Thermodynamics Of Rotating Ideal Quantum Gases In A Harmonic Trap

One of the essential problems in condensed matter physics is to understand the behaviors of rotating quantum gases (including Bose and Fermi gases). Cold atomic gases can be "spined up" either by confining them in the rotating frame, or by introducing a synthetic magnetic field. Previous researches mainly focused on the quantum vortices in superfluid phase caused by the rotation. More recently, thermodynamics of rotating quantum gases in normal state also merit a renewed attention. In this thesis, we investigate the thermodynamics of rotating ideal Bose and Fermi gases by analytical and numerical calculations, especially emphasizing on the major differences of Bose-Einstein condensation (BEC) critical temperature, magnetism and the spatial distribution of particle flow in rotating frame and synthetic magnetic field. Furthermore, we will compare the results of Bose gases with those of Fermi system in order to gain a deeper insight into the two types of quantum gases.This dissertation mainly concentrates on the following topics.First, we discuss the influences of the spatial dimension on BEC critical temperature and specific heat of ideal Bose gases trapped in any dimensional harmonic trap. Our calculation shows that the curve of the specific heat versus temperature in a two-dimensional system is continuous, while it appears an obvious BEC transition in a three-dimensional system. The specific heat of both systems tends to their respective classical values in the high-temperature limit. We also study the thermodynamics of idea Bose and Fermi gases in a power-law potential.Second, we numerically modify the thermodynamics of harmonically trapped rotating ideal Bose gases within truncated-summation approach (TSA). This method takes into account the discrete nature of energy levels, rather than the summation over single-particle energy levels by an integral in semi-classical approximation (SCA). Our results show that Bose gases in rotating frame exhibit much stronger dependence on rotation frequency than those in synthetic magnetic field. Consequently, BEC can be more easily suppressed in rotating frame than in synthetic magnetic field. The diamagnetism turns significantly stronger below the BEC critical temperature, which coincides with our intuition in superconductivity.Then, we investigate the BEC transition and the magnetic properties of harmonically trapped charged ideal spin-1 bosons confined in a magnetic field. The BEC critical temperature shows upgrade then descending tendency as the magnetic field becomes strong, which is a result of the competition between the spin and charge degrees of freedom. Charged spin-1 Bose gases present a crossover from diamagnetism to paramagnetism as the spin Lande factor increases, which is completely distinct from the spinless case.Next, we calculate the chemical potential, the magnetization and the spatial distribution of fermion flow of harmonically trapped rotating ideal Fermi gases. And then, we compare those with conventional electrons in the condensed matter. It indicates that the magnetization and fermion flow distribution of rotating ideal Fermi gases exhibit an analogous de Hass-van Alphen type oscillation in the conventional electron systems. These results provide a theoretical support for the current experiment of mimicking the magnetic field effect by use of rotating cold atoms.At last, we study the quantum oscillations of magnetization and particle flow in rotating ideal Bose and Fermi gases under a two-dimensional harmonic trap. It is found that the characteristic dependencies of the magnetization on rotary modes are clearly reflected in the spatial distribution of the particle flow. This de Hass-van Alphen type oscillation is regarded as the characteristic behavior of a constrained Fermi system.