Study On Mechanical Stability And Phase Separation Of A Two-Component Bose Gas At Finite Temperature

In this thesis,we use the Hartree-Fock mean field theory and Popov theory to study the phase diagram of a two-component Bose gas with pseudo-spin half at finite temperature.When the intra-spin interaction is repulsive,and the inter-spin interaction is attractive,the system has rotational symmetry in the ?x-?y plane and transverse magnetization occurs.A first-order phase transition occurs.The uniform system suffers mechanical instability within a certain range of medium density,resulting in phase separation between the normal phase and the Bose Einstein condensate phase.This phase transition is similar to the classical gas-liquid phase transition,and the isothermal curve of the equation of state is characterized shows characteristic plateau at the saturated pressure.In a harmonic trap,the phase separation phenomenon can be revealed by the jump of the density distribution.The amplitude of density jump depends on the temperature and the interaction strength.For a given temperature T?O,the isothermal curve approaches to that of the ideal classical gas,when the phase space density n?T3?1(where ?T is the thermal wavelength).In this case,the atoms have no spin polarization,we call equillbrium state as non-polarized normal phase(here after referred to as normal phase).With the increase of the density,the system appears transverse magnetization and enters the ferromagnetic phase.Since the condition of thermodynamic stability is not met,the system collapses and the collapse temperature is related to the inter-spin attractions.Analytically,we find that the ferromagnetic phase transition precedes to the BEC transition.In the Bose condensed phase,as the number density of particles increases,the number of thermal atoms decreases,and the transverse magnetization almost comes from the contribution of condensed particles,eventually approach to one.We derive the formula of the compressibility in both normal phase and condensed phase of the system according to the thermodynamic stability condition.The trend of the gas equation-of-state can be qualitatively judged.According to the principle of minimum free energy in thermodynamics,it is found that the isothermal pressure curve exhibits an isobaric platform,and as a result phase separation occurs.Finally,the equation of state curves of the whole system are given,and the phase diagram of the system is obtained.Then,within local density approximation we investigate the phase separation of the two-component Bose gas in the harmonic trap.The atomic cloud shows the shell structure,the density distribution has a jump at the phase boundary,and the amplitude of the jump is related to the interaction intensity and temperature.In a reasonable temperature range,the density distribution of the atom clouds can be observed experimentally using the in-situ imaging technique.When the intra-spin repulsions and inter-spin attractions are almost cancelled out,the Hartree-Fock theory cannot correctly describe the phase diagram.To fix this problem we use Popov theory to reexamine the equation-of-state.Popov theory includes the Lee-Huang-Yang correction stemming from quantum fluctuations.From the thermodynamic relation,we find the compressibility remains to be positive when the repulsive interaction and the attractive interaction nave the equal strength.Comparing the phase diagrams have the equal strength by the two theories,the predictions are very close when the absolute values of the two scattering length ratios are around 0.8,which implies the phase separation in this region is mainly driven by the thermal fluctuations.