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Italian Quantum Information Science Conference (IQIS)

The Friedrichs-Lee Model and Its Singular Coupling Limit

Davide Lonigro, Paolo Facchi A, Marilena Ligabò

Published Jun 24, 2019• 4 citations
Publisher

Abstract

Lee’s field-theoretical model describes the interaction between a qubit and a structured bosonic field. We study the mathematical properties of the Hamiltonian of the single-excitation sector of the theory, including a possibly “singular” qubit-field coupling (i.e., mediated by a non-square integrable form factor). This result allows for a rigorous description of qubit-field interactions in many physically interesting systems and may be extended to higher-excitation sectors of the theory.

DOI
10.3390/proceedings2019012017

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Lee’s field-theoretical model describes the interaction between a qubit and a structured bosonic field. We study the mathematical properties of the Hamiltonian of the single-excitation sector of the theory, including a possibly “singular” qubit-field coupling (i.e., mediated by a non-square integrable form factor). This result allows for a rigorous description of qubit-field interactions in many physically interesting systems and may be extended to higher-excitation sectors of the theory.
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The author investigates both analytically and numerically the one-point Friedrichs model characterised by a lambda V interaction with a non-square integrable coupling function nu ( omega )=1. If the continuous spectrum of the Hamiltonian is bounded from below, a form factor is needed to avoid some infinities and the solution of the spontaneous emission problem then depends on an unphysical cut-off parameter alpha . This solution is denoted by p(t; alpha ) and its alpha -dependence may be removed by taking the appropriate limit. Indeed, the author finds that: Lim alpha to infinity p(t; alpha )=1. If, on the contrary, the continuous spectrum is assumed to be infinite, then no form factor is needed and p(t)=exp(-2 pi lambda 2 t).
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.
Generalized spin-boson (GSB) models describe the interaction between a quantum mechanical system and a structured boson environment, mediated by a family of coupling functions known as form factors. We propose an extension of the class of GSB models, which can accommodate non-normalizable form factors provided that they satisfy a weaker growth constraint, thus accounting for a rigorous description of a wider range of physical scenarios; we also show that such “singular” GSB models can be rigorously approximated by GSB models with normalizable form factors. Furthermore, we discuss in greater detail the structure of the spin-boson model with a rotating wave approximation: for this model, the result is improved via a nonperturbative approach that enables us to further extend the class of admissible form factors as well as to compute its resolvent and characterize its self-adjointness domain.