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

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.

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

“…The exact nonlinear evolution operator (4) does not conserve the inner product and, as a result, is not symplectic. 34 Furthermore, the energy is not conserved because the state dependence of the Hamiltonian makes it implicitly time-dependent. However, we demonstrated that the exact nonlinear evolution operator conserves the norm and is time-reversible.…”

“…However, we demonstrated that the exact nonlinear evolution operator conserves the norm and is time-reversible. 34…”

“…27,28 To solve this NL-TDSE with cubic nonlinearity, the explicit second-order splitoperator algorithm is frequently used [29][30][31][32] because it is efficient, and geometric. 33 However, we recently showed 34 that the success of the explicit split-operator algorithm in solving this NL-TDSE is due to its simple nonlinearity and, therefore, is rather an exception than a rule. Indeed, for other nonlinearities, the algorithm becomes time-irreversible and very inefficient because its accuracy decreases to the first order in the time step.…”

“…Indeed, for other nonlinearities, the algorithm becomes time-irreversible and very inefficient because its accuracy decreases to the first order in the time step. 34 As we have recently demonstrated 34 an example of such a NL-TDSE, on which the split-operator algorithm loses accuracy and time reversibility, is provided by local control theory [35][36][37][38][39][40][41][42][43][44] (LCT). LCT is a technique that aims at controlling the expectation value of a specified a) Electronic mail: julien.roulet@epfl.ch b) Electronic mail: jiri.vanicek@epfl.ch operator by computing, from the state, an electric field that will either increase or decrease the chosen expectation value.…”

“…To overcome the limitations of the explicit splitoperator algorithm applied to general NL-TDSEs, in our previous work we developed 34 high-order integrators by symmetrically composing the implicit midpoint method. These integrators are applicable to the general nonlinear Schrödinger equation with both separable and nonseparable Hamiltonians and, in contrast to the explicit split-operator algorithm, are efficient, while preserving the geometric properties of the exact solution.…”