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.

© 2001 Optical Society of America

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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. 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, 1687–1690 (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. Rzążewski, “Fourth statistical ensemble for the Bose-Einstein condensate,” Phys. Rev. Lett. 79, 1789–1792 (1997).
    [Crossref]
  6. M. Gajda and K. Rzążewski, “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 types 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. Rzążewski, “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. Kocharovsky, V.V. Kocharovsky, and M.O. Scully, “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 SSSR166, 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. Góral, M. Gajda, and K. Rzążewski, “Multi-mode dynamics of a coupled ultracold atomic-molecular system,” preprint cond-mat/0006192, http://xxx.lanl.gov/abs/cond-mat/0006192

2000 (3)

V.V. Kocharovsky, V.V. Kocharovsky, and M.O. Scully, “Condensate statistics in interacting and ideal dilute Bose gases,” Phys. Rev. Lett. 84, 2306–2309 (2000).
[Crossref] [PubMed]

R. Graham, “Condensate fluctuations in finite Bose-Einstein condensates at finite temperature,” Phys. Rev. A 62, 023609 (2000).
[Crossref]

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

1999 (4)

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]

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]

Z. Idziaszek, M. Gajda, P. Navez, M. Wilkens, and K. Rzążewski, “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]

1998 (3)

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

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]

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]

1997 (8)

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

M. Gajda and K. Rzążewski, “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]

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]

M. Wilkens and C. Weiss, “Particle number fluctuations in an ideal Bose gas,” J. Mod. Opt. 44, 1801–1814 (1997).
[Crossref]

S. Grossmann and M. Holthaus, “Maxwell’s Demon at work: Two types of Bose condensate fluctuations in power-law traps,” Opt. Express 1, 262–271 (1997), http://www.opticsexpress.org/oearchive/source/2288.htm
[Crossref] [PubMed]

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]

1996 (1)

H. D. Politzer, “Condensate fluctuations of a trapped, ideal Bose gas,” Phys. Rev. A 54, 5048–5054 (1996).
[Crossref] [PubMed]

1995 (2)

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]

1980 (1)

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]

Anderson, M.H.

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]

Andrews, M.R.

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]

Bitouk, D.

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

Bradley, C.C.

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, 1687–1690 (1995) and Erratum 79, 1170(E) (1997).
[Crossref] [PubMed]

Burnett, K.

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.

Chirikov, B.V.

F.M. Izrailev and B.V. Chirikov, “Statistical properties of a nonlinear string,” Dokl. Akad. Nauk SSSR166, 57–59 (1966) [Sov. Phys. Dokl.11, 30–32 (1966)].

Cooper, J.

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]

Cornell, E.A.

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]

Dalfovo, F.

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]

Davis, K.B.

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]

Davis, M.J.

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.

Durfee, D.S.

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]

Eberly, J.H.

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]

Ensher, J.R.

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]

Fermi, E.

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.

Fetter, A.L.

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

Fried, D.G.

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]

Gajda, M.

Z. Idziaszek, M. Gajda, P. Navez, M. Wilkens, and K. Rzążewski, “Fluctuations of the weakly interacting Bose-Einstein condensate,” Phys. Rev. Lett. 82, 4376–4379 (1999).
[Crossref]

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

M. Gajda and K. Rzążewski, “Fluctuations of Bose-Einstein condensate,” Phys. Rev. Lett. 78, 2686–2689 (1997).
[Crossref]

K. Góral, M. Gajda, and K. Rzążewski, “Multi-mode dynamics of a coupled ultracold atomic-molecular system,” preprint cond-mat/0006192, http://xxx.lanl.gov/abs/cond-mat/0006192

Gardiner, C.W.

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]

Giorgini, S.

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]

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]

Góral, K.

K. Góral, M. Gajda, and K. Rzążewski, “Multi-mode dynamics of a coupled ultracold atomic-molecular system,” preprint cond-mat/0006192, http://xxx.lanl.gov/abs/cond-mat/0006192

Graham, R.

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]

Greytak, T.J.

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]

Grossmann, S.

Holland, M.

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

Holthaus, M.

Hulet, R.G.

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, 1687–1690 (1995) and Erratum 79, 1170(E) (1997).
[Crossref] [PubMed]

Idziaszek, Z.

Z. Idziaszek, M. Gajda, P. Navez, M. Wilkens, and K. Rzążewski, “Fluctuations of the weakly interacting Bose-Einstein condensate,” Phys. Rev. Lett. 82, 4376–4379 (1999).
[Crossref]

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

Izrailev, F.M.

F.M. Izrailev and B.V. Chirikov, “Statistical properties of a nonlinear string,” Dokl. Akad. Nauk SSSR166, 57–59 (1966) [Sov. Phys. Dokl.11, 30–32 (1966)].

Jaksch, D.

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]

Ketterle, W.

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]

Killian, T.C.

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]

Kleppner, D.

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]

Kocharovsky, V.V.

V.V. Kocharovsky, V.V. Kocharovsky, and M.O. Scully, “Condensate statistics in interacting and ideal dilute Bose gases,” Phys. Rev. Lett. 84, 2306–2309 (2000).
[Crossref] [PubMed]

V.V. Kocharovsky, V.V. Kocharovsky, and M.O. Scully, “Condensate statistics in interacting and ideal dilute Bose gases,” Phys. Rev. Lett. 84, 2306–2309 (2000).
[Crossref] [PubMed]

Kurn, D.M.

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]

Landhuis, D.

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]

Lewenstein, M.

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

Matthews, M.R.

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]

Meier, F.

F. Meier and W. Zwerger, “Anomalous condensate fluctuations in strongly interacting superfluids,” Phys. Rev. A 60, 5133–5135 (1999).
[Crossref]

Mewes, M.-O.

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]

Morgan, S.A.

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.

Moss, S.C.

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]

Narozhny, N.B.

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]

Navez, P.

Z. Idziaszek, M. Gajda, P. Navez, M. Wilkens, and K. Rzążewski, “Fluctuations of the weakly interacting Bose-Einstein condensate,” Phys. Rev. Lett. 82, 4376–4379 (1999).
[Crossref]

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

Pasta, J.

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.

Pitaevskii, L. P.

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]

Pitaevskii, L.P.

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]

Politzer, H. D.

H. D. Politzer, “Condensate fluctuations of a trapped, ideal Bose gas,” Phys. Rev. A 54, 5048–5054 (1996).
[Crossref] [PubMed]

Rzazewski, K.

Z. Idziaszek, M. Gajda, P. Navez, M. Wilkens, and K. Rzążewski, “Fluctuations of the weakly interacting Bose-Einstein condensate,” Phys. Rev. Lett. 82, 4376–4379 (1999).
[Crossref]

M. Gajda and K. Rzążewski, “Fluctuations of Bose-Einstein condensate,” Phys. Rev. Lett. 78, 2686–2689 (1997).
[Crossref]

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

K. Góral, M. Gajda, and K. Rzążewski, “Multi-mode dynamics of a coupled ultracold atomic-molecular system,” preprint cond-mat/0006192, http://xxx.lanl.gov/abs/cond-mat/0006192

Sackett, C.A.

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, 1687–1690 (1995) and Erratum 79, 1170(E) (1997).
[Crossref] [PubMed]

Sanchez-Mondragon, J.J.

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]

Scully, M.O.

V.V. Kocharovsky, V.V. Kocharovsky, and M.O. Scully, “Condensate statistics in interacting and ideal dilute Bose gases,” Phys. Rev. Lett. 84, 2306–2309 (2000).
[Crossref] [PubMed]

Stringari, S.

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]

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]

Tollett, J.J.

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, 1687–1690 (1995) and Erratum 79, 1170(E) (1997).
[Crossref] [PubMed]

Ulam, S.

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.

van Druten, N.J.

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]

Villain, P.

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

Walecka, J.D.

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

Walser, R.

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

Weiss, C.

Wieman, C.E.

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]

Wilkens, M.

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

M. Wilkens and C. Weiss, “Particle number fluctuations in an ideal Bose gas,” J. Mod. Opt. 44, 1801–1814 (1997).
[Crossref]

Williams, J.

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

Willmann, L.

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]

Zoller, P.

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]

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]

Zwerger, W.

F. Meier and W. Zwerger, “Anomalous condensate fluctuations in strongly interacting superfluids,” Phys. Rev. A 60, 5133–5135 (1999).
[Crossref]

J. Mod. Opt. (1)

M. Wilkens and C. Weiss, “Particle number fluctuations in an ideal Bose gas,” J. Mod. Opt. 44, 1801–1814 (1997).
[Crossref]

Opt. Express (2)

Phys. Rev. A (7)

H. D. Politzer, “Condensate fluctuations of a trapped, ideal Bose gas,” Phys. Rev. A 54, 5048–5054 (1996).
[Crossref] [PubMed]

F. Meier and W. Zwerger, “Anomalous condensate fluctuations in strongly interacting superfluids,” Phys. Rev. A 60, 5133–5135 (1999).
[Crossref]

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

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

Phys. Rev. Lett (1)

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]

Phys. Rev. Lett. (9)

Z. Idziaszek, M. Gajda, P. Navez, M. Wilkens, and K. Rzążewski, “Fluctuations of the weakly interacting Bose-Einstein condensate,” Phys. Rev. Lett. 82, 4376–4379 (1999).
[Crossref]

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

V.V. Kocharovsky, V.V. Kocharovsky, and M.O. Scully, “Condensate statistics in interacting and ideal dilute Bose gases,” Phys. Rev. Lett. 84, 2306–2309 (2000).
[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]

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. Rzążewski, “Fourth statistical ensemble for the Bose-Einstein condensate,” Phys. Rev. Lett. 79, 1789–1792 (1997).
[Crossref]

M. Gajda and K. Rzążewski, “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]

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]

Rev. Mod. Phys. (1)

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]

Science (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]

Other (8)

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. Góral, M. Gajda, and K. Rzążewski, “Multi-mode dynamics of a coupled ultracold atomic-molecular system,” preprint cond-mat/0006192, http://xxx.lanl.gov/abs/cond-mat/0006192

F.M. Izrailev and B.V. Chirikov, “Statistical properties of a nonlinear string,” Dokl. Akad. Nauk SSSR166, 57–59 (1966) [Sov. Phys. Dokl.11, 30–32 (1966)].

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.

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, 1687–1690 (1995) and Erratum 79, 1170(E) (1997).
[Crossref] [PubMed]

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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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