论文标题
非线性时间依赖性schrödinger方程的时间可逆和稳定的高阶集成符：局部控制理论的应用
Time-reversible and norm-conserving high-order integrators for the nonlinear time-dependent Schrödinger equation: Application to local control theory
论文作者
论文摘要
显式拆分操作算法已广泛用于求解线性，还用于求解非线性时间依赖性的schrödinger方程。当应用于非线性Gross-Pitaevskii方程式时，该方法保持时间可逆，稳定并保留其在时间步长的二阶精度。但是，该算法不适用于所有类型的非线性Schrödinger方程。确实，我们证明了局部控制理论是一种分子状态量子控制的技术，它转化为具有更一般的非线性的非线性schrödinger方程，因此，显式分裂操作算法会失去时间的可逆性和效率（因为它仅具有第一阶准确度）。同样，梯形规则（曲柄 - 尼科尔森方法）虽然时间可逆，但并不能保存非线性schrödinger方程传播的状态的规范。为了克服这些问题，我们提出了适用于一般时间依赖的非线性schrödinger方程的高阶几何积分器，也适用于不可分割的哈密顿人。这些集成剂基于隐式中点方法的对称组成，既是规范的且具有时间可逆的。积分器的几何特性在分析中得到证实，并在视网膜二维模型的局部控制上进行了数值证明。对于高度准确的计算，高阶积分器更有效。例如，对于$ 10^{ - 9} $的波函数错误，使用八阶算法在二阶隐式中点方法和梯形规则上产生$ 48 $倍的加速，以及$ 400000 $ fold-fold-fold-fold-fold速度比显式分配器算法algorithm。
The explicit split-operator algorithm has been extensively used for solving not only linear but also nonlinear time-dependent Schrödinger equations. When applied to the nonlinear Gross-Pitaevskii equation, the method remains time-reversible, norm-conserving, and retains its second-order accuracy in the time step. However, this algorithm is not suitable for all types of nonlinear Schrödinger equations. Indeed, we demonstrate that local control theory, a technique for the quantum control of a molecular state, translates into a nonlinear Schrödinger equation with a more general nonlinearity, for which the explicit split-operator algorithm loses time reversibility and efficiency (because it has only first-order accuracy). Similarly, the trapezoidal rule (the Crank-Nicolson method), while time-reversible, does not conserve the norm of the state propagated by a nonlinear Schrödinger equation. To overcome these issues, we present high-order geometric integrators suitable for general time-dependent nonlinear Schrödinger equations and also applicable to nonseparable Hamiltonians. These integrators, based on the symmetric compositions of the implicit midpoint method, are both norm-conserving and time-reversible. The geometric properties of the integrators are proven analytically and demonstrated numerically on the local control of a two-dimensional model of retinal. For highly accurate calculations, the higher-order integrators are more efficient. For example, for a wavefunction error of $10^{-9}$, using the eighth-order algorithm yields a $48$-fold speedup over the second-order implicit midpoint method and trapezoidal rule, and $400000$-fold speedup over the explicit split-operator algorithm.
