Quantum criticality from spectral collapse in the two-photon Rabi model

Abstract: Spectral collapse in the two-photon quantum Rabi model (tpQRM) has long been regarded as incompatible with quantum criticality due to the absence of a vanishing excitation gap. We show that, in the anisotropic tpQRM, spectral collapse constitutes a genuine continuous quantum phase transition governed by a single soft mode. The excitation gap within the same parity closes as $ε_{sp} \sim |g - g_c|^{zν}$ with $zν= 1/2$, placing the system in the same universality class as the standard QRM, while the gap between different parities reflects symmetry-induced level splitting rather than a critical excitation. This soft mode defines a unique energy scale that controls both equilibrium and nonequilibrium properties, including macroscopic observables, quantum Fisher information, and Kibble-Zurek dynamics. These results establish spectral collapse as an experimentally accessible realization of quantum criticality in a few-body system and demonstrate that universality is fully determined by the soft-mode structure rather than by microscopic details.

• The paper demonstrates that spectral collapse in the two-photon Rabi model drives quantum criticality through a dominant soft mode.
• It employs an adiabatic approximation to derive universal scaling laws for equilibrium gaps, photon number divergence, and quadrature fluctuations.
• The findings provide rigorous insights for quantum simulation, sensor engineering, and nonlinear quantum optics in light–matter systems.

Quantum Criticality and Spectral Collapse in the Two-Photon Rabi Model

Introduction

The paper addresses quantum criticality originating from spectral collapse in the two-photon quantum Rabi model (tpQRM), with a focus on the adiabatic approximation, the nature of critical excitations, and universal scaling behaviors. The tpQRM extends canonical quantum light–matter interaction models by incorporating quadratic (two-photon) couplings, and exhibits distinctive spectral and dynamical phenomena, such as collapse of discrete spectra into continuous bands at a critical coupling. The analysis covers both the anisotropic ($r \neq 1$) and isotropic ($r = 1$) regimes, delineates microscopic mechanisms for collapse, and establishes scaling laws for equilibrium and nonequilibrium observables.

Asymptotic Exactness of the Adiabatic Approximation

The adiabatic approximation (AA) provides an asymptotically exact low-energy description near the critical line in the tpQRM. The model's Hamiltonian includes anisotropic two-photon terms and parity symmetry, allowing decomposition into squeezing-transformed photonic subsectors. Introducing the squeezing parameter $\theta$, the effective frequency $\beta = \sqrt{1 - g^2 / g_c^2}$ vanishes at the critical coupling $g_c$, reducing the system to coupled harmonic pseudomodes.

A systematic hierarchy of corrections shows that off-diagonal couplings are parametrically suppressed as $\beta \to 0$, and the AA captures the emergent soft mode physics controlling quantum criticality. Eigenstates in the AA neglect these suppressed couplings, and resulting eigenenergies reveal two distinct excitation gaps:

• Intra-parity gap $$\epsilon_{\mathrm{sp}} \propto \beta \propto |g - g_c|^{z\nu}$$ with $z\nu = 1/2$.
• Inter-parity gap $$\epsilon_{\mathrm{dp}} \propto \beta^{5/2} \propto |g - g_c|^{\mu}$$ with $\mu = 5/4$.

Perturbative analysis confirms that AA remains valid at criticality ($r = 1$0), and off-diagonal corrections to eigenstates and energies decay with higher exponents, yielding a controlled hierarchy. This demonstrates universal scaling governed by a single soft mode, robust against anisotropy.

Spectral Collapse Mechanism in Isotropic tpQRM

For the isotropic case ($r = 1$1), the tpQRM Hamiltonian simplifies in canonical phase space, exposing the collapse point as the disappearance of harmonic confinement ($r = 1$2). At $r = 1$3, the effective potential flattens along one quadrature.

At $r = 1$4, the model decouples into position and momentum sectors; eigenstates become non-normalizable, marking the appearance of a continuous spectrum above $r = 1$5, eliminating discrete bound states and resulting in spectral collapse. For finite $r = 1$6, the effective potential acquires an inverse-square tail supporting an infinite tower of bound states accumulating at the threshold, exhibiting a geometric sequence analogous to Efimov scaling [zulli_universal_2025].

Parity symmetry at collapse, $r = 1$7, implies double degeneracy, while intra-parity level spacing is dictated by the soft mode, confirming the soft mode as sole critical excitation.

Scaling of Observables and Quantum Metrology

Critical scaling of observables within AA is controlled fully by the intra-parity gap $r = 1$8. The photon number diverges as $r = 1$9, quadrature fluctuations scale as $\theta$0, and the atomic order parameter (polarization) decays as $\theta$1.

Quantum Fisher Information (QFI) shows maximal divergence permitted for pure Hermitian systems, $\theta$2, highlighting extreme sensitivity to parameter variations near criticality. All static and metrological observables obey universal scaling laws dictated by the soft mode, confirming the tpQRM’s universality class according to quantum critical dynamics [ashhab_superradiance_2013] [hwang_quantum_2015] [ying_critical_2025].

Kibble-Zurek Scaling and Nonequilibrium Dynamics

Under slow linear quenches across the critical point, residual excitation energy is governed by the same soft gap $\theta$3. Adiabatic perturbation theory yields leading-order transitions within the same parity sector, suppressed further by parity selection rules.

Kibble-Zurek scaling analysis leads to the relationship:

$\theta$4

with $\theta$5, giving $\theta$6. This direct correspondence between equilibrium gap scaling and nonequilibrium dynamics confirms dynamical universality derived from the equilibrium soft mode.

Implications and Outlook

The spectral collapse mechanism and critical scaling in the tpQRM present a microscopic foundation for universality in quantum phase transitions driven by non-linear light–matter couplings. The essential conclusions—parametric suppression of off-diagonal AA corrections near criticality, dominance of the soft mode in both equilibrium and dynamical scaling, and precise quantum metrological behavior—apply throughout the anisotropic parameter regime. These findings provide rigorous quantitative insight for future quantum simulation, sensor engineering, and the study of nonlinear quantum optics in circuit QED, trapped ions, and hybrid platforms [wang_strong_2025] [chen_experimental_2021] [lv_quantum_2018]. The results suggest extensions to higher-order multiphoton Rabi models and exploration of universality in finite-component systems.

Conclusion

This paper establishes that quantum criticality in the two-photon Rabi model emerges via spectral collapse and is governed universally by a single soft mode. The adiabatic approximation is rigorously justified near criticality, with analytic and numerical scaling results encompassing equilibrium gaps, critical observables, and dynamical quench protocols. The theoretical framework supports ongoing and future experimental realizations of quantum phase transitions in nonlinear light–matter systems, and refines understanding of universality in quantum critical phenomena.