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Solutions to the mixed quantum Rabi model

Liwei Duan, You-Fei Xie, Qing‐Hu Chen

Published Jul 7, 2018• 2 citations
OpenAlex

Abstract

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 $4 \times 4$ determinants responsible for the exact solutions are derived. These so-called $G$-functions with pole structure can be reduced to the previous ones in the unmixed models. The zeros of $G$-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 $1/2$. 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.

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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.
The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4 × 4 determinants responsible for the exact solutions are derived. These so-called G-functions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of G-functions reproduce completely the regular spectra. The exceptional eigenvalues can also be 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. All energy levels only collapse to the lower one, which diverges negatively. The level crossings in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM due to absence of parity symmetry. In the weak two-photon coupling regime, the mixed QRM is equivalent to an one-photon QRM with an effective positive bias, suppressed photon frequency and enhanced one-photon coupling, which may pave a highly efficient and economic way to access the deep-strong one-photon coupling regime.
Abstract The analytically exact solutions to the two-photon Rabi–Stark model are given by using the Bogoliubov operators approach. Transcendental functions responsible for the exact solutions are derived. The zeros of the transcendental functions reproduce completely the regular spectra. The first-order quantum phase transitions are also detected by the pole structure of the derived transcendental functions, similar to the one-photon Rabi–Stark model, different from the two-photon Rabi model where the energy levels for the ground-state and the first excited state do not cross. The spectra collapse characteristics resemble those in two-photon Rabi model. Some low lying levels can split off from the collapse energy, which is different from the one-photon Rabi–Stark model in which all levels collapse.
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.
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.
Abstract The quantum Rabi–Stark model, where the linear dipole coupling and the nonlinear Stark-like coupling are present on an equal footing, are studied within the Bogoliubov operators approach. Transcendental functions responsible for the exact solutions are derived in a compact way, much simpler than previous ones obtained in the Bargmann representation. The zeros of transcendental functions reproduce completely the regular spectra. In terms of the explicit pole structure of these functions, two kinds of exceptional eigenvalues are obtained and distinguished in a transparent manner. Very interestingly, a first-order quantum phase transition indicated by level crossing of the ground state and the first excited state is induced by the positive nonlinear Stark-like coupling, which is however absent in any previous isotropic quantum Rabi models. When the absolute value of the nonlinear coupling strength is equal to twice the cavity frequency, this model can be reduced to an effective quantum harmonic oscillator, and solutions are then obtained analytically. The spectra collapse phenomenon is observed at a critical coupling, while below this critical coupling, infinite discrete spectra accumulate into a finite energy from below.
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.