Quantum and electromagnetic propagation with the conjugate symmetric Lanczos method

Area of Science:

• Quantum Mechanics
• Computational Physics
• Numerical Analysis

Background:

• The time-dependent Schrodinger equation governs quantum systems.
• Efficient numerical methods are crucial for solving complex quantum dynamics problems.
• Existing methods like Chebyshev and Lanczos have limitations in computational cost and storage.

Purpose of the Study:

• Introduce the conjugate symmetric Lanczos (CSL) method for solving the time-dependent Schrodinger equation.
• Develop a simple, efficient, and accurate time-domain algorithm for quantum dynamics.
• Explore the applicability of the CSL method to various physical systems.

Main Methods:

• The CSL method utilizes a low-order polynomial expansion of the quantum propagator.
• It leverages the time-reversal symmetry of the Schrodinger equation.
• Forward solutions are obtained by complex conjugating backward expansion coefficients.

Main Results:

• The CSL method demonstrates efficiency comparable to or better than existing iterative methods.
• It requires fewer matrix-vector products than the Chebyshev method.
• The algorithm shows accuracy comparable to the short iterative Lanczos method with reduced storage.

Conclusions:

• The CSL method is a powerful and efficient tool for time-dependent quantum dynamics.
• It is applicable to various systems, including harmonic and anharmonic oscillators, and electromagnetic pulse propagation.
• Corrections to the CSL algorithm may be needed for non-Hermitian systems.