| In this paper,we mainly study the superfluidity and BCS-BEC crossover of ultra-cold atomic fermion gases in mixed dimensions with population imbalance by using the pairing fluctuation theory of the G0G scheme(the basic element of scattering T matrix,pair susceptibility is composed of noninteracting Green's function G0 and interacting Green 's function G).There are two kinds of fermions with the same nucleus(equal mass)in the system,one of which is bound by the optical lattice potential in the z direction,we call it the lattice component,while the other is not affected by it,we call it the three-dimensional component.This is also the origin of the mixed dimension.Both the mixed dimensions and population imbalance will lead to the mismatch of Fermi surface,and the interaction of the two factors will bring more novel phenomena.In Chapter 1,we will briefly introduce the BCS theory and BCS-BEC crossover theory,and simply compare the differences between the BCS regime(weak cou-pling)and the BEC regime(strong coupling).In Chapter 2,we use the mathematical language—Green's function to de-scribe the BCS theory,and give the expression of T matrix and the completed equations used to describing the superfluidity under the scheme of GOG.On this basis,we consider the case with finite temperature,where the contribution of pairs with non-zero momentum can not be ignored.After a clever approximation,we introduce the pseudogap,so that we can use the same mathematical form to describe the BCS-BEC crossover theory.In Chapter 3,we will apply the theory in Chapter 2 to describe the mixed dimensional system.By adding the particle number polarization equation,we get the complete equations of the system in this paper.By solving the equations,we can get the related results of superfluidity in the mixed dimensional system with population imbalance.In Chapter 4,we systematically study the superfluidity behavior of the mixed dimensional and population imbalanced system,here,we define the p as ration of the difference of particle number density of the lattice component and the 3D component,kF as the Fermi momentum of the 3D component,d as the lattice constant,t as the hopping integral,EF=kF2/2m.As a comparison,we give the results of the conventional three-dimensional population imbalanced system and the mixed dimensional system without population imbalance,and find that there are more novel behaviors under the joint effect of the mixed dimensions and population imbalance.Except for the horizontal splitting of Tc curve where superfluidity disappears in the middle coupling strength area(re-entrant super-fluidity),we also find the vertical splitting of Tc curve which corresponds to the intermediate temperature superfluidity.As p increases,the appearance of vertical splitting is before the appearance of horizontal splitting.We find this two kind of splitting is related to restriction of mobility of pairs in z direction.Besides,when kFd=4,-0.111(?)p(?)-0.098,there is two asymptote constant value of Tc in the deep BEC regime,and when p<pc=-0.111,there is no superfluidi-ty.Then we find that,as the p increases,there is a smaller d where best Fermi surface match occurs.Moreover,for a large |p|=-p,superfluidity disappears when d in a middle value range.Furthermore,we find that larger t can reduce the mismatch of Fermi surface caused by the mixed dimensions and population imbalance-Finally,we show the zero temperature phase diagram in p-1/kFa plane at kFd=1,t/EF=0.05.In general,these results are quite different from those in pure three-dimensional free space or three-dimensional lattice,which represents new quantum phenomena.This study is an important attempt for the quantum simulation of condensed matter system using ultra-cold atom Fermi gas,and also has important academic value for understanding the complex quantum physical phenomena of mixed dimensional system and guiding the corresponding experimental research.In Chapter 5,we summarize this paper and look forward to the future work. |
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