Auxiliary matrix formalism for interaction representation transformations, optimal control and spin relaxation theories

TL;DR

This paper introduces an auxiliary matrix exponential method that simplifies and enhances the efficiency of various complex theoretical calculations in spin dynamics, including average Hamiltonian theory, relaxation, and quantum control.

Contribution

It presents a unified auxiliary matrix formalism that improves computational efficiency and simplicity for multiple interaction representation transformations and related theories.

Findings

More efficient than matrix factorization methods in spin dynamics.

Exhibits better complexity scaling with Hamiltonian dimension.

Simplifies calculations in average Hamiltonian, relaxation, and quantum control theories.

Abstract

Auxiliary matrix exponential method is used to derive simple and numerically efficient general expressions for the following, historically rather cumbersome and hard to compute, theoretical methods: (1) average Hamiltonian theory following interaction representation transformations; (2) Bloch-Redfield-Wangsness theory of nuclear and electron relaxation; (3) gradient ascent pulse engineering version of quantum optimal control theory. In the context of spin dynamics, the auxiliary matrix exponential method is more efficient than methods based on matrix factorizations and also exhibits more favourable complexity scaling with the dimension of the Hamiltonian matrix.