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Fluctuation and Noise Letters

Shock Waves in Bose–Einstein Condensation Under Gaussian White Noise

Eren Tosyalı

Published May 15, 2018• 2 citations
Publisher

Abstract

We investigate the Gross–Pitaevskii equation with the tilted bichromatical optical lattice potential for finding the dynamics of a Bose–Einstein condensate system under the Gaussian white noise. We construct the Poincare sections of system based on the relations between the system parameters and solution behaviors to understand how its shock wave like dynamic could be affected by the noise. Also the hierarchical cluster analysis method investigation of the system is presented.

DOI
10.1142/s021947751850027x

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We investigate the Gross–Pitaevskii equation with the tilted bichromatical optical lattice potential for finding the dynamics of a Bose–Einstein condensate system under the Gaussian white noise. We construct the Poincare sections of system based on the relations between the system parameters and solution behaviors to understand how its shock wave like dynamic could be affected by the noise. Also the hierarchical cluster analysis method investigation of the system is presented.
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.
Dynamical properties of the Bose–Einstein condensate in a double-well potential subject to Gaussian white noise are investigated by numerically solving the time-dependent Gross–Pitaevskii equation. The Gaussian white noise is used to describe influence of the random environmental disturbance on the double-well condensate. Dynamical evolutions from three different initial states, the Josephson oscillation state, the running phase and π-mode macroscopic quantum self-trapping states, are considered. It is shown that the system is rather robust with respect to the weak noise whose strength is small and change rate is high. If the evolution time is sufficiently long, the weak noise will finally drive the system to evolve from high-energy states to low-energy states, but in a manner rather different from the energy-dissipation effect. In the presence of strong noise with either large strength or slow change rate, the double-well condensate may exhibit very irregular dynamical behaviours.
Abstract The problem of wave breaking during its propagation in the Bose–Einstein condensate to a stationary medium is considered for the case when the initial profile at the breaking instant can be approximated by a power function of the form (–x)1/n. The evolution of the wave is described by the Gross–Pitaevskii equation so that a dispersive shock wave is formed as a result of breaking; this wave can be represented using the Gurevich–Pitaevskii approach as a modulated periodic solution to the Gross–Pitaevskii equation, and the evolution of the modulation parameters is described by the Whitham equations obtained by averaging the conservation laws over fast oscillations in the wave. The solution to the Whitham modulation equations is obtained in closed form for n = 2, 3, and the velocities of the dispersion shock wave edges for asymptotically long evolution times are determined for arbitrary integers n > 1. The problem considered here can be applied for describing the generation of dispersion shock waves observed in experiments with the Bose–Einstein condensate.