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arXiv:2610.21556

Finite-Frequency Boson-Number Bounds Across the Quantum Rabi Transition

Razmik G. Unanyan, Michael Fleischhauer, Nikolay V. Vitanov

quant-phSubmitted Oct 15, 2026• Announced Oct 16, 2026• 0 citations
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Abstract

We consider the quantum Rabi model, a fundamental model of quantum optics describing a two-level system coupled to a bosonic mode, including counter-rotating terms. In the limit of large separation between the two-level transition frequency and the bosonic mode frequency, the model exhibits a quantum phase transition at a critical coupling strength. Although the model is exactly solvable, simple analytic expressions for important physical quantities are generally unavailable. We derive rigorous upper and lower bounds for the ground-state energy and boson occupation number across the quantum phase transition at finite frequency ratios. Our results characterize the subcritical, critical, and supercritical regimes and establish the scaling behavior of the boson occupation near the critical point. Exact numerical diagonalization demonstrates the tightness of the derived bounds, while the resulting finite-frequency inequalities provide experimentally testable predictions for trapped-ion quantum simulators.

Comments
11 pages, 3 figures

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We consider the quantum Rabi model, a fundamental model of quantum optics describing a two-level system coupled to a bosonic mode, including counter-rotating terms. In the limit of large separation between the two-level transition frequency and the bosonic mode frequency, the model exhibits a quantum phase transition at a critical coupling strength. Although the model is exactly solvable, simple analytic expressions for important physical quantities are generally unavailable. We derive rigorous upper and lower bounds for the ground-state energy and boson occupation number across the quantum phase transition at finite frequency ratios. Our results characterize the subcritical, critical, and supercritical regimes and establish the scaling behavior of the boson occupation near the critical point. Exact numerical diagonalization demonstrates the tightness of the derived bounds, while the resulting finite-frequency inequalities provide experimentally testable predictions for trapped-ion quantum simulators.
We investigate the two-mode quantum Rabi model (QRM) describing the interaction between a two-level atom and a two-mode cavity field. The quantum phase transitions are found when the ratio η of transition frequency of atom to frequency of cavity field approaches infinity. We apply the Schrieffer–Wolff (SW) transformation to derive the low-energy effective Hamiltonian of the two-mode QRM, thus yielding the critical point and rich phase diagram of quantum phase transitions. The phase diagram consists of four regions: a normal phase, an electric superradiant phase, a magnetic superradiant phase and an electromagnetic superradiant phase. The quantum phase transition between the normal phase and the electric (magnetic) superradiant phase is of second order and associates with the breaking of the discrete Z 2 symmetry. On the other hand, the phase transition between the electric superradiant phase and the magnetic superradiant phase is of first order and relates to the breaking of the continuous U(1) symmetry. Several important physical quantities, for example the excitation energy and average photon number in the four phases, are derived. We find that the excitation spectra exhibit the Nambu–Goldstone mode. We calculate analytically the higher-order correction and finite-frequency exponents of relevant quantities. To confirm the validity of the low-energy effective Hamiltonians analytically derived by us, the finite-frequency scaling relation of the averaged photon numbers is calculated by numerically diagonalizing the two-mode quantum Rabi Hamiltonian.
Abstract We explore an extended quantum Rabi model describing the interaction between a two-mode bosonic field and a three-level atom. Quantum phase transitions of this few degree of freedom model is found when the ratio η of the atom energy scale to the bosonic field frequency approaches infinity. An analytical solution is provided when the two lowest-energy levels are degenerate. According to it, we recognize that the phase diagram of the model consists of three regions: one normal phase and two superradiant phases. The quantum phase transitions between the normal phase and the two superradiant phases are of second order relating to the spontaneous breaking of the discrete Z 2 symmetry. On the other hand, the quantum phase transition between the two different superradiant phases is discontinuous with a phase boundary line relating to the continuous U (1) symmetry. For a large enough but finite η , the scaling function and critical exponents are derived analytically and verified numerically, from which the universality class of the model is identified.
We consider the Rabi Hamiltonian, which exhibits a quantum phase transition (QPT) despite consisting only of a single-mode cavity field and a two-level atom. We prove QPT by deriving an exact solution in the limit where the atomic transition frequency in the unit of the cavity frequency tends to infinity. The effect of a finite transition frequency is studied by analytically calculating finite-frequency scaling exponents as well as performing a numerically exact diagonalization. Going beyond this equilibrium QPT setting, we prove that the dynamics under slow quenches in the vicinity of the critical point is universal; that is, the dynamics is completely characterized by critical exponents. Our analysis demonstrates that the Kibble-Zurek mechanism can precisely predict the universal scaling of residual energy for a model without spatial degrees of freedom. Moreover, we find that the onset of the universal dynamics can be observed even with a finite transition frequency.
In this paper, we analyze the quantum criticality of the Rabi-Stark model at finite ratios of the qubit and cavity frequencies in terms of the energy gap and the order parameter, as well as the fidelity, if the Stark coupling strength is the same as the cavity frequency. The critical exponents are derived analytically. The energy gap and the length critical exponents are different from those in the quantum Rabi model and the Dicke model. A finite-size scaling analysis for the order parameter and the fidelity susceptibility is also performed. The universal scaling behaviors are demonstrated, and several finite-size exponents then can be extracted. Furthermore, universal critical behavior also can be established in terms of the bosonic Hilbert space truncation number, and the corresponding critical scaling exponents are found. Interestingly, the critical correlation length exponents in terms of the photonic truncation number as well as the equivalently effective length scales are different in the Rabi-Stark model and the quantum Rabi model, suggesting they belong to different universality classes. The second-order quantum phase transition is convincingly corroborated in the Rabi-Stark model at finite frequency ratios; by contrast, it emerges only at the infinite frequency ratio in the original quantum Rabi model without the Stark coupling.
The second-order quantum phase transition and quench dynamics in the two-mode Rabi model are investigated. We propose a diagonalization approach of jointly using a beam-splitter operator and a squeezed operator for each effective low-energy Hamiltonian when the ratio of the qubit transition frequency to the oscillator frequency approximates infinity. Eigenenergy and eigenstate of the normal and superradiant phases in the system are analytically derived, demonstrating the second-order quantum phase transition at a critical point. This critical point means that the requirement of coupling strength between the qubit and two oscillators is largely loosened by joint effect of two modes when compared with the standard Rabi model. The universal scaling between residual energy and quench time from ${\ensuremath{\tau}}^{\ensuremath{-}\frac{1}{3}}$ to ${\ensuremath{\tau}}^{\ensuremath{-}2}$ is confirmed.
As the simplest and most fundamental model describing the interaction between light and matter, a breakdown in the rotating wave approximation of the Rabi model leads to phase transition versus coupling strength when the frequency of the qubit greatly surpasses that of the oscillator. In addition to the phase transition revealed in the ground state, we show that the dynamics of physical quantities can reflect such a phase transition for this model. In addition to the excitation of the bosonic field in the ground state, we show that the witness of inseparability (entanglement), mutual information, quantum Fisher information, and the variance of cavity quadrature can be employed to detect the phase transition in quench. We also reveal the negative impact of temperature on checking the phase transition by quench. This model can be implemented using trapped ions, superconducting artificial atoms coupled bosonic modes, and quantum simulations. By reflecting the phase transition in a fundamental quantum optics model without imposing the thermodynamic limit, this work offers an idea to explore phase transitions by nonequilibrium process for open quantums.