Effective Hamiltonians in Cavity and Waveguide QED from Transition-Operator Diagrammatic Perturbation Theory

TL;DR

This paper introduces a diagrammatic perturbation theory-based adiabatic-elimination method for deriving effective Hamiltonians in cavity and waveguide QED, applicable to complex multilevel systems.

Contribution

It offers a systematic, higher-order, transition-centric approach that overcomes limitations of existing techniques for multiphoton processes in dispersive QED regimes.

Findings

Enables explicit construction of higher-order effective Hamiltonians.

Applicable to multilevel systems and multiple qubits.

Provides a practical toolbox for multiphoton processes.

Abstract

We propose an adiabatic-elimination formalism in the dispersive regime based on a transition-centric perturbation theory. The perturbative expansion is recast into a diagrammatic framework, while adiabatic elimination is implemented through controlled projections onto transition subspaces. Our approach applies systematically at arbitrary perturbation order, and is suited to multilevel systems and multiple qubits in both cavity and waveguide quantum electrodynamics. It ultimately enables the explicit construction of effective higher-order Hamiltonians while bypassing important limitations of existing techniques, thereby providing a practical toolbox for multiphoton processes in the dispersive regime.