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.
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.
Abstract In this work, we investigate the modulational instability of plane wave solutions within a modified Gross–Pitaevskii equation framework. The equation features cubic and quartic nonlinearity. It models the behaviour of quasi-one-dimensional Bose–Einstein condensates in symmetric Bose-Bose mixtures of ultra-dilute cold atoms. The cases of equal amplitudes in both species, as well as of unequal amplitudes are studied. Our study demonstrates the pivotal role of the competition between mean-field attractions and quantum fluctuation-induced repulsions. This competition significantly affects the emergence and evolution of modulational instability. By employing linear stability analysis, we identify the essential conditions that lead to modulational instability. We find that the stability of plane wave solutions significantly depends on the interaction among system parameters. Further development of the instability leads to the fragmentation of the BEC into a chain of quantum droplets. We calculated the number of quantum droplets generated during the nonlinear stage of the instability. Our analytical results are corroborated by numerical simulations of the modified quasi-1D Gross–Pitaevskii equation. These simulations vividly depict the formation, interaction, and coalescence of droplets during the nonlinear phase of modulational instability. The investigation shows that the linear stability analysis of the modified Gross–Pitaevskii equation, considering quantum fluctuations, precisely predicts the initial stage of modulational instability phenomena across different domains of parameter space. In contrast, the later nonlinear stage, studied through numerical simulations, reveals the formation and nonintegrable dynamics of quantum droplets.
In this work, we investigate the modulational instability of plane wave solutions within a modified Gross-Pitaevskii equation framework. The equation features cubic and quartic nonlinearity. It models the behaviour of quasi-one-dimensional Bose-Einstein condensates in symmetric Bose-Bose mixtures of ultra-dilute cold atoms. Our study demonstrates the pivotal role of the competition between mean-field attractions and quantum fluctuation-induced repulsions. This competition significantly affects the emergence and evolution of modulational instability. By employing linear stability analysis, we identify the essential conditions that lead to modulational instability. We find that the stability of plane wave solutions significantly depends on the interaction among system parameters. Further development of the instability leads to the fragmentation of the BEC into a chain of quantum droplets. We calculated the quantity of quantum droplets generated during the nonlinear phase of the instability. Our analytical results are corroborated by numerical simulations of the modified quasi-1D Gross-Pitaevskii equation. These simulations vividly depict the formation, interaction, and coalescence of droplets during the nonlinear phase of modulational instability. The investigation shows that linear stability analysis of the modified Gross-Pitaevskii equation, considering quantum fluctuations, precisely forecasts modulational instability phenomena across different domains of parameter spaces.
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.
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.
The magnetically induced Richtmyer-Meshkov (RM) instability in a two-component Bose-Einstein condensate (BEC) is investigated. We construct and study analytical models describing the development of the instability at both the linear and nonlinear stages. The models indicate interesting features of the instability: surface tension implies departure from the linear growth of modes and separation of droplets, which are qualitatively different from the traditional RM case of classical gases, and the trapping potential affects the later stages of the instability. We perform numerical simulations of the instability in a trapped BEC using the Gross-Pitaevskii equation and compare the simulation results to the model predictions.
Modulational instability conditions for the generation of localized structures in the context of matter waves in Bose-Einstein condensates are investigated analytically and numerically. The model is based on a modified Gross-Pitaevskii equation, which account for the energy dependence of the two-body scattering amplitude. It is shown that the modified term due to the quantum fluctuations modify significantly the modulational instability gain. Direct numerical simulations of the full modified Gross-Pitaevskii equation are performed, and it is found that the modulated plane wave evolves into a train of pulses, which is destroyed at longer times due to the effects of quantum fluctuations.