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

High-order geometric integrators for the local cubic variational Gaussian wavepacket dynamics

Gaussian wavepacket dynamics has proven to be a useful semiclassical approximation for quantum simulations of high-dimensional systems with low anharmonicity. Compared to Heller's original local harmonic method, the variational Gaussian wavepacket dynamics is more accurate, but much more difficult to apply in practice because it requires evaluating the expectation values of the potential energy, gradient, and Hessian. If the variational approach is applied to the local cubic approximation of the potential, these expectation values can be evaluated analytically, but they still require the costly third derivative of the potential. To reduce the cost of the resulting local cubic variational Gaussian wavepacket dynamics, we describe efficient high-order geometric integrators, which are symplectic, time-reversible, and norm-conserving. For small time steps, they also conserve the effective energy. We demonstrate the efficiency and geometric properties of these integrators numerically on a multidimensional, nonseparable coupled Morse potential.

An implicit split-operator algorithm for the nonlinear time-dependent Schrödinger equation

The explicit split-operator algorithm is often used for solving the linear and nonlinear time-dependent Schrödinger equations. However, when applied to certain nonlinear time-dependent Schrödinger equations, this algorithm loses time reversibility and second-order accuracy, which makes it very inefficient. Here, we propose to overcome the limitations of the explicit split-operator algorithm by abandoning its explicit nature. We describe a family of high-order implicit split-operator algorithms that are norm-conserving, time-reversible, and very efficient. The geometric properties of the integrators are proven analytically and demonstrated numerically on the local control of a two-dimensional model of retinal. Although they are only applicable to separable Hamiltonians, the implicit split-operator algorithms are, in this setting, more efficient than the recently proposed integrators based on the implicit midpoint method.