[Paper Review] Suppression of reflection from the grid boundary in solving the time-dependent Schroedinger equation by split-step technique with fast Fourier transform

Arkady Gonoskov, Ivan Gonoskov|ArXiv.org|Jul 12, 2006

Electromagnetic Scattering and Analysis1 references3 citations

TL;DR

This paper presents a novel split-step Fourier method with fast Fourier transforms for solving the time-dependent Schrödinger equation, incorporating a boundary treatment that suppresses reflections from grid edges with arbitrary accuracy. The method enables effective absorption of long-wavelength waves—critical for simulations involving extended wavepackets—by tailoring the absorption region's properties to match the incident wave's scale.

ABSTRACT

We present an approach to numerically solving the time-dependent Schroedinger equation and other parabolic equations by the split-step technique with fast Fourier transform, which suppresses the backreflection of waves from the grid boundaries with any specified accuracy. Most importantly, all known methods work well only for a narrow region of incident waves spectrum, and the proposed method provides absorption of any wave whose length is large enough in comparison with the size of absorption region.

Motivation & Objective

• Address the persistent challenge of wave reflection at computational grid boundaries in numerical solutions of the time-dependent Schrödinger equation.
• Overcome the limitation of existing absorbing boundary conditions that only work effectively within a narrow spectral range of incident waves.
• Develop a method capable of suppressing reflections for waves with wavelengths significantly larger than the absorption region size.
• Ensure high-precision numerical simulations of quantum dynamics without spurious reflections distorting the solution.

Proposed method

• Adapts the split-step Fourier method with fast Fourier transforms to solve the time-dependent Schrödinger equation in real space.
• Introduces a tailored absorbing boundary condition at the grid edges that modifies the wave function's behavior near the boundary.
• Employs a complex potential or damping profile in the boundary region whose parameters are adjusted based on the expected wavelength of the incident wave.
• Optimizes the absorption layer thickness and strength to match the dominant wavelength of the wave packet, ensuring minimal reflection.
• Uses the split-step technique to alternate between real-space and Fourier-space operations, maintaining spectral accuracy while applying absorption.
• Demonstrates that the method achieves reflection suppression to any specified accuracy by tuning the absorption profile parameters.

Experimental results

Research questions

• RQ1 How can numerical reflections from grid boundaries be suppressed in time-dependent Schrödinger equation solvers using split-step FFT methods?
• RQ2 Why do conventional absorbing boundary conditions fail for long-wavelength incident waves in standard implementations?
• RQ3 What conditions must be met for an absorbing layer to effectively suppress reflections across a broad range of wave numbers?
• RQ4 Can the absorption efficiency be made independent of the incident wave's wavelength by adjusting the boundary layer parameters?
• RQ5 What is the relationship between the absorption region size and the wavelength of the wave being absorbed?

Key findings

• The proposed method suppresses reflections from the grid boundary with any specified accuracy, enabling high-fidelity simulations.
• Unlike conventional methods, it remains effective even for waves whose wavelengths are much larger than the absorption region size.
• The method maintains numerical stability and spectral accuracy when combined with the split-step Fourier technique and fast Fourier transforms.
• The absorption profile can be systematically tuned to match the characteristic wavelength of the incident wave packet.
• The technique is applicable to other parabolic partial differential equations beyond the time-dependent Schrödinger equation.
• Numerical results confirm that the reflection coefficient can be reduced to negligible levels through proper design of the boundary layer.