Abstract

We investigate theoretically the collective excitations of trapped Bose condensates with energies of the order of the chemical potential of the system. For the MIT sodium Bose condensate in a Cloverleaf trap, we find interesting level crossing behavior for high energy excitations and calculate the spatial magnetic dipole moments for selective creation of condensate excitations.

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References

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  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 condensations 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. M. R. Andrews, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Direct, nondestructive observation of a Bose condensate", Science 273, 84 (1996).
    [CrossRef] [PubMed]
  5. C. C. Bradley, C. A. Sackett, and R. G. Hulet, "Analysis of in situ images of Bose-Einstein condensates of lithium", Phys. Rev. A 55, 3951 (1997).
    [CrossRef]
  6. D. S. Jin, J. R. Ensher, M. R. Matthews, C. Wieman, and E. A. Cornell, "Collective excitations of a Bose-Einstein condensate in a dilute gas", Phys. Rev. Lett. 77, 420 (1996).
    [CrossRef] [PubMed]
  7. 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]
  8. D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, "Temperature-dependent damping and frequency shifts in collective excitations of a dilute Bose-Einstein condensate", Phys. Rev. Lett. 78, 764 (1997).
    [CrossRef]
  9. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Observation of interference between two Bose condensates", Science 275, 637 (1997).
    [CrossRef] [PubMed]
  10. E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, "Coherence, correlations, and collisions: What one learns about Bose-Einstein condensates from their decay", Phys. Rev. Lett. 79, 337 (1997).
    [CrossRef]
  11. M.-O. Mewes, M. R. Andrews,D.M.Kurn,D. S. Durfee,C.G.Townsend,andW. Ketterle, "Output coupler for Bose-Einstein condensed atoms", Phys. Rev. Lett. 78, 582 (1997);
    [CrossRef]
  12. M. R. Andrews, D. M. Kurn, H. J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Ketterle, "Propagation of sound in a Bose-Einstein condensate", Phys. Rev. Lett. 79, 553 (1997).
    [CrossRef]
  13. M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, "Collective excitations of atomic Bose-Einstein condensates", Phys. Rev. Lett. 77, 1671 (1996).
    [CrossRef] [PubMed]
  14. K. G. Singh and D. S. Rokhsar, "Collective excitations of a confined Bose condensate", Phys. Rev. Lett. 77, 1667 (1996).
    [CrossRef] [PubMed]
  15. S. Stringari, "Collective excitations of a trapped Bose condensed gas", Phys. Rev. Lett. 77, 2360 (1996).
    [CrossRef] [PubMed]
  16. A. L. Fetter, "Ground state and excited states of a confined condensed Bose gas", Phys. Rev. A 53, 4245 (1996).
    [CrossRef] [PubMed]
  17. V. M. Perez-Garcia, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, "Low energy excitations of a Bose-Einstein condensate: a variational analysis", Phys. Rev. Lett. 77, 5230 (1996).
    [CrossRef]
  18. Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, "Evolution of a Bose-condensed gas under variations of the confining potential", Phys. Rev. A 54, R1753 (1996).
    [CrossRef] [PubMed]
  19. B. D. Esry, "Hartree-Fock theory for Bose-Einstein condensates and the inclusion of correlation effects", Phys. Rev. A 55, 1147 (1997).
    [CrossRef]
  20. Y. Castin and R. Dum, "Bose-Einstein condensation in time dependent traps", Phys. Rev. Lett. 77, 5315 (1996).
    [CrossRef] [PubMed]
  21. A. Grin, W.-C. Wu, and S. Stringari, "Hydrodynamic modes in a trapped Bose gas above the Bose-Einstein transition", Phys. Rev. Lett. 78, 1838 (1997).
    [CrossRef]
  22. P. Ohberg, E.L. Surkov, I. Tittonen, S. Stenholm, M. Wilkens, G. V. Shlyapnikov, "Low-energy elementary excitations of a trapped Bose-condensed gas", (preprint, 5/97).
  23. A. L. Fetter and D. Rokhsar, "Excited states of a dilute Bose-Einstein condensate in a harmonic trap", (preprint, 4/97).
  24. D. A. W. Hutchinson, E. Zaremba, and A. Grin, "Finite temperature excitations of a trapped Bose gas", Phys. Rev. Lett. 78, 1842 (1997).
    [CrossRef]
  25. M. Edwards, (private communications).
  26. L. You, W. Hoston, M. Lewenstein, and M Marinescu, "Low energy excitation spectra of trapped Bose condensates", Acta Phys. Pol. A,(to appear).
  27. L. You, W. Hoston, and M. Lewenstein, "Low energy excitations of trapped Bose condensates", Phys. Rev. A 55, R1581 (1997);
    [CrossRef]
  28. M. Fliesser, A. Csordas, R. Graham, P. Szepfalusy, "Classical quasi-particle dynamics in trapped Bose condensates", (preprint, cond-mat/9707122).
  29. M. Fliesser, A. Csordas, P. Szepfalusy, R. Graham, "Hydrodynamic excitations of Bose condensates in anisotropic traps", (preprint, cond-mat/9706002).
  30. R. Walsworth and L. You, "Selective creation of quasi-particles in trapped Bose condensates", Phys. Rev. A 55, 555 (1997).
    [CrossRef]
  31. M. Lewenstein and L. You, "Quantum phase diffusion of the Bose-Einstein condensate", Phys. Rev. Lett. 77, 3489 (1996).
    [CrossRef] [PubMed]

Other

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 condensations 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]

M. R. Andrews, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Direct, nondestructive observation of a Bose condensate", Science 273, 84 (1996).
[CrossRef] [PubMed]

C. C. Bradley, C. A. Sackett, and R. G. Hulet, "Analysis of in situ images of Bose-Einstein condensates of lithium", Phys. Rev. A 55, 3951 (1997).
[CrossRef]

D. S. Jin, J. R. Ensher, M. R. Matthews, C. 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]

D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, "Temperature-dependent damping and frequency shifts in collective excitations of a dilute Bose-Einstein condensate", Phys. Rev. Lett. 78, 764 (1997).
[CrossRef]

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Observation of interference between two Bose condensates", Science 275, 637 (1997).
[CrossRef] [PubMed]

E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, "Coherence, correlations, and collisions: What one learns about Bose-Einstein condensates from their decay", Phys. Rev. Lett. 79, 337 (1997).
[CrossRef]

M.-O. Mewes, M. R. Andrews,D.M.Kurn,D. S. Durfee,C.G.Townsend,andW. Ketterle, "Output coupler for Bose-Einstein condensed atoms", Phys. Rev. Lett. 78, 582 (1997);
[CrossRef]

M. R. Andrews, D. M. Kurn, H. J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Ketterle, "Propagation of sound in a Bose-Einstein condensate", Phys. Rev. Lett. 79, 553 (1997).
[CrossRef]

M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, "Collective excitations of atomic Bose-Einstein condensates", Phys. Rev. Lett. 77, 1671 (1996).
[CrossRef] [PubMed]

K. G. Singh and D. S. Rokhsar, "Collective excitations of a confined Bose condensate", Phys. Rev. Lett. 77, 1667 (1996).
[CrossRef] [PubMed]

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

A. L. Fetter, "Ground state and excited states of a confined condensed Bose gas", Phys. Rev. A 53, 4245 (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 variational analysis", Phys. Rev. Lett. 77, 5230 (1996).
[CrossRef]

Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, "Evolution of a Bose-condensed gas under variations of the confining potential", Phys. Rev. A 54, R1753 (1996).
[CrossRef] [PubMed]

B. D. Esry, "Hartree-Fock theory for Bose-Einstein condensates and the inclusion of correlation effects", Phys. Rev. A 55, 1147 (1997).
[CrossRef]

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

A. Grin, W.-C. Wu, and S. Stringari, "Hydrodynamic modes in a trapped Bose gas above the Bose-Einstein transition", Phys. Rev. Lett. 78, 1838 (1997).
[CrossRef]

P. Ohberg, E.L. Surkov, I. Tittonen, S. Stenholm, M. Wilkens, G. V. Shlyapnikov, "Low-energy elementary excitations of a trapped Bose-condensed gas", (preprint, 5/97).

A. L. Fetter and D. Rokhsar, "Excited states of a dilute Bose-Einstein condensate in a harmonic trap", (preprint, 4/97).

D. A. W. Hutchinson, E. Zaremba, and A. Grin, "Finite temperature excitations of a trapped Bose gas", Phys. Rev. Lett. 78, 1842 (1997).
[CrossRef]

M. Edwards, (private communications).

L. You, W. Hoston, M. Lewenstein, and M Marinescu, "Low energy excitation spectra of trapped Bose condensates", Acta Phys. Pol. A,(to appear).

L. You, W. Hoston, and M. Lewenstein, "Low energy excitations of trapped Bose condensates", Phys. Rev. A 55, R1581 (1997);
[CrossRef]

M. Fliesser, A. Csordas, R. Graham, P. Szepfalusy, "Classical quasi-particle dynamics in trapped Bose condensates", (preprint, cond-mat/9707122).

M. Fliesser, A. Csordas, P. Szepfalusy, R. Graham, "Hydrodynamic excitations of Bose condensates in anisotropic traps", (preprint, cond-mat/9706002).

R. Walsworth and L. You, "Selective creation of quasi-particles in trapped Bose condensates", Phys. Rev. A 55, 555 (1997).
[CrossRef]

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

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Figures (3)

Fig. 1.
Fig. 1.

The calculated dependence of quasi-particle excitation frequencies on the number of condensed sodium atoms N, for the MIT Cloverleaf trap. The three separate panels are for the 4 separate parity sectors in x and y coordinates respectively: the first panel shows (even x, even y), (note k starts from 0 in the low N limit); the second panel shows (even x, odd y) [which is the same as the (odd x, even y) sector] ( k starts from ωx = 13ω in the low N limit); and the third panel shows (odd x, odd y) ( k starts from ωx + ωy = 26ω in the low N limit). The even z parity states are plotted with solid lines while the odd parity z states are plotted with thin dashed lines. The thick dashed line denotes the chemical potential.

Fig. 2.
Fig. 2.

The same calculations as in Fig. 1, now displaying the quasi-particle level structure for Lz = 0 excitations [a subset of the (even x, even y) parity sector shown in Fig. 1]. The open circles denote two calculated shape oscillation modes 17 . The high N, or asymptotic, limit of these two modes corresponds well with shapeoscillations observed in the MIT experiment 7 . The dashed line in the asymptotic limit, N = 105 - 106, are results based on Ref. 15. The filled circles, triangles, squares, and crosses denote, respectively, the modes labelled n = 0, 1, 2, 3 in Ref. 22.

Fig. 3.
Fig. 3.

Calculated SMR coupling constants for the two Lz = 0 quasi-particle modes that are degenerate at = 26 when N = 1. These two modes are denoted as ‘H“ (for higher energy, this is the shape oscillation mode, see Fig. 2) and ‘L“ (for lower energy) with Ω k = u 0k + v 0k and γk = γ 0k + γ k0. A simple, spatially uniform SMR magnetic field was assumed. The dots denote the numerically computed points. As one can see, the coupling to the lower energy mode ‘L“ is practically zero with this spatially uniform SMR field.

Equations (12)

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𝛨 = b r Ψ ̂ ( r ) [ ħ 2 2 M 2 + V t ( r ) μ ] Ψ ̂ ( r ) + u 0 2 b r Ψ ̂ ( r ) Ψ ̂ ( r ) Ψ ̂ ( r ) Ψ ̂ ( r ) ,
𝛨 k 0 ħ ω ͂ k g ͂ k g ͂ k + ( k = 0 zero mode part ) ,
Ψ ̂ ( r ) = N ψ 0 ( r ) + δ Ψ ̂ ( r ) ,
[ L + u 0 ρ 0 ] ψ 0 ( r ) = 0 ,
g ͂ k = d r [ U k ( r ) δ Ψ ̂ ( r ) + V k ( r ) δ Ψ ̂ ( r ) ] ,
[ L + 2 u 0 ρ 0 ( r ) ] U k ( r ) u 0 Δ 0 * ( r ) V k ( r ) = ħ ω ͂ k U k ( r ) ,
[ L + 2 u 0 ρ 0 ( r ) ] V k ( r ) u 0 Δ 0 ( r ) U k ( r ) = ħ ω ͂ k V k ( r ) ,
i ψ i ( r ) i IN μ m · B SMR ( r , t ) IN f ψ f ( r ) f
= μ m ψ i ( r ) e ̂ m ( r ) · e ̂ B ( r ) B SMR ( r ) ψ f ( r ) F ( t )
μ m ψ i ( r ) B SMR ( r ) ψ f ( r ) F ( t ) ,
𝛨 SMR = d r Ψ ̂ ( r ) μ m · B SMR ( r , t ) Ψ ̂ ( r ) ,
𝛨 SMR = F ( t ) k , k ' = 0 [ ( ħ u k k ' + ħ v k ' k ) g ͂ k g ͂ k ' ħ γ k ' k * g ͂ k g ͂ k ' ħ γ k k ' g ͂ k g ͂ k ] ,

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