Dynamics Of Quantum Droplets In PT-symmetric Optical Lattices

The ultra-cold gas has given rise to many new quantum states of matter,the most famous of which is the Bose-Einstein condensate.A great deal of theoretical and experimental research has been done on the direction of Bose-Einstein condensates,such as solitons,vortices,Josephson oscillations,chaos,and symmetry breaking.Recently,a quantum droplet composed of coherent matter waves in binary BoseEinstein condensates are a new breakthrough of Bose-Einstein condensates.Compared with liquid helium,the droplets formed by binary condensates are a better quantum liquid.It is more suitable for future experiments,especially stable quantum droplets offer potential use in various applications,such as matter-wave interferometry and manipulations of quantum information.The gain and loss system expands the study of solitons from traditional optical lattices to PT-symmetric optical lattices.As soliton-like states,quantum droplets are rarely studied in PT-symmetric systems up to now.Therefore,we start from the Lee-Huang-Yang-amended Gross-Pitaevskii equation and devote to the study of the dynamics of quantum droplets in PT-symmetric optical lattice.This paper consists of the following chapters:In the first chapter,we briefly introduce the basic theory of Bose-Einstein condensate.Then,the theoretical models and conclusions of quantum droplets in three dimensions,two dimensions and one dimension are introduced,as well as the experimental realization.Finally,the research status of PT-symmetric optical lattice is introduced.In the second chapter,We introduce the numerical methods used to study the numerical solutions and stability of quantum droplets.firstly,the Gross-Pitaevskii equation is derived.Secondly,the imaginary-time-integration method used to solve the ground state wave function of quantum droplets.The direct simulations of the evolution used to simulate the stability of the quantum droplet,Stability spectra methods used in linear stability of quantum droplets,and the plane wave expansion method are introduced to calculate the band structure.Finally,the numerical solutions and stability of quantum droplets in some models are calculated by using these methods.In the third chapter,We study stability and collisions of quantum droplets forming in a binary bosonic condensate trapped in PT-symmetric optical lattices.First of all,we introduce the model,and PT-symmetric phase transition is revealed.The second,the stability of the PT-symmetric quantum droplets is studied.It is found that the stability of quantum droplets in the PT-symmetric system depends strongly on the values of the imaginary part of the PT-symmetric optical lattices,self-repulsion strength,and the condensate norm.As expected,the PT-symmetric quantum droplets are entirely unstable in the broken PT-symmetric phase.However,the PT-symmetric quantum droplets exhibit oscillatory stability with the increase of and in the unbroken PTsymmetric phase.Finally,collisions between PT-symmetric quantum droplets are considered.The collisions of droplets with unequal norms are completely different from that in free space.Besides,a stable PT-symmetric quantum droplets collides with an unstable ones tend to merge into breathers after the collision.In the fourth chapter,we summarize the main content of this paper,and look forward to the research prospect of this direction.