Abstract
Starting from an approximate microscopic model of a trapped Bose-condensed gas at finite temperatures, we derive an equation of motion for the condensate wavefunction and a quantum kinetic equation for the distribution function for the excited atoms. The kinetic equation is a generalization of our earlier work in that collisions between the condensate and non-condensate (C 12 ) are now included, in addition to collisions between the excited atoms as described by the Uehling–Uhlenbeck (C 22 ) collision integral. The continuity equation for the local condensate density contains a source term Γ 12 which is related to the C 12 collision term. If we assume that the C 22 collision rate is sufficiently rapid to ensure that the non-condensate distribution function can be approximated by a local equilibrium Bose distribution, the kinetic equation can be used to derive hydrodynamic equations for the non-condensate. The Γ 12 source terms appearing in these equations play a key role in describing the equilibration of the local chemical potentials associated with the condensate and non-condensate components. We give a detailed study of these hydrodynamic equations and show how the Landau two-fluid equations emerge in the frequency domain ωτμ ≪ τμ is a characteristic relaxation time associated with C 12 collisions. More generally, the lack of complete local equilibrium between the condensate and non-condensate is shown to give rise to a new relaxational mode which is associated with the exchange of atoms between the two components. This new mode provides an additional source of damping in the hydrodynamic regime. Our equations are consistent with the generalized Kohn theorem for the center of mass motion of the trapped gas even in the presence of collisions. Finally, we formulate a variational solution of the equations which provides a very convenient and physical way of estimating normal mode frequencies. In particular, we use relatively simple trial functions within this approach to work out some of the monopole, dipole and quadrupole oscillations for an isotropic trap.
Similar content being viewed by others
REFERENCES
F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
D. A. W. Hutchinson, E. Zaremba, and A. Griffin, Phys. Rev. Lett. 78, 1842 (1997); for anisotropic traps, see R. J. Dodd, K. Burnett, M. Edwards, and C. Clark, Phys. Rev. A 57, R32 (1998).
A. Minguzzi and M. P. Tosi, J. Phys.: Cond. Mat. 9, 10211 (1997).
S. Giorgini, Phys. Rev. A 57, 2949 (1998).
E. Zaremba, A. Griffin, and T. Nikuni, Phys. Rev. A 57, 4695 (1998); this paper is referred to as ZGN.
L. D. Landau, J. Phys. (U.S.S.R.) 5, 71 (1941).
I. K. Khalatnikov, An Introduction to the Theory of Superfluidity, W. A. Benjamin, New York (1965).
U. Eckern, J. Low Temp. Phys. 54, 333 (1984).
L. D. Landau and I. M. Khalatnikov, Dokl. Akad. Nauk SSSR 96, 469 (1954).
L. P. Pitaevskii, Sov. Phys. JETP 35, 282 (1959).
K. Miyake and K. Yamada, Prog. Theor. Phys. 56, 1689 (1976) and references therein.
T. Nikuni, E. Zaremba, and A. Griffin, Phys. Rev. Lett. 83, 10 (1999).
T. R. Kirkpatrick and J. R. Dorfman, Phys. Rev. A 28, 2576 (1983); J. Low Temp. Phys. 58, 308, 399 (1985).
N. N. Bogoliubov, Lectures on Quantum Statistics, Gordon and Breach, N.Y. (1970), Vol. 2, p. 148.
C. W. Gardiner, M. D. Lee, R. J. Ballagh, M. J. Davies, and P. Zoller, Phys. Rev. Lett. 81, 5266 (1998). References to earlier work are given here.
N. P. Proukakis, K. Burnett, and H. T. C. Stoof, Phys. Rev. A 57, 1230 (1998); see also, N. P. Proukakis and K. Burnett, J. Res. Natl. Stand. Technol. 101, 457 (1996).
H. T. C. Stoof, J. Low Temp. Phys. 114, 11 (1999).
R. Walser, J. Williams, J. Cooper, and M. Holland, Phys. Rev. A 59, 3878 (1999).
E. Zaremba, M. Bijlsma, and H. C. T. Stoof, to be published.
J. Javinainen, Phys. Rev. A 54, R3722 (1996).
A. Griffin, Phys. Rev. B 53, 9341 (1996).
S. Stringari, Phys. Rev. Lett. 77, 2360 (1996).
E. A. Uehling and G. E. Uhlenbeck, Phys. Rev. 43, 552 (1933).
L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics, Benjamin, N.Y. (1962).
T. Lopez-Arias and A. Smerzi, Phys. Rev. A 58, 526 (1998) and references therein.
S. Giorgini, L. P. Pitaevskii, and S. Stringari, J. Low Temp. Phys. 109, 309 (1997).
M. J. Bijlsma and H. T. C. Stoof, cond-mat/9902065.
A. Griffin and E. Zaremba, Phys. Rev. A 56, 4826 (1997).
K. Huang, Statistical Mechanics, Wiley, New York (1987), 2nd ed.
T. Nikuni and A. Griffin, J. Low Temp. Phys. 111, 793 (1998).
G. M. Kavoulakis, C. J. Pethick, and H. Smith, Phys. Rev. A 57, 2938 (1998).
G. Baym and C. J. Pethick, Phys. Rev. Lett. 76, 6 (1996).
D. A. Huse and E. D. Siggia, J. Low Temp. Phys. 46, 137 (1982).
P. Nozières and D. Pines, The Theory of Quantum Liquids, Addison-Wesley, Redwood City, California (1990), Vol. II.
V. B. Shenoy and T. L. Ho, Phys. Rev. Lett. 80, 3895 (1998).
D. M. Stamper-Kurn, H.-J. Miesner, S. Inouye, M. R. Andrews, and W. Ketterle, Phys. Rev. Lett. 81, 500 (1998).
C. Gay and A. Griffin, J. Low Temp. Phys. 58, 479 (1985).
J. F. Dobson, Phys. Rev. Lett. 73, 7244 (1994), and references therein.
M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, Phys. Rev. Lett. 77, 1671 (1996).
L. D. Landau and E. M. Lifshitz, Theory of Elasticity. Pergamon Press, Oxford (1970), 2nd ed.
B. W. King, M.Sc. Thesis, Queen's University, Kingston, July 1998.
A. Griffin, W. C. Wu, and S. Stringari, Phys. Rev. Lett. 78, 1838 (1997).
D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 78, 764 (1997).
T. D. Lee and C. N. Yang, Phys. Rev. 112, 1419 (1958).
P. C. Hohenberg and P. C. Martin, Ann. of Phys. (N.Y.) 34, 291 (1965).
A. Griffin, in Bose-Einstein Condensation in Atomic Gases, edited by M. Inguscio, S. Stringari, and C. Wieman, Italian Physical Society, in press; cond-mat/9901172.
M. Imamović-Tomasović and A. Griffin, Phys. Rev. A 60, 494 (1999).
H. Shi and A. Griffin, Phys. Reports 304, 1 (1998).
N. P. Proukakis, S. A. Morgan, S. Choi, and K. Burnett, Phys. Rev. A 58, 2435 (1998).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zaremba, E., Nikuni, T. & Griffin, A. Dynamics of Trapped Bose Gases at Finite Temperatures. Journal of Low Temperature Physics 116, 277–345 (1999). https://doi.org/10.1023/A:1021846002995
Issue Date:
DOI: https://doi.org/10.1023/A:1021846002995