超冷原子气体中的BEC和BCS-BEC转换
摘要
从微观层次上,玻色子和费米子具有完全不同的量子统计特征:就玻色系统而言,量子统计的效应是与Bose-Einstein-Condensation(BEC)现象的出现相关的,而在没有相互作用的费米系统中,系统在经典行为与量子简并行为之间光滑地转换;与稳定的BEC系统具有排斥相互作用不同,在费米气体中超流的出现仅仅来源于吸引相互作用的出现。为了观察这些量子统计造成的效应,我们必须获得极低温使破色与费米系统达到量子简并的条件。幸运地是,实验上已经产生了许多不同种类的超冷玻色性或者费米性的原子气体,并且在这些系统中已经成功地观察到了BEC以及Bardeen-Cooper-Schrieffer(BCS)超流。更重要的是,当我们使用Feshbach共振调节费米原子之间相互作用时,系统可以光滑地从BCS超流转换成双原子分子的BEC(这就是BCS-BEC转换的概念)。因此在一个纯粹的费米系统中我们可以同时观察到两种不同量子统计的效应。
这篇博士论文除了简单地介绍超冷原子气体的一些基本理论(包括BEC、BCS超导、BCS-BEC转换和Feshbach共振等)以及相应的实验验证之外,主要包含了如下五个方面的内容:
(1)在玻色-费米模型的框架下利用相干态路径积分的方法推导了系统的单圈有效作用量,在此基础上计算了BCS-BEC转换中系统的零温长波低频集体模式,得到了集体模式声速的一个解析表达式以及声速在BCS-BEC转换中光滑演化的结论;这种计算方法能够自洽地给出系统的态方程,因而明显不同于通常的流体动力学方法。
(2)在两流体模型和有限温度BCS-BEC转换理论的框架下,计算了系统所有的热力学量从而得到了有限温度时系统的第一声与第二声;不仅推广了以前这方面的工作到整个BCS-BEC转换,而且证明了第一声尤其是第二声光滑演化的结论。
(3)根据没有配对的粒子在动量空间中的分布,通过简单的理论分析将由单带Hubbard模型描述的不平衡超流费米气体中的breached-pair(BP)相分成了有限温度时能够稳定存在的两类:弱耦合极限下的超流-正常-超流相以及强耦合极限下的正常-超流相;依赖于进一步的数值研究,在这两类不同的BP相之间有可能发生经典相变。
(4)通过做一个规范变换和偶极近似,定性地推导了旋转Bose-Hubbard模型,并且通过研究正常态不动点的稳定性推广了旋转Bose-Hubbard模型的零温相图到有限温度的情形;另外根据系统有效的能带宽度与旋转速度的关系(Hofstadter蝴蝶),得到了系统的临界跃迁矩阵随旋转速度的改变展示谐振行为的结论。
(5)在Bogoliubov近似和前向散射近似的条件下,推导了BEC系统中单个杂质问题的等价哈密顿,并且采用Lee-Low-Pines(LLP)方法计算了杂质的有效质量、携带的虚声子数以及杂质的能量;鉴于Landau超流准则,也仔细分析了前向散射近似的合理性。
From the microscopic level,bosons and fermions possess completely different quantum statistical characters:in terms of bosons,quantum statistical effect is associated with the phenomenon of Bose-Einstein-Condensation(BEC),while for noninteracting fermionic system,its evolution is the smooth crossover between classical and quantum behavior as the temperature is dropping;Contrary to the case that repulsive interaction stabilizes BEC,the occurrence of a superfluid phase in a fermionic gas can only be due to the presence of attractive interaction.In order to look into these behaviors caused by different quantum statistics,we must get so low temperature that bosonic or fermionic system becomes degenerate.Fortunately in many kinds of ultracold atomic gas,irrespective of their constituents,we have successfully observed BEC and Bardeen-Cooper-Schrieffer (BCS) superfiuidity.What is more important is that fermionic system evolves continuously from BCS superfiuidity to BEC when attractive interaction between fermions is tuned from weak to strong coupling by Feshbach resonance(This is the so-called BCS-BEC crossover).Hence it is possible to probe into two different kinds of quantum statistical effect in a single fermionic system.
In this doctoral thesis,we firstly review some fundamental theories about BEC、BCS superconductor,BCS-BEC crossover,Feshbach resonance and corresponding experimental evidences,then we mainly discuss following questions in ultracold atomic gases:
(1) The collective modes in a uniform fermionic atomic gas with Feshbach resonance are investigated with coherent state path integral method in the frame of boson-fermion model.We mainly concentrate on the long-wavelength and low-frequency limits at T = OK and get an analytical expression for the collective modes across the whole BCS-BEC crossover.We completely recover the Anderson-Bogoliubov modes in the BCS limit and the Bogoliubov modes of the bosonic systems in the BEC limit.The numerical results show that there exists a continuous interpolation for sound velocity between BCS and BEC limits.In this way we can also solve the equation of state of the system,which makes an explicit difference from the hydrodynamics.
(2) First and second sound modes in a uniform fermionic atom gas with Feshbach resonance are investigated in the frame of two-fluid model with a microscopic calculation of thermodynamical variables at finite temperature.The analytical results for thermodynamic quantities and sound velocities in BCS and BEC limits are obtained.The numerical results show that there exists a continuous interpolation for sound velocities of first and second sound modes between BCS and BEC limits,so we testify the validity of extrapolation for sound velocities made in some early work.
(3) Based on single-band approximation for Hubbard model and only by analyzing some physical constraints,breached-pair(BP) phase in imbalance fermionic atom gases placed in a three-dimensional optical lattice is studied theoretically. According to the distribution of unpaired particles in the first Brillouin zone, the breached-pair phase is classified into two classes(Ⅰ) normal-superfluid phase and(Ⅱ) superfluid-normal-superfluid phase which are stable in strong and weak coupling limits respectively at nonzero temperature.Moreover it is possible to predict a classical phase transition between these two different BP phase although further determination still depends on numerical solution.
(4) Taking advantage of gauge transformation and dipolar approximation, we qualitatively derive rotating Bose-Hubbard model.Then finite temperature phase boundary between superfluid phase and normal state in rotating Bose-Hubbard model is analytically derived by studying the stability of normal state in rotating bosonic optical lattice.We also analytically prove that as the function of rotation angular velocity the oscillation of critical hopping matrix directly follows the upper boundary of Hofstadter butterfly.
(5) Using Bogoliubov theory and forward-scattering approximation,we derive a reduced Hamiltonian describing an impurity in BEC.By using Lee-Low-Pines (LLP) theory,we obtain the effective mass of impurity,the phonon number carried by impurity and the energy related to the existence of impurity.Moreover according to Landau superfluidity criterion,we also analyze the validity of forward-scattering approximation.
这篇博士论文除了简单地介绍超冷原子气体的一些基本理论(包括BEC、BCS超导、BCS-BEC转换和Feshbach共振等)以及相应的实验验证之外,主要包含了如下五个方面的内容:
(1)在玻色-费米模型的框架下利用相干态路径积分的方法推导了系统的单圈有效作用量,在此基础上计算了BCS-BEC转换中系统的零温长波低频集体模式,得到了集体模式声速的一个解析表达式以及声速在BCS-BEC转换中光滑演化的结论;这种计算方法能够自洽地给出系统的态方程,因而明显不同于通常的流体动力学方法。
(2)在两流体模型和有限温度BCS-BEC转换理论的框架下,计算了系统所有的热力学量从而得到了有限温度时系统的第一声与第二声;不仅推广了以前这方面的工作到整个BCS-BEC转换,而且证明了第一声尤其是第二声光滑演化的结论。
(3)根据没有配对的粒子在动量空间中的分布,通过简单的理论分析将由单带Hubbard模型描述的不平衡超流费米气体中的breached-pair(BP)相分成了有限温度时能够稳定存在的两类:弱耦合极限下的超流-正常-超流相以及强耦合极限下的正常-超流相;依赖于进一步的数值研究,在这两类不同的BP相之间有可能发生经典相变。
(4)通过做一个规范变换和偶极近似,定性地推导了旋转Bose-Hubbard模型,并且通过研究正常态不动点的稳定性推广了旋转Bose-Hubbard模型的零温相图到有限温度的情形;另外根据系统有效的能带宽度与旋转速度的关系(Hofstadter蝴蝶),得到了系统的临界跃迁矩阵随旋转速度的改变展示谐振行为的结论。
(5)在Bogoliubov近似和前向散射近似的条件下,推导了BEC系统中单个杂质问题的等价哈密顿,并且采用Lee-Low-Pines(LLP)方法计算了杂质的有效质量、携带的虚声子数以及杂质的能量;鉴于Landau超流准则,也仔细分析了前向散射近似的合理性。
From the microscopic level,bosons and fermions possess completely different quantum statistical characters:in terms of bosons,quantum statistical effect is associated with the phenomenon of Bose-Einstein-Condensation(BEC),while for noninteracting fermionic system,its evolution is the smooth crossover between classical and quantum behavior as the temperature is dropping;Contrary to the case that repulsive interaction stabilizes BEC,the occurrence of a superfluid phase in a fermionic gas can only be due to the presence of attractive interaction.In order to look into these behaviors caused by different quantum statistics,we must get so low temperature that bosonic or fermionic system becomes degenerate.Fortunately in many kinds of ultracold atomic gas,irrespective of their constituents,we have successfully observed BEC and Bardeen-Cooper-Schrieffer (BCS) superfiuidity.What is more important is that fermionic system evolves continuously from BCS superfiuidity to BEC when attractive interaction between fermions is tuned from weak to strong coupling by Feshbach resonance(This is the so-called BCS-BEC crossover).Hence it is possible to probe into two different kinds of quantum statistical effect in a single fermionic system.
In this doctoral thesis,we firstly review some fundamental theories about BEC、BCS superconductor,BCS-BEC crossover,Feshbach resonance and corresponding experimental evidences,then we mainly discuss following questions in ultracold atomic gases:
(1) The collective modes in a uniform fermionic atomic gas with Feshbach resonance are investigated with coherent state path integral method in the frame of boson-fermion model.We mainly concentrate on the long-wavelength and low-frequency limits at T = OK and get an analytical expression for the collective modes across the whole BCS-BEC crossover.We completely recover the Anderson-Bogoliubov modes in the BCS limit and the Bogoliubov modes of the bosonic systems in the BEC limit.The numerical results show that there exists a continuous interpolation for sound velocity between BCS and BEC limits.In this way we can also solve the equation of state of the system,which makes an explicit difference from the hydrodynamics.
(2) First and second sound modes in a uniform fermionic atom gas with Feshbach resonance are investigated in the frame of two-fluid model with a microscopic calculation of thermodynamical variables at finite temperature.The analytical results for thermodynamic quantities and sound velocities in BCS and BEC limits are obtained.The numerical results show that there exists a continuous interpolation for sound velocities of first and second sound modes between BCS and BEC limits,so we testify the validity of extrapolation for sound velocities made in some early work.
(3) Based on single-band approximation for Hubbard model and only by analyzing some physical constraints,breached-pair(BP) phase in imbalance fermionic atom gases placed in a three-dimensional optical lattice is studied theoretically. According to the distribution of unpaired particles in the first Brillouin zone, the breached-pair phase is classified into two classes(Ⅰ) normal-superfluid phase and(Ⅱ) superfluid-normal-superfluid phase which are stable in strong and weak coupling limits respectively at nonzero temperature.Moreover it is possible to predict a classical phase transition between these two different BP phase although further determination still depends on numerical solution.
(4) Taking advantage of gauge transformation and dipolar approximation, we qualitatively derive rotating Bose-Hubbard model.Then finite temperature phase boundary between superfluid phase and normal state in rotating Bose-Hubbard model is analytically derived by studying the stability of normal state in rotating bosonic optical lattice.We also analytically prove that as the function of rotation angular velocity the oscillation of critical hopping matrix directly follows the upper boundary of Hofstadter butterfly.
(5) Using Bogoliubov theory and forward-scattering approximation,we derive a reduced Hamiltonian describing an impurity in BEC.By using Lee-Low-Pines (LLP) theory,we obtain the effective mass of impurity,the phonon number carried by impurity and the energy related to the existence of impurity.Moreover according to Landau superfluidity criterion,we also analyze the validity of forward-scattering approximation.
引文
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