变分法研究一维Bose-Fermi系统的稳定性

Variational-method analysis of stability of Bose-Fermi mixture

利用能量泛函变分法研究了一维Bose-Fermi系统稳定基态的存在条件.根据Bose-Fermi系统的Lagrange量可以得到三维Bose-Fermi体系所满足的非线性动力学方程组.当外势阱的横向囚禁频率远大于轴向囚禁频率时,体系可以当作一维模型来处理.从描述三维体系的动力学方程可以得到描述一维体系的动力学方程,选取适当的无量纲参数,可以对一维动力学方程组进行无量纲处理,得到数值计算和理论分析中常用到的无量纲方程.选择高斯型试探解(简单孤立子解),利用能量泛函变分法得到一维Bose-Fermi体系稳定的高斯型孤立子存在条件.分析了两种特殊情况下孤立子能够稳定存在的区域以及原子数的临界条件,最后得出了一般情况下稳定基态存在时临界散射长度与原子数以及波包宽度之间的关系.

In one-dimensional trapped Bose-Fermi mixture,described by time-dependent one-dimensional nonlinear equations that are derived from a three-dimensional Bose-Fermi system,we study the effect of atom-interactions on stability using a Gaussian Variational approach.We investigate the stable and the unstable conditions as functions of the atoms number and s-wave scattering length.We find that the interaction between the different species of atoms has significant effect on the stability of Bose-Fermi mixture.We also give critical conditions of the atoms number and s-wave scattering length for both special and general cases.