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arXiv:2603.08834

Spectral transitions in some Rabi models

Grzegorz Świderski, Lech Zieliński

Published Mar 9, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2603.08834

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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.
Abstract 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.
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Spectral collapse, the transition from a discrete to a continuous spectrum, is a characteristic in quantum Rabi models. We explore this phenomenon in the two-photon quantum Rabi model using optical phase space, and we find that, in the so-called degenerate qubit regime, the collapse is similar to the transition from a harmonic to an inverted oscillator with the free-particle potential as a critical transition point. In the degenerate qubit regime, we construct Dirac-normalizable eigenfunctions with well-defined parity for the model. In the general model, we use parity to diagonalize the system in the qubit basis and numerically find that the qubit frequency does not change the critical point where spectral collapse occurs. We numerically confirm the existence of an exceptional state at the critical coupling, and we argue its analytic provenance from both a Born-Oppenheimer and a variational approximation.
ABSTRACT We investigate the spectral properties and quantum criticality of the two‐photon Rabi–Stark model. Using the exact solution of this model, we rigorously derive a condition for complete spectral collapse, where all bound states vanish. In this case, the energy gap closes at a critical coupling, signaling a continuous quantum phase transition. The corresponding gap exponent differs from those in both the one‐photon Rabi–Stark model and the quantum Rabi model, suggesting a distinct universality class. While in the general case, an infinite number of discrete bound states exist when spectral collapse occurs and the energy gap remains open. By mapping to an inverse square potential well, these bound levels approach the threshold energy exponentially. Our results offer new insights into novel spectral phenomena in nonlinear quantum Rabi models, with potential implications for experimental realizations in circuit quantum electrodynamics and trapped ion systems.
We have investigated the eigenenergy spectrum of the two-photon Rabi model at the critical coupling, particularly the special feature "spectral collapse", by means of an elementary quantum mechanics approach. The eigenenergy spectrum is found to consist of both a set of discrete energy levels and a continuous energy spectrum. Each of these eigenenergies has a two-fold degeneracy corresponding to the spin degree of freedom. The discrete eigenenergy spectrum has a one-to-one mapping with that of a particle in a "Lorentzian function" potential well, and the continuous energy spectrum can be derived from the scattering problem associated with a potential barrier. The number of bound states available at the critical coupling is determined by the energy difference between the two atomic levels so that the extent of the "spectral collapse" can be monitored in a straightforward manner.