An efficient preconditioned conjugate-gradient solver for a two-component dipolar Bose-Einstein condensate

TL;DR

This paper introduces a highly efficient preconditioned conjugate-gradient method for computing ground states of binary dipolar Bose-Einstein condensates, significantly improving convergence speed and robustness over traditional methods.

Contribution

The authors develop a novel preconditioned nonlinear conjugate-gradient solver that enforces energy descent and demonstrates superior performance in simulating dipolar BECs across various regimes.

Findings

Reduces iteration counts by 10-100 times compared to imaginary-time evolution.

Achieves robust convergence in droplet, stripe, and supersolid phases.

Attains slightly lower energies, indicating better metastability handling.

Abstract

We develop a preconditioned nonlinear conjugate-gradient solver for ground states of binary dipolar Bose-Einstein condensates within the extended Gross-Pitaevskii equation including Lee-Huang-Yang corrections. The optimization is carried out on the product-of-spheres normalization manifold and combines a manifold-preserving analytic line search, derived from a second-order energy expansion and validated along the exact normalized path, with complementary Fourier-space kinetic and real-space diagonal (Hessian-inspired) preconditioners. The method enforces monotonic energy descent and exhibits robust convergence across droplet, stripe, and supersolid regimes while retaining spectrally accurate discretizations and FFT-based evaluation of the dipolar term. In head-to-head benchmarks against imaginary-time evolution on matched grids and tolerances, the solver reduces iteration counts by one to two orders of magnitude and overall time-to-solution, and it typically attains slightly lower energies, indicating improved resilience to metastability. We reproduce representative textures and droplet-stability windows reported for dipolar mixtures. These results establish a reliable and efficient tool for large-scale parameter scans and phase-boundary mapping, and for quantitatively linking numerically obtained metastable branches to experimentally accessible states.