A hidden analytic structure of the Rabi model

Abstract

The Rabi model describes the simplest interaction between a cavity mode with a frequency $\omega_c$ and a two-level system with a resonance frequency $\omega_0$. 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 $\Delta=\omega_0/(2\omega_c)=0$, 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 $\Delta$ leads to non-classical discrete orthogonal polynomials $\phi_{k}(\upepsilon)$ 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 $\phi_{N}(\upepsilon)$ of at least the degree $N=n+n_t$. The value of $n_t>0$, which is slowly increasing with $n$, depends on the required precision. For instance, $n_t\simeq 26$ for $n=1000$ and dimensionless interaction constant $\kappa=0.2$, if double precision is required. Although we can rigorously prove our results only for dimensionless interaction constant $\kappa< 1$, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for $\kappa\ge 1$.Comment: 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 mode

Magnetic spheres in microwave cavities

“…Also the rotating-wave approximation (RWA) (usually but not necessarily assumed in the TC model) is speaking applicable when the coupling ratio g/ω c 1, where ω c is the microwave cavity mode frequency. We may define different coupling regimes [24,25], viz. (i) strong coupling (SC) when 0.01 < g/ω c 0.1, (ii) ultrastrong coupling (USC) [26] when g/ω c 0.1, (iii) or even deep strong coupling (DSC) g/ω c ≈ 1 [27].…”

The quantum Rabi model: solution and dynamics

“…Variational approaches have also been applied [79,80], most recently in a polaron-antipolaron context [81,82]. We mention also a continued fraction and three-term recurrence relation approach which has been developed to calculate the energy spectrum using an F -function [56,73,74,75]. This is different to Braak's Gfunction, but works in a similar fashion.…”

“…The problem of finding the energy eigenvalues is reduced to the diagonalisation of an infinite tridiagonal matrix [68], which can be done numerically by truncation of the matrix to finite order. Some other approaches are series expansions [69] and continued fractions [70,71,72,73,74,75]. Various other computational schemes have been discussed (see, e.g., [76] and references therein) with the necessary aim to be effective in the ultrastrong and deep strong coupling regimes.…”

The asymmetric quantum Rabi model in the polaron picture

“…This model is well known as the Rabi model [25], which has attracted much attention in the past few years. On the one hand, the exact solutions of this model [26,27] and its cousins [28][29][30][31] have been obtained only recently, which marks an important step toward the understanding of the mathematical structure of this model by proving its integrability [26,[32][33][34][35][36]. On the other hand, the experimental progress [37][38][39][40][41][42][43] toward significant enhancement of the light-matter coupling strength raises the request to consider a full quantum Rabi model in order to explain experimental observations [42,43].…”

“…It was shown that all these methods are unable to capture the correct physics in the intermediate coupling strengths, especially in the small ω (harmonic oscillator frequency) limit. Apart from the double precision problem in eigenstate calculations [27,36] and the careful attention one might need to pay to the criterion and parameter regime for quantum integrability [30,[33][34][35], it also might be worthwhile to recognize that the basic energy scales involved in the Rabi model compete with each other in a subtle way [55]. All these factors might contribute to the fact that the physics of the quantum Rabi model still has not been fully explored.…”