2013/05/31 by Alexander Moroz · 1 citation
Mathematics · Physics and Astronomy · #Combinatorics #Dimensionless quantity #Eigenvalues and eigenvectors #Lattice (music) #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Omega #Orthogonal polynomials #Physics #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Quantum optics and atomic interactions #cond-mat.mes-hall #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1016/j.aop.2013.10.014
published as Ann. Phys. (N.Y.) 340, 252-266 (2014) · 10 pages, 3 figures - the amended versions gives more emphasis on the role played by discrete orthogonal polynomials in solving the Rabi model. New subsection IV.E summarizes open problems required to generalize the classical discrete Charlier polynomials describing the displaced harmonic oscillator into non-classical discrete polynomials describing the full Rabi model
arxiv created 2013/09/22 · openalex publication_date 2013/11/06 · arxiv updated 2013/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The Rabi model describes the simplest interaction between a cavity mode with a frequency ωc and a two-level system with a resonance frequency ω0. It is shown here that the spectrum of the Rabi model coincides with the support of the discrete Stieltjes integral measure in the orthogonality relations of recently introduced orthogonal polynomials. The exactly solvable limit of the Rabi model corresponding to Δ=ω0/(2ωc)=0, which describes a displaced harmonic oscillator, is characterized by the discrete Charlier polynomials in normalized energy \upepsilon, which are orthogonal on an equidistant lattice. A non-zero value of Δ leads to non-classical discrete orthogonal polynomials ϕk(\upepsilon) and induces a deformation of the underlying equidistant lattice. The results provide a basis for a novel analytic method of solving the Rabi model. The number of ca. \em 1350 calculable energy levels per parity subspace obtained in double precision (cca 16 digits) by an elementary stepping algorithm is up to two orders of magnitude higher than is possible to obtain by Braak's solution. Any first n eigenvalues of the Rabi model arranged in increasing order can be determined as zeros of ϕN(\upepsilon) of at least the degree N=n+nt. The value of nt>0, which is slowly increasing with n, depends on the required precision. For instance, nt≃ 26 for n=1000 and dimensionless interaction constant κ=0.2, if double precision is required. Although we can rigorously prove our results only for dimensionless interaction constant κ< 1, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for κ≥ 1.