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arXiv:cond-mat/0610366 · Physical Review A

Condition for the existence of complex modes in a trapped Bose-Einstein condensate with a highly quantized vortex

Eriko Fukuyama, M. Mine, Masahiko Okumura, Tomoka Sunaga, Y. Yamanaka

Published Oct 9, 2007• 6 citations
arXivHTMLPDF

Abstract

We consider a trapped Bose-Einstein condensate (BEC) with a highly quantized vortex. For the BEC with a doubly, triply, or quadruply quantized vortex, the numerical calculations have shown that the Bogoliubov-de Gennes equations, which describe the fluctuation of the condensate, have complex eigenvalues. In this paper, we obtain the analytic expression of the condition for the existence of complex modes, using the method developed by Rossignoli and Kowalski [R. Rossignoli and A. M. Kowalski, Phys. Rev. A 72, 032101 (2005)] for the small coupling constant. To derive it, we make the two-mode approximation. With the derived analytic formula, we can identify the quantum numbers of the complex modes for each winding number of the vortex. Our result is consistent with those obtained by the numerical calculation in the case that the winding number is two, three, or four. We prove that the complex modes always exist when the condensate has a highly quantized vortex.

DOI
10.1103/physreva.76.043608

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Publisher's Note: Condition for the existence of complex modes in a trapped Bose-Einstein condensate with a highly quantized vortex [Phys. Rev. A76, 043608 (2007)]

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Received 11 October 2007DOI:https://doi.org/10.1103/PhysRevA.76.049904©2007 American Physical Society

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We consider a trapped Bose-Einstein condensate (BEC) with a highly quantized vortex. For the BEC with a doubly, triply, or quadruply quantized vortex, the numerical calculations have shown that the Bogoliubov-de Gennes equations, which describe the fluctuation of the condensate, have complex eigenvalues. In this paper, we obtain the analytic expression of the condition for the existence of complex modes, using the method developed by Rossignoli and Kowalski [R. Rossignoli and A. M. Kowalski, Phys. Rev. A 72, 032101 (2005)] for the small coupling constant. To derive it, we make the two-mode approximation. With the derived analytic formula, we can identify the quantum numbers of the complex modes for each winding number of the vortex. Our result is consistent with those obtained by the numerical calculation in the case that the winding number is two, three, or four. We prove that the complex modes always exist when the condensate has a highly quantized vortex.
The condition for the appearance of dynamical instability of the Bose-condensed system, characterized by the emergence of complex eigenvalues in the Bogoliubov--de Gennes equations, is studied analytically. It is concluded that the degeneracy between a positive-norm eigenmode and a negative-norm one is essential for the emergence of complex modes. Based on the conclusion, we justify the two-mode approximation applied in our previous work [E. Fukuyama et al., Phys. Rev. A 76, 043608 (2007)], in which we analytically studied the condition for the existence of complex modes when the condensate has a highly quantized vortex.
The Bogoliubov-de Gennes equations are used for a number of theoretical works to describe quantum and thermal fluctuations of trapped Bose-Einstein condensates. We consider the case in which the condensate has a highly quantized vortex. It is known that these equations have complex eigenvalues in this case. We give the complete set including a pair of complex modes whose eigenvalues are complex conjugates to each other. The expansion of the quantum fields which represent neutral atoms in terms of the complete set brings the operators associated with the complex modes, which are simply neither bosonic nor fermionic ones. The eigenstate of the Hamiltonian is given. Introducing the notion of the physical states, we discuss the instability of the condensates in the context of Kubo’s linear response theory.
The splitting process of quadruply charged vortices into less charged vortices is studied at zero temperature. We solve the Bogoliubov-de Gennes (BdG) equation numerically and classify the excitations by angular momentum l relative to the condensates. Unstable modes having complex eigenvalues appear when l = 2-6. It is found that two excitations are coupled with each other when their total energy and angular momentum are zero and yield a pair of complex eigenvalues. The decay process is also numerically simulated by solving the time-dependent Gross-Pitaevskii equation. The behavior of a condensate shows good agreement with the results of BdG. The quadruply charged vortex splits into four singly quantized vortices, which are arranged as a line, triangle, and square according to the angular momentum of the excited eigenmode. We also observe the effect of nonlinearity.
The Bogoliubov equations are solved for a three-dimensional Bose-Einstein condensate containing a doubly quantized vortex, trapped in a harmonic potential. Complex frequencies, signifying dynamical instability, are found for certain ranges of parameter values. The existence of alternating windows of stability and instability, respectively, is explained qualitatively and quantitatively using variational calculus and direct numerical solutions. It is seen that the windows of stability disappear in the limit of a cigar-shaped condensate, which is consistent with recent experimental results on the lifetime of a doubly quantized vortex in that regime.
The dynamics and interaction of quantized vortices in Bose-Einstein condensates (BECs) are investigated by using the two-dimensional Gross-Pitaevskii equation (GPE) with/without an angular momentum rotation term. If all vortices have the same winding number, they would rotate around the trap center but never collide. In contrast, if the winding numbers are different, their interaction highly depends on the initial distance between vortex centers. The analytical results are presented to describe the dynamics of the vortex centers when β=0. While if β≠0, there is no analytical result but some conclusive numerical findings are provided for the further understanding of vortex interaction in BECs. Finally, the dynamic laws describing the relation of vortex interaction in nonrotating and rotating BECs are presented.