### Abstract

The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo's linear response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross-Pitaevskii equation.

Original language | English |
---|---|

Pages (from-to) | 2327-2349 |

Number of pages | 23 |

Journal | Annals of Physics |

Volume | 322 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2007 Oct |

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### Keywords

- Bose-Einstein condensation
- Indefinite metric
- Instability
- Quantized vortex
- Quantum field theory

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

**Quantum field theoretical description of unstable behavior of trapped Bose-Einstein condensates with complex eigenvalues of Bogoliubov-de Gennes equations.** / Mine, Makoto; Okumura, Masahiko; Sunaga, Tomoka; Yamanaka, Yoshiya.

Research output: Contribution to journal › Article

*Annals of Physics*, vol. 322, no. 10, pp. 2327-2349. https://doi.org/10.1016/j.aop.2007.01.008

}

TY - JOUR

T1 - Quantum field theoretical description of unstable behavior of trapped Bose-Einstein condensates with complex eigenvalues of Bogoliubov-de Gennes equations

AU - Mine, Makoto

AU - Okumura, Masahiko

AU - Sunaga, Tomoka

AU - Yamanaka, Yoshiya

PY - 2007/10

Y1 - 2007/10

N2 - The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo's linear response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross-Pitaevskii equation.

AB - The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo's linear response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross-Pitaevskii equation.

KW - Bose-Einstein condensation

KW - Indefinite metric

KW - Instability

KW - Quantized vortex

KW - Quantum field theory

UR - http://www.scopus.com/inward/record.url?scp=34548088431&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548088431&partnerID=8YFLogxK

U2 - 10.1016/j.aop.2007.01.008

DO - 10.1016/j.aop.2007.01.008

M3 - Article

AN - SCOPUS:34548088431

VL - 322

SP - 2327

EP - 2349

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

IS - 10

ER -