Abstract

We study a system of interacting bosons at zero temperature in an atomic trap. Using wave function that models the ground state of interacting bosons we examine the concepts of the order parameter, off-diagonal order and coherence of the system. We suggest that the coherence length becomes much smaller than the size of the system if the number of trapped particles exceeds a certain limit. This behavior is related to the unavoidable existence of two different length scales – one determined by the external potential and the second one depending on the two-body forces.

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References

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  1. D. Kleppner, "A Beginner's Guide to the Atom Laser," Phys. Today 8, 11-13 (1997).
    [CrossRef]
  2. M. R. Andrews et. al. "Observation of interference between two Bose condensates," Science 275, 637-641 (1997).
    [CrossRef] [PubMed]
  3. J. Stenger et al. "Bragg spectroscopy of a Bose Einstein condensate," Phys. Rev. Lett. 82, 4569-4573 (1998).
    [CrossRef]
  4. E. W. Hagle et al. "Measurement of coherence of a Bose Einstein condensate," Phys. Rev. Lett. 83, 312-315 (1999).
    [CrossRef]
  5. I. Bloch, T. W. H�nsch, and T. Esslinger, "Measurement of the spatial coherence of a trapped Bose gas at the phase transition," Nature 403, 166-170 (2000).
    [CrossRef] [PubMed]
  6. O. Penrose, "On the quantum mechanics of helium II," Phil. Mag. 42, 1373-1377 (1951).
  7. O. Penrose, L. Onsager, "Bose Einstein condensation and liquid helium," Phys. Rev. 104, 576-584 (1956).
    [CrossRef]
  8. S. T. Beliaev, "Application of the method of quantum field theory to a system of bosons," J. Exp. Theor. Phys. (USSR) 34, 417-432 (1958).
  9. C.N. Yang, "Concept of off diagonal long range order and quantum phases of liquid He and of superconductors," Rev. Mod. Phys. 34, 694-704 (1962).
    [CrossRef]
  10. J. Javanainen and S. M. Yoo, "Quantum Phase of a Bose Einstein Condensate with an Arbitrary Number of Atoms," Phys. Rev. Lett. 76, 161-164 (1996).
    [CrossRef] [PubMed]
  11. S. M .Barnett , K. Burnett, J.A. Vaccaro, "Why a condensate can be thought of as having a definite phase," J. Res. Natl. Inst. Stan. 101 593-600 (1996).
    [CrossRef]
  12. M. Zaluska-Kotur, M. Gajda, A. Orlowski, and J. Mostowski, "Soluble model of many interacting quantum particles in a trap," Phys. Rev. A 61, 033613-8 (2000).
    [CrossRef]
  13. M.Gajda, M. Zaluska-Kotur, and J. Mostowski, "Destruction of a Bose Einstein condensate by strong interactions," J. Phys. B: At. Mol. Opt. Phys. 33 4003-4016 (2000).
    [CrossRef]
  14. R. P. Feynman, "The Feynman lectures on physics" vol. III, (Addison Wesley, 1965).
  15. F. Dalfovo, S. Giorgini, L. Pitaevskii, S. Stringari, "Theory of Bose Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999).
    [CrossRef]
  16. K.Huang, "Statistical Mechanics," (Wiley, New York, 1987).
  17. D. F. Walls, "Evidence for the quantum nature of light," Nature 280, 451 (1979).
    [CrossRef]
  18. R. J. Dodd, C. W. Clark, M. Edwards, and K. Burnett, "Characterizing the coherence of Bose Einstein condensates and atom lasers," Opt. Express 1, 284-292 (1997). http://www.opticsexpress.org/oearchive/source/2369.htm
    [CrossRef] [PubMed]
  19. R. J. Glauber, "Quantum Optic and photon statistics" in Quantum Optics and Electronics, C. De Witt, A. Blandin, and C. Cohen Tannoudji, eds. (Gordon and Breach, New York, 1965).
  20. R. M. Ziff, G. E. Uhlenbeck, M. Kac, "The ideal Bose Einstein gas, revisited", Phys. Rep. 32C, 169-248 (1977).
    [CrossRef]
  21. C. J. Pethick and L. P. Pitaevskii, "On the criterion for Bose Einstein condensaton for particles in trap," preprint cond mat/0004187. http://xxx.lanl.gov/abs/cond mat/0004187

Other

D. Kleppner, "A Beginner's Guide to the Atom Laser," Phys. Today 8, 11-13 (1997).
[CrossRef]

M. R. Andrews et. al. "Observation of interference between two Bose condensates," Science 275, 637-641 (1997).
[CrossRef] [PubMed]

J. Stenger et al. "Bragg spectroscopy of a Bose Einstein condensate," Phys. Rev. Lett. 82, 4569-4573 (1998).
[CrossRef]

E. W. Hagle et al. "Measurement of coherence of a Bose Einstein condensate," Phys. Rev. Lett. 83, 312-315 (1999).
[CrossRef]

I. Bloch, T. W. H�nsch, and T. Esslinger, "Measurement of the spatial coherence of a trapped Bose gas at the phase transition," Nature 403, 166-170 (2000).
[CrossRef] [PubMed]

O. Penrose, "On the quantum mechanics of helium II," Phil. Mag. 42, 1373-1377 (1951).

O. Penrose, L. Onsager, "Bose Einstein condensation and liquid helium," Phys. Rev. 104, 576-584 (1956).
[CrossRef]

S. T. Beliaev, "Application of the method of quantum field theory to a system of bosons," J. Exp. Theor. Phys. (USSR) 34, 417-432 (1958).

C.N. Yang, "Concept of off diagonal long range order and quantum phases of liquid He and of superconductors," Rev. Mod. Phys. 34, 694-704 (1962).
[CrossRef]

J. Javanainen and S. M. Yoo, "Quantum Phase of a Bose Einstein Condensate with an Arbitrary Number of Atoms," Phys. Rev. Lett. 76, 161-164 (1996).
[CrossRef] [PubMed]

S. M .Barnett , K. Burnett, J.A. Vaccaro, "Why a condensate can be thought of as having a definite phase," J. Res. Natl. Inst. Stan. 101 593-600 (1996).
[CrossRef]

M. Zaluska-Kotur, M. Gajda, A. Orlowski, and J. Mostowski, "Soluble model of many interacting quantum particles in a trap," Phys. Rev. A 61, 033613-8 (2000).
[CrossRef]

M.Gajda, M. Zaluska-Kotur, and J. Mostowski, "Destruction of a Bose Einstein condensate by strong interactions," J. Phys. B: At. Mol. Opt. Phys. 33 4003-4016 (2000).
[CrossRef]

R. P. Feynman, "The Feynman lectures on physics" vol. III, (Addison Wesley, 1965).

F. Dalfovo, S. Giorgini, L. Pitaevskii, S. Stringari, "Theory of Bose Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999).
[CrossRef]

K.Huang, "Statistical Mechanics," (Wiley, New York, 1987).

D. F. Walls, "Evidence for the quantum nature of light," Nature 280, 451 (1979).
[CrossRef]

R. J. Dodd, C. W. Clark, M. Edwards, and K. Burnett, "Characterizing the coherence of Bose Einstein condensates and atom lasers," Opt. Express 1, 284-292 (1997). http://www.opticsexpress.org/oearchive/source/2369.htm
[CrossRef] [PubMed]

R. J. Glauber, "Quantum Optic and photon statistics" in Quantum Optics and Electronics, C. De Witt, A. Blandin, and C. Cohen Tannoudji, eds. (Gordon and Breach, New York, 1965).

R. M. Ziff, G. E. Uhlenbeck, M. Kac, "The ideal Bose Einstein gas, revisited", Phys. Rep. 32C, 169-248 (1977).
[CrossRef]

C. J. Pethick and L. P. Pitaevskii, "On the criterion for Bose Einstein condensaton for particles in trap," preprint cond mat/0004187. http://xxx.lanl.gov/abs/cond mat/0004187

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Figures (1)

Fig. 1.
Fig. 1.

Probability distributions: (i) of a one particle detection – red line; (ii) conditional probability density for detection of the second particle provided that the first one has been found at position x=0.002 – green line. The total number of particles is N=100000 and the interaction parameter κ=1.25.

Equations (7)

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Ψ ( x 1 , , x N ) = ( 1 π ) 3 4 exp [ 1 2 x CM 2 ] ( ω π ) 3 4 ( N 1 ) exp [ ω 2 ( i = 1 N x i 2 x CM 2 ) ] ,
ρ s ( x 1 , , x s ; x 1 , , x s ) = λ i ( s ) [ ϕ i ( s ) ( x 1 , , x s ) ] * ϕ i ( s ) ( x 1 , , x s ) ,
g 1 ( x , y ) = ρ 1 ( x , y ) ρ 1 ( x ; x ) ρ 1 ( y ; y ) .
g 2 ( x , y ) = ρ 2 ( x , y ; x , y ) ρ 1 ( x ; x ) ρ 1 ( y ; y ) .
g 2 ( x , y ) = g 1 ( x , y ) g 1 ( x , y ) .
P ( x CM ) = 1 2 π exp [ ( x CM δ R ) 2 ] .
P ( x CM ) = 1 2 π exp [ x CM 2 ] .

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