Ensemble qubit controllability with a single control via adiabatic and rotating wave approximations

TL;DR

This paper rigorously analyzes the combined use of adiabatic and rotating wave approximations for controlling quantum systems with a single control, demonstrating ensemble controllability and quantifying robustness and fidelity.

Contribution

It provides a theoretical justification for cascading approximations in quantum control and extends ensemble controllability results to single-control scenarios.

Findings

Quantifies the uncertainty interval for resonance frequency control.

Shows control of an ensemble of spin systems with a single real-valued control.

Provides high-order averaging techniques for fidelity and robustness analysis.

Abstract

In the physics literature it is common to see the rotating wave approximation andthe adiabatic approximation used "in cascade" to justify the use of chirped pulses for two-level quantum systems driven by one external field, in particular when the resonance frequency of the system is not known precisely. Both approximations need relatively long time and are essentially based on averaging theory of dynamical systems. Unfortunately, the two approximations cannot be done independently since, in a sense, the two time scales interact. The purpose of this paper is to study how the cascade of the two approximations can be justified and how large becomes the final time as the fidelity goes to one, while preserving the robustness of the adiabatic strategy. Our first result, based on high-order averaging techniques, gives a precise quantification of the uncertainty interval of the resonance frequency for which the population inversion works. As a byproduct of this result, we prove that it is possible to control an ensemble of spin systems by a single real-valued control, providing a non-trivial extension of a celebrated result of ensemble controllability with two controls by Khaneja and Li.