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arXiv:2101.12396 · Physical Review A

Multiple ground-state instabilities in the anisotropic quantum Rabi model

Xiang-You Chen, Liwei Duan, Daniel Braak, Qing‐Hu Chen

Published Apr 9, 2021• 24 citations
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Abstract

In this work, the anisotropic variant of the quantum Rabi model with different coupling strengths of the rotating and counterrotating wave terms is studied by the Bogoliubov operator approach. The anisotropy preserves the parity symmetry of the original model. We derive the corresponding $G$-function, which yields both the regular and exceptional eigenvalues. The exceptional eigenvalues correspond to the crossing points of two energy levels with different parities, and they are doubly degenerate. We find analytically that the ground state and the first excited state can cross several times, indicating multiple first-order phase transitions as a function of the coupling strength. These crossing points are related to manifest parity symmetry of the Hamiltonian, in contrast to the level crossings in the asymmetric quantum Rabi model, which are caused by a hidden symmetry.

DOI
10.1103/physreva.103.043708

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•Jul 7, 2018•2 citations
In this work, the anisotropic variant of the quantum Rabi model with different coupling strengths of the rotating and counterrotating wave terms is studied by the Bogoliubov operator approach. The anisotropy preserves the parity symmetry of the original model. We derive the corresponding GGG-function, which yields both the regular and exceptional eigenvalues. The exceptional eigenvalues correspond to the crossing points of two energy levels with different parities, and they are doubly degenerate. We find analytically that the ground state and the first excited state can cross several times, indicating multiple first-order phase transitions as a function of the coupling strength. These crossing points are related to manifest parity symmetry of the Hamiltonian, in contrast to the level crossings in the asymmetric quantum Rabi model, which are caused by a hidden symmetry.
In this work, the anisotropic quantum Rabi model with different coupling strengths of the rotating-wave and counter-rotating wave terms is studied by using two kinds of extended coherent states (ECS). By the first kind of ECS, we can derive a so-called GGG-function, by which both the regular and exceptional solutions can be given. The exceptional solution are just corresponding to the crossing points of two energy levels with different parities, so is doubly degenerate. By the second kind of ECS, a general scheme for the eigensolutions is derived analytically in a unified way. The zero-order approximation is just the adiabatic approximation, and the first-order approximation is actually a generalized rotating-wave approximation. The algebraic formulae for the eigensolutions are given explicitly in two approximations. The generalized rotating-wave approximations work well in a wide range of two different coupling strengths and the qubit detunings.
Abstract Since the realization of the strong coupling between light and matter in experimental setups, the quantum Rabi model and its generalized models describing the interaction between the boson field and the two-level system have attracted extensive interest again. The study of anisotropic generalized Rabi models enables us to better understand the novel physical properties of the interaction between light and matter in the ultra-strong and deep-strong coupling regions. In this work, the two-photon anisotropic Rabi-Stark model (tpARSM) is analytically solved by using the Bogoliubov operator approach and the su(1, 1) Lie algebra. We derive the G-function, whose zeros give the regular spectrum of the system. By studying the pole structure of the G-function and the coefficients in the function, exceptional solutions, including the first-order quantum phase transition points, doubly degenerate exceptional solutions and nondegenerate exceptional solutions, are obtained. By discussing the spectral structure, we give the conditions for the first-order quantum phase transition of tpARSM. Furthermore, we find that the property that all of the lowest doubly degenerate crossing points in the two-photon Rabi-Stark model have the same energy only holds for the special case of the tpARSM in which the anisotropy parameter is equal to 1. Finally, from the perspective of first-order quantum phase transitions, concise conditions for the ground state energy level to collapse to or escape from the collapse point for the tpARSM are presented. A good understanding of the tpARSM will lay a good foundation for studying the extended two-photon systems involving multiple levels and multiple bosonic modes, and even the relevant open quantum systems.
Various quantum phase transitions in the anisotropic Rabi-Stark model with both the nonlinear Stark coupling and the linear dipole coupling between a two-level system and a single-mode cavity are studied in this work. The first-order quantum phase transitions are detected by the level crossing of the ground-state and the first-excited state with the help of the pole structure of the transcendental functions derived by the Bogoliubov operators approach. As the nonlinear Stark coupling is the same as the cavity frequency, this model can be solved by mapping to an effective quantum oscillator. All energy levels close at the critical coupling in this case, indicating continuous quantum phase transitions. The critical gap exponent is independent of the anisotropy as long as the counter-rotating wave coupling is present, but essentially changed if the counter-rotating wave coupling disappears completely. It is suggested that the gapless Goldstone mode excitations could appear above a critical coupling in the present model in the rotating-wave approximation.
PTPT\mathcal {PT}PT‐Symmetric Quantum Rabi Model: Solutions and Exceptional Points
Abstract The ‐symmetric non‐Hermitian quantum Rabi model (QRM) with imaginary coupling is solved using the Bogoliubov operators approach. A transcendental function responsible for the exact solutions is derived, with its zeros yielding the regular spectrum. Two types of intersections: the exceptional points (EPs), which are well‐studied in the non‐Hermitian system; and another arising from doubly degenerate states due to the conserved QRM parity, which is well‐known in the Hermitian QRM, are found. These intersections are identified through this transcendental function. EPs emerge between pairs of adjacent excited energy levels, shifting toward lower coupling strengths as energy levels increase, and can also be predicted by a generalized rotating‐wave approximation approach. The fidelity susceptibility diverges to negative infinity at the EPs, consistent with recent findings in non‐Hermitian systems, while it diverges to positive infinity at the doubly degenerate points. The EPs are further confirmed by the vanishing c‐product in the biorthogonal basis.
The anisotropic two-photon quantum Rabi model is studied using the Bogoliubov operator approach. Doubly degenerate exceptional states are identified using analytical methods. By adjusting the position of the last exceptional point belonging to two adjacent energy levels, we derive a condition for the absence of the discrete spectrum at the critical coupling where ``spectral collapse'' occurs. In this special case, the spectrum becomes fully continuous above a threshold energy, with no bound states existing, whereas the ground state remains gapped in the general case. More interestingly, we rigorously find that a finite number of bound states exist between two anisotropy-dependent critical atomic frequencies, with infinitely many bound states beyond this frequency regime. In this manner, all issues in the two-photon quantum Rabi model are resolved.
The analytically exact solutions to the mixed quantum Rabi model including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4×44 \times 44×4 determinants responsible for the exact solutions are derived. These so-called GGG-functions with pole structure can be reduced to the previous ones in the unmixed models. The zeros of GGG-functions reproduce completely the regular spectra. The exceptional eigenvalues can be also obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point 1/21/21/2. Surprisingly, all energy levels only collapse to the lower one, which diverges negatively. All the level crossings in the unmixed models are relaxed to avoided crossings in the present mixed model due to absence of parity symmetry. Very interestingly, the effective one-photon coupling is enhanced in the presence of the two-photon coupling, which paves a highly efficient and economic way to access the deep-strong coupling regime even though the original one-photon coupling is weak.