Abstract

The cooling mechanisms for laser cooling of atoms in optical molasses have been investigated experimentally. A significant simplification over the usual three-dimensional geometry has been obtained by studying the optical molasses in one or two dimensions only. By proper choice of polarizations the behavior of a pure two-level system as well as the more complicated effects of polarization gradients on laser cooling of a multilevel atom were observed.

© 1989 Optical Society of America

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  1. P. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988);see also P. 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]
  2. Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).See also Y. Shevy, D. S. Weiss, and S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.
    [CrossRef] [PubMed]
  3. J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 1990.
  4. S. Chu, D. S. Weiss, P. J. Ungar, and Y. Shevy, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 636.
  5. P. J. Ungar, D. S. Weiss, E. Riis, and S. Chu, J. Opt. Soc. Am. B 6, this issue(1989), our companion theoretical paper.
    [CrossRef]
  6. J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
    [CrossRef]
  7. J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 608 (1981).
    [CrossRef]
  8. J. Gordon and A. Ashkin, Phys. Rev. 21, 1606 (1980).
    [CrossRef]
  9. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 577–582.
  10. V. G. Minogin, Sov. Phys. JETP 52, 1032 (1980).
  11. We use the Gordon–Ashkin formalism7 to obtain the theoretical temperatures. We make the approximation that the atoms are uniformly distributed in the 1D standing wave and numerically spatial average 〈f〉 and Dp, which are calculated at each point according to the gradient of the 1D electric field.
  12. The number of atoms confined in the initial 3D molasses drops considerably as we tune closer to the line. In order to obtain a sufficient signal-to-noise ratio at detunings closer than ∼1.5 Γ we load the 3D molasses farther away from the line and shift the laser frequency over just before the 1D measurement starts.
  13. Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989);A. Aspect, E. Arimondo, R. Kasier, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
    [CrossRef]
  14. C. Cohen-Tannoudji, Laboratoire de Spectroscopie Hertzienne, Ecole Normale Supérieure, et Collège de France, 24, rue Lhomond, F-75231 Paris Cedex 05, France, 24, rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).
  15. S. Chu, M. G. Prentiss, A. E. Cable, and J. E. Bjorkholm, in Laser Spectroscopy VII, W. Persson and S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.
  16. See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap.14.

1989 (4)

Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).See also Y. Shevy, D. S. Weiss, and S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.
[CrossRef] [PubMed]

P. J. Ungar, D. S. Weiss, E. Riis, and S. Chu, J. Opt. Soc. Am. B 6, this issue(1989), our companion theoretical paper.
[CrossRef]

J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
[CrossRef]

Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989);A. Aspect, E. Arimondo, R. Kasier, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

1988 (1)

P. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988);see also P. 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]

1981 (1)

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 608 (1981).
[CrossRef]

1980 (2)

J. Gordon and A. Ashkin, Phys. Rev. 21, 1606 (1980).
[CrossRef]

V. G. Minogin, Sov. Phys. JETP 52, 1032 (1980).

Arimondo, E.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 1990.

Ashkin, A.

J. Gordon and A. Ashkin, Phys. Rev. 21, 1606 (1980).
[CrossRef]

Aspect, A.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 1990.

Baer, T.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 608 (1981).
[CrossRef]

Bjorkholm, J. E.

S. Chu, M. G. Prentiss, A. E. Cable, and J. E. Bjorkholm, in Laser Spectroscopy VII, W. Persson and S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

Cable, A. E.

S. Chu, M. G. Prentiss, A. E. Cable, and J. E. Bjorkholm, in Laser Spectroscopy VII, W. Persson and S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

Castin, Y.

Chu, S.

P. J. Ungar, D. S. Weiss, E. Riis, and S. Chu, J. Opt. Soc. Am. B 6, this issue(1989), our companion theoretical paper.
[CrossRef]

Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).See also Y. Shevy, D. S. Weiss, and S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.
[CrossRef] [PubMed]

S. Chu, D. S. Weiss, P. J. Ungar, and Y. Shevy, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 636.

S. Chu, M. G. Prentiss, A. E. Cable, and J. E. Bjorkholm, in Laser Spectroscopy VII, W. Persson and S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

Cohen-Tannoudji, C.

J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
[CrossRef]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 1990.

C. Cohen-Tannoudji, Laboratoire de Spectroscopie Hertzienne, Ecole Normale Supérieure, et Collège de France, 24, rue Lhomond, F-75231 Paris Cedex 05, France, 24, rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

Dalibard, J.

Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989);A. Aspect, E. Arimondo, R. Kasier, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
[CrossRef]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 1990.

Flannery, B. P.

See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap.14.

Gordon, J.

J. Gordon and A. Ashkin, Phys. Rev. 21, 1606 (1980).
[CrossRef]

Gould, P. L.

P. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988);see also P. 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]

Hall, J. L.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 608 (1981).
[CrossRef]

Hollberg, L.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 608 (1981).
[CrossRef]

Kaiser, R.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 1990.

Lett, P.

P. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988);see also P. 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]

Metcalf, H. J.

P. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988);see also P. 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]

Minogin, V. G.

V. G. Minogin, Sov. Phys. JETP 52, 1032 (1980).

Phillips, W. D.

P. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988);see also P. 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.

S. Chu, M. G. Prentiss, A. E. Cable, and J. E. Bjorkholm, in Laser Spectroscopy VII, W. Persson and S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

Press, W. H.

See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap.14.

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 577–582.

Riis, E.

P. J. Ungar, D. S. Weiss, E. Riis, and S. Chu, J. Opt. Soc. Am. B 6, this issue(1989), our companion theoretical paper.
[CrossRef]

Robinson, H. G.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 608 (1981).
[CrossRef]

Salomon, C.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 1990.

Shevy, Y.

Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).See also Y. Shevy, D. S. Weiss, and S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.
[CrossRef] [PubMed]

S. Chu, D. S. Weiss, P. J. Ungar, and Y. Shevy, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 636.

Teukolsky, S. A.

See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap.14.

Ungar, P. J.

P. J. Ungar, D. S. Weiss, E. Riis, and S. Chu, J. Opt. Soc. Am. B 6, this issue(1989), our companion theoretical paper.
[CrossRef]

Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).See also Y. Shevy, D. S. Weiss, and S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.
[CrossRef] [PubMed]

S. Chu, D. S. Weiss, P. J. Ungar, and Y. Shevy, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 636.

Vansteenkiste, N.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 1990.

Vetterling, W. T.

See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap.14.

Wallis, H.

Watts, R. N.

P. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988);see also P. 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]

Weiss, D. S.

Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).See also Y. Shevy, D. S. Weiss, and S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.
[CrossRef] [PubMed]

P. J. Ungar, D. S. Weiss, E. Riis, and S. Chu, J. Opt. Soc. Am. B 6, this issue(1989), our companion theoretical paper.
[CrossRef]

S. Chu, D. S. Weiss, P. J. Ungar, and Y. Shevy, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 636.

Westbrook, C. I.

P. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988);see also P. 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]

Appl. Phys. Lett. (1)

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 608 (1981).
[CrossRef]

J. Opt. Soc. Am. B (3)

Phys. Rev. (1)

J. Gordon and A. Ashkin, Phys. Rev. 21, 1606 (1980).
[CrossRef]

Phys. Rev. Lett. (2)

P. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988);see also P. 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]

Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).See also Y. Shevy, D. S. Weiss, and S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. G. Minogin, Sov. Phys. JETP 52, 1032 (1980).

Other (8)

We use the Gordon–Ashkin formalism7 to obtain the theoretical temperatures. We make the approximation that the atoms are uniformly distributed in the 1D standing wave and numerically spatial average 〈f〉 and Dp, which are calculated at each point according to the gradient of the 1D electric field.

The number of atoms confined in the initial 3D molasses drops considerably as we tune closer to the line. In order to obtain a sufficient signal-to-noise ratio at detunings closer than ∼1.5 Γ we load the 3D molasses farther away from the line and shift the laser frequency over just before the 1D measurement starts.

C. Cohen-Tannoudji, Laboratoire de Spectroscopie Hertzienne, Ecole Normale Supérieure, et Collège de France, 24, rue Lhomond, F-75231 Paris Cedex 05, France, 24, rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

S. Chu, M. G. Prentiss, A. E. Cable, and J. E. Bjorkholm, in Laser Spectroscopy VII, W. Persson and S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap.14.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 1990.

S. Chu, D. S. Weiss, P. J. Ungar, and Y. Shevy, in Atomic Physics 11, S. Haroche, ed. (World Scientific, Singapore, 1989), p. 636.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 577–582.

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Figures (14)

Fig. 1
Fig. 1

Schematic of the optical setup for the experiment. Electro-optic and acousto-optic modulators are signified by e.o. and a.o., respectively. The laser frequency, νL, is tuned with a.o3, which permits a tuning range of ∼50 MHz about the atomic transition. The other optical modulators provide additional flexibility in choosing the frequency of the probe and slowing beams.

Fig. 2
Fig. 2

The laser beams as they enter the vacuum shown from above, along with a probe-detection schematic. The vertical molasses beam is not shown. Dashed lines represent laser beams below the plane of the horizontal molasses beams. The axial probe is 21 mm below the molasses center, and the transverse probe is 4 mm below and 11 mm to the side. The length of the transverse probe beam imaged is along the bisector of the molasses beam directions. This off-center imaging is mandated by our detection geometry, while the location to the side and slightly below the molasses center yields sensitivity to a broad range of transverse temperatures. A 5–20-mW/cm2 clearing beam helps to clear stray atoms away from the region above the axial probe before the atoms are dropped. pmt’s denote photomultiplier tubes.

Fig. 3
Fig. 3

Sample of TOF signals for σ+σ+ (two-level) polarization, along with their least-squares fits. The times show how long the atoms have spent in 1D molasses before being dropped.

Fig. 4
Fig. 4

(a) Measured axial temperature of atoms in 1D molasses with uniform polarization at various stages of equilibration. The curves are exponential fits with time constants 113 and 174 μsec for σ+σ+ and πxπx, respectively. (b) Measured transverse temperatures, shown with linear fits. The molasses intensity is 2.8 mW/cm2 per beam, the carrier detuning is −20.5 MHz, and the sideband is tuned 38 MHz below the F = 1 to F = 2 transition.

Fig. 5
Fig. 5

Measured equilibrated temperatures in 1D molasses with σ+σ+ (two-level) polarization, for a carrier intensity of 1.7 mW/cm2 per beam. The curve is a numerical calculation of the temperatures at this intensity made using the two-level theory.

Fig. 6
Fig. 6

Measured equilibrium temperature and twice the cooling rate α/m, shown as functions of 1D molasses intensity (per beam). The molasses approaches its equilibrium temperature with a theoretical time constant of m/2α. The curves are the corresponding theoretical predictions. In addition to the systematic uncertainties discussed in the text, which affect only equilibrium temperatures, there is an uncertainty in determining the temperature and the equilibration rate from curves like those shown in Fig. 4(a). These uncertainties are estimated to be 5% and 15%, respectively, based on numbers yielded by the exponential fitting routine. As is also discussed in the text, the intensity sampled by the atoms may be as much as 30% higher than the average intensity shown here. This will bring the experimental α/m values into closer agreement with the theoretical curve. The carrier detuning is −20.5 MHz for all the data used in this figure.

Fig. 7
Fig. 7

Allowed transitions between the 3S1/2 and 3P3/2 levels in sodium. The numbers between the ground and excited states are proportional to the squares of the matrix elements.

Fig. 8
Fig. 8

Temperature of atoms equilibrated in πxπx molasses as a function of the detuning of the rf sideband. The zero point corresponds to the sideband’s being shifted into resonance with the 3S1/2, F = 1 to 3P3/2, F = 2 transition. The carrier is held fixed at −27 MHz with an intensity of 7.6 mW/cm2 per beam, which is eight times the intensity of the sideband. Doppler cooling and heating to the red and the blue of the three resonances is evident.

Fig. 9
Fig. 9

Data taken with σ+σ+ polarization (1.4 mW/cm2 per beam, −20.5-MHz detuning) and a static magnetic field applied perpendicular to the beam propagation axis. (a) The equilibrated temperature of the remaining cold peak as a function of the size of the magnetic field, B. The variation of the temperature is within the experimental uncertainty. (b) The temperature of the hot peak. (c) The fraction of atoms that is determined to be associated with the cold peak.

Fig. 10
Fig. 10

TOF signals for molasses with polarization gradients along with the fitted curves. The times show how long the atoms have spent in 1D molasses before being dropped. (a) Corkscrew polarization (σ+σ), (b) crossed linear (πxπy) polarization.

Fig. 11
Fig. 11

(a) Temperature of the cold peak in 1D molasses in the presence of polarization gradients at different stages of equilibration. (b) Temperature of the hot peak. The data are fittted to exponential curves. (c) Fraction of cold atoms determined by the fitting routine. The carrier intensity and the detuning are 3.0 mW/ cm2 per beam and −27 MHz, respectively.

Fig. 12
Fig. 12

(a) Cold peak temperature for molasses with polarization gradients as a function of intensity. (b) Fraction of cold atoms. The carrier is tuned 27 MHz below the F = 2 to F = 3 transition and the sideband 38 MHz below the F = 1 to F = 2 transition.

Fig. 13
Fig. 13

(a) Cold peak temperature as function of molasses detuning in the presence of polarization gradients. (b) Fraction of cold atoms. The laser intensity is 7.3 mW/cm2 per beam, and the sideband is tuned 38 MHz below the F = 1 to F = 2 transition throughout the scan.

Fig. 14
Fig. 14

Dependence of the velocity distributions on the rf sideband frequency in the presence of polarization gradients. The zero-frequency point corresponds to the sideband light’s being in resonance with the 3S1/2, F = 1 to 3P3/2, F = 2 transition. The carrier is held fixed at −27 MHz with an intensity of 7.6 mW/cm2, which is eight times the intensity of the sideband. (a) Temperature of the cold peak. (b) Temperature of the hot peak. (c) Fraction of atoms as determined by the fitting routine.

Equations (10)

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

d n = G ( r , t ; r 0 ) d 3 r = P [ v = ( r r 0 ) / t ] d 3 υ ,
G ( r , t ; r 0 ) = 1 t 3 P ( v = r r 0 t ) .
G ( r , t ; r 0 ) = ( M 2 π k T t 2 ) 3 / 2 exp ( M | r r 0 | 2 / 2 k T t 2 ) ,
n t = kTt M 2 n .
n ( r , t ) = G ( r , t ; r 0 ) n 0 ( r 0 ) d 3 r 0 = exp ( r 2 / 2 σ c 2 ) ( 2 π σ c 2 ) 3 / 2 ,
n ( r , t ) = A ( λ ) sin ( λ r ) r exp ( λ 2 k T t 2 / 2 M ) d λ ,
G ( r , t ; r 0 ) = i exp [ ( r i r 0 i ) 2 / 2 σ g i 2 ] ( 2 π σ g i 2 ) 1 / 2 ,
n ( r , t ) = i exp [ ( r i r d i ) 2 / 2 σ c i 2 ] ( 2 π σ c i 2 ) 1 / 2 ,
TOF = d x exp { 1 2 [ x 2 σ c x 2 + ( x x p ) 2 σ p x 2 ] } [ ( 2 π σ c x 2 ) ( 2 π σ p x 2 ) ] 1 / 2 d z × exp ( 1 2 { [ z ( υ z t g t 2 / 2 ) ] σ c z 2 + ( z z p ) 2 σ p z 2 } ) [ ( 2 π σ c z 2 ) ( 2 π σ p z 2 ) ] 1 / 2 × y i y 2 d y exp { 1 2 y 2 σ c y 2 } ( 2 π σ c y 2 ) 1 / 2 ,
TOF = 1 2 ( erf y 2 2 σ c y erf y 1 2 σ c y ) × exp { 1 2 [ x p 2 σ c x 2 + σ p x 2 + z d 2 σ c z 2 + σ p z 2 ] } { [ 2 π ( σ c x 2 + σ p x 2 ) ] [ 2 π ( σ c z 2 + σ p z 2 ) ] } 1 / 2 ,

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