Jiong Li, Jun-Ling Wang, Qing-Hu Chen, and Hai-Qing Lin of Zhejiang University report that spectral collapse, long considered incapable of establishing quantum criticality due to a persisting lowest excitation gap, can indeed signal a continuous quantum phase transition. Their work demonstrates this occurs at a specific qubit frequency within the anisotropic two-photon quantum Rabi model, governed by a single soft mode.

The researchers found the same-parity gap, set by the vanishing effective oscillator frequency, closes with an exponent of zν = 1/2, placing the system in the same universality class as the standard quantum Rabi model and establishing spectral collapse as a potential path to experimentally accessible quantum criticality.

Two-Photon Rabi Model & Spectral Collapse Origin

Anisotropic interactions within the two-photon quantum Rabi model (tpQRM) enable quantum criticality at a single-qubit level, challenging previous assumptions about the conditions required for this phenomenon. For years, spectral collapse, where discrete energy levels merge into a continuum, was considered insufficient to establish quantum criticality because the lowest excitation gap typically remained finite under standard conditions. This limitation suggested that observing a true quantum phase transition via spectral collapse was unlikely, but new work demonstrates this is not universally true.

The research reveals a continuous quantum phase transition occurs at a specific qubit frequency, Δ_c = (1-r)/(1 + r), where ‘r’ defines the anisotropy of the two-photon interactions and 0 < r < 1. This critical condition is crucial; spectral collapse alone does not guarantee a transition, but when coupled with this specific qubit frequency, it initiates a change in the system’s behavior.

The team modeled this behavior using a Hamiltonian incorporating the qubit frequency, bosonic mode frequency, two-photon coupling strength, and anisotropy parameter, with ℏ = ω = 1 within the equation. This precise modeling allowed for the identification of conditions where the discrete spectrum terminates at a continuum threshold, signaling the phase transition. The study identifies the same-parity excitation as the “soft mode” driving this transition, with its associated gap serving as the characteristic energy scale.

This means the system’s behavior is governed by fluctuations in this specific excitation, rather than simply the closing of the lowest excitation gap, a previously held belief. The emergence of a lower, different-parity gap arises from symmetry-induced splitting within the system, a detail that further clarifies the underlying physics. Parity symmetry plays a critical role in restricting the dynamics of the system.

It excludes the different-parity excitation from contributing to the quantum Fisher information response, effectively confining Kibble-Zurek dynamics, a theory describing the formation of topological defects during a phase transition, to excitations within the ground-state parity sector. This restriction simplifies the analysis and reinforces the identification of the same-parity excitation as the primary driver of the transition. These findings establish spectral collapse as a viable pathway toward experimentally accessible quantum criticality in few-body systems. Two-photon interactions, which can be realized in circuit QED and trapped-ion platforms, offer a promising means of achieving this control.

While previous explorations of continuous quantum phase transitions required idealized limits or infinite frequency ratios, this work demonstrates that anisotropy can induce such transitions even at the single-qubit level. The authors note that “Although this transition has been experimentally explored and connected to quantum critical phenomena, it relies on an idealized limit,” emphasizing the significance of their results in bringing this phenomenon closer to practical realization. This work shifts the focus from simply observing spectral collapse to understanding how the structure of the soft mode dictates the universality class of the resulting quantum criticality.

Anisotropic tpQRM Defines Quantum Criticality via Soft Modes

The long-held assumption that spectral collapse could not definitively indicate quantum criticality, due to the persistence of a finite lowest excitation gap, has been challenged by new work on the anisotropic two-photon quantum Rabi model (tpQRM). Researchers demonstrated that spectral collapse, under specific conditions, does indeed mark a continuous quantum phase transition driven by a single, defining soft mode. This transition occurs at a particular qubit frequency, a detail crucial to observing the effect and distinguishing it from generic spectral collapse scenarios.

This is a significant departure from previous understandings, suggesting that the arrangement of excitations, rather than a simple gap closure, is the key determinant of critical behavior. The analysis reveals that as the system approaches criticality, the mean photon occupation diverges, scaling as ∝ |g − gc|−1/2, reflecting increasingly strong squeezing of the effective oscillator.

Experimental confirmation of these scaling relations, using numerical diagonalization, aligns quantitatively with the analytical approximations employed by the researchers. In the isotropic limit, where anisotropy parameter r equals 1 and the critical detuning Δc is zero, the different-parity gap vanishes entirely, while the same-parity gap continues to define the characteristic energy scale.

The implications extend beyond theoretical models, offering a potential route for realizing and studying quantum criticality in physical platforms like circuit QED and trapped-ion systems. The team’s results demonstrate that the anisotropic tpQRM exhibits the same critical behavior across a range of anisotropies, from 0 to 1, further solidifying its place within the established framework of quantum criticality.

Same-Parity Gap Governs Critical Exponents: ν = 1/4, z = 2

Exact diagonalization confirms exponents of zν = 1/2 for the same-parity gap and μ = 5/4 for the different-parity gap, revealing a distinct scaling behavior compared to the standard QRM which is in the same universality class. This disparity arises because the same-parity gap is directly linked to the vanishing effective oscillator frequency, while the different-parity gap stems from symmetry-induced splitting within the system.

The QFI, a sensitive probe of quantum criticality, reveals that only excited states sharing the same parity as the ground state contribute, restricting Kibble-Zurek dynamics to excitations within that specific parity sector. The analysis demonstrates that the low-energy spectrum exhibits an alternating parity structure near the collapse point, naturally separating these two distinct excitation gaps. Ground-state quadrature scaling further defines the critical exponents, with fluctuations in the quadratures diverging proportionally to, where ν = 1/4.

This divergence directly reflects the squeezed-oscillator structure of the low-energy state, a key indicator of the transition. Combining this with the relationship zν = 1/2 yields a dynamic critical exponent of z = 2. The findings demonstrate that the same-parity gap, not simply the lowest gap, governs both equilibrium and nonequilibrium critical behavior, ultimately determining the system’s universality class. This work clarifies that the soft-mode structure, rather than the mere closing of an energy gap, is the defining factor in establishing quantum criticality.

Z_4 Symmetry Classifies Low-Energy Spectrum & Parity Sectors

This symmetry, detailed in recent work, organizes these sectors into even- and odd-photon subspaces, providing a natural framework for classifying excitations and understanding the model’s critical behavior. Within each subspace, a residual $Z_{2}$ symmetry further distinguishes states by parity, creating a clear separation in the low-energy spectrum and influencing which excitations dominate responses to external stimuli. This classification is not merely organizational; parity symmetry fundamentally restricts the dynamics near the spectral collapse point.

The researchers demonstrate that the different-parity excitation is excluded from the quantum Fisher information (QFI) response because where Π is the $Z_{4}$ symmetry generator and H is the Hamiltonian. The contrasting behavior of the same- and different-parity gaps is central to understanding how spectral collapse can signal a continuous quantum phase transition.

The diagonal adiabatic approximation (AA) is important for analyzing the low-energy spectrum under these critical conditions. Asymptotically exact for the low-energy spectrum, the AA accurately reproduces the levels as they approach the continuum threshold $E_{c}$ = −1/2 as g → $g_{c}$ . However, off-diagonal couplings between different squeezed-photon manifolds become important when δ ≠ 0, requiring a precise understanding of the excitation hierarchy. The team focused on the q = 1/4 subspace containing the ground state to clarify which excitation controls singular critical responses, ultimately confirming that the lowest same-parity excitation dictates the critical behavior.

This structure is not simply a consequence of the model’s parameters but a fundamental feature arising from the interplay between symmetry and dynamics. The researchers note that these two gaps exhibit “distinct asymptotic scalings and play fundamentally different roles in the critical behavior,” solidifying the importance of parity classification.

Finite Coupling Enables Accessible Quantum Criticality in Few-Body Systems

The lowest excitation gap, long considered an impediment to achieving quantum criticality via spectral collapse, does not preclude a continuous quantum phase transition under specific conditions. This finding challenges the previously held belief that a finite excitation gap inherently prevented spectral collapse from signaling true quantum criticality, opening avenues for exploration in few-body systems. This refined understanding moves beyond simply observing spectral collapse to pinpointing the specific excitation responsible for driving the transition and defining its energy scale.

The work establishes a potential route to experimentally accessible quantum criticality, a feat previously limited by the need for idealized conditions. While experimental exploration of this transition has occurred, it historically relied on these idealized limits; realizing criticality in few-body systems under practical conditions has remained a challenge. The quantum Fisher information diverges at Δ = $Δ_{c}$ , confirming the sensitivity of this probe to quantum criticality.

Beyond the conceptual insight, this work identifies spectral collapse, under critical conditions, as a mechanism for universal critical behavior at finite coupling and finite qubit frequency. This provides, in principle, an experimentally accessible route to quantum criticality in few-body light-matter systems. The findings may also enable the preparation of strongly squeezed states and enhanced quantum metrology in nonlinear light-matter platforms operating in the ultrastrong-coupling regime.

👉 More information
🗞 Quantum Criticality from Spectral Collapse in the Two-Photon Rabi Model
✍️ Jiong Li, Jun-Ling Wang, Qing-Hu Chen and Hai-Qing Lin
🧠 DOI: http://link.aps.org/doi/10.1103/ctp9-k77q

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