Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory

TL;DR

This paper introduces an exact analytical method based on algebraic graph theory for solving the time-evolution of quantum spin systems with arbitrary waveforms, outperforming traditional numerical techniques.

Contribution

It develops a novel formalism using the path-sum method and algebraic graph theory to obtain exact solutions for quantum spin dynamics under arbitrary time-dependent waveforms.

Findings

Method provides exact solutions for various quantum spin systems.

Outperforms conventional numerical methods in accuracy and efficiency.

Successfully applied to systems under chirped pulses.

Abstract

A general approach is presented that offers exact analytical solutions for the time-evolution of quantum spin systems during parametric waveforms of arbitrary functions of time. The proposed method utilises the \emph{path-sum} method that relies on the algebraic and combinatorial properties of walks on graphs. A full mathematical treatment of the proposed formalism is presented, accompanied by an implementation in \textsc{Matlab}. Using computation of the spin dynamics of monopartite, bipartite, and tripartite quantum spin systems under chirped pulses as exemplar parametric waveforms, it is demonstrated that the proposed method consistently outperforms conventional numerical methods, including ODE integrators and piecewise-constant propagator approximations.