# Dynamical instability induced by the zero mode under symmetry breaking external perturbation

## Abstract

A complex eigenvalue in the Bogoliubov–de Gennes equations for a stationary Bose-Einstein condensate in the ultracold atomic system indicates the dynamical instability of the system. We also have the modes with zero eigenvalues for the condensate, called the zero modes, which originate from the spontaneous breakdown of symmetries. Although the zero modes are suppressed in many theoretical analyses, we take account of them in this paper and argue that a zero mode can change into one with a pure imaginary eigenvalue by applying a symmetry breaking external perturbation potential. This emergence of a pure imaginary mode adds a new type of scenario of dynamical instability to that characterized by the complex eigenvalue of the usual excitation modes. For illustration, we deal with two one-dimensional homogeneous Bose–Einstein condensate systems with a single dark soliton under a respective perturbation potential, breaking the invariance under translation, to derive pure imaginary modes. - Highlights: • Zero modes are important but ignored in many theories for the cold atomic system. • We discuss the zero mode under symmetry breaking potential in this system. • We consider the zero mode of translational invariance for a single dark soliton. • We show that it turns into anmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22447579

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics; Journal Volume: 347; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOSE-EINSTEIN CONDENSATION; EIGENVALUES; ONE-DIMENSIONAL CALCULATIONS; SOLITONS; SYMMETRY BREAKING

### Citation Formats

```
Takahashi, J., E-mail: phyco-sevenface@asagi.waseda.jp, Nakamura, Y., E-mail: nakamura@aoni.waseda.jp, and Yamanaka, Y., E-mail: yamanaka@waseda.jp.
```*Dynamical instability induced by the zero mode under symmetry breaking external perturbation*. United States: N. p., 2014.
Web. doi:10.1016/J.AOP.2014.05.004.

```
Takahashi, J., E-mail: phyco-sevenface@asagi.waseda.jp, Nakamura, Y., E-mail: nakamura@aoni.waseda.jp, & Yamanaka, Y., E-mail: yamanaka@waseda.jp.
```*Dynamical instability induced by the zero mode under symmetry breaking external perturbation*. United States. doi:10.1016/J.AOP.2014.05.004.

```
Takahashi, J., E-mail: phyco-sevenface@asagi.waseda.jp, Nakamura, Y., E-mail: nakamura@aoni.waseda.jp, and Yamanaka, Y., E-mail: yamanaka@waseda.jp. Fri .
"Dynamical instability induced by the zero mode under symmetry breaking external perturbation". United States.
doi:10.1016/J.AOP.2014.05.004.
```

```
@article{osti_22447579,
```

title = {Dynamical instability induced by the zero mode under symmetry breaking external perturbation},

author = {Takahashi, J., E-mail: phyco-sevenface@asagi.waseda.jp and Nakamura, Y., E-mail: nakamura@aoni.waseda.jp and Yamanaka, Y., E-mail: yamanaka@waseda.jp},

abstractNote = {A complex eigenvalue in the Bogoliubov–de Gennes equations for a stationary Bose-Einstein condensate in the ultracold atomic system indicates the dynamical instability of the system. We also have the modes with zero eigenvalues for the condensate, called the zero modes, which originate from the spontaneous breakdown of symmetries. Although the zero modes are suppressed in many theoretical analyses, we take account of them in this paper and argue that a zero mode can change into one with a pure imaginary eigenvalue by applying a symmetry breaking external perturbation potential. This emergence of a pure imaginary mode adds a new type of scenario of dynamical instability to that characterized by the complex eigenvalue of the usual excitation modes. For illustration, we deal with two one-dimensional homogeneous Bose–Einstein condensate systems with a single dark soliton under a respective perturbation potential, breaking the invariance under translation, to derive pure imaginary modes. - Highlights: • Zero modes are important but ignored in many theories for the cold atomic system. • We discuss the zero mode under symmetry breaking potential in this system. • We consider the zero mode of translational invariance for a single dark soliton. • We show that it turns into an anomalous or pure imaginary mode.},

doi = {10.1016/J.AOP.2014.05.004},

journal = {Annals of Physics},

number = ,

volume = 347,

place = {United States},

year = {Fri Aug 15 00:00:00 EDT 2014},

month = {Fri Aug 15 00:00:00 EDT 2014}

}