Abstract
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 only has 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 a 400 000-fold speedup over the explicit split-operator algorithm.
Highlights
Nonlinear time-dependent Schrödinger equations contain, by definition, Hamiltonians that depend on the quantum state
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
We show the derivation of the approximate explicit split-operator algorithm for the nonlinear Schrödinger equation, explain how it loses time reversibility, and briefly describe the dynamic Fourier method
Summary
Nonlinear time-dependent Schrödinger equations contain, by definition, Hamiltonians that depend on the quantum state. The explicit second-order split-operator algorithm, commonly used for the linear timedependent Schrödinger equation, is a great alternative, as it conserves, in some cases, the geometric properties of the exact solution and has spectral accuracy in space. This algorithm cannot be used for all types of nonlinear time-dependent Schrödinger equations. Would be very inefficient for highly accurate calculations and could not be used at all if exact time reversibility were important Because this failure of the explicit splitting algorithm in LCT is generic, while its success in the Gross–Pitaevskii equation is rather an exception, it is desirable to develop efficient high-order geometric integrators suitable for a general nonlinear time-dependent Schrödinger equation. IV, we numerically verify the convergence and geometric properties of the integrators by controlling, using LCT, either the population or energy transfer in a two-state two-dimensional model of retinal.
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