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arXiv:2510.20201

Strong-coupling asymptotics for anisotropic Rabi model

Masao Hirokawa, Fumio Hiroshima, D. Lee

Published Oct 23, 2025• 0 citations
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Abstract

The anisotropic quantum Rabi model provides a continuous interpolation between the quantum Rabi model and the Jaynes-Cummings model. In this paper, we study the norm-resolvent limit of the anisotropic quantum Rabi model as the coupling strength tends to infinity. After subtracting an explicit quadratic renormalization term, we identify the limiting Hamiltonian, which determines the asymptotic spectral structure of the model. Our result provides a unified description connecting the Jaynes-Cummings model and the full quantum Rabi model in the strong-coupling limit, thereby revealing the dominant role of the counter-rotating interaction.

DOI
10.48550/arxiv.2510.20201

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The anisotropic quantum Rabi model provides a continuous interpolation between the quantum Rabi model and the Jaynes-Cummings model. In this paper, we study the norm-resolvent limit of the anisotropic quantum Rabi model as the coupling strength tends to infinity. After subtracting an explicit quadratic renormalization term, we identify the limiting Hamiltonian, which determines the asymptotic spectral structure of the model. Our result provides a unified description connecting the Jaynes-Cummings model and the full quantum Rabi model in the strong-coupling limit, thereby revealing the dominant role of the counter-rotating interaction.
The anisotropic Rabi model, which was proposed recently, differs from the original Rabi model: the rotating and counter-rotating terms are governed by two different coupling constants. This feature allows us to vary the counter-rotating interaction independently and explore the effects of it on some quantum properties. In this paper, we eliminate the counter-rotating terms approximately and obtain the analytical energy spectrums and wavefunctions. These analytical results agree well with the numerical calculations in a wide range of the parameters including the ultrastrong coupling regime. In the weak counter-rotating coupling limit we find out that the counter-rotating terms can be considered as the shifts to the parameters of the Jaynes-Cummings model. This modification shows the validness of the rotating-wave approximation on the assumption of near-resonance and relatively weak coupling. Moreover, the analytical expressions of several physics quantities are also derived, and the results show the break-down of the U(1)-symmetry and the deviation from the Jaynes-Cummings model.
We investigate the quantum phase transition of the anisotropic quantum Rabi model, in which the rotating and counterrotating terms are allowed to have different coupling strengths. The model interpolates between two known limits with distinct universal properties. Through a combination of analytic and numerical approaches, we extract the phase diagram, scaling functions, and critical exponents, which determine the universality class at finite anisotropy (identical to the isotropic limit). We also reveal other interesting features, including a superradiance-induced freezing of the effective mass and discontinuous scaling functions in the Jaynes-Cummings limit. Our findings are extended to the few-body quantum phase transitions with N>1 spins, where we expose the same effective parameters, scaling properties, and phase diagram. Thus, a stronger form of universality is established, valid from N=1 up to the thermodynamic limit.
In this manuscript, we analyze the mechanism of the superradiant phase transition in the anisotropic Rabi model under the classical oscillator limit using the pattern picture. By expanding the anisotropic Rabi model Hamiltonian in operator space, we obtained three patterns, and we find that the phase transition arises from the competition between patterns. The difficulty in achieving the classical oscillator limit motivates our investigation into the quantum phase transition within a parametrically-driven Jaynes-Cummings model. This parametrically-driven Jaynes-Cummings model can reproduce the dynamics of an ultrastrong-coupling anisotropic Rabi model in a squeezed-light frame. According to the eigenenergies and eigenstates of the normal and superradiant phases of this equivalent anisotropic Rabi model, we find that the excitation energy of the normal phase and the superradiant phase vanishes at the critical point. The photon number becomes infinite beyond the critical point. These results indicate that the system undergoes a superradiant phase transition at the critical point.
Abstract The quantum Rabi model captures fundamental aspects of light--matter interaction, where symmetry dictates both spectra and dynamics. Over the past years, experiments have explored many of its nonperturbative properties, but have mostly focused on the isotropic limit, where rotating and counterrotating processes are locked together, leaving the broader symmetry landscape largely unexplored. Here we realize a programmable anisotropic quantum Rabi model in a superconducting processor, with independent control of the rotating and counterrotating couplings (g1,g2)(g_1,g_2)(g1​,g2​) and of a transverse bias ε\varepsilonε. Continuous anisotropy tuning, combined with a duality mapping, gives access to the full parameter space from the Jaynes-Cummings to the anti-Jaynes-Cummings limits. In the deep-strong-coupling regime, we show that anisotropy reconstructs the spectrum and turns complete collapse-revival dynamics into incomplete revivals even near degeneracy. With adiabatic state preparation and joint tomography, we resolve an anisotropy-induced ground-state parity switch, a crossing that has no analogue in the isotropic model. We further observe selective tunnelling associated with hidden symmetry in biased Rabi models and track its anisotropic displacement within the same device. These results establish a controllable route to engineering nonperturbative light--matter Hamiltonians, where symmetry, spectrum, and dynamics can be programmed independently.
Anisotropic quantum Rabi model is a generalization of quantum Rabi model, which allows its rotating and counter-rotating terms to have two different coupling constants. It provides us with a fundamental model to understand various physical features concerning quantum optics, solid-state physics, and mesoscopic physics. In this paper, we propose an experimental feasible scheme to implement anisotropic quantum Rabi model in a circuit quantum electrodynamics system via periodic frequency modulation. An effective Hamiltonian describing the tunable anisotropic quantum Rabi model can be derived from a qubit-resonator coupling system modulated by two periodic driving fields. All effective parameters of the simulated system can be adjusted by tuning the initial phases, the frequencies and the amplitudes of the driving fields. We show that the periodic driving is able to drive a coupled system in dispersive regime to ultrastrong coupling regime, and even deep-strong coupling regime. The derived effective Hamiltonian allows us to obtain pure rotating term and counter-rotating term. Numerical simulation shows that such effective Hamiltonian is valid in ultrastrong coupling regime, and stronger coupling regime. Moreover, our scheme can be generalized to the multi-qubit case. We also give some applications of the simulated system to the Schrödinger cat states and quantum gate generalization. The presented proposal will pave a way to further study the stronger anisotropic Rabi model whose coupling strength is far away from ultrastrong coupling and deep-strong coupling regimes in quantum optics.