Feshbach Resonance In ²³Na⁴⁰K Mixture And Phase Transitions Of Spin-1 Bosons In Optical Lattice

Ultracold quantum simulation is a very hot topic in modern physics research,many scientists are contributing to the field and great progress has been made in recent two decades.This thesis will focus on some important problems in ultracold simulation,including Feshbach resonance in Na-K mixture gas,energy level transitions in diatomic molecule and phase transition process of spin-1 bosons trapped in optical lattice.The thesis is divided into three parts bases on the contents.The first part is about Feshbach resonance in Na-K mixture gas,we present the basic priniple and calculating method of Feshbach resonance,especially for coupled-channel Feshbach problem,we introduce Log-Derivative vari-step propagation method to do the calculation.More-over,we have included the coupling of different partial-wave components?different L?which is neglected in other studies.Then given the experimental data of Feshbach reso-nance measured in ²³Na⁴⁰K mixture,we adjust the Born-Oppenheimer potentials of the ground statesX1?+ and a3?:of ²³Na⁴⁰K molecule,the adjusted potential energy curve is also very important in the calculation of scattering length and Feshbach resonances in other Na-K isotopes collisions.The second part describes the energy level structure,including electronic and vi-brational energy levels,of diatomic molecule.Then we give the methods of calculating transition properties between molecule energy levels and fitting potential energy curves from molecular spectral data.Finally,we apply the methods to solve the potential en-ergy curves of excited states of NaK molecule and calculate the Franck-Condon factor between ground state and vibrational levels in excited states.The third part focuses on bosonic system in optical lattice.We use Bose-Hubbard model to describe such system,and superfluid-Mott insulator phase transition process is solved with several methods.We pay particularly attention on spin-1 bosons,besides superfluid-Mott insulator phase transition process,we study the magnetic phase transi-tions within Mott insulator phase.We find that standard basis operator?SBO?method is particularly suitable for calculating these phase transition processes,the calculted result is very close to that obtained with Monte-Carlo simulation method.