A Reduction-Based Strategy for Optimal Control of Bose-Einstein Condensates

TL;DR

This paper introduces a novel control approach for Bose-Einstein Condensates that reduces a complex PDE problem to a low-dimensional ODE system, enabling efficient optimization and excitation minimization.

Contribution

It presents a reduction-based control method combining Galerkin expansion, CRAB, and hybrid optimization to improve state preparation in BECs.

Findings

Significant reduction in excitations in the full GPE system.

Effective control strategy using reduced models and hybrid optimization.

Potential for more efficient BEC state preparation methods.

Abstract

Applications of Bose-Einstein Condensates (BEC) often require that the condensate be prepared in a specific complex state. Optimal control is a reliable framework to prepare such a state while avoiding undesirable excitations, and, when applied to the time-dependent Gross-Pitaevskii Equation (GPE) model of BEC in multiple space dimensions, results in a large computational problem. We propose a control method based on first reducing the problem, using a Galerkin expansion, from a PDE to a low-dimensional Hamiltonian ODE system. We then apply a two-stage hybrid control strategy. At the first stage, we approximate the control using a second Galerkin-like method known as CRAB to derive a finite-dimensional nonlinear programming problem, which we solve with a differential evolution (DE) algorithm. This search method then yields a candidate local minimum which we further refine using a variant of gradient descent. This hybrid strategy allows us to greatly reduce excitations both in the reduced model and the full GPE system.