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Personne# Julien Roulet

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Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applicatio

Algorithme génétique

Les algorithmes génétiques appartiennent à la famille des algorithmes évolutionnistes. Leur but est d'obtenir une solution approchée à un problème d'optimisation, lorsqu'il n'existe pas de méthode exa

Algorithme

thumb|Algorithme de découpe d'un polygone quelconque en triangles (triangulation).
Un algorithme est une suite finie et non ambiguë d'instructions et d’opérations permettant de résoudre une classe de

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Many physical and chemical reactions are driven by nonadiabatic processes, which imply the breakdown of the celebrated Born-Oppenheimer approximation. To understand these processes, experimentalists employ spectroscopic techniques. However, the obtained results are difficult to decipher, and accurate molecular quantum dynamics simulations are used to interpret the results.The second-order split-operator algorithm is one of the most popular numerical methods for simulating the nonadiabatic quantum dynamics because it is explicit, easy to implement, and it preserves many geometric properties of the exact solution. However, the second-order accuracy of this algorithm makes it unaffordable if very accurate results are needed, as tiny time steps are required. To remedy this lack of efficiency, we use composition methods to generate higher-order split-operator algorithms.Although compositions methods increase the accuracy of the standard split-operator algorithm to arbitrary even orders of convergence, the efficiency of the obtained algorithms is still questioned because the computational cost per time step increases drastically with the order of convergence. Therefore, using one- and three-dimensional models of NaI and pyrazine, respectively, we investigate the convergence, efficiency, and geometric properties of these high-order integrators and find that they are, for accurate simulations, more efficient than the standard split-operator algorithm while still preserving the same geometric properties. Besides employing these integrators for simulating the nonadiabatic quantum dynamics, we also explore quantum control and, more specifically, local control theory. This technique uses the instantaneous dynamics of the system to compute an electric field, which interacts with the system in order to drive the state in a desired direction. Because the electric field is obtained from the state itself, we demonstrate that this technique translates into a nonlinear time-dependent Schrödinger equation. Although it is geometric and second-order accurate for simple nonlinearities, the standard split-operator algorithm loses its time-reversal symmetry and second-order accuracy when employed for more complicated nonlinear time-dependent Schrödinger equations. One example of the latter is the one appearing in local control theory.We demonstrate that this lack of generality of the standard split-operator algorithm occurs due to its explicit nature. Thus, we propose two strategies to overcome this issue: First, we completely abandon the split-operator algorithm and present a numerical method based on the implicit midpoint method instead. Second, we make the standard split-operator algorithm implicit, which avoids abandoning the split-operator algorithm altogether. The accuracy and geometric properties of both strategies are then numerically verified using a two-dimensional model of retinal, a molecule whose photochemistry triggers the first event in the biological process of vision. The results demonstrate that both approaches yield second-order methods that preserve all the geometric properties of the exact solution. Because the developed integrators are symmetric, we further improve their accuracy and efficiency using composition methods.

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.

2021The 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.

2021