一维相互作用量子气体的基态和热力学性质
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摘要
超冷原子实验在以下几个方面取得了突破性的进展:依据Feshbach共振原理实现了对粒子间相互作用强度的调控,在磁光阱中和原子芯片上实现了准一维玻色爱因斯坦凝聚,并且制备出由玻色费米混合物组成的量子气体。这些成果使得超冷原子气体成为研究一维强关联系统各种量子多体效应的理想平台。本文简要介绍了一维量子系统的基本理论模型,包括平均场理论,Tonks-Girardeau气体,Bethe ansatz方法,重点研究了双势阱中玻色气体的基态性质和有限温度下准一维玻色费米混合物的热力学性质。
首先,我们研究束缚在中间带有δ势垒的无限深方势阱中的玻色气体的基态性质。利用玻色子与费米子的对应关系构建了Tonks-Girardeau气体的基态波函数,研究势垒两边的关联特性以及这种特性和粒子数目宇称的关系;并且运用精确对角化方法计算在任意势垒高度和原子间相互作用强度下系统基态的密度分布,占据数分布,动量分布等。计算表明动量次峰峰值随着势垒的升高而增大,随着原子间相互作用强度的增大而减小。
其次,基于热力学Bethe ansatz方法研究低温下由玻色费米混合物组成的准一维量子气体在简谐势阱中的热力学性质。运用热力学Yang-Yang方程和局域密度近似,数值计算玻色子和费米子的空间密度分布,重点讨论了费米子数目、粒子间相互作用强度对玻色子密度分布的影响。结果表明玻色费米相分离需要非常低的温度和很强的粒子间相互作用,为相关的实验提供比较依据。
Recent years have witnessed great development of laser cooling and optical trapping technology,with remarkable achievements in several aspects, e.g.manipulation of the interaction strength between particles according to Feshbach resonance technique,the Bose-Einstein Condensation of quasi-one-dimensional Bose gas trapped in magnetic and optical potential and atom chip,strongly interacting Tonks-Girardeau gases,and the Bose-Fermi mixture which rarely occur in nature.These progresses make ultra-cold atoms a popularly investigated platform for various many-body effects in strongly correlated one-dimensional systems.This dissertation will introduce several theories for one-dimensional systems,including the Mean-Field theory,the Tonks-Girardeau gases,and the exact solution with Bethe ansatz.We then devote to theoretical investigation of the ground state properties of an interacting few-boson system in a splited hard-wall potential and the thermodynamics of quasi-one-dimensional Bose-Fermi mixture at finite temperature.
First of all,we carry out a detailed examination of the ground-state properties of a few-boson system in a one-dimensional hard-wall potential with aδsplit in the center.In the Tonks-Girardeau limit with infinite repulsion between particles,we use the Bose-Fermi mapping to construct the exact N-particle ground-state wave function,which allows us to study the correlation properties accurately.For the general case with finite interparticle interaction,the exact diagonalization method is exploited to study the ground-state density distribution,occupation number distribution,and momentum distribution for variable interaction strengths and barrier heights. The secondary peaks in the momentum distribution reveal the interference between particles on the two sides of the split,which is more prominent for large barrier strength and small interaction strength.
Secondly,we investigate theoretically the behavior of one-dimensional interacting Bose-Fermi mixture at finite temperature in the scheme of thermodynamic Bethe ansatz.Combining the Yang-Yang thermodynamic formalism with local density approximation in a harmonic trap,we calculate the density distribution of bosons,which is mainly influenced by the number of Fermions and interacting strength between particles.It is shown that Bose-Fermi phase separation needs even lower temperature and very large interacting strength between particles,which will provide comparative materials for relative experiments.
首先,我们研究束缚在中间带有δ势垒的无限深方势阱中的玻色气体的基态性质。利用玻色子与费米子的对应关系构建了Tonks-Girardeau气体的基态波函数,研究势垒两边的关联特性以及这种特性和粒子数目宇称的关系;并且运用精确对角化方法计算在任意势垒高度和原子间相互作用强度下系统基态的密度分布,占据数分布,动量分布等。计算表明动量次峰峰值随着势垒的升高而增大,随着原子间相互作用强度的增大而减小。
其次,基于热力学Bethe ansatz方法研究低温下由玻色费米混合物组成的准一维量子气体在简谐势阱中的热力学性质。运用热力学Yang-Yang方程和局域密度近似,数值计算玻色子和费米子的空间密度分布,重点讨论了费米子数目、粒子间相互作用强度对玻色子密度分布的影响。结果表明玻色费米相分离需要非常低的温度和很强的粒子间相互作用,为相关的实验提供比较依据。
Recent years have witnessed great development of laser cooling and optical trapping technology,with remarkable achievements in several aspects, e.g.manipulation of the interaction strength between particles according to Feshbach resonance technique,the Bose-Einstein Condensation of quasi-one-dimensional Bose gas trapped in magnetic and optical potential and atom chip,strongly interacting Tonks-Girardeau gases,and the Bose-Fermi mixture which rarely occur in nature.These progresses make ultra-cold atoms a popularly investigated platform for various many-body effects in strongly correlated one-dimensional systems.This dissertation will introduce several theories for one-dimensional systems,including the Mean-Field theory,the Tonks-Girardeau gases,and the exact solution with Bethe ansatz.We then devote to theoretical investigation of the ground state properties of an interacting few-boson system in a splited hard-wall potential and the thermodynamics of quasi-one-dimensional Bose-Fermi mixture at finite temperature.
First of all,we carry out a detailed examination of the ground-state properties of a few-boson system in a one-dimensional hard-wall potential with aδsplit in the center.In the Tonks-Girardeau limit with infinite repulsion between particles,we use the Bose-Fermi mapping to construct the exact N-particle ground-state wave function,which allows us to study the correlation properties accurately.For the general case with finite interparticle interaction,the exact diagonalization method is exploited to study the ground-state density distribution,occupation number distribution,and momentum distribution for variable interaction strengths and barrier heights. The secondary peaks in the momentum distribution reveal the interference between particles on the two sides of the split,which is more prominent for large barrier strength and small interaction strength.
Secondly,we investigate theoretically the behavior of one-dimensional interacting Bose-Fermi mixture at finite temperature in the scheme of thermodynamic Bethe ansatz.Combining the Yang-Yang thermodynamic formalism with local density approximation in a harmonic trap,we calculate the density distribution of bosons,which is mainly influenced by the number of Fermions and interacting strength between particles.It is shown that Bose-Fermi phase separation needs even lower temperature and very large interacting strength between particles,which will provide comparative materials for relative experiments.
引文
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