Time minimal trajectories for two-level quantum systems with two bounded controls

TL;DR

This paper investigates the minimum time control problem for a two-level quantum system driven by two bounded external fields, using optimal control techniques without the Rotating Wave Approximation, and provides a detailed synthesis of optimal trajectories.

Contribution

It offers a characterization of candidate optimal trajectories for the time-minimal control problem on the Bloch sphere without relying on common approximations, including numerical illustrations.

Findings

Complete optimal synthesis is achieved near the initial point.

Numerical simulations illustrate properties of candidate trajectories.

Results apply when controls have equal bounds and are small relative to energy level differences.

Abstract

In this paper we consider the minimum time population transfer problem for a two level quantum system driven by {\em two} external fields with bounded amplitude. The controls are modeled as real functions and we do not use the Rotating Wave Approximation. After projection on the Bloch sphere, we tackle the time-optimal control problem with techniques of optimal synthesis on 2-D manifolds. Based on the Pontryagin Maximum Principle, we characterize a restricted set of candidate optimal trajectories. Properties on this set, crucial for complete optimal synthesis, are illustrated by numerical simulations. Furthermore, when the two controls have the same bound and this bound is small with respect to the difference of the two energy levels, we get a complete optimal synthesis up to a small neighborhood of the antipodal point of the starting point.