Superradiant Phase Transition Overview

• Superradiant phase transition is a quantum phase transition where collective coupling between quantum emitters and bosonic fields produces macroscopic coherence at zero temperature.
• The mechanism extends the Dicke model by incorporating multi-level atoms, squeezed light, and complex symmetry-breaking effects that modify critical couplings.
• Experimental realizations in atomic, solid-state, and network systems highlight its practical impact, advancing quantum devices and coherent phase control.

A superradiant phase transition (SRPT) is a quantum phase transition in which cooperative coupling between many quantum emitters (such as atoms or spins) and one or more bosonic modes (typically quantized electromagnetic fields or magnons) leads to the emergence of macroscopic coherences in both matter and field subsystems, even at zero temperature. This phenomenon generalizes the concept introduced by the Dicke model into a broad variety of physical systems and interaction settings, with distinctive features arising from the nature of the constituents, the symmetries of the Hamiltonian, and constraints arising from microscopic physics such as diamagnetic terms and sum rules.

1. Hamiltonian Structure and Model Foundations

The prototypical Hamiltonian underpinning the superradiant phase transition is the Dicke-type model, which in a standard two-level, single-mode realization takes the form

$$H = \omega_c a^{\dagger} a + \frac{\omega_a}{2} \sum_{j=1}^N \sigma_z^{(j)} + \frac{g}{\sqrt{N}} (a^{\dagger} + a) \sum_{j=1}^N \sigma_x^{(j)} + H_{A^2}$$

where $a^{\dagger}, a$ are bosonic creation and annihilation operators, $\sigma_{\alpha}^{(j)}$ are Pauli matrices for atom $j$, and $H_{A^2}$ denotes possible diamagnetic ($A^2$) or other non-linear contributions arising from the minimal coupling Hamiltonian. The SRPT occurs when the coupling $g$ surpasses a critical value $g_c = \sqrt{\omega_c \omega_a}/2$ (in the ideal Dicke model).

Generalizations studied in the literature extend this paradigm to:

• Multi-level atoms: e.g., lambda systems interacting with two field modes, where cross-coupling and diamagnetic effects are parameterized and constrain the phase diagram (Hayn et al., 2012).
• Spin-boson models with two-photon or squeezed-light terms: resulting in two-photon superradiant transitions and tunable symmetry breaking (Garbe et al., 2017, Zhu et al., 2019).
• Fermionic and bosonic matter models: addressing the influence of quantum statistics, e.g., across the BEC-BCS crossover (Chen et al., 2014).
• Network and lattice systems: where site connectivity, degree distributions, and multimode/multichannel couplings redefine the collective enhancement and transition temperature (Bazhenov et al., 2020, Bazhenov et al., 22 Jul 2025).
• Solid-state quantum materials: e.g., magnonic Dicke models arising from exchange-coupled magnetic subsystems with no diamagnetic counterpart (Bamba et al., 2020, Kim et al., 3 Jan 2024).

2. Symmetry Breaking and Order Parameters

The SRPT is associated with a spontaneous breaking of a discrete or continuous symmetry:

• In two-level Dicke models, a $\mathbb{Z}_2$ parity symmetry is broken when the field acquires a macroscopic coherent amplitude $\langle a \rangle \ne 0$.
• In lambda or multimode models, independent or combined parity operators protect (or break) degeneracies, and the loss of independent parities in the presence of diamagnetic or cross-coupling terms leads to a unique superradiant phase with simultaneous macroscopic occupations (Hayn et al., 2012).
• In systems with squeezed light or two-photon terms, the symmetry can be tuned between U(1), $\mathbb{Z}_2$, or more general sectors, affecting the order and nature of the transition (Zhu et al., 2019).

The relevant order parameters typically include:

• The mean value of the bosonic field, $\langle a_n \rangle$.
• Mean collective atomic operators, e.g., via Holstein–Primakoff bosonic representations, mapping atomic population differences or coherences to auxiliary bosonic fields.
• In lattice and network models, weighted sums over degrees or network positions, reflecting the distributed nature of the collective behavior (Bazhenov et al., 2020, Bazhenov et al., 22 Jul 2025).

3. Phase Transition Characteristics and Classification

The critical properties of the SRPT depend on both microscopic details and system symmetries:

• Order of the transition: In basic Dicke/Rabi models and lambda-systems, both first and second order transitions can occur. Cross-coupling, diamagnetic terms, or TRK sum rule constraints may drive the transition first order (order parameters jump), particularly in atomic realizations (Hayn et al., 2012). In squeezed-light driven models, both orders occur and a tricritical point separates the two regimes (Zhu et al., 2019).
• Critical coupling and normal modes: Explicit expressions for critical couplings, e.g., $g_c = \sqrt{\omega_c \omega_a}/2$, and normal mode frequencies are derived for each model, sometimes involving collective enhancements, e.g., $g \propto \sqrt{N} g_0$ modified by topological features in network systems (Bazhenov et al., 22 Jul 2025).
• Mean-field vs. quantum character: The transition is mean-field in the Dicke, Rabi, Tavis–Cummings, and Jaynes–Cummings models, with quantum fluctuations vanishing in the thermodynamic limit for Dicke/Rabi and being strictly zero (due to excitation conservation) for TC/JC (Larson et al., 2016).
• Role of quantum statistics: In Fermi gases, the interplay of Fermi surface nesting and Pauli blocking leads to strongly density-dependent thresholds and statistics crossovers between BCS and BEC behavior (Chen et al., 2014).
• Influence of microscopic constraints: Diamagnetic (A$^2$) terms, enforced by gauge invariance, and the TRK sum rule generally suppress the transition in two-level systems, but may permit it in multilevel or band systems if certain symmetry or coupling configurations are realized (Hayn et al., 2012, Guerci et al., 2020).

4. Extensions, Topological and Many-Body Effects

SRPTs have been advanced as a unifying mechanism in a variety of many-body and topological settings:

• Electronic band and lattice systems: The Peierls substitution in tight-binding models allows the photon field to induce momentum-dependent superradiant transitions. In nested Fermi surfaces or band-touching models, the SRPT can lead to time-reversal symmetry breaking and emergent topologically protected modes (e.g., Dirac points, non-trivial Chern numbers, and edge states) (Guerci et al., 2020, Su et al., 4 Mar 2025).
• Network-induced phenomena: In complex quantum networks, the critical temperature and coupling strength for the SRPT are parametrically enhanced by network statistics (first and second moments of the degree distribution). The presence of hubs and multimode structure leads to multichannel superstrong coupling, raising transition temperatures and enabling robust control of quantum coherence (Bazhenov et al., 2020, Bazhenov et al., 22 Jul 2025).
• Magnonic and hybrid materials: In ErFeO₃, Fe magnons assume the role of the photonic mode, with Er spins as two-level atoms. The absence of a diamagnetic term in the Fe–Er exchange facilitates an equilibrium magnonic SRPT, as evidenced by kinks and softening in the hybridized spin-magnon modes under magnetospectroscopy (Bamba et al., 2020, Kim et al., 3 Jan 2024).

5. Experimental Realizations, Detection, and Applications

Experimental evidence for SRPTs has been reported and proposed across several platforms:

• Atomic and ion-cavity QED: Observations of superradiant transitions in cavity-confined atomic ensembles, with first and second order transitions modulated by detuning, squeezing, or dissipation (Zhu et al., 2019, Ferioli et al., 2022, 2207.13285).
• Solid-state systems: Terahertz and gigahertz spectroscopies in ErFeO₃ demonstrate magnonic SRPT signatures, circumventing the no-go theorem and enabling direct observation of ground-state order parameter changes (Kim et al., 3 Jan 2024).
• Complex networks and metrological devices: Artificial photonic networks (e.g., photonic crystal microstructures) with engineered connectivity provide routes to high-temperature SRPTs, scalable quantum devices, and optically controllable phase switches (Bazhenov et al., 22 Jul 2025).
• Quantum thermodynamical devices: In the quantum Rabi model, the SRPT can be detected through singularities in thermal conductance as a function of coupling, enabling quantum heat valves and sensitive nano-thermal management (Yamamoto et al., 20 Feb 2025).
• Finite-time detection techniques: Advanced Lee–Yang theory has been applied for extracting nonanalytic critical points from finite-time photon counting statistics, opening a practical route for detection in open quantum systems (Brange et al., 22 May 2024).

6. Implications, Theoretical Insights, and Outlook

The SRPT provides a rich paradigm for macroscale quantum coherence, symmetry breaking, and critical phenomena in both equilibrium and driven–dissipative many-body systems:

• It generalizes phase transitions beyond conventional symmetry-breaking frameworks, as exemplified by non-equilibrium transitions without traditional order parameter structure but with well-defined local observables (Ferioli et al., 2022).
• The interplay between microscopic mechanisms (multi-level structures, band topology, quantum statistics, and network topology) and observable macroscopic order (field coherence, polarization, phase switching) allows precise control of phase boundaries, orders of transition, and critical properties.
• These insights contribute to the design of quantum materials, devices for quantum information processing at elevated temperatures, optically addressable phase switches, and quantum simulators probing collective and topological quantum phenomena in controlled settings.

Superradiant phase transitions thus serve as a central organizing concept connecting collective light–matter interactions, many-body physics, and modern quantum technology platforms.