Loading a Bose-Einstein Condensate onto an Optical Lattice: an Application of Optimal Control Theory to The Non Linear Schr\"odinger Equation

TL;DR

This paper extends Optimal Control Theory to the nonlinear Schrödinger equation and demonstrates its application in controlling phase development in a Bose-Einstein condensate during optical lattice loading.

Contribution

It introduces a formal framework and algorithm for applying OCT to the NLSE, enabling precise control of BEC dynamics under nonlinear conditions.

Findings

Successful formulation of OCT for NLSE

Controlled phase development in BEC using optimized trap adjustments

First rigorous application of OCT to nonlinear Schrödinger equation

Abstract

Using a set of general methods developed by Krotov [A. I. Konnov and V. A. Krotov, Automation and Remote Control, {\bf 60}, 1427 (1999)], we extend the capabilities of Optimal Control Theory to the Nonlinear Schr\"odinger Equation (NLSE). The paper begins with a general review of the Krotov approach to optimization. Although the linearized version of the method is sufficient for the linear Schr\"odinger equation, the full flexibility of the general method is required for treatment of the nonlinear Schr\"odinger equation. Formal equations for the optimization of the NLSE, as well as a concrete algorithm are presented. As an illustration, we consider a Bose-Einstein condensate initially at rest in a harmonic trap. A phase develops across the BEC when an optical lattice potential is turned on. The goal is to counter this effect and keep the phase flat by adjusting the trap strength. The problem is formulated in the language of Optimal Control Theory (OCT) and solved using the above methodology. To our knowledge, this is the first rigorous application of OCT to the Nonlinear Schr\"odinger equation, a capability that is bound to have numerous other applications.