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arXiv:1802.01671 · Physical review. E

Beating effects of vector solitons in Bose-Einstein condensates

Li-Chen Zhao

Published Jun 5, 2018• 36 citations
arXivHTMLPDF

Abstract

We study the beating effects of solitons in multicomponent coupled Bose-Einstein condensate systems. Our analysis indicates that the period of beating behavior is determined by the energy eigenvalue difference in the effective quantum well induced by solitons, and the beating pattern is determined by the eigenstates of a quantum well, which are involved in the beating behavior. We show that the beating solitons correspond to linear superpositions of eigenstates in some quantum wells, and the correspondence relations are identical for solitons in both an attractive interaction and a repulsive interaction condensate. This provides a possible way to understand the beating effects of solitons for attractive and repulsive interaction cases in a unified way, based on the knowledge of quantum eigenstates. Moreover, our results demonstrate many different beating patterns for solitons in multicomponent coupled condensates, in sharp contrast to the beating dark soliton reported before. The beating behavior can be used to test the eigenvalue differences in certain quantum wells, and more abundant beating patterns are expected to exist in more component-coupled systems.

DOI
10.1103/physreve.97.062201

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We study the beating effects of solitons in multicomponent coupled Bose-Einstein condensate systems. Our analysis indicates that the period of beating behavior is determined by the energy eigenvalue difference in the effective quantum well induced by solitons, and the beating pattern is determined by the eigenstates of a quantum well, which are involved in the beating behavior. We show that the beating solitons correspond to linear superpositions of eigenstates in some quantum wells, and the correspondence relations are identical for solitons in both an attractive interaction and a repulsive interaction condensate. This provides a possible way to understand the beating effects of solitons for attractive and repulsive interaction cases in a unified way, based on the knowledge of quantum eigenstates. Moreover, our results demonstrate many different beating patterns for solitons in multicomponent coupled condensates, in sharp contrast to the beating dark soliton reported before. The beating behavior can be used to test the eigenvalue differences in certain quantum wells, and more abundant beating patterns are expected to exist in more component-coupled systems.
We investigate nondegenerate bound-state solitons systematically in multicomponent Bose-Einstein condensates, through developing the Darboux transformation method to derive exact soliton solutions analytically. In particular, we show that bright solitons with nodes correspond to the excited bound states in effective quantum wells, in sharp contrast to the bright solitons and dark solitons reported before (which usually correspond to ground state and free state, respectively). We further demonstrate that bound-state solitons with nodes are induced by incoherent superposition of solitons in different components. Moreover, we reveal that the interactions between these bound-state solitons are usually inelastic, caused by the incoherent interactions between solitons in different components and the coherent interactions between solitons in the same component. Additionally, the detailed spectral stability analysis demonstrates the stability of nondegenerate bound-state solitons. The bound-state solitons can be used to study many different physical problems, such as beating dynamics, spin-orbit coupling effects, quantum fluctuations, and even quantum entanglement states.
We analyze the dynamics of two-component vector solitons, namely, dark-bright solitons, via the variational approximation in Bose-Einstein condensates. The system is described by a vector nonlinear Schr\"odinger equation appropriate to multicomponent Bose-Einstein condensates. The variational approximation is based on a hyperbolic tangent (hyperbolic secant) for the dark (bright) component, which leads to a system of coupled ordinary differential equations for the evolution of the ansatz parameters. We obtain the oscillation dynamics of two-component dark-bright solitons. Analytical calculations are performed for same-width components in the vector soliton, and numerical calculations extend the results to arbitrary widths. We calculate the binding energy of the system and find it to be proportional to the intercomponent coupling interaction and numerically demonstrate the breakup or unbinding of a dark-bright soliton. Our calculations explore observable eigenmodes, namely, the internal oscillation eigenmode and the Goldstone eigenmode. We find analytically that the number of atoms in the bright component is required to be less than the number of atoms displaced by the dark soliton in the other component in order to find the internal oscillation eigenmode of the vector soliton and support the existence of the dark-bright soliton. This outcome is confirmed by numerical results. Numerically, we find that the oscillation frequency is amplitude independent. For dark-bright solitons in 87Rb^{87}\mathrm{Rb}87Rb we find that the oscillation frequency range is 90 to 405 Hz and therefore observable in multicomponent Bose-Einstein condensate experiments.
Wave properties of solitons in a two-component Bose–Einstein condensate are investigated in detail. We demonstrate that dark solitons in one of components admit interference and tunneling behavior, in sharp contrast to the scalar dark solitons and vector dark solitons. Analytic analyses of interference properties show that spatial interference patterns are determined by the relative velocity of solitons, while temporal interference patterns depend on the velocities and widths of two solitons, differing from the interference properties of scalar bright solitons. Especially, for an attractive interactions system, we show that interference effects between the two dark solitons can induce some short-time density humps (whose densities are higher than background density). Moreover, the maximum hump value is remarkably sensitive to the variation of the solitons’ parameters. For a repulsive interactions system, the temporal-spatial interference periods of dark–bright solitons have lower limits. Numerical simulation results suggest that interference patterns for the dark–bright solitons are more robust against noises than bright–dark solitons. These explicit interference properties can be used to measure the velocities and widths of solitons. It is expected that these interference behaviors can be observed experimentally and can be used to design matter wave soliton interferometer in vector systems.
We investigate the interactions between components of various vector solitons in Bose—Einstein condensates by means of the least action principle, and derive the effective potentials for different vector solitons, which indicate that the interactions are of short range, and may be repulsive or attractive decided by the different intra- and inter-species interactions in such a system. In the case of attraction, the two solitons will oscillate about and pass through each other around the equilibrium state. The comparison of analytical results with mumertical simulation is presented.
We demonstrate that the model of a spatially nonuniform two-component Bose-Einstein condensates (BEC) featuring the helicoidal spin-orbit coupling (SOC), gives rise to dark-bright soliton complexes characterized by spatiotemporal periodic oscillations in each component. These solitons are formed by the superposition of dark and bright ones, and exhibit a beating state over time and a striped state across space, earning them the designation of beating stripe solitons. Our analysis demonstrates that helicoidal SOC significantly affects the formation and dynamical properties of these solitons, also serving as the primary driver for spin oscillations. Through the nonlinear superposition of the beating stripe solitons, a range of intricate scenarios of the interaction between multiple solitons emerges, including head-on collisions, bound states, and parallel states.
Abstract We investigate oscillatory soliton pairs including the bright–bright solitons, dark–dark solitons, bright–dark solitons and beating solitons (beating soliton refers to a type of soliton characterized by components that are each composed of both bright and dark solitons, and the overall density is equivalent to that of a single soliton) in the spin-orbit coupled spin-1/2 Bose–Einstein condensates. Two important effects that are the spin orbit coupling (SOC) and Raman detuning resonance on these solitons are studied. The results show that the intensity of SOC affects the spatial oscillation of solitons, while Raman detuning resonance affects the temporal oscillation of solitons. Moreover, the oscillatory soliton pairs and their background are affected by gain/loss. Under different gain/loss modulator, the periodic, parabolic, dromion type and line solitons are constructed for bright–bright pairs, and the periodic and parabolic background planes are presented for dark–dark soliton pairs, as well as the bound state solitons on periodic and kink type background planes are obtained for bright–dark soliton pairs. For one-beating soliton pairs, two types of beating solitons including the petal and breather-like are discovered by adjusting the wave number. For two-beating soliton pairs, the interaction between different combinations of the two types of beating solitons mentioned above, especially the X- and Y-type beating solitons, are presented by controlling two wave numbers. Furthermore, we verify the stability of soliton pairs through numerical simulation and the physics-informed neural network methods.