 2 1 ( R 2 + A 2 ) ,
b 3 b 1 = ( 5 6 ) 4 A 2 3 R 2 ( R 2 + A 2 ) 2 .
α io 4.2 δ Γ 5 Γ 2 + 4 δ 2 k 2
α sis k 2 3 δ 20 Γ ( 3 δ 2 + 14 Γ 2 δ 2 + Γ 2 ) .
r z 2 0.17 Γ g J μ B ( d B / d z ) k ( Ω Γ ) 2 ( 5 Γ 2 + 4 δ 2 ) Γ 2 + 4 δ 2 ) .
r z 2 Γ g μ B ( d B / d z ) k 1 Ω 2 ( δ 2 + Γ 2 / 4 ) 3 2 δ 2 Γ 2 .
n 12 [ g μ B ( d B / d z ) I S 2 k c Γ 2 ] ( Ω 2 / 2 + δ 2 + Γ 2 / 4 ) 3 2 ( 2 Ω 2 + Γ 2 ) ( 5 Γ 2 + 4 δ 2 ) δ Γ ( Ω 2 / 2 ) 2 ,
12 g μ B ( d B / d z ) I S 2 k c Γ 2 = 4.6 × 10 9 atoms / cm 3 = 1 ( 7.1 λ ) 3 .
k υ , μ B B < Ω 2 / 6 δ Γ ,
w = I 1 I 2 I 1 + I 2
I 1 = ( 1 + w ) I 1 + I 2 2 ,
I 2 = ( 1 w ) I 1 + I 2 2 .
f = 120 17 k δ Γ k υ 5 Γ 2 + 4 δ 2 + 10 17 Γ s 0 w k .
B = g J μ B ( 5 Γ 2 + 4 δ 2 Γ 2 + 4 δ 2 ) Ω 2 6 δ w .
w Γ 2 + 4 δ 2 5 Γ 2 + 4 δ 2
1 at large detunings .
f = k γ ( s 1 s 2 )
= α υ κ z + k Γ w s 0 ,
B = Γ g μ B ( δ 2 + Γ 2 / 4 2 δ Γ ) w .
w = I 1 I 2 I 1 + I 2 = t 1 cos 2 ( 2 θ ) t 2 sin 2 ( 2 θ ) t 1 cos 2 ( 2 θ ) + t 2 sin 2 ( 2 θ ) ,
B w = Γ 6 g F μ B d Ω 2 δ Γ ,
d = ( 5 Γ 2 + 4 δ 2 Γ 2 + 4 δ 2 ) .

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