## Studies of two-species Bose-Einstein condensation

Optics Express, Vol. 2, Issue 8, pp. 330-337 (1998)

http://dx.doi.org/10.1364/OE.2.000330

Acrobat PDF (371 KB)

### Abstract

We describe our recent progress on the investigation of two-species Bose-Einstein condensation. From a theoretical analysis we show that there is a new rich phenomenology associated with two-species Bose-Einstein condensates which does not exist in a single-species condensate. We then describe results of a numerical model of the evaporative cooling process of a trapped two-species gas.

© Optical Society of America

## 1. Introduction

^{1–31. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observations of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198 (1995). [CrossRef] [PubMed] }has opened the field of weakly-interacting degenerate Bose gases. During the past two and a half years, substantial experimental and theoretical progress has been made on the study of the properties of this new state of matter. Indeed, the physics of trapped diluted condensates has emerged as one of the most exciting fields of physics in this decade. Recently, the remarkable experimental realization of a condensate mixture composed of two spin states of

^{87}Rb

^{44. C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, “Production of two overlapping Bose-Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 586 (1997). [CrossRef] }has prompted significant interest in the physics of a new class of quantum fluids: the two-species Bose-Einstein condensate (TBEC)

^{5–125. Tin-Lun Ho and V. B. Shenoy, “Binary mixtures of Bose condensates of alkali atoms,” Phys. Rev. Lett. 77, 3276 (1996). [CrossRef] [PubMed] }. Multi-component and, particularly, multi-species condensates offer new degrees of freedom, which give rise to a rich set of new phenomena that do not exist in a one-species condensate. Furthermore, the TBEC offers new and interesting experimental challenges.

*and*which are experimentally accessible. Finally, we describe results derived from a numerical model of the evaporative cooling process developed to help determine the optimal strategy for cooling into the doubly condensed phase.

^{1313. J. Shaffer and N. P. Bigelow, “Two-species trap experiments,” Opt. Photonics News Supp. 6:7, 47 (1995).}to be less desirable due to large inter-species trap loss rates. We stress, however, that many of our results are generalizable to other atomic mixtures, spin-state mixtures and Fermi-Bose mixtures.

## 2. Properties of the TBEC: some theoretical predictions

^{55. Tin-Lun Ho and V. B. Shenoy, “Binary mixtures of Bose condensates of alkali atoms,” Phys. Rev. Lett. 77, 3276 (1996). [CrossRef] [PubMed] ,66. B. D. Esry, Chris H. Greene, James P. Burke Jr., and John L. Bohn, “Hartree-Fock theory for double condensates,” Phys. Rev. Lett. 78, 3594 (1995). [CrossRef] ,1010. H. Pu and N. P. Bigelow, “Properties of two-species Bose condensates,” Phys. Rev. Lett. 801130 (1998). [CrossRef] ,1111. H. Pu and N. P. Bigelow, “Collective excitations, metastability and nonlinear response of a trapped two-species Bose-Einstein condensate,” Phys. Rev. Lett. 801134 (1998). [CrossRef] }:

*ψ*

_{i}(

*r*,

*t*) denotes the macroscopic condensate wave function for species

*i*, with

*r*being the radial coordinate.

*N*

_{i},

*m*

_{i}and

*ω*

_{i}are particle number, mass and trap frequency, respectively. The interaction between particles are described by a self-interaction term

*U*

_{i}= 4

*πħ*

^{2}

*a*

_{i}/

*m*

_{i}and a term that corresponds to the interaction between different species

*U*

_{12}= 2

*πħ*

^{2}

*a*

_{12}/

*m*(with m being the reduced mass of the two species), where

*a*

_{i}is the scattering length of species

*i*and

*a*

_{12}between species 1 and 2. The time-independent form of the nonlinear Schrödinger equations are obtained by replacing the left hand sides of Eqs. (1),(2) with

*μ*

_{i}

*ψ*

_{i}(

*r*) (

*i*=1,2), with

*μ*

_{i}being the chemical potential.

*ψ*

_{1}and

*ψ*

_{2}can be obtained

^{1010. H. Pu and N. P. Bigelow, “Properties of two-species Bose condensates,” Phys. Rev. Lett. 801130 (1998). [CrossRef] }by solving the coupled GPEs (Eqs. (1)and (2)) iteratively. For negative

*a*

_{12}(i.e., attractive inter-species interaction), both wave functions are compressed as compared to the case of two independent condensates (i.e.,

*a*

_{12}= 0). If the magnitude of

*a*

_{12}becomes larger than a certain value, the condensates mixture will eventually collapse. By contrast, for small positive

*a*

_{12}, two coupled, interpenetrating condensates are formed and each of the individual condensate wavefunctions become somewhat flattened due to the mutual repulsion between the species. However, for large positive

*a*

_{12}, the ground state is no longer necessarily a mixture of two overlapping condensates. Instead, the system “phase separates” into two distinct condensates

^{55. Tin-Lun Ho and V. B. Shenoy, “Binary mixtures of Bose condensates of alkali atoms,” Phys. Rev. Lett. 77, 3276 (1996). [CrossRef] [PubMed] ,1010. H. Pu and N. P. Bigelow, “Properties of two-species Bose condensates,” Phys. Rev. Lett. 801130 (1998). [CrossRef] }: one forming a core at the center of the trap and the other forming a surrounding shell. Fig. 1 illustrates the ground state density distribution of a Na-Rb condensates mixture at different values of

*a*

_{12}. At large

*a*

_{12}, we see a phase separated TBEC with a Rb core and a Na shell. Another interesting phenomenon arising from a large repulsive inter-species interaction is the possibility of the formation of a metastable state of the TBEC

^{1111. H. Pu and N. P. Bigelow, “Collective excitations, metastability and nonlinear response of a trapped two-species Bose-Einstein condensate,” Phys. Rev. Lett. 801134 (1998). [CrossRef] }. Our simulations (see below) show that, in the Na-Rb system, the Na condensate will form prior to the Rb condensate such that the Rb atoms condense in the presence of the repulsive Na condensate core into a metastable shell around the Na. However, from previous calculations, we know that the more stable state in this case should be comprised of a Rb core and a Na shell as shown in Fig. 1. We investigate the mechanical stability of this Na-core/Rb-shell system by externally perturbing the trapping potential and find that it is indeed not unconditionally stable: under a sufficiently strong external perturbation it will make a

*macroscopic quantum jump*to the more stable Rb-core/Na-shell system. These macroscopic metastable states arise from the inter-species interactions and hence are unique for the multi-component condensates.

^{1414. D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Collective Excitations of a Bose-Einstein condensate in a dilute gas,” Phys. Rev. Lett. 77, 420 (1996). [CrossRef] [PubMed] ,1515. M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Collective excitations of a Bose-Einstein condensate in a magnetic trap,” Phys. Rev. Lett. 77, 988 (1996). [CrossRef] [PubMed] }and theoretically calculated

^{16–1916. Mark Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and Charles W. Clark, “Collective excitations of atomic Bose-Einstein condensates,” Phys. Rev. Lett. 77, 1671 (1996). [CrossRef] [PubMed] }, and good agreement has been found between the two. We have generalized the standard Bogoliubov-Hartree theory

^{2020. A. L. Fetter, “Nonuniform states of an imperfect Bose gas,” Ann. Phys. (N.Y.) 70, 67 (1972). [CrossRef] }for one-species BEC to the case of the TBEC

^{1111. H. Pu and N. P. Bigelow, “Collective excitations, metastability and nonlinear response of a trapped two-species Bose-Einstein condensate,” Phys. Rev. Lett. 801134 (1998). [CrossRef] }and find that inter-species coupling dramatically modifies the excitation spectrum. We identified two types of isotropic breathing modes: in-phase and out-of-phase modes. We have also found that for large repulsive coupling, some non-isotropic modes possess imaginary frequencies indicating that the TBEC is unstable

^{77. Elena V. Goldstein and Pierre Meystre, “Quasiparticle instabilities in multicomponent atomic condensates,” Phys. Rev. A , 55, 2935 (1997). [CrossRef] }. Recently, Öhberg

^{2121. Patrik Öhberg and Stig Stenholm,“Hartree-Fock treatment of the two-component Bose-Einstein condensate,” Phys. Rev. A 57, 1272 (1998). [CrossRef] }showed that, under these conditions, a symmetry-breaking state is more stable and hence might represent the true ground state. However, that work is done for a 2d condensate with a very small number of particles, and the relative energy difference between the symmetric and un-symmetric states is only a few percent. More detailed studies with realistic parameters in a 3d trap is needed for the full understanding the dynamics of the transition between a symmetric and non-symmetric state. The recent observation of Feshbach resonance

^{2222. S. Inouye, M. R. Andrews, J. Stenger, H. -J. Meisner, D. M. Stamper-Kurn, and W. Ketterle, “Observation of Feshback resonance in a Bose-Einstein condensate,” Nature 392, 151 (1998). [CrossRef] }provides us with the exciting possibility of tuning the value of

*a*

_{12}and studying such transitions experimentally. The instability induced by large repulsive coupling is reminiscent of the cross-phase modulation (XPM) instability in nonlinear optics

^{2323. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press Inc., New York, 1989).}. In fact, the GPEs for BEC are very similar to the nonlinear Schrödinger equations describing wave propagation inside optical fibers. When one light field is present, a modulation instability occurs in the case of anomalous group velociy dispersion, analogous to the negative scattering length instability for a one-species BEC. When two light fields co-propagate, instability can be induced by XPM in both the anomalous- and normal-dispersion regimes. Such modulation instability can lead to the break up of intense cw radiation into ultrashort pulses and formation of solitons. Extensive work in the field of nonlinear optics may then help us understand the physics of BEC, such as the detailed dynamics of how quantum fluctuations will affect a TBEC with imaginary modes.

^{2424. D. S. Rokhsar, “Vortex stability and persistent currentsin trapped Bose gases,” Phys. Rev. Lett. 79, 2164 (1997). [CrossRef] }. In the case of a Rb-Na TBEC, one may produce a system comprised of a vortex-free Rb condensate at the center of the trap surrounded by a Na condensate in a vortex state such that a repulsive inter-species interaction may prevent the existence of the bound state and hence, stabilize the system. A detailed study of vortices in the TBEC is currently under way. In the above, we have briefly discussed some novel properties of the TBEC. There are still many open questions. More theoretical investigations are in progress to deepen our understanding of this unique macroscopic quantum system. Of equal importance is the realization of an experiment that will test these theoretical predictions.

## 3. Models of forced evaporative cooling in the Na-Rb system

^{2828. G.A. Bird, “Molecular Gas Dynamics,” Claredon Press, Oxford, (1976).}which is very similar to that employed by Wu, Arimondo and Foot to study evaporative cooling and Bose condensation in the single species case

^{2929. H. Wu, E. Arimondo, and C. J. Foot, “Dynamics of evaporative cooling for Bose-Einstein condensation,” Phys. Rev. A 56, 560 (1997). [CrossRef] }. The method can be described as follows: two atomic samples, each at a given temperature, are generated and accordingly accommodated by a harmonic trap (with possibly different characteristic frequencies). The samples are then separated into cells according to their position. Collisions are carried out at each cell for a time

*δt*, much smaller than the average collision time. These are hard-sphere collisions such that pairs of atoms with high relative velocity have higher probability of collision. After that, the atoms are allowed to evolve in the trap with their post-collision velocities during the same time

*δt*. Evaporative cooling is modeled by requiring that atoms with a distance from the center of the trap larger than

*R*

_{in}exp(-

*t*/

*τ*) are ejected. We used

*R*

_{in}=0.4 cm and

*τ*= 12 s.

^{2222. S. Inouye, M. R. Andrews, J. Stenger, H. -J. Meisner, D. M. Stamper-Kurn, and W. Ketterle, “Observation of Feshback resonance in a Bose-Einstein condensate,” Nature 392, 151 (1998). [CrossRef] }collisions. From our result, the sodium loss rate due to the collision with rubidium is

*β*

^{*}

*n*

_{Rb}~ 0.1/s (with a density of

*n*

_{Rb}= 8 × 10

^{8}cm

^{-3}). And using a relative velocity of ~ 15 cm/s, we obtain the estimated Na-Rb cross section

*σ*

_{Na-Rb}= 1.8 × 10

^{-12}cm

^{2}. However, it should be noted that, even though this estimate is the best that can be inferred from the available data from two-species experiments, the true value of the inter-species cross-section is dependent on the yet undetermined inter-species scattering length. If the actual cross-section is much smaller than our estimate, the evaporative cooling process may need to be carried out at a slower rate in order to allow for efficient thermalization of the atomic samples during the evaporation process. As a first step in this investigation, we have modeled the thermalization of the two atomic clouds assuming that they are loaded into themagnetic trap with each species at its respective Doppler limited temperature (240

*μ*K for Na and 120

*μ*K for Rb). As shown in Fig. 3, we find that thermal relaxation occurs in a few hundred ms. Naturally, larger the inter-species cross section

*σ*

_{Na-Rb}is, the faster the system reaches its equilibrium.

*n*

^{2929. H. Wu, E. Arimondo, and C. J. Foot, “Dynamics of evaporative cooling for Bose-Einstein condensation,” Phys. Rev. A 56, 560 (1997). [CrossRef] }. Hence the penetration into the critical BEC boundary (defined by

*n*

^{2929. H. Wu, E. Arimondo, and C. J. Foot, “Dynamics of evaporative cooling for Bose-Einstein condensation,” Phys. Rev. A 56, 560 (1997). [CrossRef] }) is currently being utilized. Moreover, we are also taking full advantage of our multi-species capability to model the physics of sympathetic cooling, formation of Fermi-Bose mixtures and mixtures of Bose gases with various combinations of scattering lenghts.

## 4. Conclusions and prospects

## Acknowledgements

## References

1. | M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observations of Bose-Einstein condensation in a dilute atomic vapor,” Science |

2. | K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. |

3. | C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. |

4. | C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, “Production of two overlapping Bose-Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. |

5. | Tin-Lun Ho and V. B. Shenoy, “Binary mixtures of Bose condensates of alkali atoms,” Phys. Rev. Lett. |

6. | B. D. Esry, Chris H. Greene, James P. Burke Jr., and John L. Bohn, “Hartree-Fock theory for double condensates,” Phys. Rev. Lett. |

7. | Elena V. Goldstein and Pierre Meystre, “Quasiparticle instabilities in multicomponent atomic condensates,” Phys. Rev. A , |

8. | Th. Busch, J. I. Cirac, V. M. Perez-Garcia, and P. Zoller, “Stability and collective excitations of a two-component Bose-Einstein condensed gas: a moment approach,” Phys. Rev. A , |

9. | C. K. Law, H. Pu, N. P. Bigelow, and J. H. Eberly, “Stability signature in two-species dilute Bose-Einstein condensates,” Phys. Rev. Lett. |

10. | H. Pu and N. P. Bigelow, “Properties of two-species Bose condensates,” Phys. Rev. Lett. |

11. | H. Pu and N. P. Bigelow, “Collective excitations, metastability and nonlinear response of a trapped two-species Bose-Einstein condensate,” Phys. Rev. Lett. |

12. | C. K. Law, H. Pu, N. P. Bigelow, and J. H. Eberly, “Quantum phase diffusion of a two-component dilute Bose-Einstein condensate,” accepted by Phys. Rev. A. |

13. | J. Shaffer and N. P. Bigelow, “Two-species trap experiments,” Opt. Photonics News Supp. |

14. | D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Collective Excitations of a Bose-Einstein condensate in a dilute gas,” Phys. Rev. Lett. |

15. | M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Collective excitations of a Bose-Einstein condensate in a magnetic trap,” Phys. Rev. Lett. |

16. | Mark Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and Charles W. Clark, “Collective excitations of atomic Bose-Einstein condensates,” Phys. Rev. Lett. |

17. | S. Stringari, “Collective excitations of a trapped Bose-condensed gas,” Phys. Rev. Lett. |

18. | Y. Castin and R. Dum, “Bose-Einstein condensates in time dependent traps,” Phys. Rev. Lett. |

19. | V. M. Perez-Garcia, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, “Low energy excitations of a Bose-Einstein condensate: a time-dependent variational analysis,” Phys. Rev. Lett. |

20. | A. L. Fetter, “Nonuniform states of an imperfect Bose gas,” Ann. Phys. (N.Y.) |

21. | Patrik Öhberg and Stig Stenholm,“Hartree-Fock treatment of the two-component Bose-Einstein condensate,” Phys. Rev. A |

22. | S. Inouye, M. R. Andrews, J. Stenger, H. -J. Meisner, D. M. Stamper-Kurn, and W. Ketterle, “Observation of Feshback resonance in a Bose-Einstein condensate,” Nature |

23. | G. P. Agrawal, |

24. | D. S. Rokhsar, “Vortex stability and persistent currentsin trapped Bose gases,” Phys. Rev. Lett. |

25. | M. Lewenstein and L. You, “Quantum phase diffusion of a Bose-Einstein condensate,” Phys. Rev. Lett. |

26. | R. Ejnisman, Y. E. Young, P. Rudy, and N. P. Bigelow, to be submitted. |

27. | We have recently learned that V.Bagnato’s group in São Carlos, Brazil, has also been investigating a Na-Rb MOT. |

28. | G.A. Bird, “Molecular Gas Dynamics,” Claredon Press, Oxford, (1976). |

29. | H. Wu, E. Arimondo, and C. J. Foot, “Dynamics of evaporative cooling for Bose-Einstein condensation,” Phys. Rev. A |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(140.3320) Lasers and laser optics : Laser cooling

**ToC Category:**

Focus Issue: Collective phenomena in trapped atoms and ions

**History**

Original Manuscript: February 1, 1998

Published: April 13, 1998

**Citation**

Renato Ejnisman, Han Pu, York Young, Nicholas Bigelow, and C. Law, "Studies of two-species Bose-Einstein condesation," Opt. Express **2**, 330-337 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-8-330

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### References

- M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, "Observations of Bose-Einstein condensation in a dilute atomic vapor," Science 269, 198 (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, "Bose-Einstein condensation in a gas of sodium atoms," Phys. Rev. Lett. 75, 3969 (1995). [CrossRef] [PubMed]
- C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, "Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions," Phys. Rev. Lett. 75, 1687 (1995). [CrossRef] [PubMed]
- C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, "Production of two overlapping Bose-Einstein condensates by sympathetic cooling," Phys. Rev. Lett. 78, 586 (1997). [CrossRef]
- Tin-Lun Ho and V. B. Shenoy, "Binary mixtures of Bose condensates of alkali atoms," Phys. Rev. Lett. 77, 3276 (1996). [CrossRef] [PubMed]
- B. D. Esry, Chris H. Greene, James P. Burke, Jr., and John L. Bohn, "Hartree-Fock theory for double condensates," Phys. Rev. Lett. 78, 3594 (1995). [CrossRef]
- Elena V. Goldstein and Pierre Meystre, "Quasiparticle instabilities in multicomponent atomic condensates," Phys. Rev. A, 55, 2935 (1997). [CrossRef]
- Th. Busch, J. I. Cirac, V. M. Perez-Garcia, and P. Zoller, "Stability and collective excitations of a two-component Bose-Einstein condensed gas: a moment approach," Phys. Rev. A, 56, 2978 (1997). [CrossRef]
- C. K. Law, H. Pu, N. P. Bigelow, and J. H. Eberly, "Stability signature in two-species dilute Bose-Einstein condensates," Phys. Rev. Lett. 79, 3105 (1997). [CrossRef]
- H. Pu and N. P. Bigelow, "Properties of two-species Bose condensates," Phys. Rev. Lett. 80 1130 (1998). [CrossRef]
- H. Pu and N. P. Bigelow, "Collective excitations, metastability and nonlinear response of a trapped two-species Bose-Einstein condensate," Phys. Rev. Lett. 80 1134 (1998). [CrossRef]
- C. K. Law, H. Pu, N. P. Bigelow, and J. H. Eberly, "Quantum phase diffusion of a two-component dilute Bose-Einstein condensate," accepted by Phys. Rev. A.
- J. Shaffer and N. P. Bigelow, "Two-species trap experiments," Opt. Photonics News Supp. 6:7, 47 (1995).
- D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, "Collective Excitations of a Bose-Einstein condensate in a dilute gas," Phys. Rev. Lett. 77, 420 (1996). [CrossRef] [PubMed]
- M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, "Collective excitations of a Bose-Einstein condensate in a magnetic trap," Phys. Rev. Lett. 77, 988 (1996). [CrossRef] [PubMed]
- Mark Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and Charles W. Clark, "Collective excitations of atomic Bose-Einstein condensates," Phys. Rev. Lett. 77, 1671 (1996). [CrossRef] [PubMed]
- S. Stringari, "Collective excitations of a trapped Bose-condensed gas," Phys. Rev. Lett. 77, 2360 (1996). [CrossRef] [PubMed]
- Y. Castin and R. Dum, "Bose-Einstein condensates in time dependent traps," Phys. Rev. Lett. 77, 5315 (1996). [CrossRef] [PubMed]
- V. M. Perez-Garcia, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, "Low energy excitations of a Bose-Einstein condensate: a time-dependent variational analysis," Phys. Rev. Lett. 77, 5320 (1996). [CrossRef] [PubMed]
- A. L. Fetter, "Nonuniform states of an imperfect Bose gas," Ann. Phys. (N.Y.) 70, 67 (1972). [CrossRef]
- Patrik Oehberg and Stig Stenholm,"Hartree-Fock treatment of the two-component Bose-Einstein condensate," Phys. Rev. A 57, 1272 (1998). [CrossRef]
- S. Inouye, M. R. Andrews, J. Stenger, H. -J. Meisner, D. M. Stamper-Kurn, and W. Ketterle, "Observation of Feshback resonance in a Bose-Einstein condensate," Nature 392, 151 (1998). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press Inc., New York, 1989).
- D. S. Rokhsar, "Vortex stability and persistent currentsin trapped Bose gases," Phys. Rev. Lett. 79, 2164 (1997). [CrossRef]
- M. Lewenstein and L. You, "Quantum phase diffusion of a Bose-Einstein condensate," Phys. Rev. Lett. 77, 3489 (1996). [CrossRef] [PubMed]
- R. Ejnisman, Y. E. Young, P. Rudy and N. P. Bigelow, to be submitted.
- We have recently learned that V.Bagnato's group in S~ao Carlos, Brazil, has also been investigating a Na-Rb MOT.
- G.A.Bird, "Molecular Gas Dynamics," Claredon Press, Oxford, (1976).
- H. Wu, E. Arimondo and C. J. Foot, "Dynamics of evaporative cooling for Bose-Einstein condensation," Phys. Rev. A 56, 560 (1997). [CrossRef]

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