Study Of The Dynamics Of Knots In Bose-Einstein Condensates

Knots and links are topological structures that play an important role in classical fluids and superflows,and they are closely related to the physical properties of real systems.Such topological defects and their dynamics have attracted extensive research interest and have important applications in many branches of physics,chemistry,biology and other disciplines.The trefoil knot,Solomon link,and the DNA double helix are well-known knot or link structures.Although these topologies arise spontaneously in some everyday phenomena,these randomly generated structures are usually in a chaotic background,which makes it impossible to isolate them from the environment and to manipulate and study their physical properties.Generating topological excitation with simple structures,such as vortex ring,in a controlled manner in the laboratory is a challenging task.At present,most of the research focuses on how to untie knots with complex structures into knots with simple structures,but the research on whether the evolution from simple structure to complex structure can be achieved controllably is still missing.Because of the complexity of viscous fluids,it is difficult to quantitatively study the physical properties of knots.Bose-Einstein condensates provide an ideal platform for the study of quantum knot dynamics due to their superfluid properties and artificial manipulability.Based on the mean field theory,this dissertation studies the controllable generation of knot and link excitations with complex topological structures,dynamics and factors affecting their topological evolution paths in three dimensional Bose-Einstein condensates using vortex lines and vortex rings as building blocks.In this dissertation,the topological evolution path from vortex lines to vortex rings in condensates is firstly studied.It is found that there is a qualitative relationship between the initial distance of the vortex dipole and the transverse to longitudinal ratio of the generated vortex ring.A perfect vortex ring can be produced by the appropriate initial distance.On this basis,we further investigate the collective excitation behavior of the condensate with a vortex ring excitation,and find the relationship between the amplitude of the condensate center of mass oscillation and the initial radius of the vortex ring and the initial parameters of the condensate.Three independent oscillation modes are proposed.By introducing the Kelvin wave perturbations,a vortex ring is deformed,and the effect of this deformation on the collective excitation of the condensate is studied.We use the combination of vortex line and vortex ring to generate a hopfion in the dynamic process and study its dynamic characteristics.The dynamic behavior of the vortex ring in the hopfion is further analyzed,and the influence of the initial parameters on the velocity of the vortex ring is discussed.Then,we study the topological evolution of the systems with two rings based on a single ring system.We propose for the first time the path of constructing various complex torus knots and links by vortex rings,and find that the Kelvin wave perturbation has a great influence on the evolution of the system.Different types of topological excitation(knot or link)can be generated by controlling the parity of the wave number of Kelvin waves.We also analyze the influences of various initial parameters of the system,such as the initial radius,relative distance,relative angle and Kelvin wave intensity of the vortex ring,on the stability of the generated knot and link,and further find a method to realize multiple topological evolution paths by controlling different parameters.The variation of the transverse to longitudinal ratio with time in the process of forming the knot is analyzed by using the moment of inertia tensor.This study reveals that there exist abundant topological transfer pathways in superfluids,which is of great significance for the further design of topological excitations with more complex structures or chemical and biological molecules with specific functions,as well as for the deeper understanding of the mechanisms of turbulent systems.