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Progress of Theoretical Physics

Quantum Phase Coordinate as a Zero-Mode in Bose-Einstein Condensed States

H. Matsumoto, S. Sakamoto

Published Apr 1, 2002• 28 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.107.679

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Journal of the Physical Society of Japan•Jan 1, 2003•0 citations
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.
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 unperturbed Hamiltonian for the Bose-Einstein condensate, which includes not only the first and second powers of the zero mode operators but also the higher ones, is proposed to determine a unique and stationary vacuum at zero temperature. From the standpoint of quantum field theory, it is done in a consistent manner that the canonical commutation relation of the field operator is kept. In this formulation, the condensate phase does not diffuse and is robust against the quantum fluctuation of the zero mode. The standard deviation for the phase operator depends on the condensed atom number with the exponent of $\ensuremath{-}1/3$, which is universal for both homogeneous and inhomogeneous systems.