Fourth-order algorithms for solving the imaginary-time Gross-Pitaevskii equation in a rotating anisotropic trap

Area of Science:

• Quantum mechanics
• Computational physics
• Nonlinear dynamics

Background:

• The Gross-Pitaevskii equation describes Bose-Einstein condensates and other quantum systems.
• Accurate and efficient numerical methods are crucial for simulating these systems.
• Existing second-order algorithms often require small time steps, limiting computational speed.

Purpose of the Study:

• To develop novel, high-order numerical algorithms for the Gross-Pitaevskii equation.
• To improve the efficiency and accuracy of simulations for trapped quantum systems.
• To introduce a systematic approach for creating advanced factorization schemes.

Main Methods:

• Implementation of the exact density matrix for a rotating anisotropic harmonic trap.
• Derivation of fourth-order algorithms utilizing forward, positive time step factorization.
• Application of time-dependent factorization schemes for nonlinear equation solving.

Main Results:

• A class of very fast and accurate fourth-order algorithms for imaginary time evolution.
• Convergence at time-step sizes an order-of-magnitude larger than conventional second-order methods.
• Demonstration of a systematic approach for devising advanced numerical algorithms.

Conclusions:

• The derived fourth-order algorithms offer significant speed and accuracy improvements.
• The use of exact density matrix and specific factorization schemes enables higher-order accuracy.
• This methodology provides a robust framework for solving complex nonlinear equations in physics.