Quantum Resonance Advances Scattering Analysis with Exceptional Points and Unbounded Hamiltonians

The behaviour of quantum systems near exceptional points, where standard physical properties break down, attracts growing interest as researchers explore non-Hermitian physics in platforms like optics and superconducting circuits. Okuto Morikawa from RIKEN’s iTHEMS, alongside Shoya Ogawa and Soma Onoda from Kyushu University, investigate how geometric phases, a fundamental concept in quantum mechanics, manifest around these exceptional points within more realistic, unbounded quantum systems. The team tackles a long-standing challenge by applying the complex scaling method, a technique that transforms resonant states into discrete energy levels, allowing them to directly observe the coalescence of resonant and scattering states into an exceptional point. This work establishes a crucial link between the mathematical framework of non-Hermitian physics and the well-established theory of quantum resonances, offering new insights into the behaviour of open quantum systems and potentially paving the way for novel quantum technologies.

Geometric Phase and Quantum Resonance near EPs

Non-Hermitian operators and exceptional points (EPs) are gaining prominence in quantum mechanics, offering potential for advancements in quantum sensing and metrology. This work investigates the geometric phase acquired by a quantum state evolving around an EP, using the complex scaling method to analyze non-Hermitian systems and clarify the relationship between the geometric phase and quantum resonance. The research details how the adiabatic evolution of a quantum state near an EP can be analyzed, transforming the non-Hermitian Hamiltonian into an effectively Hermitian one to calculate eigenvalues and eigenvectors. The team demonstrates that the geometric phase accumulated around an EP directly reflects the quantum resonance occurring at that point, exhibiting a characteristic logarithmic divergence as the system approaches the EP, mirroring the resonance amplitude.

This understanding reveals that the geometric phase is a fundamental property arising from the topology of the energy surface near the EP, offering a novel perspective on the interplay between geometric phases and quantum resonances in non-Hermitian quantum systems. Exceptional points are now routinely observed in systems like optical resonators and superconducting qubits, but their foundations in realistic scattering problems remain less clear. Researchers addressed how the geometric phase associated with encircling an EP should be formulated when the underlying states are quantum resonances within a one-dimensional scattering model, employing the complex scaling method to realize resonance poles as discrete eigenvalues. This constructs scenarios where resonant and scattering states coalesce into an EP, embedded within the continuum spectrum, and the team analysed the self-orthogonality near the EP and the Berry phase, providing a connection between theoretical frameworks.

Resonances and Exceptional Points in Quantum Systems

This research connects non-Hermitian spectral theory with the established understanding of resonances, addressing a gap in how non-Hermitian operators function in infinite-dimensional quantum systems. The team investigated exceptional points within a one-dimensional scattering model, employing a complex scaling method to analyze the behavior of resonant states and define the geometric phases associated with encircling an exceptional point when the underlying states are resonances embedded within a continuous energy spectrum. By introducing a discretization of the continuous spectrum using momentum bins, researchers could treat both resonant and continuum states on a more equal footing, observing how a resonance merges with the continuum to form an EP. They highlighted that at the EP, the usual biorthogonal normalization breaks down, and the resonant and continuum eigenstates become degenerate, consistent with the appearance of a Jordan block in the Hamiltonian’s matrix representation.

A central finding is the calculation of the geometric phase acquired by the resonant wave function as a parameter is rotated around the EP, revealing a 4π periodicity, meaning the wave function only returns to its original state after a 4π rotation. This work provides a more complete picture of EPs by showing how they arise naturally in the context of scattering theory and resonances, which are inherently infinite-dimensional problems. The discovery of the 4π periodicity of the geometric phase provides a direct link between the mathematical description of EPs and the physical behavior of resonant states, offering a way to experimentally probe their existence and having implications for understanding open quantum systems and the dynamics of unstable states.

Resonances and Exceptional Points in Scattering Systems

This research establishes a connection between non-Hermitian spectral theory and the established understanding of resonances, addressing a gap in how non-Hermitian operators function in infinite-dimensional quantum systems. The team investigated exceptional points, which arise in non-Hermitian systems, within the context of a one-dimensional scattering model, employing a complex scaling method to analyze the behavior of resonant states. This approach allowed them to define how geometric phases, typically associated with encircling an exceptional point, should be understood when the underlying states are resonances embedded within a continuous energy spectrum. The results demonstrate that resonant and scattering states can coalesce into an exceptional point within the complex energy plane, and the researchers thoroughly analyzed the self-orthogonality of wave functions in the vicinity of these points, alongside the associated Berry phase. While the study successfully bridges theoretical frameworks, the authors acknowledge that their analysis focuses on a specific one-dimensional model and may not directly extend to more complex, multi-dimensional scattering scenarios. Future work, they suggest, could explore the implications of these findings for understanding non-Hermitian phenomena in broader physical systems, including those exhibiting non-Hermitian skin effects and complex open quantum systems.