The Quantum Phase Transition Of Low-dimension Ultracold Atom

Bose-Einstein condensation (BEC) is that a large number of bosons occupied in one quantum energy when the temperature of system reach to the critical temperature. The reason is that the bosons follow the boson statistical distribution which is corresponding with the Fermi statistical distribution. The Fermi gas obey the Fermi statistical distribution in which there is only one Fermi in one quantum energy and this determinates that the Fermi gas can not formed the condensate. With the development of the experiment condition and the improving of the laser cooling and trapping techniques, it became true to trap and cool the dilute gas in the experiment. It is a big progress of the successful application of the techniques to get the degenerate gas and atom BEC. Two Fermi atoms with the opposite spin can form a cooper pair and it will be formed a kind of condensate when lower the temperature which is called Barden-Cooper-Schrieffer (BCS) in which is a kind of weak attractive interaction. The atom BEC is a kind of weak repelling interaction. The Feshbach resonance can change the interaction between two Fermi atoms by adjusting the strength of the magnetic field. It is possible to achieve the BCS-BEC crossover. The BCS-BEC crossover has attracted intense interest in which there are challenging research of the physical quantity and the quantum transition. The research of ultracold quantum gases is the cross point of many disciplines. It has significant importance from both the theoretical viewpoint and technological viewpoint.In this article, we mainly talk about the quantum phase transition of low-dimension ultracold atom. The results presented here include these aspects:In chapter one, we introduce the theoretical and technological development process of the BEC in which the experiment condition keep update on new technology, especially the trapping and cooling technology, now the laser is used to trapping the Fermi atom in the experiment. The theory and the development process of Feshbach resonance are useful to degenerate Fermi gas. We introduce the physical quantity and the transition of the development of the ultracold atom and BCS-BEC crossover. The experimental research of the strong correlation Fermi gas is investigated in this chapter. We know the different transition phases were decided by the different experimental subjects in the BCS-BEC crossover. In the last of this chapter, we research the physical quantity of the crossover.In chapter two, we involve the two components Fermi gas system with unequal mass and chemical potential in free two-dimension space at zero temperature. Because of the asymmetric two-component, there will be caused the Sarma phase. It is a gapless excited energy in the transition. We analytically investigate the gap and binding energy by the functional path integral method which is a new result. We can obtain the phase diagram of this system by analyzing the result.In chapter three, we involve the three components Fermi gas system with equal mass and unequal chemical potential in free two-dimension space at zero temperature. In this system, only two components are paired and one is left unpaired. The method is the same with chapter two. We get the phase diagram of the BCS-BEC crossover and analyzing the importance of the third component for the system.In chapter four, we first introduce the three components Fermi gas with equal mass and chemical potential and obtain the condensate fraction of this system. Then using the analytical solution in the chapter two, we get the diagram of the condensate fraction and the binding energy in the two component system with unequal mass and chemical potential. We also can obtain the diagram of the condensate fraction and the binding energy in the three component system by using the same method of the two component system. We compare the three diagram of three different systems and get the results about the effect of the unpaired component and the unequal chemical potential.