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Open Physics

Chaotic control problem of BEC system based on Hartree–Fock mean field theory

Yang Shen, Meng Xu

Published Jan 1, 2024• 3 citations
Publisher

Abstract

Abstract Due to the difficulty of studying nonlinear quantum systems and the unique composition of Bose–Einstein condensate (BEC) systems, BECs face significant difficulties in solving dynamic analysis and chaotic control problems. Therefore, Hartree–Fock mean field theory is introduced to study the chaotic characteristics, control, and synchronization issues of BEC systems loaded on optical lattices. First, the stability and chaos of BECs in optical lattices were analyzed. Subsequently, constant shift method and activation control were introduced based on the Gross–Pitaevskii equation to achieve control and synchronization of the BEC system. Second, based on the Lyapunov exponent theory, offset parameters are added to BEC chaotic control to achieve control of particle density. Finally, based on the stability theory of linear systems, a control term is introduced to achieve variable analysis of the system’s drive–response system, ensuring that chaotic systems with different initial conditions can still achieve good synchronization and anti-synchronization control. The chaotic problem of BEC system was analyzed using numerical and theoretical methods in the experiment. The effect of adjusting the parameters of the BEC system under the constant shift method is significant. The system exhibits a chaotic state under the Lyapunov exponent, which is mainly concentrated between [3.4, 4.5], demonstrating good system stability. When the offset constant range is [4.21, 5.67], the maximum Lyapunov exponent value is below 0. In the problem of chaotic synchronization, adding activation control causes the system’s time series to exhibit anti-synchronization with spatiotemporal variable variation, while adding control terms leads the system to tend towards synchronization and anti-synchronization with time evolution. The analysis of chaotic control problems in BEC systems can provide reference value and theoretical basis for the dynamic research of quantum physics and related nonlinear systems.

DOI
10.1515/phys-2023-0196

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Abstract Due to the difficulty of studying nonlinear quantum systems and the unique composition of Bose–Einstein condensate (BEC) systems, BECs face significant difficulties in solving dynamic analysis and chaotic control problems. Therefore, Hartree–Fock mean field theory is introduced to study the chaotic characteristics, control, and synchronization issues of BEC systems loaded on optical lattices. First, the stability and chaos of BECs in optical lattices were analyzed. Subsequently, constant shift method and activation control were introduced based on the Gross–Pitaevskii equation to achieve control and synchronization of the BEC system. Second, based on the Lyapunov exponent theory, offset parameters are added to BEC chaotic control to achieve control of particle density. Finally, based on the stability theory of linear systems, a control term is introduced to achieve variable analysis of the system’s drive–response system, ensuring that chaotic systems with different initial conditions can still achieve good synchronization and anti-synchronization control. The chaotic problem of BEC system was analyzed using numerical and theoretical methods in the experiment. The effect of adjusting the parameters of the BEC system under the constant shift method is significant. The system exhibits a chaotic state under the Lyapunov exponent, which is mainly concentrated between [3.4, 4.5], demonstrating good system stability. When the offset constant range is [4.21, 5.67], the maximum Lyapunov exponent value is below 0. In the problem of chaotic synchronization, adding activation control causes the system’s time series to exhibit anti-synchronization with spatiotemporal variable variation, while adding control terms leads the system to tend towards synchronization and anti-synchronization with time evolution. The analysis of chaotic control problems in BEC systems can provide reference value and theoretical basis for the dynamic research of quantum physics and related nonlinear systems.
Chaos control of a Bose–Einstein condensate (BEC) loaded into a moving optical lattice with attractive interaction is investigated on the basis of Lyapunov stability theory. Three methods are designed to control chaos in BEC. As a controller, a bias constant, periodic force, or wavelet function feedback is added to the BEC system. Numerical simulations reveal that chaotic behavior can be well controlled to achieve periodicity by regulating control parameters. Different periodic orbits are available for different control parameters only if the maximal Lyapunov exponent of the system is negative. The abundant effect of chaotic control is also demonstrated numerically. Chaos control can be realized effectively by using our proposed control strategies.
The spatial structure of a Bose–Einstein condensate (BEC) loaded into an optical lattice potential is investigated and the spatially chaotic distributions of the condensates are revealed. A method of chaos control with linear feedback is presented in this paper. By using the method, we propose a scheme of controlling chaotic behavior in a BEC with atomic mirrors. The results of the computer simulation show that controlling the chaos into the stable states could be realized by adjusting the coefficient of feedback only if the maximum Lyapunov exponent of the system is negative.
Based on the nonlinear characteristics of Bose-Einstein condensed body in moving optical lattices and Lyapunov stability theory,and the combination with the sine coupling chaotic synchronization method,chaotic synchronization of Einstein condensed body in moving optical lattices was realized.With the help of theoretical analysis and numerical calculation,the authors proved the effectiveness of the method on the sine coupling chaotic synchronization and respectively realized the chaotic synchronization of the single variable coupling and double variable coupling.On the basis of the numerical calculation to determine the parameters range in which the chaotic synchronization can be realized,we proved that chaotic synchronization double variable coupling is faster than single variable coupling and small coupling coefficient can achieve chaotic synchronization.We also analysed the relationship between the synchronization time and the coupling coefficient.
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 chaotic dynamic characteristic in Bose-Einstein Condensate (BEC) system of a 1D tilted optical superlattice potential with attractive interaction is investigated in this paper. The spatial evolution of chaos was shown numerically by resolving Gross-Pitaevskii (G-P) equation for the system with the fourth Runge-Kutta(RK) algorithm. Numerical analysis reveals that as the tilt or the amplitude of the optical superlattice potential is increased the chaos in the BEC system increases. These elements make the chaotic system more unstable and the phase-space orbit becomes more chaotic. The chaotic system can be effectively controlled to a stable periodic orbit through adjusting the amplitude of the optical superlattice potential and initial condition. Controlling chaos can also be realized by spatial constant bias in the BEC system of a 1D tilted optical superlattice potential with attractive interaction. Phase orbits are suppressed gradually then the chaotic states of the BEC system are converted into period one through quais-period.