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Chinese Physics Letters

Modulational Instability of Trapped Two-Component Bose–Einstein Condensates*

Jian-Wen Zhou, X. X. Li, Rui Gao, Wen-Shan Qin, Hao-Hao Jiang, Taotao Li, Ju-Kui Xue

Published Sep 1, 2019• 3 citations
Publisher

Abstract

Abstract The modulational instability of two-component Bose–Einstein condensates (BECs) under an external parabolic potential is discussed. Based on the trapped two-component Gross–Pitaevskill equations, a time-dependent dispersion relation is obtained analytically by means of the modified lens-type transformation and linear stability analysis. It is shown that a modulational unstable time scale exists for trapped two-component BECs. The modulational properties—which are determined by the wave number, external trapping parameter, intra- and inter-species atomic interactions—are modified significantly. The analytical results are confirmed by direct numerical simulation. Our results provide a criterion for judging the occurrence of instability of the trapped two-component BECs in experiment.

DOI
10.1088/0256-307x/36/9/090302

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Abstract The modulational instability of two-component Bose–Einstein condensates (BECs) under an external parabolic potential is discussed. Based on the trapped two-component Gross–Pitaevskill equations, a time-dependent dispersion relation is obtained analytically by means of the modified lens-type transformation and linear stability analysis. It is shown that a modulational unstable time scale exists for trapped two-component BECs. The modulational properties—which are determined by the wave number, external trapping parameter, intra- and inter-species atomic interactions—are modified significantly. The analytical results are confirmed by direct numerical simulation. Our results provide a criterion for judging the occurrence of instability of the trapped two-component BECs in experiment.
We investigate analytically and numerically the modulational instability of a Bose-Einstein condensate with both two- and three-body interatomic interactions and trapped in an external parabolic potential. Analytical investigations performed lead us to establish an explicit time-dependent criterion for the modulational instability of the condensate. The effects of the potential as well as of the quintic nonlinear interaction are studied. Direct numerical simulations of the Gross-Pitaevskii equation with two- and three-body interactions describing the dynamics of the condensate agree with the analytical predictions.
Abstract We demonstrate the existence of modulational instability (MI) in both trapped miscible and immiscible two component Bose–Einstein condensates. The study is addressed theoretically and numerically in the framework of one-dimensional coupled Gross–Pitaevskii equations incorporating intra- and interspecies cubic-quintic nonlinearities with higher-order ones. Using the time-dependent variational approach, we derive the new Euler-Langrange equations for the time evolution of the phase and amplitude of the modulational perturbation as well as the effective potential and the instability criteria of the system. We examine the effects of higher order nonlinearities on the instability dynamics of the condensates. We show that the modulational properties of the chosen wave numbers are significantly modified. Direct numerical simulations run by the split step Fourier method confirm the analytical predictions.
The stability of the ground state of two-component Bose–Einstein condensates (BECs) loaded into the central well of an axially symmetric Bessel lattices (BLs) potential with attractive or repulsive atoms interactions is studied using the time-dependent Gross–Pitaevskii equation (GPE). By using the variational method, we find that stable ground state of two-component BECs can exist in BLs. The BLs's depth and the intra-species atom interaction play an important role in the stability of ground state. The collapse of two-component BECs in BLs is also studied and a collapse condition for trapped two-component BECs is obtained. It is shown that the two-component BECs exhibit rich collapse dynamics. That is, the two-component BECs can collapse in the system with both intra- and inter-attractive, or with intra-attractive and inter-repulsive, or with intra-repulsive and inter-attractive atom interactions. Furthermore, the control of the collapse of the two-component BECs in BLs is discussed in detail. The stability diagram of the ground state in parameter space is obtained. The results show that the collapse of two-component BECs can be controlled by temporal modulation of the atom interaction.
The modulational instability (MI) of Bose–Einstein condensates based on a modified Gross–Pitaevskii equation (GPE) which takes into account quantum fluctuations and a shape-dependent term, trapped in an external time-dependent complex potential is investigated. The external potential consists of an expulsive parabolic background with a complex potential and a gravitational field. The theoretical analysis uses a modified lens-type transformation which converts the modified GPE into a modified form without an explicit spatial dependence. A MI criterion and a growth rate are explicitly derived, both taking into account quantum fluctuations and the parameter related to the feeding or loss of atoms in the condensate which significantly affect the gain of instability of the condensate. Direct numerical simulations of the modified GPE show convincing agreements with analytical predictions. In addition, our numerical results also reveal that the gravitational field has three effects on the MI: (i) the deviation backward or forward of solitons trains, (ii) the enhancement of the appearance of the MI and (iii) the reduction of the lifetime of pulses. Moreover, numerical simulations proved that it is possible to control the propagation of the generated solitons trains by a proper choice of parameters characterizing both the loss or feeding of atoms and the gravitational field, respectively.
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.
By means of the time-dependent variational approach, we study the modulational instability of Bose–Einstein condensates (BECs), with both two- and three-body interatomic interactions, trapped in an external parabolic potential. Within this framework, we derive and analyse ordinary differential equations for the explicit time evolution of the amplitude and phase of modulational perturbation. The effect of the trapping coefficients as well as the quintic nonlinear interaction on the dynamics of the BEC are examined. Numerical simulations are carried out in order to support our theoretical findings.