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.

© 2001 Optical Society of America

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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. Andrewset. al. “Observation of interference between two Bose condensates,” Science 275, 637–641 (1997).
    [Crossref] [PubMed]
  3. J. Stengeret al. “Bragg spectroscopy of a Bose-Einstein condensate,” Phys. Rev. Lett. 82, 4569–4573 (1998).
    [Crossref]
  4. E. W. Hagleyet al. “Measurement of coherence of a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 312–315 (1999).
    [Crossref]
  5. I. Bloch, T. W. Hänsch, 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 and 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, and J.A. Vaccaro, “Why a condensate can be thought of as having a definite phase,” J. Res. Natl. Inst. Stan. 101593–600 (1996).
    [Crossref]
  12. M. ZaFluska-Kotur, M. Gajda, A. OrFlowski, 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. ZaFluska-Kotur, and J. Mostowski, “Destruction of a Bose-Einstein condensate by strong interactions,” J.Phys.B: At. Mol. Opt. Phys. 334003–4016 (2000).
    [Crossref]
  14. R. P. Feynman, “The Feynman lectures on physics” vol. III, (Addison-Wesley, 1965).
  15. F. Dalfovo, S. Giorgini, L. Pitaevskii, and 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,” Optics 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, and 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

2000 (3)

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[Crossref] [PubMed]

M. ZaFluska-Kotur, M. Gajda, A. OrFlowski, and J. Mostowski, “Soluble model of many interacting quantum particles in a trap,” Phys. Rev. A 61, 033613–8 (2000).
[Crossref]

M. Gajda, M. ZaFluska-Kotur, and J. Mostowski, “Destruction of a Bose-Einstein condensate by strong interactions,” J.Phys.B: At. Mol. Opt. Phys. 334003–4016 (2000).
[Crossref]

1999 (2)

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

E. W. Hagleyet al. “Measurement of coherence of a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 312–315 (1999).
[Crossref]

1998 (1)

J. Stengeret al. “Bragg spectroscopy of a Bose-Einstein condensate,” Phys. Rev. Lett. 82, 4569–4573 (1998).
[Crossref]

1997 (3)

D. Kleppner, “A Beginner’s Guide to the Atom Laser,” Phys. Today,  8, 11–13 (1997).
[Crossref]

M. R. Andrewset. al. “Observation of interference between two Bose condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

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

1996 (2)

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, and J.A. Vaccaro, “Why a condensate can be thought of as having a definite phase,” J. Res. Natl. Inst. Stan. 101593–600 (1996).
[Crossref]

1979 (1)

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

1962 (1)

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]

1958 (1)

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).

1956 (1)

O. Penrose and L. Onsager, “Bose-Einstein condensation and liquid helium,” Phys. Rev. 104, 576–584 (1956).
[Crossref]

1951 (1)

O. Penrose, “On the quantum mechanics of helium II,” Phil. Mag. 42, 1373–1377 (1951)

Andrews, M. R.

M. R. Andrewset. al. “Observation of interference between two Bose condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

Barnett, S. M.

S. M. Barnett, K. Burnett, and J.A. Vaccaro, “Why a condensate can be thought of as having a definite phase,” J. Res. Natl. Inst. Stan. 101593–600 (1996).
[Crossref]

Beliaev, S. T.

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).

Bloch, I.

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[Crossref] [PubMed]

Burnett, K.

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

S. M. Barnett, K. Burnett, and J.A. Vaccaro, “Why a condensate can be thought of as having a definite phase,” J. Res. Natl. Inst. Stan. 101593–600 (1996).
[Crossref]

Clark, C.W.

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

Dalfovo, F.

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

Dodd, R.J.

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

Edwards, M.

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

Esslinger, T.

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[Crossref] [PubMed]

Feynman, R. P.

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

Gajda, M.

M. ZaFluska-Kotur, M. Gajda, A. OrFlowski, and J. Mostowski, “Soluble model of many interacting quantum particles in a trap,” Phys. Rev. A 61, 033613–8 (2000).
[Crossref]

M. Gajda, M. ZaFluska-Kotur, and J. Mostowski, “Destruction of a Bose-Einstein condensate by strong interactions,” J.Phys.B: At. Mol. Opt. Phys. 334003–4016 (2000).
[Crossref]

Giorgini, S.

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

Glauber, R. J.

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).

Hagley, E. W.

E. W. Hagleyet al. “Measurement of coherence of a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 312–315 (1999).
[Crossref]

Hänsch, T. W.

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[Crossref] [PubMed]

Huang, K.

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

Javanainen, J.

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]

Kac, M.

R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, Phys. Rep.32C, 169–248 (1977).
[Crossref]

Kleppner, D.

D. Kleppner, “A Beginner’s Guide to the Atom Laser,” Phys. Today,  8, 11–13 (1997).
[Crossref]

Mostowski, J.

M. ZaFluska-Kotur, M. Gajda, A. OrFlowski, and J. Mostowski, “Soluble model of many interacting quantum particles in a trap,” Phys. Rev. A 61, 033613–8 (2000).
[Crossref]

M. Gajda, M. ZaFluska-Kotur, and J. Mostowski, “Destruction of a Bose-Einstein condensate by strong interactions,” J.Phys.B: At. Mol. Opt. Phys. 334003–4016 (2000).
[Crossref]

Onsager, L.

O. Penrose and L. Onsager, “Bose-Einstein condensation and liquid helium,” Phys. Rev. 104, 576–584 (1956).
[Crossref]

OrFlowski, A.

M. ZaFluska-Kotur, M. Gajda, A. OrFlowski, and J. Mostowski, “Soluble model of many interacting quantum particles in a trap,” Phys. Rev. A 61, 033613–8 (2000).
[Crossref]

Penrose, O.

O. Penrose and L. Onsager, “Bose-Einstein condensation and liquid helium,” Phys. Rev. 104, 576–584 (1956).
[Crossref]

O. Penrose, “On the quantum mechanics of helium II,” Phil. Mag. 42, 1373–1377 (1951)

Pethick, C. J.

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

Pitaevskii, L.

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

Pitaevskii, L. P.

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

Stenger, J.

J. Stengeret al. “Bragg spectroscopy of a Bose-Einstein condensate,” Phys. Rev. Lett. 82, 4569–4573 (1998).
[Crossref]

Stringari, S.

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

Uhlenbeck, G.E.

R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, Phys. Rep.32C, 169–248 (1977).
[Crossref]

Vaccaro, J.A.

S. M. Barnett, K. Burnett, and J.A. Vaccaro, “Why a condensate can be thought of as having a definite phase,” J. Res. Natl. Inst. Stan. 101593–600 (1996).
[Crossref]

Walls, D. F.

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

Yang, C.N.

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]

Yoo, S. M.

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]

ZaFluska-Kotur, M.

M. Gajda, M. ZaFluska-Kotur, and J. Mostowski, “Destruction of a Bose-Einstein condensate by strong interactions,” J.Phys.B: At. Mol. Opt. Phys. 334003–4016 (2000).
[Crossref]

M. ZaFluska-Kotur, M. Gajda, A. OrFlowski, and J. Mostowski, “Soluble model of many interacting quantum particles in a trap,” Phys. Rev. A 61, 033613–8 (2000).
[Crossref]

Ziff, R.M.

R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, Phys. Rep.32C, 169–248 (1977).
[Crossref]

J. Exp. Theor. Phys. (USSR) (1)

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).

J. Res. Natl. Inst. Stan. (1)

S. M. Barnett, K. Burnett, and J.A. Vaccaro, “Why a condensate can be thought of as having a definite phase,” J. Res. Natl. Inst. Stan. 101593–600 (1996).
[Crossref]

J.Phys.B: At. Mol. Opt. Phys. (1)

M. Gajda, M. ZaFluska-Kotur, and J. Mostowski, “Destruction of a Bose-Einstein condensate by strong interactions,” J.Phys.B: At. Mol. Opt. Phys. 334003–4016 (2000).
[Crossref]

Nature (2)

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

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[Crossref] [PubMed]

Optics Express (1)

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

Phil. Mag. (1)

O. Penrose, “On the quantum mechanics of helium II,” Phil. Mag. 42, 1373–1377 (1951)

Phys. Rev. (1)

O. Penrose and L. Onsager, “Bose-Einstein condensation and liquid helium,” Phys. Rev. 104, 576–584 (1956).
[Crossref]

Phys. Rev. A (1)

M. ZaFluska-Kotur, M. Gajda, A. OrFlowski, and J. Mostowski, “Soluble model of many interacting quantum particles in a trap,” Phys. Rev. A 61, 033613–8 (2000).
[Crossref]

Phys. Rev. Lett. (3)

J. Stengeret al. “Bragg spectroscopy of a Bose-Einstein condensate,” Phys. Rev. Lett. 82, 4569–4573 (1998).
[Crossref]

E. W. Hagleyet al. “Measurement of coherence of a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 312–315 (1999).
[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]

Phys. Today (1)

D. Kleppner, “A Beginner’s Guide to the Atom Laser,” Phys. Today,  8, 11–13 (1997).
[Crossref]

Rev. Mod. Phys. (2)

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]

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

Science (1)

M. R. Andrewset. al. “Observation of interference between two Bose condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

Other (5)

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

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, and 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

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

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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)

Equations on this page are rendered with MathJax. Learn more.

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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