Reconstruction of Hamiltonians from given time evolutions

TL;DR

This paper introduces a systematic method to determine Hamiltonians that produce a specified quantum state evolution, using group theory and the von Neumann equation, with illustrative examples.

Contribution

It presents a novel approach to solve the inverse problem of quantum dynamics by leveraging the adjoint action of the unitary group on Hermitian matrices.

Findings

Method successfully reconstructs Hamiltonians for given evolutions

Applicable to both pure and mixed quantum states

Illustrated with concrete examples

Abstract

In this paper we propose a systematic method to solve the inverse dynamical problem for a quantum system governed by the von Neumann equation: to find a class of Hamiltonians reproducing a prescribed time evolution of a pure or mixed state of the system. Our approach exploits the equivalence between an action of the group of evolution operators over the state space and an adjoint action of the unitary group over Hermitian matrices. The method is illustrated by two examples involving a pure and a mixed state.