论文标题
基于时间依赖的量子解决方案
Quantum-based solution of time-dependent complex Riccati equations
论文作者
论文摘要
使用Wei-norman理论，我们获得了时间相关的复杂riccati方程（TDCRE），作为时间相关（TD）汉密尔顿人所描述的量子系统的时间进化运算符（TEO）的解决方案，这些系统是$ \\\hrak {su}（1,1）$，$ \ Mathak and Mathak and Mathak and Mathak and Mathak and Mathak and Matherffrak and and Mathfrak {suffrak and} $ \ \ \ c}的线性组合$ \ mathfrak {so}（2,1）$ lie代数。在这些量子系统的时间演化中，我们将tdcre递归解决，作为概括性的持续分数，这对于数值实现是最佳的，并为TEO的单位性建立了必要和足够的条件，以分解为分解的表示。量子系统的遗传对称性可以通过对TDCRE的简单检查来识别，从而使有效的量子汉密尔顿人与之相关，因为我们为Bloch-Riccati方程式显示了Hamiltonian的hamiltonian与Lie Algebra $ $ \ Mathfrak {su}（su}（2）$的Bloch-Riccati方程相对应。作为一种应用，也是作为一致性测试，我们将解决方案与Bloch-Riccati方程的分析溶液进行了比较，考虑到由复杂的双曲线固定脉冲驱动的狂热频率产生自旋反转，显示出极好的一致性。
Using the Wei-Norman theory we obtain a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems described by time-dependent (TD) Hamiltonians that are linear combinations of the generators of the $\mathfrak{su}(1,1)$, $\mathfrak{su}(2)$ and $\mathfrak{so}(2,1)$ Lie algebras. Using a recently developed solution for the time evolution of these quantum systems we solve the TDCRE recursively as generalized continued fractions, which are optimal for numerical implementations, and establish the necessary and sufficient conditions for the unitarity of the TEO in the factorized representation. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE, allowing effective quantum Hamiltonians to be associated with it, as we show for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a generic TD system of the Lie algebra $\mathfrak{su}(2)$. As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation considering the Rabi frequency driven by a complex hyperbolic secant pulse generating spin inversion, showing an excellent agreement.
