Time-optimal control fields for quantum systems with multiple avoided crossings

TL;DR

This paper develops a method to find time-optimal control fields for quantum systems with multiple avoided crossings, revealing simple control shapes and analytical solutions that scale with system size.

Contribution

It introduces a robust, spectrum-based guess for optimal control in multi-crossing quantum systems and derives simple, analytical control fields at the quantum speed limit.

Findings

Control fields are simple and parameterized.

Optimal control fields scale with system size.

Analytical solutions for the full evolution are obtained.

Abstract

We study time-optimal protocols for controlling quantum systems which show several avoided level crossings in their energy spectrum. The structure of the spectrum allows us to generate a robust guess which is time-optimal at each crossing. We correct the field applying optimal control techniques in order to find the minimal evolution or quantum speed limit (QSL) time. We investigate its dependence as a function of the system parameters and show that it gets proportionally smaller to the well-known two-level case as the dimension of the system increases. Working at the QSL, we study the control fields derived from the optimization procedure, and show that they present a very simple shape, which can be described by a few parameters. Based on this result, we propose a simple expression for the control field, and show that the full time-evolution of the control problem can be analytically solved.