非线性含时薛定谔方程的时间可逆与范数守恒高阶积分器：在局部控制理论中的应用
Time-reversible and norm-conserving high-order integrators for the nonlinear time-dependent Schrödinger equation: Application to local control theory
AI总结：
针对局部控制理论中一般非线性薛定谔方程现有算法失去时间可逆性或范数守恒的问题，提出了基于隐式中点法对称复合的时间可逆且范数守恒的高阶几何积分器，并在视黄醛模型中实现了显著的计算加速。
摘要：
显式分裂算符算法已被广泛用于求解线性和非线性含时薛定谔方程。应用于非线性 Gross-Pitaevskii 方程时，该方法保持时间可逆、范数守恒，且在时间步长上具有二阶精度。然而，该算法并不适用于所有类型的非线性薛定谔方程。事实上，我们证明了局部控制理论（一种用于分子态量子控制的技术）可转化为具有更一般非线性的非线性薛定谔方程，对于该方程，显式分裂算符算法失去了时间可逆性和效率（因为它仅有一阶精度）。类似地，梯形规则（即 Crank-Nicolson 方法）虽然时间可逆，却不能守恒由非线性薛定谔方程演化状态的范数。为克服这些问题，我们提出了适用于一般含时非线性薛定谔方程且适用于不可分哈密顿量的高阶几何积分器。这些基于隐式中点法对称复合的积分器，兼具范数守恒和时间可逆性。我们解析证明了这些积分器的几何性质，并在视黄醛二维模型的局部控制上进行了数值演示。对于高精度计算，高阶积分器效率更高。例如，对于 $10^{-9}$ 的波函数误差，使用八阶算法比二阶隐式中点法和梯形规则快 48 倍，比显式分裂算符算法快 400000 倍。
英文摘要：
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.
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