Exploring the top and bottom of the quantum control landscape

TL;DR

This paper investigates the control landscape of finite quantum systems, extending control parameters to include Hamiltonian structure, and presents algorithms to explore control level sets at landscape extrema, with applications to robust and rapid control solutions.

Contribution

It introduces a novel approach to control landscapes by including Hamiltonian structure as controls and develops algorithms to explore level sets at landscape extrema.

Findings

Existence of level sets at landscape bottom and top

Algorithms for exploring control solutions at landscape extrema

Numerical demonstrations on simple quantum systems

Abstract

A controlled quantum system possesses a search landscape defined by the target physical objective as a function of the controls. This paper focuses on the landscape for the transition probability Pif between the states of a finite level quantum system. Traditionally, the controls are applied fields; here we extend the notion of control to also include the Hamiltonian structure, in the form of time independent matrix elements. Level sets of controls that produce the same transition probability value are shown to exist at the bottom Pif = 0.0 and top Pif = 1.0 of the landscape with the field and/or Hamiltonian structure as controls. We present an algorithm to continuously explore these level sets starting from an initial point residing at either extreme value of Pif . The technique can also identify control solutions that exhibit the desirable properties of (a) robustness at the top and (b) the ability to rapidly rise towards an optimal control from the bottom. Numerical simulations are presented to illustrate the varied control behavior at the top and bottom of the landscape for several simple model systems.