Zero-mode in Bose-Einstein condensed states

Yusuke Nakamura, Junichi Takahashi, Y. Yamanaka

Takahashi Jun'ichi, Yuto Hayaki, Y. Yamanaka

H. Matsumoto, S. Sakamoto

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.

Ostrovskaya, Yuri S. Kivshar, Lisak +3 more

Junichi Takahashi, Yusuke Nakamura, Y. Yamanaka

Investigating the Bogoliubov-de Gennes equation, widely used in the analysis of Bose-Einstein condensate in neutral atomic gases, we reveal some aspect of the interplay between the zero and complex modes. The complex mode implies that the system is dynamically unstable. The zero modes originate from the spontaneous symmetry breakdown, but their roles are not clear because they cause the infrared divergence. In this paper, we deal with the system of a dark soliton in Bose-Einstein condensate which has two zero modes corresponding to the spontaneously broken U(1) gauge and translational symmetries, and show that the zero and its adjoint modes, associated with the spontaneous breakdown of translational symmetry, turn into pure imaginary modes under the perturbation which explicitly breaks translational symmetry.

Junichi Takahashi, Yuto Hayaki, Y. Yamanaka

Zero-mode operators associated with a spontaneous breakdown of internal symmetry are often suppressed and their quantum fluctuations are neglected owing to the absence of a definite criterion for determining the ground state and infrared divergences in physical quantities. Because the zero-mode fluctuations for systems with finite-size and spatially inhomogeneity cannot be neglected, we have proposed a novel treatment of the zero-mode for finite systems, which helps determine the unique ground state and evaluate the physical quantities without divergence. Following the proposed treatment, we consider trapped systems with two-component Bose–Einstein condensates. The results obtained demonstrate that the interaction strength between the two components considerably affect the dynamical properties, illustrating the importance of the zero-mode for finite systems.