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

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.