Abstract
We consider the Bogoliubov vacuum state in the number-conserving Bogoliubov theory proposed by Castin and Dum [Phys. Rev. A 57, 3008 (1998)]. We show that, in the particle representation, the vacuum can be written in a simple diagonal form. The vacuum state can describe the stationary N-particle ground state of a condensate in a trap, but it can also represent a dynamical state when, for example, a Bose-Einstein condensate initially prepared in the stationary ground state is subject to a time-dependent perturbation. In both cases the diagonal form of the Bogoliubov vacuum can be obtained by basically diagonalizing the reduced single-particle density matrix of the vacuum. We compare N-body states obtained within the Bogoliubov theory with the exact ground states in a 3-site Bose-Hubbard model. In this example, the Bogoliubov theory fails to accurately describe the stationary ground state in the limit when N → ∞ but a small fraction of depleted particles is kept constant.
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F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
M. H. Anderson, J. R. Ensher, M. R. Matthews, et al., Science 269, 198 (1995); K. B. Davis, M.-O. Mewes, M. R. Andrews, et al., Phys. Rev. Lett. 75, 3969 (1995).
Y. Castin, in Les Houches Session LXXII, Coherent Atomic Matter Waves 1999, Ed. by R. Kaiser, C. Westbrook, and F. David (Springer, New York, 2001).
H. Nozieres and D. Pines, The Theory of Quantum Liquids (Addison Wesley, New York, 1990).
M. D. Girardeau and R. Arnowitt, Phys. Rev. 113, 755 (1959).
M. D. Girardeau, Phys. Rev. A 58, 775 (1998).
C. W. Gardiner, Phys. Rev. A 56, 1414 (1997).
Y. Castin and R. Dum, Phys. Rev. A 57, 3008 (1998).
J. Dziarmaga and K. Sacha, Phys. Rev. A 67, 033608 (2003).
J. Dziarmaga, Z. P. Karkuszewski, and K. Sacha, J. Phys. B: At. Mol. Opt. Phys. 36, 1217 (2003).
J. Dziarmaga and K. Sacha, J. Phys. B: At. Mol. Opt. Phys. (in press).
V. I. Yukalov, private communication.
J. Javanainen, Phys. Rev. A 60, 4902 (1999).
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Original Text © Astro, Ltd., 2006.