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LARGE EIGENVALUES OF THE INTENSITY-DEPENDENT RABI MODEL

Rana Alsayed Houssein, Lech Zieliński

Published Apr 1, 2024• 0 citations
OpenAlex

Abstract

We prove that the spectrum of the intensity-dependent Rabi Hamiltonian consists of two eigenvalue sequences (E_m^+), (E_m^-), m=0, 1, 2, 3, ... , satisfying a three-term asymptotic formula with the remainder estimate O(m^{−1} ln m) when m tends to infinity. Using the three leading terms of this asymptotic formula, one obtains anapproximation analogous to the generalized rotating-wave approximation for large eigenvalues of the usual quantum Rabi model.

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We prove that the spectrum of the intensity-dependent Rabi Hamiltonian consists of two eigenvalue sequences (E_m^+), (E_m^-), m=0, 1, 2, 3, ... , satisfying a three-term asymptotic formula with the remainder estimate O(m^{−1} ln m) when m tends to infinity. Using the three leading terms of this asymptotic formula, one obtains anapproximation analogous to the generalized rotating-wave approximation for large eigenvalues of the usual quantum Rabi model.
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The generalized rotating-wave approximation (GRWA) is presented for the two-qubit quantum Rabi model. The analytical expressions in the zeroth-order approximation recover the previous adiabatic ones. The counter-rotating-wave terms can be eliminated by performing the first-order corrections. An effective solvable Hamiltonian with the same form as the ordinary RWA one is then obtained, giving very accurate eigenvalues and eigenstates. Energy levels in the present GRWA are in accordance with the numerical exact diagonalization ones in a wide range of coupling strengths. The population dynamics in the GRWA are in quantitative agreement with the numerical results and exhibit the absence of collapses clearly, revealing the effects of the counter-rotating wave.
We consider the transition from discrete to continuous spectrum which occurs in some quantum Rabi models. We present recent developments of the subordinacy theory and their applications to study the spectral properties of the intensity-dependent Rabi model, the anisotropic two-photon Rabi model, and the two-photon Rabi-Stark model for the whole range of parameters. Our analysis allows us to detect and locate the continuous spectrum. Moreover, we prove the absence of eigenvalues in the interior of the continuous spectrum and the absence of a singular spectrum for all models considered.