Collective Breathing Mode Of 1D Fermi Gas At Finite Temperature

In this paper, we use Newton’s method to solve the Gaudin-Yang model of coupled thermodynamic Bethe-ansatz equations in the homogenous system. The relevant thermodynamic properties of the system are obtained. The breath-ing mode of the collective oscillating frequency in the one-dimensional harmonic trap Gaudin-Yang model was obtained by using Landau’s two-fluid theory and variational method.In the first two chapters we introduce the background and significance of the research for cold atoms and one-dimensional strongly correlated systems, then we introduce the basic concepts of cold-atom related research and the progress in one-dimensional strongly correlated systems by following our research collective excitation breathing mode of 1D Fermi gas at finite temperature.At the first aspect, we study thermodynamic properties of Gaudin-Yang model at finite temperature in one-dimensional homogeneous system. By solving the thermodynamic Bethe-ansatz equations using Newton’s method, we get the thermodynamic quantities, such as density, specific heat, entropy, compressibility, Luttinger parameters depending on temperature and chemical potential. We get the systems phases, the classic regime,critical regime and the quantumn regime. The phase transition boundary area may be determined by the specific heat peak line.At the second aspect, we introduce the background of experiments and the significance of hydrodynamics method for solving the collective oscillations of one-dimensional strongly correlated system, we get the breathing mode frequen-cy of Gaudin-Yang mode in the harmonic potential by variational method. Dipole mode frequency and breathing mode frequency are obtained. We get the distri-bution of particle number density by using local density approximation(LDA), and the frequency of breathing mode.Finally, we make a summarize and outlook follow-up.