光晶格中超流费米气体的能带结构和Bloch波
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摘要
目前，实验室开展了很多关于光晶格中超流费米气体的研究。几们可以通过Feshbach共振技术来调控原子之间的相互作用，从而能使费米气体由分子BEc态过度到超流BCS态。与此同时实验室也实现了光晶格中的超流费米气体，在这个超流体系中，几们紧随实验在理论上也开展了许多的研究，比如，Bloch震荡，超流一绝缘转变，集体激发，以及热力学性质和稳定性问题的讨论，同时也讨论了在一维和二维光晶格中带隙孤子及其激发问题，以及在三维光晶格中自囚禁，稳定运动的孤子和呼吸子等问题。
论文的第二章讨论了一维光晶格中费米气体的能带和相直Bloch波的稳定性，在流体力学理论和两模近似条件下，我们得到了在第一布里渊区边界处Bloch波发生能量不稳定性和动力学不稳定性的条件，也发现了当原子数密度超过一定的临界密度时，在Bcs—BEc整个渡越区，最低的一条能带在第一布里渊区边界处将出现能带的燕尾结构。但是能带结构和Bl。ch波的稳定性在整个跨越区受原子间相互作用的影响很大，与此同时受到原子密度，以及晶格参数的影响，同时我们还解析得到了能带出现燕尾结构和能量发生不稳定性临界密度的表达式。
论文的第三章在周期性势场和平均场流体力学的框架下，讨论了一维光晶格中超流费米气体能隙孤子和能隙波的存在性和稳定性。我们展现了囚禁在光晶格中的费米气体在第一，第二带隙中各种类型的孤子，它们当中有基本孤子和基本孤子的组合孤子，以及第二带隙当中的更基本孤子f在一个晶格中有两个方向相异的峰)。我们发现这种孤子在Bcs—BEc整个区域都是存在的，利用线性稳定性分析证明更基本孤子及其组合态的稳定性在BEc段和幺正区是不一样的，但是基本孤子及其组合态不论在BEc段还是在幺正极限情况下部是稳定的。同时也发现在Bcs—BEc整个渡越区，存在Bloch波和相直的带隙波相吻合的现象，带隙孤子可以看成是Bl。ch波的一个片段。最后我们也揭示了囚禁在一维光晶格中超流费米气体带隙波f多峰带隙孤子)和自囚禁现象之间的连带关系。
最后在论文的第四章我们总结了全文并对超冷费米气体的研究作了展望。
Currently, experiments on ultracold Fermi gases are rapidly advancing toward therealization of super?uid states in optical lattices. The Fermi gases can be changed contin-uously from a Bose-Einstein condensate(BEC) to a BCS super?uid by using the Feshbachresonance technique. Now, the super?uidity of ultracold fermions in optical lattices hasbeen established. In such a super?uid system, people have given many theoretical e?ortson the crossover from the BEC state to the BCS state in optical lattices, such as the Blochoscillations, the super?uid-insulator transition, the collective excitations, the thermody-namic properties, and the stability. The intrinsically localized excitations [i.e., gap solitonin one- (1D) and two-dimensional (2D) optical lattices, self-trapping, and stable-movingsoliton and breather in three-dimensional (3D) optical lattices] are also presented.
In the first part, Bloch band and stabilities of Bloch waves of super?uid Fermi gasesin one-dimensional periodic optical lattices are discussed. Within the hydrodynamicaltheory and the two-mode approximation, the Bloch band structure, the energetic anddynamical instabilities of Bloch waves at the first Brilliouin zone are presented. Theresults show that, when the atom density is beyond a critical value, a loop structurein the Bloch band at the zone edge is developed along the BCS-BEC crossover. TheBloch band structure and the stabilities of Bloch waves are modified dramatically whenthe system crosses from the BCS side to the BEC side, and they can be adjusted to therequired characteristics by changing the atom s interaction (with the Feshbach resonancetechnique), the atom density, and the lattice parameters. The analytical expressions ofthe critical atom density for exciting the loop structure and maintaining the stabilities ofBloch waves are obtained.
Secondly, the existence and stability of gap soliton(GS) and gap wave of super?uidFermi gases in 1D periodic optical lattices are discussed. Within the framework of mean-field- hydrodynamic model and with a periodic potential. We present several GS speciesin first two band gaps of the optical lattices induced spectrum. These include fundamentalgap solitons(FGSs) and their bound states, as well as a subfundamental gap soliton in thesecond gap, featuring two peaks of opposite signs in a single well of the periodic potential.We found this kind of gap soliton exist both in the BEC limit and at unitary regime.Linear-stability analysis demonstrate that the stability property are di?erent between inthe BEC limit and at the unitary regime. However, fundamental solitons in first bandgap and second band gap are always stable not only in BEC limit but also at unitaryregime. Lastly, our results show that there also exists a composition relation betweenBloch waves and GSs or gap waves in super?uid Fermi gases like that in Bose system, and GSs is viewed as a segment of the Bloch waves. Furthermore, we also reveal thatself-trapping e?ect in one-dimensional optical lattices can be associated with the existenceof a localized gap state(gap wave) in super?uid Fermi gas.
Finally, we summarize the main result of this paper and give an outlook of the futurein this field.
论文的第二章讨论了一维光晶格中费米气体的能带和相直Bloch波的稳定性，在流体力学理论和两模近似条件下，我们得到了在第一布里渊区边界处Bloch波发生能量不稳定性和动力学不稳定性的条件，也发现了当原子数密度超过一定的临界密度时，在Bcs—BEc整个渡越区，最低的一条能带在第一布里渊区边界处将出现能带的燕尾结构。但是能带结构和Bl。ch波的稳定性在整个跨越区受原子间相互作用的影响很大，与此同时受到原子密度，以及晶格参数的影响，同时我们还解析得到了能带出现燕尾结构和能量发生不稳定性临界密度的表达式。
论文的第三章在周期性势场和平均场流体力学的框架下，讨论了一维光晶格中超流费米气体能隙孤子和能隙波的存在性和稳定性。我们展现了囚禁在光晶格中的费米气体在第一，第二带隙中各种类型的孤子，它们当中有基本孤子和基本孤子的组合孤子，以及第二带隙当中的更基本孤子f在一个晶格中有两个方向相异的峰)。我们发现这种孤子在Bcs—BEc整个区域都是存在的，利用线性稳定性分析证明更基本孤子及其组合态的稳定性在BEc段和幺正区是不一样的，但是基本孤子及其组合态不论在BEc段还是在幺正极限情况下部是稳定的。同时也发现在Bcs—BEc整个渡越区，存在Bloch波和相直的带隙波相吻合的现象，带隙孤子可以看成是Bl。ch波的一个片段。最后我们也揭示了囚禁在一维光晶格中超流费米气体带隙波f多峰带隙孤子)和自囚禁现象之间的连带关系。
最后在论文的第四章我们总结了全文并对超冷费米气体的研究作了展望。
Currently, experiments on ultracold Fermi gases are rapidly advancing toward therealization of super?uid states in optical lattices. The Fermi gases can be changed contin-uously from a Bose-Einstein condensate(BEC) to a BCS super?uid by using the Feshbachresonance technique. Now, the super?uidity of ultracold fermions in optical lattices hasbeen established. In such a super?uid system, people have given many theoretical e?ortson the crossover from the BEC state to the BCS state in optical lattices, such as the Blochoscillations, the super?uid-insulator transition, the collective excitations, the thermody-namic properties, and the stability. The intrinsically localized excitations [i.e., gap solitonin one- (1D) and two-dimensional (2D) optical lattices, self-trapping, and stable-movingsoliton and breather in three-dimensional (3D) optical lattices] are also presented.
In the first part, Bloch band and stabilities of Bloch waves of super?uid Fermi gasesin one-dimensional periodic optical lattices are discussed. Within the hydrodynamicaltheory and the two-mode approximation, the Bloch band structure, the energetic anddynamical instabilities of Bloch waves at the first Brilliouin zone are presented. Theresults show that, when the atom density is beyond a critical value, a loop structurein the Bloch band at the zone edge is developed along the BCS-BEC crossover. TheBloch band structure and the stabilities of Bloch waves are modified dramatically whenthe system crosses from the BCS side to the BEC side, and they can be adjusted to therequired characteristics by changing the atom s interaction (with the Feshbach resonancetechnique), the atom density, and the lattice parameters. The analytical expressions ofthe critical atom density for exciting the loop structure and maintaining the stabilities ofBloch waves are obtained.
Secondly, the existence and stability of gap soliton(GS) and gap wave of super?uidFermi gases in 1D periodic optical lattices are discussed. Within the framework of mean-field- hydrodynamic model and with a periodic potential. We present several GS speciesin first two band gaps of the optical lattices induced spectrum. These include fundamentalgap solitons(FGSs) and their bound states, as well as a subfundamental gap soliton in thesecond gap, featuring two peaks of opposite signs in a single well of the periodic potential.We found this kind of gap soliton exist both in the BEC limit and at unitary regime.Linear-stability analysis demonstrate that the stability property are di?erent between inthe BEC limit and at the unitary regime. However, fundamental solitons in first bandgap and second band gap are always stable not only in BEC limit but also at unitaryregime. Lastly, our results show that there also exists a composition relation betweenBloch waves and GSs or gap waves in super?uid Fermi gases like that in Bose system, and GSs is viewed as a segment of the Bloch waves. Furthermore, we also reveal thatself-trapping e?ect in one-dimensional optical lattices can be associated with the existenceof a localized gap state(gap wave) in super?uid Fermi gas.
Finally, we summarize the main result of this paper and give an outlook of the futurein this field.
引文
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[3] J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, et al. J nature 435, 1047–1051(2005).
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