Effective Spin-Chain Model For Strongly Interacting One-Dimensional Atomic Gases

One of the fundamental and meaningful theoretical models in the cold atom research area is the strongly interacting one-dimensional(1D)quantum gas.Early stage studies of cold gases in optical lattices witnessed the absence of some quantum phenomena due to the limitations in the optical lattices.For instance,although short-range antiferromagnetism has been verified in dimerized lattices,the Néel long-range order in two-component Fermi gases has not yet been observed.In the experiments of Jochim's team,they realized a Heisenberg antiferromagnetic spin chain in the vicinity of a scattering resonance by trapping ultracold 6Li atoms in a harmonic trap.Later,it was proposed that the Néel AFM may be realized in a spin-1/2 Fermi gas with p-wave interactions in a 1D harmonic trap.In this thesis,we derive the effective spin-chain model for 1D strongly interacting atomic gases by means of the technique of Bose-Fermi mapping.We calculate the spin density distribution and energy spectrum of two components Fermi system by the exact diagonalization of the Hamiltonian matrix in a 1D harmonic oscillator and square potential well,respectively.The spin density distributions for balanced spin chain are the same at ground state,but not for an unbalanced spin chain.The change of external potential gives rise to a shift of density peak position and energy spectrum.Furthermore,applying a B-field gradient in the spin chain results in an alternating distribution of spin density distribution for balanced spin chain at ground state,indicating the antiferromagnetic order.The gradient destroys the spatial symmetry of spin density distribution for an unbalanced spin chain at ground state,and breaks the degeneracy for system with infinite interaction.The high tunability of ultracold atomic systems offers the fascinating possibility to realize an effective spin chain without the need for an optical lattice.The model of the effective chain is not only valid for bosons and fermions,but also valid for systems with more than two components.The model offers an alternative simple scenario for the study of 1D quantum magnetism in cold gases.