Time-reversible and norm-conserving high-order integrators for the nonlinear time-dependent Schr\"{o}dinger equation: Application to local control theory

• URL: http://arxiv.org/abs/2006.16902v3
• Date: Tue, 6 Apr 2021 09:14:47 GMT
• Title: Time-reversible and norm-conserving high-order integrators for the nonlinear time-dependent Schr\"{o}dinger equation: Application to local control theory
• Authors: Julien Roulet, Ji\v{r}\'i Van\'i\v{c}ek
• Abstract summary: We present high-order geometric suitable for general time-dependent nonlinear Schr"odinger equations. These compositions, based on the symmetric midpoint implicit method, are both norm-conserving and time-reversible.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The explicit split-operator algorithm has been extensively used for solving not only linear but also nonlinear time-dependent Schr\"{o}dinger 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 Schr\"{o}dinger equations. Indeed, we demonstrate that local control theory, a technique for the quantum control of a molecular state, translates into a nonlinear Schr\"{o}dinger 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 Schr\"{o}dinger equation. To overcome these issues, we present high-order geometric integrators suitable for general time-dependent nonlinear Schr\"{o}dinger 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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