OSA's Digital Library

Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 2, Iss. 8 — Apr. 13, 1998
  • pp: 330–337
« Show journal navigation

Studies of two-species Bose-Einstein condensation

R. Ejnisman, H. Pu, Y. E. Young, N. P. Bigelow, and C. K. Law  »View Author Affiliations


Optics Express, Vol. 2, Issue 8, pp. 330-337 (1998)
http://dx.doi.org/10.1364/OE.2.000330


View Full Text Article

Acrobat PDF (371 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

The realization of Bose-Einstein condensation (BEC) in dilute alkali vapors1–3

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 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 87Rb4

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. 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)5–12

5. 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.

The investigation of a TBEC requires progress on several different fronts. On the theoretical side, our effort is rooted in the study of the properties of a set of coupled non-linear Schrödinger equations which can be used to describe the TBEC within the mean field limit. Of equal importance is our development of an experimental strategy for preparing a high-density laser cooled atomic mixture, for loading magnetic trap where evaporativecooling can be applied to form a condensate. We first discuss our contributions to the theoretical description of the TBEC with an emphasis on those phenomenological features which distinguish the TBEC from the single component BEC 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.

In this paper we focus our discussion on the particular system composed of a mixture of sodium (Na) and rubidium (Rb) atoms. We have selected this system because (a) both species have been successfully Bose condensed and (b) because other candidate systems, such as a sodium-cesium mixture, have already been found by us13

13. 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

In the following, we give a brief description of the novel properties of a TBEC confined in an isotropic spherical potential at zero temperature. Throughout our discussion, we assume that the effect of gravity can be compensated for by choosing appropriate atomic species and spin states or by using the proper magnetic/optical trapping fields. Neglecting gravitational effects not only simplifies our calculations, it allows us to focus on the more essential intrinsic couplings within the TBEC.

Figure 1. Density distribution of a Na-Rb TBEC for different values of a 12. In our calculations, we take 87Rb as species 1 and 23Naas species 2, with scattering lengths 6 and 3 nm, respectively. N 1 = N 2 = 103. The trapping frequencies are: ω 1 = 2π × 160 Hz and ω 2 = 2π × 310 Hz. [Media 1]

At zero temperature, the self-consistent nonlinear Schrödinger equations, known as Gross-Pitaevskii equations (GPEs), for a TBEC may be written as5

5. Tin-Lun Ho and V. B. Shenoy, “Binary mixtures of Bose condensates of alkali atoms,” Phys. Rev. Lett. 77, 3276 (1996). [CrossRef] [PubMed]

,6

6. 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]

,10

10. H. Pu and N. P. Bigelow, “Properties of two-species Bose condensates,” Phys. Rev. Lett. 801130 (1998). [CrossRef]

,11

11. 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]

:

ψ1rtt=[ħ22m12+12m1ω12r2+N1U1ψ12+N2U12ψ22]ψ1,
(1)
ψ2rtt=[ħ22m22+12m2ω22r2+N2U2ψ22+N1U12ψ12]ψ2,
(2)

where ψi (r,t) denotes the macroscopic condensate wave function for species i, with r being the radial coordinate. Ni , mi and ωi are particle number, mass and trap frequency, respectively. The interaction between particles are described by a self-interaction term Ui = 4πħ 2 ai /mi and a term that corresponds to the interaction between different species U 12 = 2πħ 2 a 12/m (with m being the reduced mass of the two species), where ai 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 μiψi (r) (i=1,2), with μi being the chemical potential.

Figure 2. Density at the center of the trap as functions of time under a sinusoid modulation of the trapping potential. The units for time, length and density are: 1ω1,ξ=ħ2m1ω1 and ξ -3, respectively. a 12 = 9.6 nm. Other parameters are the same as in Fig. 1. A macroscopic quantum jump occurs at about t=25.

One of the fundamental properties of the confined condensate lies in the nature of the collective excitations. Excitation frequencies of a single-species Rb or Na condensate have been experimentally measured14

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. 77, 420 (1996). [CrossRef] [PubMed]

,15

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. 77, 988 (1996). [CrossRef] [PubMed]

and theoretically calculated16–19

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. 77, 1671 (1996). [CrossRef] [PubMed]

, and good agreement has been found between the two. We have generalized the standard Bogoliubov-Hartree theory20

20. 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 TBEC11

11. 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 unstable7

7. Elena V. Goldstein and Pierre Meystre, “Quasiparticle instabilities in multicomponent atomic condensates,” Phys. Rev. A , 55, 2935 (1997). [CrossRef]

. Recently, Öhberg21

21. 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 resonance22

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 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 optics23

23. 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.

As an important property of superfluidity, quantum vortices in alkali BEC have attracted several theoretical investigations. In his recent work, Rokhsar argued that a vortex state is unstable in a one-species condensate due to the presence of a bound core state24

24. 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

Figure 3. Thermal relaxation of a mixture of Na and Rb atoms. Trap frequencies for Na: ωx = ωy = 2×ωz = 100×2πrad/s; for Rb: : ωx = ωy = 2×ωz = 52×2πrad/s. Other parameters are: densities nRb = nNa = 8 × 108 cm-3, intra-species cross-sections σNa = 2 × 10-12 cm2, σRb = 9 × 10-12 cm2.

In order to perform the simulations, we used an estimate for the cross-section between Na and Rb obtained from our experimental studies of the inter-species22

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 392, 151 (1998). [CrossRef]

collisions. From our result, the sodium loss rate due to the collision with rubidium is β * nRb ~ 0.1/s (with a density of nRb = 8 × 108 cm-3). And using a relative velocity of ~ 15 cm/s, we obtain the estimated Na-Rb cross section σNa-Rb = 1.8 × 10-12cm2. 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 μK for Na and 120 μK 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 σNa-Rb is, the faster the system reaches its equilibrium.

The results of the evaporative cooling for the Na-Rb system are shown in Fig. 4. In Fig. 4(a), we see that after 50s (approximately 500 average collision times), more than 1% of the initial number of atoms remain in the trap. In Fig. 4(b), we note that due to the inter-species interaction, the system remains in equilibrium while evaporation takes place (except during the first second, where sympathetic cooling takes place). Fig. 4(c) displays the time dependence of the size of the samples together with the RF cut-off modeled in the manner described above. In Fig. 4(d), we show the time evolution of phase-space density (nλdB3) for the Na-Rb system undergoing evaporative cooling.

Figure 4. Evaporative cooling of a sample containing initially an equal number of Na and Rb atoms. (a) Fraction of atoms remaining as the evaporation takes place; (b) evolution temperature of the Na and Rb atoms; (c) average radius of the samples together with the cut-off imposed by the RF-field; (d) phase space density (nλdB3) for each sample. The dashed line indicates the BEC border. Inter-species cross-section σNa-Rb = 1.8×10-12 cm2. Other parameters are the same as in Fig. 3.

The model discribed here assumes purely classical hard-sphere collisions and does not take into account the (1+N) Bose-Einstein enhancement factor29

29. 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 nλdB3=2.61) should be considered as a conservative estimate. In sum, our simulations clearly indicate that evaporative cooling has the power to increase phase-space density enough to expect that TBEC formation will occur for such conditions. In the animation (Fig. 5), the evaporative cooling path can be followed in phase space (temperature vs. density) so that the effectiveness of the process can be fully appreciated.

In order to more realistically study the formation of the two-species condensate, a version of the code that includes Bose statistics (which affect the collisional rate29

29. 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

In summary, we have discussed some of the properties of the TBEC. We have shown that the nonlinear coupling between the two components gives rise to a rich set of new phenomena. We also modeled a two-species evaporative cooling process, indicating that it can lead to the formation of a TBEC. The territory of two inter-penetrating quantum fluids is still wide open and we believe that many exciting and unexpected new phenomena are waiting to be discovered.

Figure 5. Evolution of the Na and Rb samples in phase space (density vs. temperature). The lines indicate the BEC boundary for Na and Rb. Parameters are the same as in Fig. 4. [Media 2]

Acknowledgements

The experiments mentioned in this paper have benefited from the crucial work of M. Banks, J. Janis, P. Rudy and S. B. Weiss. Aspects of the theoretical work were carried out in collaboration with J. H. Eberly. This work was supported by the National Science Foundation, the David and Lucile Packard Foundation and the Army Research Office. RE is grateful for the financial support from CNPq during part of this work.

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 269, 198 (1995). [CrossRef] [PubMed]

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. 75, 3969 (1995). [CrossRef] [PubMed]

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. 75, 1687 (1995). [CrossRef] [PubMed]

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. 78, 586 (1997). [CrossRef]

5.

Tin-Lun Ho and V. B. Shenoy, “Binary mixtures of Bose condensates of alkali atoms,” Phys. Rev. Lett. 77, 3276 (1996). [CrossRef] [PubMed]

6.

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]

7.

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

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 , 56, 2978 (1997). [CrossRef]

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. 79, 3105 (1997). [CrossRef]

10.

H. Pu and N. P. Bigelow, “Properties of two-species Bose condensates,” Phys. Rev. Lett. 801130 (1998). [CrossRef]

11.

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]

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. 6:7, 47 (1995).

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. 77, 420 (1996). [CrossRef] [PubMed]

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. 77, 988 (1996). [CrossRef] [PubMed]

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. 77, 1671 (1996). [CrossRef] [PubMed]

17.

S. Stringari, “Collective excitations of a trapped Bose-condensed gas,” Phys. Rev. Lett. 77, 2360 (1996). [CrossRef] [PubMed]

18.

Y. Castin and R. Dum, “Bose-Einstein condensates in time dependent traps,” Phys. Rev. Lett. 77, 5315 (1996). [CrossRef] [PubMed]

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. 77, 5320 (1996). [CrossRef] [PubMed]

20.

A. L. Fetter, “Nonuniform states of an imperfect Bose gas,” Ann. Phys. (N.Y.) 70, 67 (1972). [CrossRef]

21.

Patrik Öhberg and Stig Stenholm,“Hartree-Fock treatment of the two-component Bose-Einstein condensate,” Phys. Rev. A 57, 1272 (1998). [CrossRef]

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 392, 151 (1998). [CrossRef]

23.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press Inc., New York, 1989).

24.

D. S. Rokhsar, “Vortex stability and persistent currentsin trapped Bose gases,” Phys. Rev. Lett. 79, 2164 (1997). [CrossRef]

25.

M. Lewenstein and L. You, “Quantum phase diffusion of a Bose-Einstein condensate,” Phys. Rev. Lett. 77, 3489 (1996). [CrossRef] [PubMed]

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 56, 560 (1997). [CrossRef]

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


Sort:  Journal  |  Reset  

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 269, 198 (1995). [CrossRef] [PubMed]
  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. 75, 3969 (1995). [CrossRef] [PubMed]
  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. 75, 1687 (1995). [CrossRef] [PubMed]
  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. 78, 586 (1997). [CrossRef]
  5. Tin-Lun Ho and V. B. Shenoy, "Binary mixtures of Bose condensates of alkali atoms," Phys. Rev. Lett. 77, 3276 (1996). [CrossRef] [PubMed]
  6. 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]
  7. Elena V. Goldstein and Pierre Meystre, "Quasiparticle instabilities in multicomponent atomic condensates," Phys. Rev. A, 55, 2935 (1997). [CrossRef]
  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, 56, 2978 (1997). [CrossRef]
  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. 79, 3105 (1997). [CrossRef]
  10. H. Pu and N. P. Bigelow, "Properties of two-species Bose condensates," Phys. Rev. Lett. 80 1130 (1998). [CrossRef]
  11. 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]
  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. 6:7, 47 (1995).
  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. 77, 420 (1996). [CrossRef] [PubMed]
  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. 77, 988 (1996). [CrossRef] [PubMed]
  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. 77, 1671 (1996). [CrossRef] [PubMed]
  17. S. Stringari, "Collective excitations of a trapped Bose-condensed gas," Phys. Rev. Lett. 77, 2360 (1996). [CrossRef] [PubMed]
  18. Y. Castin and R. Dum, "Bose-Einstein condensates in time dependent traps," Phys. Rev. Lett. 77, 5315 (1996). [CrossRef] [PubMed]
  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. 77, 5320 (1996). [CrossRef] [PubMed]
  20. A. L. Fetter, "Nonuniform states of an imperfect Bose gas," Ann. Phys. (N.Y.) 70, 67 (1972). [CrossRef]
  21. Patrik Oehberg and Stig Stenholm,"Hartree-Fock treatment of the two-component Bose-Einstein condensate," Phys. Rev. A 57, 1272 (1998). [CrossRef]
  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 392, 151 (1998). [CrossRef]
  23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press Inc., New York, 1989).
  24. D. S. Rokhsar, "Vortex stability and persistent currentsin trapped Bose gases," Phys. Rev. Lett. 79, 2164 (1997). [CrossRef]
  25. M. Lewenstein and L. You, "Quantum phase diffusion of a Bose-Einstein condensate," Phys. Rev. Lett. 77, 3489 (1996). [CrossRef] [PubMed]
  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 S~ao 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 56, 560 (1997). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Multimedia

Multimedia FilesRecommended Software
» Media 1: MOV (123 KB)      QuickTime
» Media 2: MOV (598 KB)      QuickTime

« Previous Article

OSA is a member of CrossRef.

CrossCheck Deposited