Geometric phase of exceptional point as quantum resonance in complex scaling method

TL;DR

This work addresses how to formulate geometric phases around exceptional points in genuine quantum scattering with unbounded Hamiltonians. By employing the complex scaling method and a momentum-bin discretization of the complex-scaled continuum, the authors realize resonance poles as discrete non-Hermitian eigenstates and identify EPs as coalescences of resonances with the rotated continuum. They demonstrate self-orthogonality and a Berry-phase-like geometric phase when encircling a resonance pole, including a multi-sheet branch structure driven by continuum discretization, yielding a nontrivial rotation in parameter space. The study provides a bridge between non-Hermitian spectral theory and resonance theory in scattering, offering a framework for analyzing decay and memory effects in open quantum systems and suggesting directions for future work on Stokes topology and non-Markovian dynamics near EP-like regimes.

Abstract

Non-Hermitian operators and exceptional points (EPs) are now routinely realized in few-mode systems such as optical resonators and superconducting qubits. However, their foundations in genuine scattering problems with unbounded Hamiltonians remain much less clear. In this work, we address how the geometric phase associated with encircling an EP should be formulated 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 and the Berry phase. Our results provide a bridge between non-Hermitian spectral theory and the traditional theory of quantum resonances.

Figures (5)

• Figure 1: Schematic energy-plane picture of complex scaling. The complex-scaled continuum spectrum is rotated by $-2\theta$ (blue) from the positive real axis. We set $\theta_n<\theta<\theta_{n+1}$. Resonance poles at $E_{n+1,n+2\dots}^{\mathrm{R}}$ appear below the rotated cut, and Gamow states at $E_{0,1,\dots,n}^{\mathrm{R}}$ become regularized as discrete states.
• Figure 2: $\lambda_{0,1}^\pm$ as functions of $\theta$. $\frac{\beta^2\hbar^2}{4m}=1$. $\lambda$ should be above the blue curve which denotes $\lambda_0^+$ ($\lambda$ should be larger than it), or be below the orange curve which is $\lambda_0^-$ ($\lambda$ should be smaller than it). The green and red curves correspond to $\lambda_1^+$ and $\lambda_1^-$, respectively, and hence $\lambda$ should be inside the region between these curves. Eventually, we find that $\lambda$ is larger than $\lambda_0^+$ and smaller than $\lambda_1^+$ as long as $\theta$ is not too large.
• Figure 3: Changing $\lambda$ and deformation of energy Riemann sheet. Left: For a moderate value of $\lambda$, the resonant spectrum is completely isolated. Right: Near $\lambda_0^{+}$, the resonance pole is embedded into the continuum spectrum.
• Figure 4: The resonance wave function as a function of $\lambda$ around $\lambda_{\mathrm{bp}}$. Given $\lambda$, we can see the behavior of $\psi_{\mathrm{CSM}}(\lambda)$ with $E_{0}^{\mathrm{R}}(\lambda)$. We consider the $\lambda$-contour around $\lambda_{\mathrm{bp}}$ as depicted by the blue circle. Below the red line, $\psi_{\mathrm{CSM}}$ is convergent and a pseudo-bound state. Above it, the wave function is not regularized by CSM, so $\psi_{\mathrm{CSM}}$ diverges; this region is a kind of branch plane. At intersections between the red line and blue curve, the resonance is embedded into the continuum spectrum, and its wave function is identical to the scattering solution.
• Figure 5: The resonance and continuum wave functions in the complex energy plane. The contour around $E_{\mathrm{bp}}$ is depicted by the blue circle. Below the red line, i.e., in the region A, $\psi_{\mathrm{CSM}}^{\mathrm{R}}$ is well-defined. Otherwise, in the region B, we focus on $\psi_{\mathrm{CSM}}^{\mathrm{conti}}$. At intersections between the red line and blue curve, say $\lambda^{+}$ and $\lambda^{-}$, the resonance is identical to the continuum state $\psi_{\mathrm{CSM}}^{\mathrm{conti}}$ along the limit of $\lambda$ as the left/right eigenfunction. We set the argument of $\alpha$ as the right panel so that Moiseyev's ansatz \ref{['eq:moiseyev']} becomes $E(\lambda^{-})$ when $\phi=0$.