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Communications in Theoretical Physics

Chaotic and regular spatial structures of Bose–Einstein condensates with a spatially modulated atom-atom interaction and without an external trapping potential*

Li Fei, Wenwu Li

Published Mar 18, 2025• 1 citation
Publisher

Abstract

Abstract We investigate the chaotic and regular spatial structures of Bose–Einstein condensates (BECs) with a spatially modulated atom-atom interaction and without an external trapping potential. A BEC with a spatially modulated atom-atom interaction is equivalent to being constrained by a nonlinear optical lattice. Theoretical analyses show the existence of a steady atomic current in the BEC with a spatially varying phase. Under perturbative conditions, the Melnikov chaos criteria of BECs with a spatially varying phase and a constant one are theoretically obtained, respectively. When the perturbative conditions cannot be satisfied, for a repulsive BEC with a spatially varying phase, numerical simulations demonstrate that changing the initial condition can eliminate the chaotic spatial structure and then the system transitions into a biperiodic spatial structure. Increasing the chemical potential can result in a transition from the biperiodic spatial structure to a single-periodic spatial structure. For an attractive BEC with a spatially varying phase, numerical simulations show that decreasing the chemical potential can lead to a high atomic density, but when the wave number of the laser inducing the optical Feshbach resonance exceeds a critical value, the atomic density falls back to a finite range. Regardless of whether the BEC has a spatially varying phase or a constant one, modulating the laser wave number can effectively suppress the chaotic spatial structure in the BEC and then force it into a regular spatial structure.

DOI
10.1088/1572-9494/adc18b

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Abstract We investigate the chaotic and regular spatial structures of Bose–Einstein condensates (BECs) with a spatially modulated atom-atom interaction and without an external trapping potential. A BEC with a spatially modulated atom-atom interaction is equivalent to being constrained by a nonlinear optical lattice. Theoretical analyses show the existence of a steady atomic current in the BEC with a spatially varying phase. Under perturbative conditions, the Melnikov chaos criteria of BECs with a spatially varying phase and a constant one are theoretically obtained, respectively. When the perturbative conditions cannot be satisfied, for a repulsive BEC with a spatially varying phase, numerical simulations demonstrate that changing the initial condition can eliminate the chaotic spatial structure and then the system transitions into a biperiodic spatial structure. Increasing the chemical potential can result in a transition from the biperiodic spatial structure to a single-periodic spatial structure. For an attractive BEC with a spatially varying phase, numerical simulations show that decreasing the chemical potential can lead to a high atomic density, but when the wave number of the laser inducing the optical Feshbach resonance exceeds a critical value, the atomic density falls back to a finite range. Regardless of whether the BEC has a spatially varying phase or a constant one, modulating the laser wave number can effectively suppress the chaotic spatial structure in the BEC and then force it into a regular spatial structure.
The dynamical behaviors of Bose-Einstein condensates (BECs) depend largely on the nonlinear interactions between BEC atoms. The advancement of experimental techniques enables the rapid and effective modulation of the nonlinear interactions through Feshbach resonance technique. At present, both the nonlinear time-varying interaction and nonlinear space-varying interaction have been realized, respectively, thus making it possible to simultaneously modulate the nonlinear interactions in time and space through the combination of techniques. It will provide more options to conduct various studies by manipulating the BECs. Therefore, BECs with time- and space-varying interactions must possess unique advantages in studying BEC dynamics. This paper focuses on the chaotic spatiotemporal dynamics of BECs with nonlinear time- and space-varying interactions in moving optical lattices. When the intensities of the moving optical lattice potential and the modulation of the nonlinear interaction are small, the system satisfies the perturbation conditions and the Melnikov-function method is used in the theoretical analyses to obtain the Melnikov spatiotemporal chaotic criterion of the system. When the system does not meet the perturbation conditions, numerical simulations show that for a BEC with an attractive atomic interaction, increasing the modulation intensity of the nonlinear interaction can deepen the degree of spatiotemporal chaos in the system. In certain parameter regions, the modulation frequency of the nonlinear interaction can have a significant influence on the spatiotemporal dynamical behavior of the system. Further numerical research results show that larger chemical potentials can suppress the spatiotemporal chaos not only in the attractive BEC but also in the repulsive BEC. Based on the above research results, spatiotemporal chaos in BEC system can be avoided or triggered off in experiments as needed.
The spatial structure of a collisionally inhomogeneous Bose-Einstein condensate (BEC) in an optical lattice is studied. A spatially dependent current with an explicit analytic expression is found in the case with a spatially dependent BEC phase. The oscillating amplitude of the current can be adjusted by a Feshbach resonance, and the intensity of the current depends heavily on the initial and boundary conditions. Increasing the oscillating amplitude of the current can force the system to pass from a single-periodic spatial structure into a very complex state. But in the case with a constant phase, the spatially dependent current disappears and the Melnikov chaotic criterion is obtained via a perturbative analysis in the presence of a weak optical lattice potential. Numerical simulations show that a strong optical lattice potential can lead BEC atoms to a state with a chaotic spatial distribution via a quasiperiodic route.
In this paper, we study the spatially chaotic distribution of atoms in a Bose-Einstein condensate system, trapped in an asymmetric periodic potential. For a constant phase of condensate, without atom currents in the system, the space distributed structure of condensated atoms can be described by an undamped Duffing equation with double drivers. Through theoretical analyses, the Mel'nikov chaotic criterion for the system with a repulsive interatomic interaction is presented. Numerical simulations show that an increasing chemical potential can exert considerable suppression on the chaotic distribution of condensated atoms and even completely eliminate chaos. For a system with an attractive interatomic interaction, under some specific parametric conditions, adjusting the ratio between optical lattice potential amplitudes will force the condensated atoms from a periodic state into a spatially chaotic distribution; with the increase of chemical potential, the spatially chaotic distribution is completely suppressed.
Abstract We study the spatial structure of a Bose–Einstein condensate (BEC) with a space-dependent s-wave scattering length in a combined trap. There exists a space-dependent nonlinear atomic current in the system. The atomic current has an important influence on the spatial structure of the BEC. Research findings reveal that a large chemical potential can effectively suppress the chaotic spatial structure in the BEC system. Due to the large chemical potential, a strong atomic current is necessary to make the system lose its periodic spatial structure and lead the system into a chaotic spatial structure. But when the atomic current intensity exceeds a critical value, the chaotic spatial structure of the BEC will be completely eliminated and the system will always be kept in a series of single-periodic states as the atomic current becomes stronger. For a very weak atomic current, the spatial structure of the BEC is very sensitive to the intensity of the atomic current and a very small change of the intensity can dramatically change the spatial structure of the BEC. The effects of the combined trap parameters on the spatial structure of BECs are also discussed.
This paper studies the regular and chaotic spatial distribution of Bose-Einstein condensed atoms with a space-dependent nonlinear interaction.There exists a steady atomic current in the system due to the space-dependent phase of condensate.For the system with a positive chemical potential and repulsive interatomic interaction,we construct the general solution of the 1st-order equation,whose boundedness conditions contain the famous Mel'nikov chaotic criterion.When the system doesn't satisfy the perturbation conditions,numerical simulations reveal that increasing the atomic current can destroy the spatial symmetry of the distributional structure of condensed atoms,whether the condensed atoms in a chaotic or regular distribution.
We study the spatiotemporal dynamics of Bose-Einstein condensates (BECs) with spatiotemporally varying two- and three-body interactions and without an external trapping potential. The cubic and quintic terms in the Gross-Pitaevskii (GP) equation play the roles of external trapping potentials. Spatiotemporal modulation of the two- and three-body interactions can be of great importance in BEC field. But it is the subject of relatively fewer studies. Through theoretical analyses we obtain the equilibrium points (centers and saddles in phase space) of the unperturbed repulsive and attractive cases. With the heteroclinic solution of the unperturbed repulsive system, we theoretically construct the general solution of the 1st-order perturbed equation of the system. By using the boundedness conditions of the general solution we obtain the Melnikov chaos criterion predicting the existence of Smale-horseshoe chaos in the system. With a set of system parameters satisfied the Melnikov chaos criterion, numerical simulations show that the system is in a chaotic state. For BECs that do not satisfy the Melnikov chaos criterion, numerical simulations demonstrate that the modulating frequency can be an effective controlling parameter for the spatiotemporal dynamics of the BECs.