Time-reversible high-order integrators for the nonlinear time-dependent Schrödinger equation: Application to local control theory

Local control theory is a technique for controlling the evolution of a molecular state with an electric field, whose amplitude is computed, using the current molecular state, in order to increase (or decrease) the expectation value of a chosen operator. Because the electric field depends on the molecular state, the time-dependent Schr\"odinger equation becomes nonlinear, which is often ignored in related studies that use a na\"ive implementation of the split-operator algorithm. To capture the nonlinearity, we present here high-order time-reversible integrators for the general time-dependent Schr\"odinger equation. These integrators are based on the symmetric compositions of the implicit midpoint method and, therefore, are norm-preserving, symmetric, time-reversible, and unconditionally stable. In contrast to split-operator algorithms, the proposed algorithms are also applicable to Hamiltonians nonseparable into a position- and momentum-dependent terms. The geometric properties of the integrators are proven analytically and demonstrated numerically on the local control of a two-dimensional model of retinal. Efficiency analysis shows that, for highly accurate calculations, the higher-order integrators are more efficient. For example, for an error of $10^{-9}$, a 160000-fold speedup is observed when using the sixth-order method instead of the elementary explicit Euler method.