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

Julien Roulet1, Jiří Vaníček1

  • 1Laboratory of Theoretical Physical Chemistry, Institut des Sciences et Ingénieries Chimiques, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland.

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|December 2, 2021
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Summary
This summary is machine-generated.

New implicit split-operator algorithms overcome limitations in solving nonlinear Schrödinger equations. These novel methods offer improved efficiency, accuracy, and time reversibility for complex quantum dynamics simulations.

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Area of Science:

  • Quantum mechanics
  • Computational physics
  • Numerical analysis

Background:

  • The explicit split-operator algorithm is widely used for solving time-dependent Schrödinger equations.
  • This explicit method suffers from loss of time reversibility and second-order accuracy for certain nonlinear problems, reducing efficiency.
  • Existing numerical methods may not adequately preserve crucial physical properties like norm conservation.

Purpose of the Study:

  • To develop novel numerical algorithms for solving nonlinear time-dependent Schrödinger equations.
  • To address the inefficiencies and loss of accuracy associated with the explicit split-operator method.
  • To introduce implicit integrators that maintain desirable geometric properties.

Main Methods:

  • Development of a family of high-order implicit split-operator algorithms.
  • Analytical proofs of the geometric properties (norm-conserving, time-reversible) of the new integrators.
  • Numerical demonstration of the algorithms' performance on a two-dimensional model of retinal control.

Main Results:

  • The proposed implicit split-operator algorithms are norm-conserving and time-reversible.
  • These algorithms exhibit high-order accuracy and significantly improved efficiency compared to explicit methods.
  • Numerical simulations confirm the analytical findings and demonstrate practical applicability.

Conclusions:

  • Implicit split-operator algorithms offer a superior alternative to explicit methods for specific nonlinear Schrödinger equations.
  • These new integrators provide a robust and efficient approach for simulating quantum systems with separable Hamiltonians.
  • The developed methods enhance the simulation of complex quantum dynamics, such as molecular control.