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

The explicit split-operator algorithm (ESOA) has been extensively used for solving linear and nonlinear time-dependent Schrödinger equations (NLTDSEs).¹ When applied to the Gross-Pitaevskii equation, the method remains time-reversible, norm-preserving, and retains its second-order accuracy in the time step.² However, this algorithm is not suitable for all types of NLTDSE. Indeed, we demonstrate that local control theory, a technique for the quantum control of a molecular state, translates into a NLTDSE with a more general nonlinearity, for which the ESOA loses time reversibility and has only first-order accuracy, becoming very inefficient. To overcome these issues, we present high-order integrators for general NLTDSEs which preserve the geometric properties³ of the exact flow and are more efficient than the ESOA.

[1] J. Comp. Phys. 47, 412 (1982)
[2] Comp. Phys. Comm. 184, 2621 (2013)
[3] E. Hairer et al., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer, 2006)