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Journal of Physics Conference Series

The Dynamic of Trapped Bose-Einstein Condensate Under Noise

Eren Tosyalı, F. Aydogmus Sen

Published Dec 1, 2018• 0 citations
Publisher

Abstract

It is known that Bose-Einstein Condansate is well described by the non-linear Schrodinger equation known as the Gross Pitaevskii Equation (GPE) with the macroscopic wave function which evaluates with time and space. Many studies have been performed on nonlinear properties in Bose-Einstein Condansate. In this study, we present some numerical results of the Gross-Pitaevskii Equation with the external potential under noise. The phase portraits and Poincaré sections of the system are simulated numerically both with and without noise.

DOI
10.1088/1742-6596/1141/1/012124

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It is known that Bose-Einstein Condansate is well described by the non-linear Schrodinger equation known as the Gross Pitaevskii Equation (GPE) with the macroscopic wave function which evaluates with time and space. Many studies have been performed on nonlinear properties in Bose-Einstein Condansate. In this study, we present some numerical results of the Gross-Pitaevskii Equation with the external potential under noise. The phase portraits and Poincaré sections of the system are simulated numerically both with and without noise.
The Gross-Pitaevskii equation (GPE) plays an important role in the description of Bose-Einstein condensates (BECs) at the mean-field level. The GPE belongs to the class of nonlinear Schr\"odinger equations which are known to feature dynamical instability and collapse for attractive nonlinear interactions. We show that the GPE with repulsive nonlinear interactions typical for BECs features chaotic wave dynamics. We find positive Lyapunov exponents for BECs expanding in periodic and aperiodic smooth external potentials, as well as disorder potentials. Our analysis demonstrates that wave chaos characterized by the exponential divergence of nearby initial wave functions is to be distinguished from the notion of nonintegrability of nonlinear wave equations. We discuss the implications of these observations for the limits of applicability of the GPE, the problem of Anderson localization, and the properties of the underlying many-body dynamics.
The Gross-Pitaevskii equation, or more generally the nonlinear Schrödinger equation, models the Bose-Einstein condensates in a macroscopic gaseous superfluid wave-matter state in ultra-cold temperature. We provide analytical study of the NLS with L2 initial data in order to understand propagation of the defocusing and focusing waves for the BEC mechanism in the presence of electromagnetic fields. Numerical simulations are performed for the two-dimensional GPE with anisotropic quadratic potentials.
We assume the macroscopic wave function of a Bose-Einstein condensate as a superposition of Gaussian wave packets, with time-dependent complex width parameters, insert it into the mean-field energy functional corresponding to the Gross-Pitaevskii equation (GPE) and apply the time-dependent variational principle. In this way the GPE is mapped onto a system of coupled equations of motion for the complex width parameters, which can be analyzed using the methods of nonlinear dynamics. We perform a stability analysis of the fixed points of the nonlinear system, and demonstrate that the eigenvalues of the Jacobian reproduce the low-lying quantum mechanical Bogoliubov excitation spectrum of a condensate in an axisymmetric trap.
The nonlinear dynamics of self-bound Bose–Einstein matter waves under the action of an attractive nonlocal and a repulsive local interaction is analyzed by means of a time-dependent variational formalism. The mean-field model described by the Gross–Pitaevskii equation (GPE) is reduced to a single second-order conservative ordinary differential equation admitting a potential function. The stable fixed points and the linear oscillation frequencies of the breather modes are determined. The chosen nonlocal interactions are a Gaussian-shaped potential and a van der Waals-like potential. The variational solutions are compared with direct numerical simulations of the GPE, for each of these nonlocalities. The spatio-temporal frequency spectra of linear waves are also determined.