【作者】 张强；

【导师】 陈启瑾；

【作者基本信息】 浙江大学 ， 理论物理， 2020， 硕士

【摘要】 本文主要利用G0G方案(即散射T矩阵的基本单元pair susceptibility由无相互作用格林函数G0和有相互作用格林函数G组成)的配对涨落理论研究了混合维度粒子数极化超冷原子费米子系统中的超流和BCS-BEC渡越现象。系统中包含两种同核(等质量)数量不等的费米子,其中一种组分受到位于z方向的光晶格势的束缚,我们称之为晶格组分,而另一种组分不受其影响,我们称之为三维组分,这也是混合维度的由来。混合维度和粒子数极化都会带来费米面的失配,两种因素的共同作用会带来更多的新奇现象。第一章,我们将简单介绍BCS理论以及BCS-BEC渡越理论,并简单对比了系统在BCS区(弱耦合)和BEC区(强耦合)的不同。第二章,我们用数学语言——格林函数来描述BCS理论,并给出了G0G方案下T矩阵的表达式以及描述超流的完备方程组,在此基础上,我们考虑有限温度的情形,此时非零动量粒子对的贡献将变得不可忽略,经过一种巧妙的近似处理,我们引入了赝能隙这一物理量,使得可以用相同的数学形式来描述BCS-BEC渡越理论。第三章,我们将应用第二章中的理论来描述混合维度系统。通过加入粒子数极化方程我们得到了本文所研究系统的完备方程组,求解该方程组,我们将可以得到混合维度粒子数极化系统的超流结果。第四章,我们系统的研究了混合维度粒子数极化系统的超流行为,这里我们定义极化率p为晶格组分和三维组分间粒子数密度的差值与总粒子数密度的比值,kF为三维组分的费米动量,d为晶格常数,t为隧穿振幅,EF=kF2/2m。作为对比我们给出了常规三维粒子数极化系统和混合维度无粒子数极化系统的结果,并且发现在维度混合和粒子数极化的共同作用下出现了更多的新奇行为。除了发现Tc曲线会左右劈裂即系统在中间耦合强度区域超流消失(可重入超流)外,我们还发现了Tc曲线会发生上下劈裂,即呈现出中间温度超流,这种现象在之前的研究中还未发现过。并且随着极化率p的增大,这种上下劈裂的方式出现在左右劈裂之前。我们发现这两种劈裂现象都和粒子对在z方向运动受限的程度有关。此外,当kFd=4,-0.111(?)p(?)-0.098时,Tc曲线在深BEC区将会出现两个渐进值,并且当p<pe=-0.111,系统将不会出现超流。随后我们发现,随着p的增大,发生最佳费米面匹配的d值将变小。另外,对于较大的|p|=-p,当kFd处于某中间值域时,系统超流消失。其次,我们发现增大t能够减缓维度混合和粒子数极化带来的费米面失配程度。最后,我们绘制了kFd=1,t/EF=0.05时p—1/kFa平面的零温相图,并且简单展示了当另一种组分也受到光晶格势能影响时的结果。总的来说,这些新的发现对于混合维度体系的量子现象复杂性以及对现有配对和超流理论的检验具有重要学术价值。第五章,我们对本文进行了总结,并展望了未来可以继续的工作。更多还原

【Abstract】 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.更多还原