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arXiv:2503.09982 · Physical Review A

General phase diagram features of superradiant phase transitions

Wen Zhao, Junlong Tian, Jie Peng

Published Apr 7, 2026• 0 citations
arXivHTMLPDF

Abstract

Various light-matter interactions lead to diverse phase diagram structures in superradiant phase transition (SPT) studies. Such systems consist of multiqubit and multimode with anisotropic couplings, one- and two-photon interactions, Stark shifts, inter-cavity hoppings, qubit-qubit interactions and so on. We find a general phase diagram feature that the origin is in normal phase (NP) and SPT happens only once along the radial direction of a chosen coupling parameter vector with the mean-field method at finite temperature. We can calculate the phase boundary and SPT properties by a concise method. We illustrate it with specific models and find SPT can be achieved in strong coupling regime by means of multimode collective behavior. We also find the disorder will shift the phase boundary. These general features facilitate SPT studies and their diverse applications.

DOI
10.1103/9l6x-4fnx

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Various light-matter interactions lead to diverse phase diagram structures in superradiant phase transition (SPT) studies. Such systems consist of multiqubit and multimode with anisotropic couplings, one- and two-photon interactions, Stark shifts, inter-cavity hoppings, qubit-qubit interactions and so on. We find a general phase diagram feature that the origin is in normal phase (NP) and SPT happens only once along the radial direction of a chosen coupling parameter vector with the mean-field method at finite temperature. We can calculate the phase boundary and SPT properties by a concise method. We illustrate it with specific models and find SPT can be achieved in strong coupling regime by means of multimode collective behavior. We also find the disorder will shift the phase boundary. These general features facilitate SPT studies and their diverse applications.
We prove that superradiant phase transitions (SPTs) of the Dicke model and its generalizations in the thermodynamic and classical oscillator limit are indeed of the same type. In this sense, we unify SPTs under both limits at zero and finite temperature. We show that the mean-field approximation for bosons is exact in both cases, and compute the structure and location of the phase transitions in parameter space using a concise analytic method. Moreover, we illustrate how SPT properties (first order, second order, or none) are related to symmetry. Finally, we uncover general features of the phase structure in the space of parameters of these models with dipolar couplings. There will be a region of normal phase in the neighborhood of the origin of the space of dipolar couplings $\stackrel{P\vec}{\ensuremath{\gamma}}$, and that generally one flows radially in this space to a superradiant phase.
The superradiant phenomenon, usually described by the Dicke model, is a hallmark of strong light-matter interaction. We explore how matter-matter interactions influence this phenomenon by performing ground-state simulations of Dicke-like models with both isotropic and anisotropic interactions. We find that Ising-type interactions produce two qualitatively distinct phase boundaries, one of which gives rise to an antiferromagnetic-normal phase connected to the superradiant regime via a first-order phase transition. Under anisotropic couplings, we uncover a strongly correlated phase where in-plane spin order coexists with superradiance, exhibiting sublinear scaling of the photon occupation per site and power-law decay of spin correlations. Furthermore, superradiance is strengthened by tuning either isotropic or anisotropic interactions, highlighting the role of intrinsic many-body correlations in shaping light-matter quantum phases.
In this doctoral thesis, the thermodynamic phases und phase transitions of a generalised Dicke model are studied and characterised. Both, finite and vanishing temperatures are considered. The Dicke model of quantum optics describes collective phenomena which occur when light interacts with a many-atom system. In the thermodynamic limit, a so-called Hepp–Lieb superradiant phase transition sets in. A superradiant phase develops for low temperatures or strong coupling between light and atoms, which is characterised by a macroscopic excitation of the light field and a spontaneous, collective polarisation of the atoms. This thesis discusses a generalised version of the Dicke model, which is described by a quantum mechanical system consisting of three-level atoms in Lambda-configuration and two modes of a resonator. By means of the Holstein–Primakoff transformation, the Hamiltonian of this system is written in terms of four interacting, non-linear oscillators, which can be linearised in the thermodynamic limit, yielding the ground-state energy as well as the low-energy excitations. The phase diagram consisting of two superradiant phases separated by continuous and first-order phase transitions is derived. In order to clarify the question whether or not the superradiant phase transition of this generalised Dicke model can be observed experimentally for real atoms, the significance of the diamagnetic term is discussed. In contrast to the original Dicke model, superradiant phase transitions are possible in principle. This is due to the first-order phase transitions. In addition, a no-go theorem for continuous superradiant phase transitions is presented. The argument is based on the Thomas–Reiche–Kuhn sum rule. Last, we study the superradiant phase transition of the generalised Dicke model at finite temperatures. Therefore, the partition sum is computed and analysed in the thermodynamic limit using Laplace’s method. At finite temperatures, the properties of the phase diagram and phases remain. However, here all phase transitions are of first order.
The Dicke model can exhibit quantum phase transition between the normal and the superradiant phases when the strength of the light-matter coupling exceeds the ultrastrong-coupling regime. However, it is challenging to observe this phase transition in practical systems due to limited coupling strength or finite two-photon A2{A}^{2}A2 terms. Here we show that by applying a periodic modulation to the frequency of the two-level systems in a standard Dicke model in the strong-coupling regime, an anisotropic Dicke model with tunable rotating and counter-rotating terms in the ultrastrong-coupling regime can be achieved. We calculate the ground state and the excitation spectrum of this model in terms of the modulation parameters. Our result shows that the superradiant phases can be observed in cavity- or circuit-quantum electrodynamics systems with strong coupling.
A new concept of topological organization of microstructures that maintain the ultrastrong coupling of two-level systems to a photon field and have the topology of a network (graph) with a power-law node degree distribution has been proposed. A phase transition to the superradiant state, which leads to the formation of two dispersion branches of polaritons and is accompanied by the appearance of a nonzero macroscopic polarization of two-level systems, has been studied within the mean field theory. It has been found that the specific behavior of such a system depends on the statistical characteristics of the network structure, more precisely, on the normalized second moment ζ≡⟨k2⟩/⟨k⟩\zeta \equiv \langle {{k}^{2}}\rangle {\text{/}}\langle k\rangle ζ≡⟨k2⟩/⟨k⟩ of the distribution of node degrees. It has been shown that the Rabi frequency can be significantly increased in the anomalous regime of the network structure, where ζ increases significantly. The multimode (waveguide) structure of the interaction between matter and field in this regime can establish a ultrastrong coupling, which is primarily responsible for the high-temperature phase transition.
In this study, the energy spectrum and thermal equilibrium states of the finite-size Dicke-Stark model were numerically obtained within the extended coherent state space by solving the dressed master equation for strongly coupled light-atom systems. The critical point of the superradiant phase transition in the infinite-size Dicke-Stark model was analytically derived using the mean-field approach and confirmed with numerical calculation. Under thermal equilibrium conditions, analyses of the negativity, zero-time-delay two-photon correlation function, and atom-spin squeezing parameters in the finite-size Dicke-Stark model reveal that as the coupling strength increases, the light field undergoes a transition from photon bunching to anti-bunching and then back to bunching. The Stark field can modulate both the maximum and minimum values of the two-photon correlation function and their corresponding coupling strengths. At low temperatures, the system exhibits entanglement and spin squeezing. As temperature rises, entanglement gradually diminishes, while strong coupling facilitates the preservation of entanglement in the system state. Atom-spin squeezing spin squeezing is highly sensitive to temperature and vanishes rapidly with increasing temperature. This work contributes to the fundamental understanding of quantum phenomena in Dicke-Stark systems.