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Physical Review A

Modulational instability of two-component Bose-Einstein condensates in a quasi-one-dimensional geometry

Thokala Soloman Raju, Prasanta Kumar Panigrahi, Kuppuswamy Porsezian

Published Mar 28, 2005• 26 citations
Publisher

Abstract

A simple treatment is presented to analyze the modulational instability of two-component Bose-Einstein condensates, described within the Gross-Pitaevskii approximation, in a quasi-one-dimensional geometry. An explicit expression for the growth rate of a purely growing modulational instability is presented and analyzed analytically.
DOI
10.1103/physreva.71.035601

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In this paper, we consider the coupled Gross-Pitaevskii (CGP) equations in a quasi-one dimensional geometry, which are governed by the dynamics of the coupled Bose Einstein condensations (BECs) in the presence of the three body interaction potential. Using the linear stability analysis (LSA), we obtain the dispersion relation for the system of coupled BECs. Analytically, we obtain a new explicit expression for the growth rate. Besides, we investigate the modulational instability (MI) criterion of constant amplitude of both uncoupled and coupled BECs for different values of the three body interaction potential. Finally, we compare our results of both uncoupled and coupled BECs in the presence and absence of three body interaction term.
In this paper, we consider the dynamic evolution of a binary mixture of a Bose-Einstein condensate taking into account the presence of dissipation inside the components. Using the introduction of the dissipative function, the modified Gross-Pitaevskii equations are obtained. These equations, in contrast to the usual Gross-Pitaevskii equations for two-component condensate, allow us to take into account the dissipation in the system. The influence of dissipative processes on the development of modulation instability in a spatially homogeneous two-component Bose-Einstein condensate is investigated. In contrast to the one-component Bose-Einstein condensate, in which modulation instability arises only when there are forces of attraction between atoms, in a two-component Bose-Einstein condensate nonlinear dynamics, leading to modulation instability is more complex. It essentially depends on the signs and values of the constant interaction of the components, which leads to a greater variety of possible scenarios for the development of modulation instability. The paper considers two cases. The first case is when repulsive forces act inside the components, and the second is when repulsive forces act in the first component, and in the second one - attractive forces. At the same time, the situation when there is a repulsion in the first component, and attraction between the particles in the second component differs significantly from the case of only positive interaction inside the components. The relations between the interaction constants that determine the development of the modulation instability turn out to be different. Given the relations between the interaction constants, taking into account dissipation processes, the occurrence of modulation instability in two-component Bose-Einstein condensates was studied, the maximum growth rate of oscillations was found, and the limits of the existence of modulation instability in the space of wave numbers were found. It is shown that the small effect of dissipation on the modulation instability in the Bose – Einstein condensate is explained not only by the smallness of the friction forces. For wave vectors corresponding to a mode with a maximum increment, the contribution of dissipation in the linear approximation with respect to the dissipative parameter is strictly zero. Thus, the condition for the development of the most rapidly growing mode of oscillations, which determines the beginning of the modulation instability, remains the same as in the nondissipative case.
Modulational instability of a uniform two-dimensional binary Bose-Einstein condensate (BEC) in the presence of quantum fluctuations is studied. The analysis is based on the coupled Gross-Pitaevskii equations. It is shown that quantum fluctuations can induce instability when the BEC density is below a threshold. The dependence of the growth rate of modulations on the BEC parameters is found. It is observed that an asymmetry of the interaction parameters and/or initial densities of the components typically decreases the growth rate. Further development of the instability results in the BEC breaking up into a set of quantum droplets. These droplets merge dynamically with each other, so the total number of droplets decreases rapidly. The rate of this decrease is evaluated numerically for different initial parameters.
We investigate nonlinear dynamics induced by the modulation instability of a two-component mixture in an atomic Bose-Einstein condensate. The nonlinear dynamics is examined using numerical simulations of the time-dependent coupled Gross-Pitaevskii equations. The unstable modulation grows from initially miscible condensates into various types of vector solitary waves, depending on the combinations of the sign of the coupling constants (intracomponent and intercomponent). We discuss the detailed features of the modulation instability, dynamics of solitary wave formation, and an analogy with the collapsing dynamics in a single-component condensate with attractive interactions.