Abstract We investigate the origin of spectral collapse occurring in nonlinear Rabi Hamiltonians with an su(1,1) coupling scheme, showing how the collapse can be triggered by the competition between the Rabi parameter g and the field frequency W . The collapse already appears in the model Hamiltonian where the atomic-energy term is absent. After showing that su(1,1) is the dynamical algebra of the Hamiltonian, we demonstrate how the occurrence of spectral collapse can be directly related to the three types of equivalence classes characterizing the structure of this algebra. We highlight how the dramatic change of the spectrum significantly affects the structure of eigenstates represented in a suitable momentum–coordinate picture.
Abstract We investigate the origin of spectral collapse occurring in nonlinear Rabi Hamiltonians with an su(1,1) coupling scheme, showing how the collapse can be triggered by the competition between the Rabi parameter g and the field frequency W . The collapse already appears in the model Hamiltonian where the atomic-energy term is absent. After showing that su(1,1) is the dynamical algebra of the Hamiltonian, we demonstrate how the occurrence of spectral collapse can be directly related to the three types of equivalence classes characterizing the structure of this algebra. We highlight how the dramatic change of the spectrum significantly affects the structure of eigenstates represented in a suitable momentum–coordinate picture.
Abstract An analytical approach is proposed to study the two-photon, two-mode and intensity-dependent Rabi models. By virtue of the su(1, 1) Lie algebra, all of them can be unified to the same Hamiltonian with Z 2 symmetry. There exist exact isolated solutions, which are located at the level crossings between different parities and correspond to eigenstates with finite dimension. Beyond the exact isolated solutions, the regular spectrum can be achieved by finding the roots of the G-function. The corresponding eigenstates are of infinite dimension. It is noteworthy that the expansion coefficients of the eigenstates present an exponential decay behavior. The decay rate decreases with increasing coupling strength. When the coupling strength tends to the spectral collapse point g → ω/2, the decay rate tends to zero which prevents the convergence of the wave functions. This work paves a way for the analysis of novel physics in nonlinear quantum optics.
Abstract The two-photon quantum Rabi model is analyzed using the Z 4 -symmetry and su ( 1 , 1 ) -algebra. We derive the G -function whose zeros give the exact eigenvalues of the Hamiltonian. The singularity structure of the G -function allows to draw conclusions about the distribution of these eigenvalues along the real axis and we derive the spectral collapse phenomenon at critical coupling g c , found numerically before. The spectrum at g c consists of a discrete and a continuous part: the ground state is always separated from the continuum by a finite excitation gap, ruling out a quantum phase transition in the usual sense. For large qubit splitting, also other low lying states split off from the continuum. However, perturbation theory predicts the vanishing of the gap to all orders, demonstrating its non-perturbative nature. We corroborate this result with a variational calculation for the ground state.
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.
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.
We present a perturbative analysis of a Rabi model where the coupling between the quantized single-mode electromagnetic field and the two-level atom depends on the field intensity. Upon modeling the matter–radiation coupling through the Holstein–Primakoff realization of algebra su(1,1), we evaluate first- and second-order eigenenergies and eigenstates both in the weak-coupling regime (atom transition frequency smaller than the coupling strength) and in the strong-coupling regime. In the first case, among various effects, we observe a quadratic dependence on the photon number of energy eigenvalues and the possible formation of level doublets. In the strong-coupling case, the perturbative analysis becomes considerably complex due to the su(1,1)-valued form of the unperturbed Hamiltonian. The critical condition for the transition to an almost continuous spectrum is found in terms of the model parameters.
Abstract We present an exact analytical solution of a generic two-mode quantum Rabi model with non-identical bosonic modes. By means of a Bogoliubov transformation, the Hamiltonian is reduced to invariant subspaces labeled by a conserved quantity, and a transcendental G-function is derived, whose zeros determine the regular energy spectrum. We show that the spectra in different subspaces are related by a simple energy shift, revealing the crucial role of mode asymmetry in shaping the spectral structure. In the symmetric limit, the model exhibits an enhanced degeneracy, including fourfold degenerate energy levels arising from the merging of invariant subspaces. The structure of exceptional solutions is analyzed in detail. We identify finite-dimensional Juddian solutions associated with level crossings, as well as special nondegenerate solutions that give rise to isolated states. The degeneracy of these solutions is shown to be controlled by the underlying symmetry of the model. Furthermore, we investigate spectral collapse in the generic two-mode quantum Rabi model. Unlike in the two-photon quantum Rabi model, the collapse energy depends explicitly on the subspace, leading to a richer collapse scenario. We demonstrate that spectral collapse originates from the vanishing spacing between adjacent poles of the G-function, and that the collapse point supports an infinite sequence of bound states. These states exhibit exponential accumulation toward the collapse energy, revealing a hierarchical structure of collapse-induced bound states.