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摘要 本文首先介绍了相关理论以及实验研究背景，然后阐述了整个计算过程并分析了所得理论结果的物理含义，最后对本文的工作进行总结。本文中我们的研究对象是在各向同性谐势阱下从 Bardeen-Cooper-Schrieffer(BCS)态到 Bose-Einstein凝聚(BEC)渡越间的三维费米气体，该费米气体的动力学行为由多方近似下的推广的Gross-Pitaevskii 方程(GGPE)来描述。我们利用耦合相位-系数变换解析求解了三维GGPE，最终求得其暗孤子解，并且没有添加任何额外的可积条件，我们发现所得的暗孤子在整个 BCS-BEC 渡越区经历了振动，而且振动周期为常数，但暗孤子的振幅随着多方指数的改变而变化，这表明了由三维 GGPE 来描述的系统具有特别的非线性性。58980
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毕业论文关键词：BEC-BCS 渡越；Gross-Pitaevskii 方程；自相似变换；孤子
Abstract We introduced the related theories and the background of the experiments in the first place. After that we performed the details of calculation and analyzed the physical meaning of the result obtained followed with a summary for the whole scheme in the end. In this paper, We studied the three-dimensional Fermi gas in an isotropic harmonic trap during the Bardeen-Cooper-Schrieffer superfluid to Bose-Einstein condensate (BCS-BEC) crossover, which is modeled by using the generalized Gross-Pitaevskii equation (GGPE) in the polytropic approximation. We analytically solved the 3D GGPE with a coupled modulus-phase transformation without introducing any additional integrability constraint, reaching the dark soliton-like solution. We find that the dark soliton identified undergoes an oscillation with a constant period over the whole BCS-BEC crossover region, although the amplitude of the dark soliton varies with polytropic index, which demonstrate the peculiar nonlinear properties for the system modeled by using the 3D GGPE.
Keywords: BEC-BCS crossover; Gross-Pitaevskii equation; soliton; self-similar transformation;
目录
第一章绪论1
1.1Bose-Einstein凝聚.1
1.2超流超导及BCS理论.3
1.3Gross-Pitaevskii方程.4
1.4Feshbach共振5
1.5本文的主要工作.6
第二章计算方法.7
2.1目标方程7
2.2坐标变换7
2.3自相似变换.8
结论.10
致谢.11
参考文献.11
第一章 绪论 过去二十年来， 超冷物理在实验和理论上都经历了巨大进步并随之吸引了大量研究人员的关注，尤其是在实验上由 Feshbach 共振技术主导的 Bose-Einstein凝聚(BEC)到 Bardeen-Cooper-Schrieffer(BCS)超流的渡越的研究。
1.1 Bose-Einstein凝聚 Bose-Einstein 凝聚是指无相互作用的 Bose 系统在温度足够低时所有粒子都将占据最低能量的量子态，这种宏观地占据能量最低量子态的现象称为Bose-Einstein 凝聚。对于无相互作用，动量为 k的Bose系统，其色散关系为： �