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Journal of the Physical Society of Japan

Role of Nambu–Goldstone Mode in Trapped Bose–Einstein Condensation

Y. Yamanaka, Masahiko Okumura, M. Mine

Published Jan 1, 2003• 0 citations
Publisher

Abstract

The system of the trapped Bose–Einstein condensation is investigated from the viewpoint of Quantum Field Theory. It is pointed out that the Nambu–Goldstone mode, which is a discrete mode due to the trapping potential, creates observable effects through fluctuations. The way how the Nambu–Goldstone mode is treated is not unique. The introduction of quantum coordinates is a possible candidate.

DOI
10.1143/jpsjs.72sc.152

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The system of the trapped Bose–Einstein condensation is investigated from the viewpoint of Quantum Field Theory. It is pointed out that the Nambu–Goldstone mode, which is a discrete mode due to the trapping potential, creates observable effects through fluctuations. The way how the Nambu–Goldstone mode is treated is not unique. The introduction of quantum coordinates is a possible candidate.
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.
We theoretically study the properties of Nambu-Goldstone bosons in an interacting single-component Bose-Einstein condensate (BEC). We first point out that the proofs of Goldstone's theorem by Goldstone, et al. [Phys. Rev. {\bf 127} (1962) 965] may be relevant to distinct massless modes of the BEC: whereas the first proof deals with the poles of the single-particle Green's function G^\hat{G}G^, the second one concerns those of the two-particle Green's function. Thus, there may be multiple Nambu-Goldstone bosons even in the single-component BEC with broken U(1) symmetry. The second mode turns out to have an infinite lifetime in the long-wavelength limit in agreement with the conventional viewpoint. In contrast, the first mode from G^\hat{G}G^, i.e., the Bogoliubov mode in the weak-coupling regime, is shown to be a "bubbling" mode fluctuating temporally out of and back into the condensate. The substantial lifetime originates from an "improper" structure of the self-energy inherent in the BEC, which has been overlooked so far and will be elucidated here, and removes various infrared divergences pointed out previously.
We show that quasi-Nambu-Goldstone (NG) modes, which play prominent roles in high energy physics but have been elusive experimentally, can be realized with atomic Bose-Einstein condensates. The quasi-NG modes emerge when the symmetry of a ground state is larger than that of the Hamiltonian. When they appear, the conventional vacuum manifold should be enlarged. Consequently, topological defects that are stable within the conventional vacuum manifold become unstable and decay by emitting the quasi-NG modes. Contrary to conventional wisdom, however, we show that the topological defects are stabilized by quantum fluctuations that make the quasi-NG modes massive, thereby suppressing their emission.
The Gross-Pitaevskii or nonlinear Schr\"odinger equation relevant to ultracold atomic gaseous Bose-Einstein condensates possesses a modified Schr\"odinger symmetry in two spatial dimensions, in the presence of a harmonic trapping potential, an (artificial) constant magnetic field (or rotation), and an (artificial) electric field of a quadratic electrostatic potential. We find that a variance and a center of a trapped gas with or without a vorticity can be regarded as massive Nambu-Goldstone (NG) modes associated with spontaneous breaking of the modified Schr\"odinger symmetry. We show that the Noether theorem for the modified Schr\"odinger symmetry gives universal equations of motion which describe exact time evolutions of the trapped gases such as a harmonic oscillation, a cyclotron motion, and a breathing oscillation with frequencies determined by the symmetry independent of the details of the system. We further construct an exact effective action for all the NG modes.
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.