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Progress of Theoretical Physics

Effects of Quantum Coordinates on Condensate Density in a Trapped Bose-Einstein Condensate

Masahiko Okumura, Y. Yamanaka

Published Feb 1, 2004• 5 citations
Publisher

Abstract

The effects of quantum coordinates associated with the Nambu-Goldstone mode in a trapped Bose-Einstein condensate are investigated at zero and finite temperatures. We develop a loop expansion and perform numerical calculations over a wide range of temperature, i.e., between zero and the critical temperatures. Our results are compared with those obtained in cases without the Nambu-Goldstone mode (the Bogoliubov approximation) and with the Nambu-Goldstone mode as a zero-energy particle mode.

DOI
10.1143/ptp.111.199

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The effects of quantum coordinates associated with the Nambu-Goldstone mode in a trapped Bose-Einstein condensate are investigated at zero and finite temperatures. We develop a loop expansion and perform numerical calculations over a wide range of temperature, i.e., between zero and the critical temperatures. Our results are compared with those obtained in cases without the Nambu-Goldstone mode (the Bogoliubov approximation) and with the Nambu-Goldstone mode as a zero-energy particle mode.
The dynamical response of a trapped Bose-Einstein condensate (BEC) is formulated consistently with quantum field theory and is numerically evaluated. We regard the BEC as a manifestation of the breaking of the global phase symmetry. Then, the Goldstone theorem implies the existence of a zero energy excitation mode (the zero-mode). We calculate the effect of the zero-mode on the response frequency and show that the contribution of the zero-mode to the first excitation mode is not so important in the parameter set realized in the existing experiment. This is the reason that experimental results can be described using the Bogoliubov prescription, although it breaks the consistency of the description in quantum field theory.
The dynamical response of a trapped Bose-Einstein condensate (BEC) is formulated consistently with quantum field theory and is numerically evaluated. We regard the BEC as a manifestation of the breaking of the global phase symmetry. Then, the Goldstone theorem implies the existence of a zero energy excitation mode (the zero-mode). We calculate the effect of the zero-mode on the response frequency and show that the contribution of the zero-mode to the first excitation mode is not so important in the parameter set realized in the existing experiment. This is the reason that experimental results can be described using the Bogoliubov prescription, although it breaks the consistency of the description in quantum field theory.
The two approaches of consistent quantum field theory for systems of the trapped Bose–Einstein condensates are known, one is the Bogoliubov–de Gennes approach and the other is the generalized Bogoliubov approach. In this paper, we investigate the relation between the two approaches and show that they are formally equivalent to each other. To do this one must carefully treat the Nambu–Goldstone mode which plays a crucial role in the condensation. It is emphasized that the choice of vacuum is physically relevant.
In Bose-Einstein condensation of dilute alkaline atomic gases, a magnetic trapped potential forces particle-excitation levels to have a discrete spectrum. This indicates that there must arise a discrete zero-mode associated with the phase symmetry to be consistent with the Goldstone theorem. It is shown that the zero-mode can be identified as a quantum phase coordinate and that its presence is required from the canonical relation of field operators. Thermal fluctuations of the quantum phase coordinate induce a T log Nc3/5T term in the free energy and a kBTNc−2/5 term in the particle number, where Nc is the number of condensed particles.