Noncommuting vector fields, polynomial approximations and control of inhomogeneous quantum ensembles

TL;DR

This paper explores how noncommuting vector fields and Lie algebra structures enable the design of pulse sequences that compensate for parameter dispersion in quantum systems, with applications in NMR spectroscopy.

Contribution

It explicitly links noncommutativity and Lie algebra properties to the development of control pulses for inhomogeneous quantum ensembles, advancing quantum control techniques.

Findings

Identifies the role of Lie algebras and non-commutativity in pulse design.

Analyzes three common dispersions in NMR: Larmor, rf-inhomogeneity, and coupling strength.

Provides insights into controlling inhomogeneous quantum systems.

Abstract

Finding control fields (pulse sequences) that can compensate for the dispersion in the parameters governing the evolution of a quantum system is an important problem in coherent spectroscopy and quantum information processing. The use of composite pulses for compensating dispersion in system dynamics is widely known and applied. In this paper, we make explicit the key aspects of the dynamics that makes such a compensation possible. We highlight the role of Lie algebras and non-commutativity in the design of a compensating pulse sequence. Finally we investigate three common dispersions in NMR spectroscopy, the Larmor dispersion, rf-inhomogeneity and strength of couplings between the spins.