Nonlinear Inverse Iterations for Spin-Orbit Coupled Quantum Gases

TL;DR

This paper develops a nonlinear inverse iteration method with spectral shifting to efficiently compute ground states of spin-orbit coupled Bose-Einstein condensates, overcoming slow convergence of traditional methods.

Contribution

It introduces a novel nonlinear inverse iteration scheme with spectral shifting for complex coupled eigenproblems in quantum gases, achieving faster convergence.

Findings

The method converges locally at a linear rate based on spectral gaps.

Adaptive shifts lead to superlinear convergence.

Numerical experiments confirm the efficiency of the proposed approach.

Abstract

This work concerns the computation of ground states of two-component spin-orbit coupled Bose-Einstein condensates (SO-coupled BECs), modelled by a coupled nonlinear eigenvalue problem of Gross-Pitaevskii type. Spin-orbit coupling gives rise to fascinating phenomena, including supersolid-like phases with spatially modulated densities. However, in such complex settings, conventional numerical approaches, such as generalized inverse iterations or gradient descent, often converge very slowly. To overcome this issue, we apply the concept of the J-method [E.~Jarlebring, S.~Kvaal, W.~Michiels. SIAM~J.~Sci.~Comput.~36-4,~2014] to construct a nonlinear inverse iteration scheme whose convergence can be accelerated through spectral shifting, analogous to techniques used for linear eigenproblems. For a fixed shift parameter, we establish local linear convergence rates determined by spectral gaps in the neighbourhood of each quasi-unique ground state. With adaptively chosen shifts, superlinear convergence is observed, which we verify through numerical experiments.