Projection operator based expansion of the evolution operator

TL;DR

This paper introduces a projection operator method to derive an alternative, more accurate expansion of the quantum evolution operator, improving short-time dynamics analysis and enabling efficient computation for finite systems.

Contribution

It presents a novel projection operator-based expansion of the evolution operator that enhances accuracy and efficiency over traditional methods.

Findings

New formula for short-time quantum dynamics

Improved accuracy over the traditional chronological exponent

Effective computation for finite-dimensional quantum systems

Abstract

The not necessarily unitary evolution operator of a finite dimensional quantum system is studied with the help of a projection operators technique. Applying this approach to the Schr\"odinger equation allows the derivation of an alternative expression for the evolution operator, which differs from the traditional chronological exponent. An appropriate choice of projection operators results in the possibility of studying the diagonal and non-diagonal elements of the evolution operator separately. The suggested expression implies a particular form of perturbation expansion, which leads to a new formula for the short time dynamics. The new kind of perturbation expansion can be used to improve the accuracy of the usual chronological exponent significantly. The evolution operator for any arbitrary time can be efficiently recovered using the semigroup properties. The method is illustrated by two examples, namely the dynamics of a three-level system in two nonresonant laser fields and the calculation of the partition function of a finite XY-spin chain.