Few-body Problems In One-dimensional Ultracold Atomic Gases

Because of the rapid advance of ultra-cold atomic experiments,the high control and tunability of ultracold atomic systems offer the possibility to study the few-body problem.Study of the few-body system is the building blocks of many-body problem.Recently ultra-cold atomic few-body systems have drawn a lot of focus of experiment research,which is becoming a hot topic.This thesis mainly studies the few-body properties of one-dimensional atomic gases,including the wavefunctions,energy spectrum,tunneling dynamics,spin-mixing dynamics and the quantum magnetism of strongly correlated quantum system.Firstly,a more reasonable trial ground state wave function is proposed for the relative motion of an interacting two-fermion system in a one-dimensional(1D)metastable trap.The relative coordinate space is divided into three regions.At the boundaries both the wave function and its first derivative are continuous and the quasi-momentum is determined by a more practical constraint condition which associates two variational parameters.The ground state energy is obtained by applying the variational principle to the expectation value of the Hamiltonian of relative motion on the trial wave function.The resulted energy and wave function show better agreement with the analytical solution than the original proposal.Secondly,we present the Bethe-ansatz type exact solution for N interating bosonic atoms in the δ-split double well and focus specially on the two-paricle case.The occupation probability of the two-atom odd-parity eigenstate shows evident dependence on the interaction,distinct from the result of traditional two-mode model.The tunneling dynamics of two atoms starting from the NOON state with infinite barrier height can be derived from the exactly solved model of a δ-barrier split double well based on a Bethe ansatz type hypothesis of the wave functions.We find that the single-particle tunneling shifts between the probability of double occupancy in the same well and that of single occupancy in different wells.Thirdly,we consider a one-dimensional trapped gas of strongly interacting few spin-1 atoms which can be described by an effective spin chain Hamiltonian.Away from the SU(3)integrable point,where the energy spectrum is highly degenerate,the rules of ordering and crossing of the energy levels and the symmetry of the eigenstates in the regime of large but finite repulsion have been elucidated.We study the spin-mixing dynamics which is shown to be very sensitive to the ratio between the two channel interactions 0 2g/ g and the effective spin chain transfers the quantum states more perfectly than the Heisenberg bilinear-biquadratic spin chain.Finally,based on the exact solution of one-dimensional Fermi gas systems with SU(n)symmetry,we demonstrate that it is possible to sort the ordering the lowest energy eigenvalues of states with all allowed permutation symmetries,which can be solely marked by certain quantum numbers in the Bethe ansatz equations.Our results go beyond the generalized Lieb-Mattis theorem,which can only compare the ordering of energy levels of states belonging to different symmetry classes if they are comparable according to the pouring principle.In the strongly interacting regime,we show that the ordering rule can be determined by an effective spin-exchange model and extend our results to the non-uniform system trapped in the harmonic potential.Our results provide a rule for ordering the high-spin SU(n)symmetric systems.