Anomalous Superradiant Phase in Light–Matter Systems

• Anomalous superradiant phases are unconventional light–matter states characterized by deviations from standard Dicke transitions, featuring unique symmetry breaking and gauge-invariant order parameters.
• They emerge in diverse models, from anisotropic Rabi dynamics to squeezed-frame Jaynes–Cummings settings, effectively bypassing traditional TRK constraints and no-go theorems.
• These phases offer practical insights for experimental realizations in circuit QED and free-space platforms, driving advances in quantum phase transitions and photonic device engineering.

An anomalous superradiant phase is a context-dependent extension of superradiance in which the ordered light–matter state departs from the standard Dicke picture of a thermodynamic-limit instability into a macroscopically occupied bosonic mode. In the literature, the anomaly may refer to superradiant criticality in a finite-component system, a phase that survives Thomas–Reiche–Kuhn and $A^2$-term constraints, a non-equilibrium transition with local order parameters but without conventional symmetry breaking, a current-carrying or topological superradiant state, or a squeezing-induced superradiant regime with complex excitation spectra and effective non-Hermitian structure (Qiu et al., 5 Nov 2025, Liu et al., 2021, Ferioli et al., 2022, Li et al., 3 Mar 2025, Wang et al., 25 Jun 2026). The term therefore denotes a family of nonstandard superradiant phenomena rather than a single universality class.

1. Conceptual scope and relation to standard superradiance

In canonical light–matter models, the normal–superradiant transition is the instability of the photon vacuum toward a state with macroscopic photonic and matter excitations. A useful baseline is the analysis of Dicke, Tavis–Cummings, quantum Rabi, and Jaynes–Cummings models, where the transition is continuous and mean-field, but quantum fluctuations are negligible in Dicke and Rabi limits and strictly zero in Tavis–Cummings and Jaynes–Cummings because conserved symmetry sectors do not mix (Larson et al., 2016). In that sense, the superradiant phase is already “anomalous” relative to textbook quantum criticality: there is a non-analytic ground state and symmetry breaking, yet the instability is not fluctuation-driven.

A second conceptual strand emphasizes that what appears as “superradiant” can depend on how radiation is defined. In arbitrary-gauge cavity QED, the unique thermodynamic transition is signaled by a macroscopic gauge-invariant polarization, while the extent to which that abnormal phase is called superradiant depends on whether longitudinal electric degrees of freedom are included in the radiative subsystem (Stokes et al., 2019). This shifts the focus from a macroscopic cavity amplitude alone to gauge-invariant polarization as the robust order parameter.

A third usage reserves “anomalous” for phases whose structure is absent in single-mode Dicke physics: finite-component superradiance, photon currents, frustration-induced criticality, Meissner-like cancellation, edge-state amplification, or $\mathcal{PT}$-broken superradiant spectra (Qiu et al., 5 Nov 2025, Li et al., 3 Mar 2025, Lu et al., 24 Jun 2026, Wang et al., 25 Jun 2026). Across these usages, the common element is a superradiant or superradiant-like ordered state whose mechanism, symmetry structure, or critical scaling differs from the standard cavity mean-field scenario.

2. Microscopic mechanisms in zero-dimensional and few-component systems

A central finite-component mechanism is realized in the anisotropic Rabi model and its parametrically driven Jaynes–Cummings realization. The anisotropic Rabi Hamiltonian,

$$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$

has the critical condition

$\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$

with the quantum phase transition at $\xi_c=1$. In the parametrically driven Jaynes–Cummings model,

$$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$

a squeezing transformation with $\tanh(2r)=\eta/\delta_c$ maps the system to an effective anisotropic Rabi model with $$\omega_{0,\mathrm{eff}}=\delta_c\,\mathrm{sech}(2r)$$, $g_1=g\cosh(2r)$, $g_2=g\sinh(2r)$, and anisotropy $\mathcal{PT}$0. The critical point becomes $\mathcal{PT}$1, with

$\mathcal{PT}$2

so parametric amplification suppresses the effective oscillator frequency while enhancing anisotropy and effective couplings (Qiu et al., 5 Nov 2025).

The microscopic origin of that transition is formulated as a competition among three exact operator-space “patterns.” In the normal phase, the ground-state energy is dominated by one pattern; near criticality that contribution decreases sharply, while the other two compensate and stabilize the superradiant solution. This pattern picture makes the anomalous aspect explicit: the system is a single-qubit, finite-component model, yet it displays superradiant criticality in the classical oscillator limit, with gap closing and divergent photon number in the closed theory (Qiu et al., 5 Nov 2025).

A distinct route to anomalous superradiance appears in interacting atomic ensembles with the diamagnetic term included. For coupled two-level atoms, attractive XY interactions lower the critical coupling to

$\mathcal{PT}$3

creating a superradiant window

$\mathcal{PT}$4

under

$\mathcal{PT}$5

The anomaly is that superradiance becomes possible while still satisfying the TRK sum rule (Liu et al., 2021).

Multilevel structure yields a related mechanism. For three-level systems, excited-level transitions redistribute oscillator strength so that TRK constraints and the diamagnetic term do not universally forbid superradiance. Ladder and $\mathcal{PT}$6 configurations can support superradiant regions compatible with TRK bounds, and the additional nonlinear coupling between branches can drive first-order superradiant transitions, which are absent in the two-level Dicke model (Baksic et al., 2012).

3. Criticality, order parameters, and symmetry structure

The order parameters of anomalous superradiant phases depend on the model. In the anisotropic Rabi and squeezed-frame parametrically driven Jaynes–Cummings settings, the key observables are the cavity amplitude $\mathcal{PT}$7, the photon number $\mathcal{PT}$8, and spin polarization. In the normal phase $\mathcal{PT}$9; in the superradiant phase of the closed effective model $$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$0, the field develops two displaced branches $$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$1, and the excitation gaps of both normal and superradiant phases vanish at criticality. The ground-state energy remains continuous, but its second derivative in coupling is discontinuous, indicating a second-order transition (Qiu et al., 5 Nov 2025).

Symmetry remains central, but not uniformly so. In Rabi-type models, the relevant broken symmetry is $$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$2 parity, yielding two degenerate displaced minima; in Tavis–Cummings and Jaynes–Cummings models, a continuous $$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$3 symmetry instead produces a Goldstone mode in the closed thermodynamic limit (Larson et al., 2016). The distinction matters because dissipation affects them differently: open Dicke and Rabi models remain critical, whereas open Tavis–Cummings and Jaynes–Cummings models lose steady-state criticality and relax to the trivial vacuum (Larson et al., 2016).

Not all anomalous superradiant transitions rely on conventional spontaneous symmetry breaking. In free space, a driven collectively decaying ensemble of two-level atoms realizes a non-equilibrium transition between a magnetized, phase-locked regime and a superradiant-emission regime, controlled by

$$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$4

The threshold is $$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$5, equivalently $$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$6. Below threshold, a collective dipole screens the drive; above threshold, cooperative spontaneous emission dominates and scales as $$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$7. The work explicitly characterizes this as a phase transition without conventional symmetry breaking but with local order parameters such as $$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$8, $$H_{\mathrm{AR}}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\xi_1(a\sigma_+ + a^\dagger \sigma_-) - \xi_2(a\sigma_- + a^\dagger \sigma_+),$$9, and $\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$0 (Ferioli et al., 2022).

4. Current-carrying, frustrated, topological, and non-Hermitian variants

Spatial structure and synthetic gauge fields generate anomalous superradiant phases whose order is encoded in currents or nontrivial band topology. In a quantum Rabi zigzag chain with staggered flux, the Meissner superradiant phase is defined by persistent counter-flowing edge currents on the even and odd legs, with vanishing net current: $\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$1 It is accompanied by anomalous gap exponents. For the frustrated $\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$2 case, one soft mode closes with $\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$3, while another closes with $\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$4 on one side and $\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$5 on the other; for the unfrustrated $\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$6 case, the two lowest modes close with $\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$7 and $\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$8. The same model also supports even-chiral, odd-chiral, and ferromagnetic superradiant phases separated by first-order boundaries (Li et al., 3 Mar 2025).

Broken time-reversal symmetry in Dicke lattices produces a related but distinct anomaly. Complex hopping makes the photon dispersion asymmetric, so the normal-phase gap can close linearly, $\xi_c=\frac{\xi_1+\xi_2}{\sqrt{\omega_0\Omega}},$9, rather than with the mean-field square root. In that anomalous normal phase, local photon number, quadrature variances, and equal-time correlations remain bounded at criticality, while frustrated superradiant phases appear on the ordered side and are separated by first-order lines (Zhao et al., 2022).

Open coupled-cavity spin systems allow a further departure from equilibrium intuition. In the spontaneous superradiant photon-current phase, there is no external pump, yet in the superradiant regime a steady current

$\xi_c=1$0

flows from a lower-frequency cavity to a higher-frequency cavity. The mechanism requires superradiance, cavity loss, and broken mode symmetry: dissipation fixes nontrivial phase relations, counter-rotating processes extract photons from the vacuum, and the steady current is balanced by losses (Qiao et al., 2024).

Topological generalizations add band inversion and edge amplification. In an SSH-like quantum Rabi array with species-dependent phases $\xi_c=1$1 and $\xi_c=1$2, the system supports three superradiant phases, including asymmetric quadrature-condensed phases with edge photons accumulating at opposite boundaries. Gap closing is anomalous relative to standard SSH because it requires not only $\xi_c=1$3 at $\xi_c=1$4 but also matching of quadrature-dependent pairing scales, including a nontrivial superradiance-induced closure at $\xi_c=1$5 (Lu et al., 24 Jun 2026).

Squeezing can also turn a Hermitian Dicke model into an effectively non-Hermitian many-body system. In the squeezing–Dicke model, strong photon and spin squeezing generate an effective $\xi_c=1$6 symmetry in the Bogoliubov problem. Between the normal phase and a conventional superradiant phase lies a dynamical superradiant phase with nonzero order parameter but a complex excitation spectrum, indicating spontaneous $\xi_c=1$7-symmetry breaking. The transition to the conventional superradiant phase occurs at an exceptional point, and an artificial magnetic flux then yields nonreciprocal amplification and unidirectional enhanced transmission (Wang et al., 25 Jun 2026).

5. No-go theorems, gauge dependence, and major controversies

A persistent controversy concerns whether anomalous superradiance truly evades standard no-go theorems or merely redefines the observable being called superradiant. For ideal non-interacting two-level atoms with the $\xi_c=1$8 term, the TRK sum rule imposes $\xi_c=1$9, which forbids the usual Dicke transition. Attractive interatomic interactions and multilevel transitions modify that conclusion by lowering the critical coupling or redistributing oscillator strength, so superradiant phases can exist without violating TRK (Liu et al., 2021, Baksic et al., 2012).

Gauge arguments sharpen that point. In arbitrary-gauge cavity QED, the unique thermodynamic transition is signaled by a macroscopic gauge-invariant polarization,

$$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$0

while the apparent macroscopic radiation displacement depends on gauge. In the Coulomb gauge the abnormal phase may appear purely ferroelectric; in the multipolar gauge the same phase may appear fully superradiant. The no-go/counter–no-go debate is therefore resolved by distinguishing gauge-invariant polarization from gauge-relative notions of radiation (Stokes et al., 2019).

Another dispute concerns whether some anomalous superradiant phases are genuine thermodynamic phases at all. The two-photon Dicke model was previously argued to host a superradiant phase with $$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$1 and finite squeezing-induced photon number, but a later analysis shows that this region is a finite-size effect. The putative superradiant window shrinks as

$$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$2

and disappears as $$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$3, while spectral collapse at $$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$4 remains (Ramírez et al., 27 Jan 2026). A plausible implication is that “anomalous” can sometimes denote an unconventional but genuine phase, and in other cases a finite-size crossover that mimics one.

The question of whether superradiant transitions are “quantum” in the strict sense remains open to interpretation. One line of work states explicitly that for Dicke and Rabi the transitions are not fluctuation-driven, while for Tavis–Cummings and Jaynes–Cummings quantum fluctuations are strictly zero; on that definition, calling them quantum phase transitions is “a matter of taste” (Larson et al., 2016). That observation underlies several later uses of “anomalous.”

6. Experimental realizations and practical limits

Circuit-QED experiments have already demonstrated a parity-broken superradiant phase with nonclassical photonic order in an effective quantum Rabi model. A slow quench across the transition produced a cat-like superradiant photonic state with $$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$5, cat size $$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$6, Wigner minima $$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$7 and $$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$8, superradiant-phase negativity $$H_{\mathrm{pJC}}=\delta_c a^\dagger a+\frac{\delta_q}{2}\sigma_z-\frac{\eta}{2}(a^{\dagger 2}+a^2)+g(a^\dagger \sigma_-+a\sigma_+),$$9, and full-output negativity $\tanh(2r)=\eta/\delta_c$0. The work itself does not use the phrase “anomalous superradiant phase,” but under the definition of superradiance with intrinsically nonclassical photonic order it qualifies as such (Zheng et al., 2022).

Parametric amplification provides a practical route to finite-component anomalous superradiance because it reproduces the classical-oscillator-limit phenomenology without requiring unattainable bare frequency ratios. The squeezed-frame mapping of the parametrically driven Jaynes–Cummings model is explicitly proposed for circuit QED and trapped ions, where two-photon drives, tunable detunings, and Jaynes–Cummings couplings are natural ingredients (Qiu et al., 5 Nov 2025). More elaborate anomalous phases, including Meissner currents, spontaneous photon currents, and topological edge amplification, are likewise tied in the literature to superconducting circuit-QED lattices, photonic or optomechanical arrays, and trapped-ion emulators (Li et al., 3 Mar 2025, Qiao et al., 2024, Lu et al., 24 Jun 2026).

Free-space platforms show that anomalous superradiant behavior is not confined to cavities. A pencil-shaped cloud of laser-cooled atoms was used to realize the driven Dicke model in free space, with $\tanh(2r)=\eta/\delta_c$1, effective atom number $\tanh(2r)=\eta/\delta_c$2, and threshold $\tanh(2r)=\eta/\delta_c$3 (Ferioli et al., 2022). A complementary theory for large two-dimensional atomic arrays predicts an abrupt switch from near-total reflection to rapidly increasing transmission at $\tanh(2r)=\eta/\delta_c$4, with $\tanh(2r)=\eta/\delta_c$5 in the large-array limit, and shows that pseudospin-breaking processes can convert the continuous cooperative transition into a first-order bistable one (Ruostekoski, 2024).

Practical realizations also delimit the concept. Closed-system theories often predict divergent photon numbers or perfectly sharp thresholds, but realistic cavity and qubit losses regularize divergences and round critical features (Qiu et al., 5 Nov 2025). In some models dissipation is destructive, as in open Tavis–Cummings and Jaynes–Cummings criticality; in others it is constitutive, as in spontaneous superradiant photon currents and free-space non-equilibrium superradiance (Larson et al., 2016, Qiao et al., 2024, Ferioli et al., 2022). That dual role of loss is one of the defining technical features of anomalous superradiant phases.