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arXiv:1504.06420 · Physical Review A

Interacting multiple zero mode formulation and its application to a system consisting of a dark soliton in a condensate

J. Takahashi, Yusuke Nakamura, Y. Yamanaka

Published Aug 18, 2015• 4 citations
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Abstract

To formulate the zero modes in a finite-size system with spontaneous breakdown of symmetries in quantum field theory is not trivial, for in the naive Bogoliubov theory, one encounters difficulties such as phase diffusion, the absence of a definite criterion for determining the ground state, and infrared divergences. An interacting zero mode formulation that has been proposed for systems with a single zero mode to avoid these difficulties is extended to general systems with multiple zero modes. It naturally and definitely gives the interactions among the quantized zero modes, the consequences of which can be observed experimentally. In this paper, as a typical example, we consider an atomic Bose-Einstein condensed system with a dark soliton that contains two zero modes corresponding to the spontaneous breakdown of the U(1) gauge and translational symmetries. Then we evaluate the standard deviations of the zero mode operators and see how the mutual interaction between the two zero modes affects them.

DOI
10.1103/physreva.92.023627

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To formulate the zero modes in a finite-size system with spontaneous breakdown of symmetries in quantum field theory is not trivial, for in the naive Bogoliubov theory, one encounters difficulties such as phase diffusion, the absence of a definite criterion for determining the ground state, and infrared divergences. An interacting zero mode formulation that has been proposed for systems with a single zero mode to avoid these difficulties is extended to general systems with multiple zero modes. It naturally and definitely gives the interactions among the quantized zero modes, the consequences of which can be observed experimentally. In this paper, as a typical example, we consider an atomic Bose-Einstein condensed system with a dark soliton that contains two zero modes corresponding to the spontaneous breakdown of the U(1) gauge and translational symmetries. Then we evaluate the standard deviations of the zero mode operators and see how the mutual interaction between the two zero modes affects them.
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.
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.
The dark soliton solution of the Gross-Pitaevskii equation in one dimension has two parameters that do not change the energy of the solution: the global phase of the condensate wave function and the position of the soliton. These degeneracies appear in the Bogoliubov theory as Bogoliubov modes with zero frequencies and zero norms. These ``zero modes'' cannot be quantized as the usual Bogoliubov quasiparticle harmonic oscillators. They must be treated in a nonperturbative way. In this paper I develop a nonperturbative theory of zero modes. This theory provides a nonperturbative description of quantum phase diffusion and quantum diffusion of soliton position. An initially well localized wave packet for soliton position is predicted to disperse beyond the width of the soliton.
The unperturbed Hamiltonian for the Bose-Einstein condensate, which includes not only the first and second powers of the zero mode operators but also the higher ones, is proposed to determine a unique and stationary vacuum at zero temperature. From the standpoint of quantum field theory, it is done in a consistent manner that the canonical commutation relation of the field operator is kept. In this formulation, the condensate phase does not diffuse and is robust against the quantum fluctuation of the zero mode. The standard deviation for the phase operator depends on the condensed atom number with the exponent of $\ensuremath{-}1/3$, which is universal for both homogeneous and inhomogeneous systems.