Universal solution to the Schrieffer-Wolff Transformation Generator

TL;DR

This paper introduces a universal, closed-form solution for the Schrieffer-Wolff transformation generator applicable to a wide range of quantum systems, including time-dependent cases, enhancing the method's generality and practical utility.

Contribution

The authors derive a systematic, closed-form, and universally applicable solution for the SWT generator, extending it to time-dependent systems with periodic perturbations.

Findings

Successfully applied to analyze dispersive shifts in anharmonic resonators

Provides a unified framework for static and dynamic perturbative systems

Enhances the applicability of SWT in quantum system analysis

Abstract

The Schrieffer-Wolff transformation (SWT) is an important perturbative method in quantum mechanics used to simplify Hamiltonians by decoupling low- and high-energy subspaces. Existing methods for implementing the SWT often lack general applicability to arbitrary perturbative systems or fail to provide a closed-form solution for the SWT generator. In this article, we present a systematic and unified framework for the SWT that addresses these shortcomings. Specifically, we derive a closed-form solution for the SWT generator that is universally applicable to any system that satisfies the conditions required for perturbative treatment. Furthermore, we extend this solution to time-dependent systems with periodic perturbations, covering all frequency regimes. The effectiveness of this approach is then demonstrated by applying it to analyze the dispersive shift of an anharmonic resonator coupled to a two-level system with time-dependent coupling.