A new basis- and integral-free approach to perturbation theory: The Schrödinger dynamics of N trapped ions in the high-intensity regime

Abstract

A novel, basis- and integral-free perturbative method for the Dyson series describing Schrödinger dynamics in general is introduced. The $N$ trapped ions in the high-intensity regime are addressed under this new approach. The approach is based on the Omega Matrix Calculus (OMC) which is rooted in the theory of partitions of natural numbers due to MacMahon. A key ingredient in our formalism comprises a new OMC representation to compute multiple integrals involving the matrix exponential, allowing for a simpler treatment using Omega calculus elimination rules and enhancing previous representations, such as the one by Francisco Neto (2020). This approach not only implies previous perturbative approaches based on divided-differences in Kalev and Hen (2021) and the matrix method in Villegas-Martínez et al. (2022), but also simplifies the derivation process and provides a closed-form expression for the $n$th term in the perturbative expansion solving previously open questions highlighted in prior works, including Villegas-Martínez et al. aforementioned work. Additionally, it reveals that the $n$th term in perturbation theory is governed by a generalized exponential function based on divided differences, which simplifies to the ordinary exponential for $n=0$. In specific cases, where $N=1$, $n=1,2$, and the interaction term is time-independent, the results are consistent with those obtained by Villegas-Martínez et al.