## Strong evaporative cooling of a trapped cesium gas

Optics Express, Vol. 2, Issue 8, pp. 323-329 (1998)

http://dx.doi.org/10.1364/OE.2.000323

Acrobat PDF (380 KB)

### Abstract

Using forced radio-frequency evaporation, we have cooled cesium atoms prepared in the sublevel *F* = -*m*_{F}
= 3 and confined in a magnetic trap. At the end of the evaporation ramp, the sample contains ~ 7000 atoms at 80 nK, corresponding to a phase space density 3 × 10^{-2}. A molecular dynamics approach, including the effect of gravity, gives a good account for the experimental data, assuming a scattering length larger than 300 Å.

© Optical Society of America

## Introduction

2. C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. **78**, 985 (1997). [CrossRef]

3. C. C. Bradley, C. A. Sackett, J. J. Tollet, and R. G. Hulet, Phys. Rev. Lett. **75**, 1687 (1995). [CrossRef] [PubMed]

4. K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. **75**, 3969 (1995). [CrossRef] [PubMed]

5. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science **275**, 737 (1997). [CrossRef]

6. E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. **79**, 337 (1997). [CrossRef]

*i.e*. cesium, which is at the basis of primary time and frequency standards, efforts to prepare a Bose-Einstein condensate have so far all foundered. We have recently measured the inelastic collision rate in an ultracold Cs sample prepared in its doubly polarized state (

*F*=

*m*

_{F}= 4) [7]. We observed a giant dipole relaxation that limited the maximal phase space density achievable by evaporative cooling to ~ 10

^{-5}. In the present paper, we therefore focus on the study of the evaporative cooling of an atomic sample of Cs atoms in the lower hyperfine sublevel

*F*= -

*m*

_{F}= 3. The atoms are confined around the magnetic field minimum of a Ioffe-Pritchard type trap. We obtain a cloud with an ultra-low temperature (80 nK), with a phase space density 3 × 10

^{-2}, two orders of magnitude below the condensation threshold. Although we could not observe Bose-Einstein condensation, this constitutes a significant achievement due to the importance of cesium for metrology.

## Experimental setup

8. A. Steane, P. Szriftgiser, P. Desbiolles, and J. Dalibard, Phys. Rev. Lett. **74**, 4972 (1995). [CrossRef] [PubMed]

^{-8}mBar of residual Cs vapor). This load, typically 10

^{8}atoms, is cooled by a short molasses phase (15 ms) and then falls down to the lower cell with a much better vacuum (residual pressure < 10

^{-10}mBar). A large fraction of the atoms (70%) is recaptured in the lower MOT and cooled down to 6

*μ*K using again a short molasses phase.

*F*= -

*m*

_{F}= 3 by applying a 500

*μ*s pulse of circularly polarized light, aligned with a magnetic field of 2 × 10

^{-4}T, and resonant with the transition 6

*S*

_{1/2},

*F*= 3 ↔ 6

*P*

_{3/2},

*F*′ = 3. An additional beam resonant with the transition 6

*S*

_{1/2},

*F*= 4 ′ 6

*P*

_{3/2},

*F*′ = 4 depumps the atoms from the

*F*= 4 sublevel. The pumping efficiency is larger than 90%. We then switch on the magnetic trap, which is generated by a 50 A current running in three identical circular coils whose axes point towards +

*x*, -

*x*, and +

*y*respectively, where

*z*denotes the vertical axis (fig. 1a). Each coil has 80 turns with an average diameter 34 mm, and it is located at 40 mm from the center

*O*of the MOT, which coincides with the center of the magnetic trap. The resulting field configuration is equivalent to a Ioffe-Pritchard type trap, with a non-zero local minimum of |B(r)| in r = 0. The leading terms in the field variations around

*O*are (

*b*′

*x*,

*B*

_{0}+ b″

*y*

^{2}/2, -

*b*′

*z*), with

*b*′ = 1.1 T/m,

*h*″ = 60 T/m

^{2}, and

*B*

_{0}= 1-1 × 10

^{-2}T.

^{-4}T, using a pair of Helmholtz coils centered on the magnetic trap (fig. 1b). After this compression, oscillation frequencies around the center of the trap are

*v*

_{x}=

*v*

_{z}= (3

*μb*′

^{2}/4

*mB*

_{0})

^{1/2}/2

*π*= 87 Hz and

*v*

_{y}= (3

*μb*″/4

*m*)

^{1/2}/2

*π*= 7 Hz where

*μ*is the Bohr magneton, and

*m*the atomic mass. At this stage, the cigar-shaped atomic cloud contains about 5 × 10

^{7}atoms, at a temperature of 120

*μ*K and a peak density 7×10

^{10}cm

^{-3}. The lifetime of the cloud due to the collisions with the residual background gas is 200 s.

*via*elastic collisions, while the particles in the high energy tail of the phase space distribution are beeing removed [9, 10

10. O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven, Phys. Rev. A **53**, 381 (1996). [CrossRef] [PubMed]

12. W. Ketterle and N.J. van Druten, Adv. At. Mol. Opt. Phys. **37**, 181 (1996). [CrossRef]

^{-6}T) induces adiabatic transitions from the trapped Zeeman substate to an untrapped substate (

*m*= -3 →

*m*= 3) . This energy selective transfer takes place on a surface

*μ*|B(r)| = 4

*hv*around the trap center. The RF frequency

*v*is ramped down linearly in 30 seconds, from 4000 kHz to an adjustable final frequency. From the final frequency

*v*

_{0}which expells all atoms (427 (±3) kHz), we determine precisely the bias field

*B*

_{0}= 1.17 (±0.01) × 10

^{-4}T, taking into account the effect of gravity (see below).

## Temperature and density measurements

*μ*s. The atoms are pumped into the

*F*= 4 sublevel using a 200

*μ*s light pulse resonant with the 6

*S*

_{1/2},

*F*= 3 ↔ 6

*P*

_{3/2},

*F*′ = 3 transition. The cloud is then briefly (80

*μ*s) illuminated by a circularly polarized imaging laser beam, propagating along the

*x*axis and resonant with the 6

*S*

_{1/2},

*F*= 4 ↔ 6

*P*

_{3/2},

*F*′ = 5 transition. The shadow produced by the cloud in this beam is imaged with a magnification 2.47 onto a CCD array (optical resolution 7

*μ*m). The image is then digitally processed to extract the column density of the cloud ∫

*n*(

*x*,

*y*,

*z*)

*dx*, where

*n*(r) is the spatial density. This quantity is well fitted by a 2D Gaussian function as expected for a thermal distribution in a harmonic trap. Assuming rotational symmetry around the

*y*-axis, we then deduce all relevant quantities such as the number of atoms

*N*and the temperature

*T*. We evaluate

*T*using only the size of the cloud along the

*y*axis, so that the optical resolution of our detection scheme is not a limit, even for the lowest measured temperature. For instance a

*rms*size along

*y*equal to 51.5

*μ*m corresponds to a cloud at

*T*= 80 nK.

*τ*, and take an image. The time of flight

*τ*is varied from 1 ms up to 13 ms, with 1 ms steps. A typical resulting set of pictures is shown in figure 2. We fit the evolution of the

*rms*size along the

*z*axis by (Δ

*v*

^{2}(

*t*-

*t*

_{0})

^{2})

^{1/2}, with the three adjustable parameters Δ

*z*

_{0}, Δ

*v*and

*t*

_{0}. From Δ

*v*we deduce the temperature

*T*=

*m*Δ

*v*

^{2}/

*k*

_{B}= 1.0

*μ*K, to be compared with the value 0.76

*μ*K, deduced from the

*y*size at

*τ*= 0. The analysis of several sets of similar data for initial conditions in the micro-Kelvin domain show that the results of these two methods agree with each other to within ± 25%.

## The role of gravity

*z*axis is now:

*mg*/(3

*b*′

*μ*) is of the order of 0.3 for our setup. The harmonic approximation around the bottom of the potential remains valid since ϵ

^{2}≪ 1. On the

*z*axis, the RF induces spin flips at

*v*

_{0}which expells all atoms is obtained by taking

*z*=

*z*

_{0}in the above formula, which leads to:

*B*

_{0}= 4

*hv*

_{0}/

*μ*when gravity can be neglected. As pointed out in [11

11. E. L. Surkov, J. T. M. Walraven, and G. V. Shlyapnikov, Phys. Rev. A **53**, 3403 (1996). [CrossRef] [PubMed]

12. W. Ketterle and N.J. van Druten, Adv. At. Mol. Opt. Phys. **37**, 181 (1996). [CrossRef]

*μ*K, the gravitational energy varies in our setup by more than 2

*k*

_{B}

*T*over the evaporation surface. In this regime the atoms only escape from the trap near the bottom (1D evaporation), which limits severely the efficiency of evaporative cooling.

## Numerical simulation of the evaporation

^{7}real atoms initially present. Each macro-atom represents

*p*real atoms, with

*p*= 2

^{13}= 8192 at the beginning of the simulation shown below. Each macro-atom has the same mass and the same magnetic moment as a Cs atom. The collisional cross-section between two macro-atoms is

*p*times larger than the collisional cross-section for two real atoms with the same velocity. Each time the number of macro-atoms becomes lower than 4000, either because of evaporation or because of losses simulating collisions with the residual gas, every macro-atom is replaced by two new macro-atoms, each of which represents only

*p*/2 atoms. If the parent macro-atom is in (

*x*,

*y*,

*z*) with velocity (

*v*

_{x},

*v*

_{y},

*v*

_{z}), one of two new macro-atoms is placed at the same point with the same velocity, and the other one is placed in (-

*x*, -

*y*,

*z*) with the velocity (-

*v*

_{x}, -

*v*

_{y},

*v*

_{z}). This duplication, which exploits the symmetry of the trap, guarantees that these two new macro-atoms will not undergo a collision with each other immediatly after the duplication process. The duplication process stops when

*p*is equal to 1.

13. H. Wu and C. Foot, J. Phys. B , **29**, L321–L328 (1996). [CrossRef]

*δt*, the position of each particle is dis-cretized with a step

*δr*, so that each particle is assigned to a cubic box of volume

*δr*

^{3}. The number of boxes is 3.5 × 10

^{5}; the size

*δr*is adjusted as the cloud cools down, so that the probability for having two particles in the same box is much smaller than 1. When two macro-atoms are found in the same box, a collision may take place between them. The probability for this collision is

*p*

*σ*(

*k*)

*v*

*δt*/

*δr*

^{3}, where

*k*and

*v*= 2

*ħk*/

*m*are the relative wave vector and velocity of the colliding particles, and

*σ*(

*k*) the collisional cross section between two real Cs atoms. The time step

*δt*is chosen such that this probability is small compared to 1. The occurence of a collision is then randomly decided. The collision is isotropic since only the

*l*= 0 partial wave contributes at these ultra-low temperatures. We put [14, 15, 16

16. M. Arndt, M. Ben Dahan, D. Guéry-Odelin, M. W. Reynolds, and J. Dalibard, Phys. Rev. Lett. **79**, 625 (1997). [CrossRef]

*a*is the scattering length. At very low relative velocity one recovers the well known limit

*σ*= 8

*πa*

^{2}, while one obtains for higher velocities the unitary limit

*σ*(

*k*) = 8

*π*/

*k*

^{2}, corresponding to the result for a zero-energy resonance [16

16. M. Arndt, M. Ben Dahan, D. Guéry-Odelin, M. W. Reynolds, and J. Dalibard, Phys. Rev. Lett. **79**, 625 (1997). [CrossRef]

## Results

*a*| = 100 Å and |

*a*| = 1000 Å. Both scattering lenghts lead to the same results until the two last seconds of the ramp. Indeed for

*t*≤ 28 s, the temperature is large enough for the unitary approximation

*σ*(

*k*) ≃ 8

*π*/

*k*

^{2}to be valid for both

*a*’s. For

*t*= 30 s, the predictions of the simulation concerning the remaining number of atoms

*N*and the final temperature

*T*are summarized in table 1. The best agreement with the experimental data is obtained for |

*a*| > 300 Å. Such a large value for the scattering length is in good agreement with recent measurements based on photoassociation experiments [17] (see also [16

16. M. Arndt, M. Ben Dahan, D. Guéry-Odelin, M. W. Reynolds, and J. Dalibard, Phys. Rev. Lett. **79**, 625 (1997). [CrossRef]

^{-7}, typical of a compressed magneto-optical trap, and it ends at 3 × 10

^{-2}, a factor 100 below Bose-Einstein condensation. The set of data between 25 and 29 seconds exhibits a clear “runaway evaporation”: the product of the density times the

*rms*velocity remains constant, although the number of atoms is divided by 7. For a velocity-independent collisional cross section, this would mean a constant collisional rate. In our case, where the cross section varies as 1/

*T*in this region (unitary limit of eq.1), the situation is even more favourable since the collisional rate increases by 4.

*t*=30 s) corresponds to a cloud whose expected transverse extension is 4

*μ*m, estimated from the measurement of the longitudinal dimension of the cloud (52

*μ*m) and from its expected ellipticity (~ 13). This is below the resolution of our optical system and indeed the measured size (~ 7

*μ*m) is larger than this expected value. To evaluate the spatial density and the phase space density, we have not corrected the transverse dimension by imposing a constant ellipticity, which leads therefore to a conservative value for the largest achieved phase space density.

## Conclusion

18. A. Kastberg, W. D. Phillips, S. L. Rolston, and R. J. C. Spreeuw, Phys. Rev. Lett. **74**, 1542 (1995). [CrossRef] [PubMed]

19. D. Boiron, A. Michaud, P. Lemonde, Y. Castin, C. Salomon, S. Weyers, K. Szymaniec, L. Cognet, and A. Clairon, Phys. Rev. A **53**, R3734 (1996). [CrossRef] [PubMed]

*t*≥ 25 s in the data set of figs. 3 and 5. This indicates that, with an optimized evaporation ramp, BEC should be reachable with more than 5000 condensed atoms [10

10. O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven, Phys. Rev. A **53**, 381 (1996). [CrossRef] [PubMed]

*a*| ≥ 250 Å. However we could not increase the phase space density above the present maximal value 3 × 10

^{-2}. Indeed we have observed for these ultra-cold clouds density dependent losses, accompanied by a heating of the atoms when the RF shield is removed. These loss processes, which we attribute to inelastic collisions between trapped atoms, are currently under study.

## Footnotes

† | Unité de recherche de l’ENS et de l’Université Pierre et Marie Curie, associée au CNRS. |

## References

1. | M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science |

2. | C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. |

3. | C. C. Bradley, C. A. Sackett, J. J. Tollet, and R. G. Hulet, Phys. Rev. Lett. |

4. | K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. |

5. | M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science |

6. | E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. |

7. | J. Söding, D. Guéry-Odelin, P. Desbiolles, G. Ferrari, and J. Dalibard, accepted for publication in Phys. Rev. Lett. |

8. | A. Steane, P. Szriftgiser, P. Desbiolles, and J. Dalibard, Phys. Rev. Lett. |

9. | H. F. Hess, Bull. Am. Phys. Soc. [2] |

10. | O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven, Phys. Rev. A |

11. | E. L. Surkov, J. T. M. Walraven, and G. V. Shlyapnikov, Phys. Rev. A |

12. | W. Ketterle and N.J. van Druten, Adv. At. Mol. Opt. Phys. |

13. | H. Wu and C. Foot, J. Phys. B , |

14. | L. D. Landau and E. M. Lifshitz, |

15. | C. J. Joachain, |

16. | M. Arndt, M. Ben Dahan, D. Guéry-Odelin, M. W. Reynolds, and J. Dalibard, Phys. Rev. Lett. |

17. | A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and P. Pillet, preprint (November 1997). |

18. | A. Kastberg, W. D. Phillips, S. L. Rolston, and R. J. C. Spreeuw, Phys. Rev. Lett. |

19. | D. Boiron, A. Michaud, P. Lemonde, Y. Castin, C. Salomon, S. Weyers, K. Szymaniec, L. Cognet, and A. Clairon, Phys. Rev. A |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(020.2070) Atomic and molecular physics : Effects of collisions

(020.7010) Atomic and molecular physics : Laser trapping

**ToC Category:**

Focus Issue: Collective phenomena in trapped atoms and ions

**History**

Original Manuscript: January 21, 1998

Published: April 13, 1998

**Citation**

D. Guery-Odelin, J. Soeding, P. Desbiolles, and Jean Dalibard, "Strong evaporative cooling of a trapped cesium gas," Opt. Express **2**, 323-329 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-8-323

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### References

- M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 1989 (1995).
- C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997). [CrossRef]
- C. C. Bradley, C. A. Sackett, J. J. Tollet, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995). [CrossRef] [PubMed]
- K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [CrossRef] [PubMed]
- M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 737 (1997). [CrossRef]
- E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 79, 337 (1997). [CrossRef]
- J. Soeding, D. Guery-Odelin, P. Desbiolles, G. Ferrari, and J. Dalibard, accepted for publication in Phys. Rev. Lett.
- A. Steane, P. Szriftgiser, P. Desbiolles, and J. Dalibard, Phys. Rev. Lett. 74, 4972 (1995). [CrossRef] [PubMed]
- H. F. Hess, Bull. Am. Phys. Soc.[2] 30, 854 (1985).
- O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven, Phys. Rev. A 53, 381 (1996). [CrossRef] [PubMed]
- E. L. Surkov, J. T. M. Walraven, and G. V. Shlyapnikov, Phys. Rev. A 53, 3403 (1996). [CrossRef] [PubMed]
- W. Ketterle and N. J. van Druten, Adv. At. Mol. Opt. Phys. 37, 181 (1996). [CrossRef]
- H. Wu and C. Foot, J. Phys. B, 29, L321-L328 (1996). [CrossRef]
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon Press, Oxford, 1977), Sect. 143.
- C. J. Joachain, Quantum collision theory, (North-Holland, Amsterdam, 1983) pp. 78-105
- M. Arndt, M. Ben Dahan, D. Guery-Odelin, M. W. Reynolds, and J. Dalibard, Phys. Rev. Lett. 79, 625 (1997). [CrossRef]
- A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and P. Pillet, preprint (November 1997).
- A. Kastberg, W. D. Phillips, S. L. Rolston, and R. J. C. Spreeuw, Phys. Rev. Lett. 74, 1542 (1995). [CrossRef] [PubMed]
- D. Boiron, A. Michaud, P. Lemonde, Y. Castin, C. Salomon, S. Weyers, K. Szymaniec, L. Cognet, and A. Clairon, Phys. Rev. A 53, R3734 (1996). [CrossRef] [PubMed]

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