Revisiting Quantum Optimal Control Theory: New Insights for the Canonical Solutions

TL;DR

This paper revises Quantum Optimal Control Theory, revealing the discontinuous nature of extremal functions and introducing new continuous solutions, thereby enhancing the theoretical understanding and potential strategies in quantum control.

Contribution

It provides a rigorous mathematical analysis of QOCT, clarifies the nature of extremal solutions, and introduces new continuous solutions for the control equations.

Findings

Extremal functions are not continuous, with the costate vanishing after measurement time.

The driving field remains continuous in optimal solutions.

New continuous solutions to the QOCT equations are identified.

Abstract

In this study, we present a revision of the Quantum Optimal Control Theory (QOCT) originally proposed by Rabitz et al (Phys. Rev. A 37, 49504964 (1988)), which has broad applications in physical and chemical physics. First, we identify the QOCT equations as the Euler-Lagrange equations of the functional associated to the control scheme. In this framework we prove that the extremal functions found by Rabitz are not continuous, as it was claimed in previous works. Indeed, we show that the costate is discontinuous and vanishes after the measurement time. In contrast, we demonstrate that the driving field is continuous. We also identify a new set of continuous solutions to the QOCT. Overall, our work provides a significant contribution to the QOCT theory, promoting a better understanding of the mathematical solutions and offering potential new directions for optimal control strategies.