Exact Nonlinear Bloch Solutions And It's Application

Recently,matter waves(Bose-Einstein Condensates) in periodic potential have attracted the attention both experimentist and theoretist,due to controllable potential and sharing many features with electron in solid.Since the nonlinear interaction between ultra-cold atoms,many novel phenomena have been reported.So far,most of the theretical works are obtained by numerical calculation for this nonlinear interaction term.Therefore,it is interesting to look for the exact solution for this system,which will be helpful to deeply understand the reported numerical works.This is goal of this thesis.We introduced the nonlinear terms into the Kronig-Penny model and found the exact stationary solutions for nonlinear Schrodinger equation in a periodic array of quantum wells.The relative problems related with the Bloch theory are investigated,based on the exact solutions.The detail contents are following:Firstly,we found a full set of exact nonlinear Bloch-like solutions to the nonlinear Scr(o|¨)dinger equation(or Gross-Pitaevskii equation) in a periodic array of quantum wells.Since these solutions can be reduced to the linear solutions for Schrodinger equation in the case of zero nonlinear interaction,it is convenient to study the effect of the nonlinear interaction on the Bolch theory.We comprehensively studied the Bloch band,the compressibility, effective mass and the sound speed as functions of both the potential depth and interatomic interaction,based on these exact nonlinear Bloch solutions. Our results shown that increasing the nonlinear parameter induces the Bloch band become wider,in contrast increasing the potential depth makes it narrow. When the potential well is high enough,the Bloch function localized in the well,and then the tight-banding approximation is availabe.The inverse of the compressibilityΚ^-1 increases linearly only for small gn,while in the case of large gn,the relation betweenΚ^-1 and gn is nonlinear.The effective mass significantly increases with the increase of potential depth and gradually decreases with the enhancement of the nonlinear interaction.The sound velocity is decreasing with increasing the potential depth,for the competition betweenΚ^-1 and gn.Secondly,we obtained the nonlinear Wannier function for this system by considering the relation between the Wannier function and the Bloch function. Further more,we investigated the properties of nonlinear Wannier functions and the basic parameters U and J of Bose-Hubbard model are calculated. The nonlinear interaction makes the Wannier functions slowly fall off as exponential law with distance compared with the linear case,and enhances the tunneling coupling between the neighbor wells.On the other hand the on-site interaction U/gn is monotonously decreased with increasing gn,but increasing the potential depth makes U increase.When the potential depth large than23E_R,the ratio U/J exceed the critical value of phase transition, the system is in Mott-insulator state.Thirdly,we have comprehensively investigated the Landau and dynamical instabilities for a Bose-Einstein condensate in a periodic array of quantum wells.In the limit of tight-binding,the analytical expressions for both Landau and dynamical instabilities are obtained and then the stability phase diagrams are shown.The region of Landau instability decreases with the increasing of the potential depth and the nonlinear parameter.The dynamical instability spreads more in the area of Landau instability.For any nonlinear parameter and potential depth,the left borderline of dynamical instability is at k=π/2.Finally,we investigated the collective excitation of Bose-Einstein condensates in a periodic array of quantum wells.The analytic expression of spectrum of the collective excitations and the excitation strength in terms of the effective mass and compressibility,are obtained in the limit of tight-binding.Comparing with the lowest energy band of the collective excitation and the lowest Bloch band,we found that the effect of the nonlinear interaction on the collective excitation is larger than on the Bloch energy band.All the excitations in the lowest band acquire the linear phonon dispersion with a finite slope at small quasi-momenta.The excitation strength towards the first band develops an oscillating behavior as a function of the momentum transfer,vanishing at even multiples of the Bragg momentum due to the presence of a phononic regime.