Exactly solvable one-qubit driving fields generated via non-linear equations

TL;DR

This paper derives exact solutions for one-qubit driving fields by transforming the quantum dynamics into non-linear equations, enabling the design of time-dependent Hamiltonians relevant for nuclear magnetic resonance.

Contribution

It introduces a method to generate exactly solvable one-qubit Hamiltonians using non-linear equations linked to the Ermakov equation, expanding control options in quantum systems.

Findings

Derived families of time-dependent Hamiltonians with exact solutions

Connected Hamiltonians to the Ermakov equation for physical interpretation

Applicable to nuclear magnetic resonance phenomena

Abstract

Using the Hubbard representation for $SU(2)$ we write the time-evolution operator of a two-level system in the disentangled form. This allows us to map the corresponding dynamical law into a set of non-linear coupled equations. In order to find exact solutions, we use an inverse approach and find families of time-dependent Hamiltonians whose off-diagonal elements are connected with the Ermakov equation. The physical meaning of the so-obtained Hamiltonians is discussed in the context of the nuclear magnetic resonance phenomenon