[Paper Review] The role of exceptional points in quantum systems

I. Rotter|arXiv (Cornell University)|Nov 2, 2010

Advanced Physical and Chemical Molecular Interactions1 references3 citations

TL;DR

This paper investigates exceptional points (EPs) in non-Hermitian quantum systems, where eigenvalues and eigenfunctions coalesce due to coupling to a continuum. It demonstrates that EPs induce dynamical phase transitions, leading to enhanced transmission, chaotic features in long-lived states, and breakdown of spectroscopic relations, with applications in open quantum and PT-symmetric systems.

ABSTRACT

In the present paper, first the mathematical basic properties of the exceptional points are discussed. Then, their role in the description of real physical quantum systems is considered. Most interesting value is the phase rigidity of the eigenfunctions which varies between 1 (for distant non-overlapping states) and 0 (at the exceptional point where the resonance states completely overlap). This variation allows the system to incorporate environmentally induced effects. In the very neighborhood of an exceptional point, the system can be described well by a conventional nonlinear Schrödinger equation. In the regime of overlapping resonances, a dynamical phase transition takes place to which all states of the system contribute: a few short-lived resonance states are aligned to the scattering states of the environment by trapping the other states. The trapped resonance states show chaotic features. Due to the alignment of a few states with the states of the environment, observable values may be enhanced. The dynamical phase transition allows us to understand some experimental results which remained puzzling in the framework of conventional Hermitian quantum physics. The effects caused by the exceptional points in physical systems allow us to manipulate them for many different applications.

Motivation & Objective

• To clarify the role of exceptional points (EPs) in real physical quantum systems, particularly in open quantum systems with continuum coupling.
• To explain why EPs—though unobservable directly—produce measurable effects through eigenvalue trajectory behavior and phase rigidity changes.
• To establish a connection between EPs and dynamical phase transitions in overlapping resonance regimes.
• To demonstrate how non-Hermitian Hamiltonians, arising from coupling to a continuum, lead to spectroscopic redistribution and state alignment with scattering states.
• To show that EPs enable systematic manipulation of quantum systems for applications, especially in PT-symmetric and open quantum systems.

Proposed method

• Analyzes a 2×2 non-Hermitian Hamiltonian model with energy-level coupling via a continuum, derived from open quantum systems with scattering wavefunctions.
• Uses exact solutions of the Schrödinger equation with non-Hermitian Hamiltonians to study eigenvalue trajectories and phase rigidity (rₖ = Aₖ⁻¹) as a function of control parameters.
• Applies the concept of avoided level crossing and double poles in the S-matrix to characterize EPs in scattering theory.
• Introduces the nonlinear Schrödinger equation in the neighborhood of EPs to describe strong state entanglement and dynamical behavior.
• Compares Hermitian and non-Hermitian regimes: Hermitian case shows rigid phases and avoided crossing; non-Hermitian case allows level crossing and phase rigidity collapse to zero at EPs.
• Analyzes the transition from discrete states to resonance states via width bifurcation, identifying a dynamical phase transition driven by environmental coupling.

Experimental results

Research questions

• RQ1 How do exceptional points influence eigenvalue trajectories and eigenfunction phases in open quantum systems with continuum coupling?
• RQ2 What physical consequences arise from the collapse of phase rigidity (rₖ → 0) at exceptional points in non-Hermitian quantum systems?
• RQ3 How does the dynamical phase transition induced by EPs alter the spectroscopic properties of resonance states and their coupling to the environment?
• RQ4 In what way do EPs lead to enhanced transmission and chaotic features in long-lived resonance states despite the absence of direct observation?
• RQ5 How does the formal equivalence between PT-symmetric optical lattices and non-Hermitian quantum mechanics provide insight into quantum system behavior?

Key findings

• Exceptional points in quantum systems are singularities where two eigenvalues and their corresponding eigenfunctions coalesce, leading to a collapse of phase rigidity from 1 to 0.
• The dynamical phase transition at EPs results in alignment of a few short-lived resonance states with the scattering continuum, reducing the number of localized states and breaking spectroscopic symmetries.
• In the regime of overlapping resonances, the real parts of eigenvalue trajectories can cross with broad states, while narrow states exhibit level repulsion and chaotic features.
• Transmission through the system is enhanced in the transition region due to partial alignment of resonance states with the environment, a signature of EP-induced effects.
• The Schrödinger equation becomes nonlinear near EPs due to non-rigid phases (rₖ < 1), leading to strong entanglement and maximal mixing at the critical point.
• The results hold for PT-symmetric systems, and the formal equivalence between PT-symmetric optics and quantum mechanics allows for new insights and experimental analogies.