A hidden analytic structure of the Rabi model

•A significantly simplified analytic solution of the Rabi model.•The spectrum is the lattice of discrete orthogonal polynomials.•Up to 1350 levels in double precision can be obtained for a given parity.•Omission of any level can be easily detected.

The Rabi model describes the simplest interaction between a cavity mode with a frequency ωcωc and a two-level system with a resonance frequency ω0ω0. 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)=0Δ=ω0/(2ωc)=0, which describes a displaced harmonic oscillator, is characterized by the discrete Charlier polynomials in normalized energy ϵϵ, which are orthogonal on an equidistant lattice. A non-zero value of ΔΔ leads to non-classical discrete orthogonal polynomials ϕk(ϵ)ϕk(ϵ) 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. 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 nn eigenvalues of the Rabi model arranged in increasing order can be determined as zeros of ϕN(ϵ)ϕN(ϵ) of at least the degree N=n+ntN=n+nt. The value of nt>0nt>0, which is slowly increasing with nn, depends on the required precision. For instance, nt≃26nt≃26 for n=1000n=1000 and dimensionless interaction constant κ=0.2κ=0.2, if double precision is required. Given that the sequence of the llth zeros xnlxnl’s of ϕn(ϵ)ϕn(ϵ)’s defines a monotonically decreasing discrete flow with increasing nn, the Rabi model is indistinguishable from an algebraically solvable model in any finite precision. Although we can rigorously prove our results only for dimensionless interaction constant κ<1κ<1, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for κ≥1κ≥1.