We develop a method for finding the time evolution of exactly solvable models by Bethe ansatz, with Hilbert space linear in excitation number. The dynamical Bethe wavefunction takes the same form as the stationary Bethe wavefunction except for time varying Bethe parameters and a complex phase prefactor. From this, we derive a set of first order nonlinear coupled differential equations for the Bethe parameters, called the dynamical Bethe equations. We find that this gives the exact solution to particular types of exactly solvable models, including the Bose-Hubbard dimer. The developed formalism allows us to demonstrate analytically that time dynamics of the detuning-quenched Bose-Hubbard dimer occurs in the subspace which dimensionality is less than that of physical Hilbert space. This allows for calculating the time dynamics within a reduced problem dimensionality.

中文翻译：

---

Bethe ansatz可解模型的时间动态

我们开发了一种方法，用于通过Bethe ansatz查找激发数为Hilbert空间线性的完全可解模型的时间演化。动态Bethe波函数采用与固定Bethe波函数相同的形式，不同之处在于时变Bethe参数和复杂的相位预因子。由此，我们推导出了一组用于Bethe参数的一阶非线性耦合微分方程，称为动态Bethe方程。我们发现这为特定类型的可完全求解模型（包括Bose-Hubbard二聚体）提供了精确解决方案。发达的形式主义使我们能够分析地证明失谐淬灭后的Bose-Hubbard二聚体的时间动力学发生在子空间，该子空间的维数小于物理希尔伯特空间的维数。