Geometric phase from encircling an exceptional point of a quantum resonance in the complex-scaling method

TL;DR

This paper investigates the geometric phase associated with encircling an exceptional point in quantum resonances using the complex-scaling method, linking non-Hermitian spectral topology with quantum scattering theory.

Contribution

It formulates the geometric phase for quantum resonances at exceptional points within a scattering framework using complex scaling, bridging non-Hermitian topology and quantum resonances.

Findings

Resonance poles can coalesce into an EP in the complex energy plane.

The study clarifies the origin of EP branch structure and geometric holonomy from resonance poles.

Connections established between non-Hermitian spectral topology and quantum scattering theory.

Abstract

Non-Hermitian operators are now routinely used to describe few-mode systems such as optical resonators and superconducting qubits, and exceptional points (EPs) are defective spectral singularities of such non-Hermitian operators. In contrast, the scattering-theoretic formulation of EP physics for unbounded Hamiltonians remains less settled. In this work, we formulate the geometric phase associated with encircling an EP when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex-scaling method, where resonance poles of the S matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic. Our results clarify how EP branch structure and geometric holonomy arise directly from resonance poles in scattering theory, thereby connecting non-Hermitian spectral topology with the traditional theory of quantum resonances.