TY - JOUR

T1 - Unitarily inequivalent vacua in Bose-Einstein condensation of trapped gases

AU - Okumura, Masahiko

AU - Yamanaka, Yoshiya

N1 - Funding Information:
The authors would like to thank Professors I. Ohba and H. Nakazato for their helpful comments and their encouragement, and Dr. M. Mine for useful discussions. The authors thank the Yukawa Institute for Theoretical Physics at Kyoto University for offering them the opportunity of discussions during the YITP workshop YITP-W-04-07 on “Thermal Quantum Field Theories and Their Applications,” which were useful to complete this work. This work was partially supported by a Grant-in-Aid for Scientific Research (C) (No. 17540364) from the Japan Society for the Promotion of Science, for Young Scientists (B) (No. 17740258) and for Priority Area Research (B) (No. 13135221) both from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The authors are grateful for the Waseda University Grant for Special Research Projects (2005).

PY - 2006/6/15

Y1 - 2006/6/15

N2 - We approach the problem of the Bose-Einstein condensation of neutral atoms in an external trap, which is a finite system in size, from the viewpoint of quantum field theory. The phenomenon is considered as a spontaneous symmetry breakdown. The main result of this paper is that the vacua for this system with finite size and finite number of trapped neutral atoms, parametrized by different phases of the order parameter, are orthogonal to each other, namely the Fock spaces associated with each vacuum are unitarily inequivalent to each other. In the proof, the zero-energy mode which is the Nambu-Goldstone mode, and is discrete due to the presence of the trap, plays a crucial role. We introduce the discrete zero-energy mode in the formulation of the generalized Bogoliubov transformation and introduce an additional infinitesimal symmetry-breaking term in the Hamiltonian to control the zero-energy mode properly.

AB - We approach the problem of the Bose-Einstein condensation of neutral atoms in an external trap, which is a finite system in size, from the viewpoint of quantum field theory. The phenomenon is considered as a spontaneous symmetry breakdown. The main result of this paper is that the vacua for this system with finite size and finite number of trapped neutral atoms, parametrized by different phases of the order parameter, are orthogonal to each other, namely the Fock spaces associated with each vacuum are unitarily inequivalent to each other. In the proof, the zero-energy mode which is the Nambu-Goldstone mode, and is discrete due to the presence of the trap, plays a crucial role. We introduce the discrete zero-energy mode in the formulation of the generalized Bogoliubov transformation and introduce an additional infinitesimal symmetry-breaking term in the Hamiltonian to control the zero-energy mode properly.

KW - Bose-Einstein condensation

KW - Spontaneous symmetry breakdown

KW - Unitarily inequivalent vacua

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U2 - 10.1016/j.physa.2005.09.064

DO - 10.1016/j.physa.2005.09.064

M3 - Article

AN - SCOPUS:33646081269

SN - 0378-4371

VL - 365

SP - 429

EP - 445

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

IS - 2

ER -