Bound states and decay dynamics in $N$-level Friedrichs model with factorizable interactions

TL;DR

This paper analytically investigates bound states and decay dynamics in an N-level Friedrichs model with factorizable interactions, revealing conditions for bound state formation and describing the system's dissipative evolution, including applications to atomic chains in photonic crystals.

Contribution

It provides explicit criteria for bound state existence and derives the system's non-Hermitian Hamiltonian dynamics within the Friedrichs model, extending understanding of decay suppression and anti-PT symmetry.

Findings

Bound states can suppress complete decay.

Derived energy-independent non-Hermitian Hamiltonian.

Observed anti-PT symmetry in atomic chain dynamics.

Abstract

Considering an $N$-level system interacting factorizably with a continuous spectrum, we derive analytical expressions for the bound states and the dynamical evolution within this single-excitation Friedrichs model by using the projection operator formalism. First, we establish explicit criteria to determine the number of bound states, whose existence suppresses the complete spontaneous decay of the system. Second, we derive the open system's dissipative dynamics, which is naturally described by an energy-independent non-Hermitian Hamiltonian in the Markovian limit. As an example, we apply our framework to an atomic chain embedded in a photonic crystal waveguide, uncovering a rich variety of decay dynamics and realizing an anti-$\mathcal{PT}$-symmetric Hamiltonian in the system's evolution.