Raman laser induced self-organization with topology in a dipolar condensate

We investigate the ground states of a dipolar Bose-Einstein condensate (BEC) subject to Raman laser induced spin-orbit coupling with mean-field theory. Owing to the interplay between spin-orbit coupling and atom-atom interactions, the BEC presents remarkable self-organization behavior and thus hosts various exotic phases including vortex with discrete rotational symmetry, stripe with spin helix, and chiral lattices with $C_4$ symmetry. The peculiar chiral self-organized array of square lattice, which spontaneously breaks both $U(1)$ and rotational symmetries, is observed when the contact interaction is considerable in comparison with the spin-orbit coupling. Moreover, we show that the Raman-induced spin-orbit coupling plays a crucial role in forming rich topological spin textures of the chiral self-organized phases by introducing a channel for atoms to turn on spin flipping between two components. The self-organization phenomena predicted here feature topology owing to spin-orbit coupling. In addition, we find long-lived metastable self-organized arrays with $C_6$ symmetry in the case of strong spin-orbit coupling. We also present a proposal to observe these predicted phases in ultracold atomic dipolar gases with laser-induced spin-orbit coupling, which may stimulate broad theoretical as well as experimental interest

Radial mises stress contour map of yielding components.

Radial mises stress contour map of yielding components.</p

Displacement of elastic surrounding rock and support.

Displacement of elastic surrounding rock and support.</p

Model parameters.

In solving the whole process of interaction between soft rock and yielding support in high-stress environments in tunnels using mechanical analysis methods, it is challenging to simultaneously satisfy both displacement coordination and static equilibrium at the contact surface between the rock and the support structure. This paper, based on the mechanical analysis of rock and rigid support, considers the impact of the circumferential installation of yielding elements on radial displacement, and proposes displacement approximation and support force approximation methods using displacement coordination and static equilibrium as approximation conditions. The study fits curves of numerical simulation results and laboratory test results of yielding elements, and attempts to directly use the laboratory test data set of yielding elements as computational data. By calculating two circular tunnel examples and comparing the effects of the trisection method, bisection method, and substitution method on the convergence of the displacement approximation method, the effectiveness of the methods proposed in this paper is verified. The research results show that the two approximation algorithms proposed in this paper have good accuracy and reliability in calculating the relative displacement of rock and yielding support structure contact surfaces, and the support force of yielding support. The bisection method outperforms the trisection and substitution methods in terms of stability and convergence. However, there are certain limitations in this study, such as the effectiveness of the algorithm may be influenced by geological conditions; the complexity of actual geological conditions may exceed the assumptions of the current rock-support mechanical analysis model.</div

Displacement of different approximation algorithms.

Displacement of different approximation algorithms.</p

Outer flange steel.

In solving the whole process of interaction between soft rock and yielding support in high-stress environments in tunnels using mechanical analysis methods, it is challenging to simultaneously satisfy both displacement coordination and static equilibrium at the contact surface between the rock and the support structure. This paper, based on the mechanical analysis of rock and rigid support, considers the impact of the circumferential installation of yielding elements on radial displacement, and proposes displacement approximation and support force approximation methods using displacement coordination and static equilibrium as approximation conditions. The study fits curves of numerical simulation results and laboratory test results of yielding elements, and attempts to directly use the laboratory test data set of yielding elements as computational data. By calculating two circular tunnel examples and comparing the effects of the trisection method, bisection method, and substitution method on the convergence of the displacement approximation method, the effectiveness of the methods proposed in this paper is verified. The research results show that the two approximation algorithms proposed in this paper have good accuracy and reliability in calculating the relative displacement of rock and yielding support structure contact surfaces, and the support force of yielding support. The bisection method outperforms the trisection and substitution methods in terms of stability and convergence. However, there are certain limitations in this study, such as the effectiveness of the algorithm may be influenced by geological conditions; the complexity of actual geological conditions may exceed the assumptions of the current rock-support mechanical analysis model.</div

Diagram of the failure form of yielding components.

(a) Failure form of slot and bolt hole. (b) Failure form of inner and outer slot steel.</p

Load-displacement curve of yielding components.

In solving the whole process of interaction between soft rock and yielding support in high-stress environments in tunnels using mechanical analysis methods, it is challenging to simultaneously satisfy both displacement coordination and static equilibrium at the contact surface between the rock and the support structure. This paper, based on the mechanical analysis of rock and rigid support, considers the impact of the circumferential installation of yielding elements on radial displacement, and proposes displacement approximation and support force approximation methods using displacement coordination and static equilibrium as approximation conditions. The study fits curves of numerical simulation results and laboratory test results of yielding elements, and attempts to directly use the laboratory test data set of yielding elements as computational data. By calculating two circular tunnel examples and comparing the effects of the trisection method, bisection method, and substitution method on the convergence of the displacement approximation method, the effectiveness of the methods proposed in this paper is verified. The research results show that the two approximation algorithms proposed in this paper have good accuracy and reliability in calculating the relative displacement of rock and yielding support structure contact surfaces, and the support force of yielding support. The bisection method outperforms the trisection and substitution methods in terms of stability and convergence. However, there are certain limitations in this study, such as the effectiveness of the algorithm may be influenced by geological conditions; the complexity of actual geological conditions may exceed the assumptions of the current rock-support mechanical analysis model.</div

Dimensions of groove surface.

In solving the whole process of interaction between soft rock and yielding support in high-stress environments in tunnels using mechanical analysis methods, it is challenging to simultaneously satisfy both displacement coordination and static equilibrium at the contact surface between the rock and the support structure. This paper, based on the mechanical analysis of rock and rigid support, considers the impact of the circumferential installation of yielding elements on radial displacement, and proposes displacement approximation and support force approximation methods using displacement coordination and static equilibrium as approximation conditions. The study fits curves of numerical simulation results and laboratory test results of yielding elements, and attempts to directly use the laboratory test data set of yielding elements as computational data. By calculating two circular tunnel examples and comparing the effects of the trisection method, bisection method, and substitution method on the convergence of the displacement approximation method, the effectiveness of the methods proposed in this paper is verified. The research results show that the two approximation algorithms proposed in this paper have good accuracy and reliability in calculating the relative displacement of rock and yielding support structure contact surfaces, and the support force of yielding support. The bisection method outperforms the trisection and substitution methods in terms of stability and convergence. However, there are certain limitations in this study, such as the effectiveness of the algorithm may be influenced by geological conditions; the complexity of actual geological conditions may exceed the assumptions of the current rock-support mechanical analysis model.</div

Supporting force and its range.

In solving the whole process of interaction between soft rock and yielding support in high-stress environments in tunnels using mechanical analysis methods, it is challenging to simultaneously satisfy both displacement coordination and static equilibrium at the contact surface between the rock and the support structure. This paper, based on the mechanical analysis of rock and rigid support, considers the impact of the circumferential installation of yielding elements on radial displacement, and proposes displacement approximation and support force approximation methods using displacement coordination and static equilibrium as approximation conditions. The study fits curves of numerical simulation results and laboratory test results of yielding elements, and attempts to directly use the laboratory test data set of yielding elements as computational data. By calculating two circular tunnel examples and comparing the effects of the trisection method, bisection method, and substitution method on the convergence of the displacement approximation method, the effectiveness of the methods proposed in this paper is verified. The research results show that the two approximation algorithms proposed in this paper have good accuracy and reliability in calculating the relative displacement of rock and yielding support structure contact surfaces, and the support force of yielding support. The bisection method outperforms the trisection and substitution methods in terms of stability and convergence. However, there are certain limitations in this study, such as the effectiveness of the algorithm may be influenced by geological conditions; the complexity of actual geological conditions may exceed the assumptions of the current rock-support mechanical analysis model.</div