Dark soliton solutions in the one-dimensional classical nonlinear Schrödinger equation have been considered to be related to the yrast states corresponding to the type-II excitations in the Lieb–Liniger model. However, the relation is nontrivial and remains unclear because a dark soliton localized in space breaks the translation symmetry, while yrast states are translationally invariant. In this work, we construct a symmetry-broken quantum soliton state and investigate the relation to the yrast states. By interpreting a quantum dark soliton as a Bose–Einstein condensation to the wave function of a classical dark soliton, we find that the quantum soliton state has a large weight only on the yrast states, which is analytically proved in the free-boson limit and numerically verified in the weak-coupling regime. By extending these results, we derive a parameter-free expression of a quantum soliton state that is written as a superposition of yrast states with Gaussian weights. The de...

中文翻译：

---

一维玻色气体中量子暗孤子的构造

一维经典非线性Schrödinger方程中的暗孤子解被认为与对应于Lieb–Liniger模型中II型激发的激态有关。但是，这种关系是不平凡的，并且仍然不清楚，因为位于空间中的暗孤子破坏了平移对称性，而仓促状态平移不变。在这项工作中，我们构造了一个对称破坏的量子孤子态，并研究了与yrast态的关系。通过将量子暗孤子解释为经典暗孤子的波函数的玻色-爱因斯坦凝聚，我们发现量子孤子态仅在厄拉斯特态具有较大权重，这在自由光子极限和在弱耦合系统中进行了数值验证。通过扩展这些结果，我们导出了量子孤子状态的无参数表达式，该表达式被写为具有高斯权重的拉斯特状态的叠加。的...