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arXiv:2109.02939 · The European Physical Journal Plus

The self-energy of Friedrichs–Lee models and its application to bound states and resonances

Davide Lonigro

Published Apr 1, 2022• 6 citations

Abstract

Abstract A system composed of two-level systems interacting with a single excitation of a one-dimensional boson field with continuous spectrum, described by a Friedrichs (or Friedrichs–Lee) model, can exhibit bound states and resonances; the latter can be characterized by computing the so-called self-energy of the model. We evaluate an analytic expression, valid for a large class of dispersion relations and coupling functions, for the self-energy of such models. Afterwards, we focus on the case of identical two-level systems, and we refine our analysis by distinguishing between dominant and suppressed contributions to the associated self-energy; we finally examine the phenomenology of bound states in the presence of a single dominant contribution.

DOI: 10.1140/epjp/s13360-022-02690-y

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Abstract A system composed of two-level systems interacting with a single excitation of a one-dimensional boson field with continuous spectrum, described by a Friedrichs (or Friedrichs–Lee) model, can exhibit bound states and resonances; the latter can be characterized by computing the so-called self-energy of the model. We evaluate an analytic expression, valid for a large class of dispersion relations and coupling functions, for the self-energy of such models. Afterwards, we focus on the case of identical two-level systems, and we refine our analysis by distinguishing between dominant and suppressed contributions to the associated self-energy; we finally examine the phenomenology of bound states in the presence of a single dominant contribution.

The Friedrichs model consists in an interaction between the bound state of a quantum system (say, an atom) and an external field, interaction which produces a resonance. This interaction intertwines the bound state with the eigenvectors of the continuous spectrum of the field Hamiltonian and depends on a coupling parameter. The resonance appears as a pole of the analytic continuation of the reduced resolvent corresponding to the interacting Hamiltonian. This model admits generalizations like, for instance, adding bound states and/or degeneracy to the field Hamiltonian. We examine what happens as the imaginary component of the eigenvalue associated with a resonant energy approaches zero.

Bound states and decay dynamics in

N

-level Friedrichs model with factorizable interactions

Considering an

N

-level system interacting factorizably with a continuous spectrum, we derive 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 decay dynamics, which is naturally described by an energy-independent non-Hermitian Hamiltonian in the Markovian limit. As an example, we apply our framework to a two-level atomic chain side-coupled to a photonic lattice, uncovering a rich variety of decay dynamics and realizing an anti-

\mathcal{PT}

-symmetric Hamiltonian in the system's evolution.

Abstract We present here a relation of different types of Friedrichs models and their use in the description and comprehension of resonance phenomena. We first discuss the basic Friedrichs model and obtain its resonance in the case that this is simple or doubly degenerated. Next, we discuss the model with N levels and show how the probability amplitude has an oscillatory behavior. Two generalizations of the Friedrichs model are suitable to introduce resonance behavior in quantum field theory. We also discuss a discrete version of the Friedrichs model and also a resonant interaction between two systems both with continuous spectrum. In an appendix, we review the mathematics of rigged Hilbert spaces.

Abstract We present here a relation of different types of Friedrichs models and their use in the description and comprehension of resonance phenomena. We first discuss the basic Friedrichs model and obtain its resonance in the case that this is simple or doubly degenerated. Next, we discuss the model with N levels and show how the probability amplitude has an oscillatory behavior. Two generalizations of the Friedrichs model are suitable to introduce resonance behavior in quantum field theory. We also discuss a discrete version of the Friedrichs model and also a resonant interaction between two systems both with continuous spectrum. In an appendix, we review the mathematics of rigged Hilbert spaces.