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Chinese Physics B

Unified approach to various quantum Rabi models with arbitrary parameters*

Xiao-Fei Dong, You-Fei Xie, Qing‐Hu Chen

Published Dec 23, 2019• 0 citations
Publisher

Abstract

A general approach is proposed to the quantum Rabi model and its several variants within the extended coherent states. The solutions to all these models including the anisotropy and the nonlinear Stark coupling are then obtained in an unified way. The essential characteristics such as the possible first-order phase transition can be detected analytically. This approach can be easily applied to the recent experiments with various tunable parameters without much additional effort, so it should be very helpful to the analysis of the experimental data.

DOI
10.1088/1674-1056/ab6555

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A general approach is proposed to the quantum Rabi model and its several variants within the extended coherent states. The solutions to all these models including the anisotropy and the nonlinear Stark coupling are then obtained in an unified way. The essential characteristics such as the possible first-order phase transition can be detected analytically. This approach can be easily applied to the recent experiments with various tunable parameters without much additional effort, so it should be very helpful to the analysis of the experimental data.
Abstract The quantum Rabi-Stark model, where the linear dipole coupling and the nonlinear Stark-like coupling is present on an equal footing, is studied within the tunable extended coherent states. The eigenvalues and eigenstates are therefore obtained exactly. Surprisingly, the entanglement entropy in the ground-state is found to jump suddenly with the coupling strength. The first-order quantum phase transition can be detected by level crossing of the ground state and the first excited state, which is however lacking in the original linear quantum Rabi model. Performing the first-order approximation in the present theory, we can derive closed-form analytical results for the ground-state. Interestingly, it agrees well with the exact solutions up to the ultra-strong coupling regime in a wide range of model parameters. The spectral collapses when the absolute value of the nonlinear coupling strength approaches to twice the cavity frequency is observed with the help of new solutions in the limits .
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.
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 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.
We present our recent studies on the quantum Rabi model (QRM). Firstly, by using a variational wave function, which facilitates to extract physics in entire parameter regime with high accuracy, we unveil a ground-state phase diagram of the QRM and argue that the main constituents are polaron and anti-polaron. Secondly, introducing an anisotropy into the QRM, in which the rotating- and counter-rotating terms are allowed to have different coupling strength, we explore that how the criticality of the model interpolates between two known limits with distinct universal properties. Through a combination of analytic and numerical approaches we compute phase diagram, scaling functions and critical exponents, and establish that the universality class at the finite anisotropy is the same as that of the isotropic limit.
In this Brief Report, we present a perturbation theory that omits the weak multiphoton process to obtain analytical solutions for the Rabi model. Under this approximation, the Rabi model can be mapped into the solvable Jaynes-Cummings-like model by choosing a proper variation parameter. As a result, both the ground- and excited-state-energy spectrums and wave functions can be derived easily. Moreover, these explicit results agree well with the direct numerical simulation in a wide range of the experimental parameters. In addition, based on our obtained energy spectrums, the recent experimental observation of the Bloch-Siegert shift in the circuit quantum electrodynamics with ultrastrong coupling can be explained perfectly. We believe that our results have potential applications in solid-state quantum-information processing.