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Punjab University Journal of Mathematics

A Numerical Treatment for the Stability of Josephson Vortices in BEC

Muhammad Irfan Qadir, Nisha Shiraz

Published Dec 31, 2017• 0 citations
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Abstract

We consider a system of a pair of quasi one dimensional BoseEinstein condensates which are coupled with each other. The waveguides of the two condensates are assumed to be parallel. The system can be characterized by two coupled nonlinear Schrodinger equations. This system admits different topological structures such as vortices. We present a numerical treatment for the stability of Josephson vortices and dark solitons in a coupled system of Bose-Einstein condensate.

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We consider a system of a pair of quasi one dimensional BoseEinstein condensates which are coupled with each other. The waveguides of the two condensates are assumed to be parallel. The system can be characterized by two coupled nonlinear Schrodinger equations. This system admits different topological structures such as vortices. We present a numerical treatment for the stability of Josephson vortices and dark solitons in a coupled system of Bose-Einstein condensate.
We study the existence and stability of the bound state Josephson vortices solution in two parallel quasi one-dimensional coupled Bose–Einstein condensates. The system can be elucidated by linearly coupled Gross–Pitaevskii equations. The purpose of this study is to investigate the effects of altering the strength of coupling between the two condensates over the stability of the bound-state Josephson vortices. It is found that the stability of bound-state Josephson vortices depends on the value of coupling strength. However, at a critical value of coupling parameter, the Josephson vortices solution transforms into a coupled dark soliton.
A system of two parallel coupled cigar-shaped Bose–Einstein condensates is considered in an effectively one-dimensional limit. The dynamics of the system is characterized by a pair of coupled nonlinear Gross–Pitaevskii equations. In particular, the existence and stability of symmetric bound states of Josephson vortices are investigated. It is realized that the symmetric bound state Josephson vortices solution persists stably in its whole domain of existence for the coupling strength. Nevertheless, the bound states solution converts into a dark soliton at a critical value of coupling parameter.
The Josephson vortex (JV) between two long parallel coupled atomic Bose-Einstein condensates (BECs) is considered analytically and numerically in the quasi-one-dimensional limit. Its closed analytical solution is presented. It is found that, upon increasing the Josephson coupling, the JV transforms spontaneously into a dark soliton and vice versa. We suggest how the JV and its transformation can be detected. We also propose that the JV can be utilized for controlled and coherent transfer of atomic BECs and as a possible mobile qubit.
It is shown that the quasi-one-dimensional Bose-Einstein condensate is experimentally accessible and rich in intriguing phenomena. We demonstrate numerically and analytically the existence, stability and perturbation-induced dynamics of all types of stationary states of the quasi-one-dimensional nonlinear Schrödinger equation for both repulsive and attractive cases. Among our results are: the connection between stationary states and solitons; creation of vortices from such states; manipulation of such states with simple phase profiles; demonstration of the fragility of the condensate phase in response to shock; and a robust stabilization of the attractive Bose-Einstein condensate.
The nonlinear Schrodinger equation models a wide variety of different physical phenomena ranging from nonlinear optics, water waves, magnetization of ferromagnets to Bose-Einstein condensates (BEC). The structure of the equation supports existence of topologically non-trivial solutions - vortices. Surprisingly, we demonstrate that the Landau-Lifshitz magnetization equation which is formally also a nonlinear Schrodinger equation does not admit such solutions. On the other hand, the contrary is true for the Gross-Pitaevskii equation which describes the mean-field approximation of BEC. We investigate stability of vortex solutions by means of a very reliable, sensitive and robust technique - the Evans function. This method, although limited to two dimensions, allows us to study rotating axisymmetric BEC for large particle numbers which can be unattainable by other means.We found a singly-quantized vortex linearly stable.The linear stability of multi-quantized vortices depends on the diluteness of a condensate, with alternating intervals of stability and instability. This work justifies previous results in the literature obtained by less reliable methods and opens up a few interesting questions.
The stability properties of dark solitons in quasi-one-dimensional Bose-Einstein condensate (BEC) loaded in a Jacobian elliptic sine potential with three-body interactions are investigated theoretically. The solitons are obtained by the Newton-Conjugate Gradient method. A stationary cubic-quintic nonlinear Schrödinger equation is derived to describe the profiles of solitons via the multi-scale technique. It is found that the three-body interaction has distinct effect on the stability properties of solitons. Especially, such a nonlinear system supports the so-called dark solitons (kink or bubble), which can be excited not only in the gap, but also in the band. The bubbles are always linearly and dynamically unstable, and they cannot be excited if the three-body interaction is absent. Both stable and unstable kinks, depending on the physical parameters, can be excited in the BEC system.