Abstract

We study the equilibrium dynamics of a weakly interacting Bose-Einstein condensate trapped in a box. In our approach we use a semiclassical approximation similar to the description of a multi-mode laser. In dynamical equations derived from a full N-body quantum Hamiltonian we substitute all creation (and annihilation) operators (of a particle in a given box state) by appropriate c-number amplitudes. The set of nonlinear equations obtained in this way is solved numerically. We show that on the time scale of a few miliseconds the system exhibits relaxation – reaches an equilibrium with populations of different eigenstates fluctuating around their mean values.

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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, "Observation of Bose Einstein Condensation in a Dilute Atomic Vapor," Science 269, 198-201 (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, 3969-3972 (1995).
    [CrossRef] [PubMed]
  3. C.C. Bradle , 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, 1687-1690 (1995) and Erratum 79, 1170(E) (1997).
    [CrossRef] [PubMed]
  4. D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner, and T.J. Greytak, "Bose Einstein condensation of atomic hydrogen," Phys. Rev. Lett. 81, 3811-3814 (1998).
    [CrossRef]
  5. P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzazewski, "Fourth statistical ensemble for the Bose Einstein condensate," Phys. Rev. Lett. 79, 1789-1792 (1997).
    [CrossRef]
  6. M. Gajda and K. Rzazewski, "Fluctuations of Bose Einstein condensate," Phys. Rev. Lett. 78, 2686-2689 (1997).
    [CrossRef]
  7. S. Grossmann and M. Holthaus, "Fluctuations of the particle number in a trapped Bose Einstein condensate," Phys. Rev. Lett. 79, 3557-3560 (1997).
    [CrossRef]
  8. S. Grossmann and M. Holthaus, "Maxwell's Demon at work: Two t pes of Bose condensate fluctuations in power law traps," Opt. Express 1, 262-271 (1997), http://www.opticsexpress.org/oearchive/source/2288.htm
    [CrossRef] [PubMed]
  9. H. D. Politzer, "Condensate fluctuations of a trapped, ideal Bose gas," Phys. Rev. A 54, 5048-5054 (1996).
    [CrossRef] [PubMed]
  10. M. Wilkens and C. Weiss, "Particle number fluctuations in an ideal Bose gas," J. Mod. Opt. 44, 1801-1814 (1997).
    [CrossRef]
  11. M. Wilkens and C. Weiss, "Particle number counting statistics in ideal Bose gases," Opt. Express 1, 272-283 (1997), http://www.opticsexpress.org/oearchive/source/2372.htm
    [CrossRef] [PubMed]
  12. S. Giorgini, L.P. Pitaevskii, and S. Stringari, "Anomalous fluctuations of the condensate in interacting Bose gases," Phys. Rev. Lett 80, 5040-5043 (1998).
    [CrossRef]
  13. Z.Idziaszek, M.Gajda, P. Navez, M. Wilkens, and K.Rzazewski, "Fluctuations of the weakly interacting Bose Einstein condensate," Phys. Rev. Lett. 82, 4376-4379 (1999).
    [CrossRef]
  14. F. Meier and W. Zwerger, "Anomalous condensate fluctuations in strongly interacting superfluids," Phys. Rev. A 60, 5133-5135 (1999).
    [CrossRef]
  15. V.V. Kocharovsk , V.V. Kocharovsk , and M.O. Scull , "Condensate statistics in interacting and ideal dilute Bose gases," Phys. Rev. Lett. 84, 2306-2309 (2000).
    [CrossRef] [PubMed]
  16. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999).
    [CrossRef]
  17. R. Graham, "Condensate fluctuations in finite Bose Einstein condensates at finite temperature," Phys. Rev. A 62, 023609 (2000).
    [CrossRef]
  18. R. Graham, "Decoherence of Bose Einstein condensates in traps at finite temperature," Phys. Rev. Lett. 81, 5262-5265 (1998).
    [CrossRef]
  19. C.W. Gardiner and P. Zoller, "Quantum kinetic theory: A quantum kinetic master equation for condensation of a weakly interacting Bose gas without a trapping potential," Phys. Rev. A 55, 2902-2921 (1997).
    [CrossRef]
  20. D. Jaksch, C.W. Gardiner, and P. Zoller, "Quantum kinetic theory. 2.Simulation of the quantum Boltzmann master equation," Phys. Rev. A 56, 575- 586 (1997).
    [CrossRef]
  21. R. Walser, J. Williams, J. Cooper, and M. Holland, "Quantum kinetic theory for a condensed bosonic gas," Phys. Rev. A 59, 3878-3889 (1999).
    [CrossRef]
  22. R. Walser, J. Williams, and M. Holland, "Reversible and irreversible evolution of a condensed bosonic gas," preprint cond mat/0004257, http://xxx.lanl.gov/abs/cond mat/0004257
  23. The case of boundary conditions different from the periodic ones (e.g. a rectangular trap) presents an interesting and challenging problem. In this case there are no universal eigenstates of a one-particle density matrix and therefore the definition of a condensate is unclear.
  24. A.L. Fetter and J.D.Walecka, Quantum theory of many-particle systems (McGraw Hill, New York, 1991).
  25. E. Fermi, J. Pasta, and S. Ulam, "Studies of Nonlinear Problems. I," in Collected Papers of Enrico Fermi (Accademia Nazionale dei Lincei and University of Chicago, Roma, 1965), Vol. II, p. 978.
  26. P. Villain and M. Lewenstein, "Fermi Pasta Ulam problem revisited with a Bose Einstein condensate," Phys. Rev. A 62, 043601 (2000).
    [CrossRef]
  27. F.M. Izrailev and B.V. Chirikov, "Statistical properties of a nonlinear string," Dokl. Akad. Nauk SSSR 166, 57-59 (1966) [Sov. Phys. Dokl. 11, 30-32 (1966)].
  28. J.H. Eberly, N.B. Narozhny, and J.J. Sanchez Mondragon, "Periodic spontaneous collapse and revival in a simple quantum model," Phys. Rev. Lett. 44, 1323-1326 (1980).
    [CrossRef]
  29. In a recent preprint M.J. Davis, S.A. Morgan, and K. Burnett, "Simulations of Bose fields at finite temperature," preprint cond-mat/0011431, http://xxx.lanl.gov/abs/cond mat/0011431, using similar methods, the authors establish a link between the energy and the temperature for temperatures below the critical region.
  30. K. G�ral, M.Gajda, and K. Rzazewski, "Multi-mode dynamics of a coupled ultracold atomic molecular system," preprint cond-mat/0006192, http://xxx.lanl.gov/abs/cond mat/0006192

Other

M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, "Observation of Bose Einstein Condensation in a Dilute Atomic Vapor," Science 269, 198-201 (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, 3969-3972 (1995).
[CrossRef] [PubMed]

C.C. Bradle , 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, 1687-1690 (1995) and Erratum 79, 1170(E) (1997).
[CrossRef] [PubMed]

D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner, and T.J. Greytak, "Bose Einstein condensation of atomic hydrogen," Phys. Rev. Lett. 81, 3811-3814 (1998).
[CrossRef]

P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzazewski, "Fourth statistical ensemble for the Bose Einstein condensate," Phys. Rev. Lett. 79, 1789-1792 (1997).
[CrossRef]

M. Gajda and K. Rzazewski, "Fluctuations of Bose Einstein condensate," Phys. Rev. Lett. 78, 2686-2689 (1997).
[CrossRef]

S. Grossmann and M. Holthaus, "Fluctuations of the particle number in a trapped Bose Einstein condensate," Phys. Rev. Lett. 79, 3557-3560 (1997).
[CrossRef]

S. Grossmann and M. Holthaus, "Maxwell's Demon at work: Two t pes of Bose condensate fluctuations in power law traps," Opt. Express 1, 262-271 (1997), http://www.opticsexpress.org/oearchive/source/2288.htm
[CrossRef] [PubMed]

H. D. Politzer, "Condensate fluctuations of a trapped, ideal Bose gas," Phys. Rev. A 54, 5048-5054 (1996).
[CrossRef] [PubMed]

M. Wilkens and C. Weiss, "Particle number fluctuations in an ideal Bose gas," J. Mod. Opt. 44, 1801-1814 (1997).
[CrossRef]

M. Wilkens and C. Weiss, "Particle number counting statistics in ideal Bose gases," Opt. Express 1, 272-283 (1997), http://www.opticsexpress.org/oearchive/source/2372.htm
[CrossRef] [PubMed]

S. Giorgini, L.P. Pitaevskii, and S. Stringari, "Anomalous fluctuations of the condensate in interacting Bose gases," Phys. Rev. Lett 80, 5040-5043 (1998).
[CrossRef]

Z.Idziaszek, M.Gajda, P. Navez, M. Wilkens, and K.Rzazewski, "Fluctuations of the weakly interacting Bose Einstein condensate," Phys. Rev. Lett. 82, 4376-4379 (1999).
[CrossRef]

F. Meier and W. Zwerger, "Anomalous condensate fluctuations in strongly interacting superfluids," Phys. Rev. A 60, 5133-5135 (1999).
[CrossRef]

V.V. Kocharovsk , V.V. Kocharovsk , and M.O. Scull , "Condensate statistics in interacting and ideal dilute Bose gases," Phys. Rev. Lett. 84, 2306-2309 (2000).
[CrossRef] [PubMed]

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

R. Graham, "Condensate fluctuations in finite Bose Einstein condensates at finite temperature," Phys. Rev. A 62, 023609 (2000).
[CrossRef]

R. Graham, "Decoherence of Bose Einstein condensates in traps at finite temperature," Phys. Rev. Lett. 81, 5262-5265 (1998).
[CrossRef]

C.W. Gardiner and P. Zoller, "Quantum kinetic theory: A quantum kinetic master equation for condensation of a weakly interacting Bose gas without a trapping potential," Phys. Rev. A 55, 2902-2921 (1997).
[CrossRef]

D. Jaksch, C.W. Gardiner, and P. Zoller, "Quantum kinetic theory. 2.Simulation of the quantum Boltzmann master equation," Phys. Rev. A 56, 575- 586 (1997).
[CrossRef]

R. Walser, J. Williams, J. Cooper, and M. Holland, "Quantum kinetic theory for a condensed bosonic gas," Phys. Rev. A 59, 3878-3889 (1999).
[CrossRef]

R. Walser, J. Williams, and M. Holland, "Reversible and irreversible evolution of a condensed bosonic gas," preprint cond mat/0004257, http://xxx.lanl.gov/abs/cond mat/0004257

The case of boundary conditions different from the periodic ones (e.g. a rectangular trap) presents an interesting and challenging problem. In this case there are no universal eigenstates of a one-particle density matrix and therefore the definition of a condensate is unclear.

A.L. Fetter and J.D.Walecka, Quantum theory of many-particle systems (McGraw Hill, New York, 1991).

E. Fermi, J. Pasta, and S. Ulam, "Studies of Nonlinear Problems. I," in Collected Papers of Enrico Fermi (Accademia Nazionale dei Lincei and University of Chicago, Roma, 1965), Vol. II, p. 978.

P. Villain and M. Lewenstein, "Fermi Pasta Ulam problem revisited with a Bose Einstein condensate," Phys. Rev. A 62, 043601 (2000).
[CrossRef]

F.M. Izrailev and B.V. Chirikov, "Statistical properties of a nonlinear string," Dokl. Akad. Nauk SSSR 166, 57-59 (1966) [Sov. Phys. Dokl. 11, 30-32 (1966)].

J.H. Eberly, N.B. Narozhny, and J.J. Sanchez Mondragon, "Periodic spontaneous collapse and revival in a simple quantum model," Phys. Rev. Lett. 44, 1323-1326 (1980).
[CrossRef]

In a recent preprint M.J. Davis, S.A. Morgan, and K. Burnett, "Simulations of Bose fields at finite temperature," preprint cond-mat/0011431, http://xxx.lanl.gov/abs/cond mat/0011431, using similar methods, the authors establish a link between the energy and the temperature for temperatures below the critical region.

K. G�ral, M.Gajda, and K. Rzazewski, "Multi-mode dynamics of a coupled ultracold atomic molecular system," preprint cond-mat/0006192, http://xxx.lanl.gov/abs/cond mat/0006192

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Figures (3)

Fig. 1.
Fig. 1.

Condensate occupation as a function of time for total energy per particle E/ħ=539Hz.

Fig. 2.
Fig. 2.

Probability distribution of the condensate (k=0 mode) population. Different colors signify different total energies per particle: blue E/ħ=2036 Hz, green E/ħ=1680 Hz, orange E/ħ=1424 Hz, yellow E/ħ=1164 Hz, red E/ħ=539 Hz.

Fig. 3.
Fig. 3.

Condensate occupation (in blue) and fluctuations (in red) versus total energy per particle.

Equations (4)

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H = d 3 r Φ p 2 2 m Φ + V g 2 d 3 r Φ Φ Φ Φ ,
Φ ( r ) = 1 V k exp ( i k · r ) a k ,
H = ξ k n 2 a k a k + 1 2 g k , k , k a k + k k a k a k a k ,
α ˙ k = ig k , k exp [ 2 i ξ ( n n ) ( n n ) t ] α k + k k α k α k .

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