Dicke ModelEdit
The Dicke model is a fundamental construct in quantum optics and many-body physics that describes how a collection of N identical two-level systems interacts collectively with a single bosonic mode, such as a mode of the electromagnetic field in a cavity. Introduced by R. H. Dicke in the mid-20th century to account for cooperative emission phenomena, the model captures how light and matter can exchange energy in a way that is more than the sum of its parts. In the limit of a large number of emitters, the model exhibits a quantum phase transition between a normal phase and a superradiant phase, a phenomenon that has driven both theoretical developments and experimental explorations in modern quantum technologies. For researchers, the Dicke model provides a clean, tunable platform to study collective light–matter coupling, symmetry breaking, and critical dynamics.
Beyond its original formulation, the Dicke model has grown into a family of related descriptions that appear across disciplines, from cavity [quantum electrodynamics] to ultracold atoms in optical resonators and to engineered quantum circuits. Its core ideas underpin how a many-body system can synchronize its response to a common field, leading to macroscopic occupation of the photonic mode and a reorganization of the atomic ensemble. The model also motivates a range of extensions—multimode couplings, inhomogeneous interactions, and driven or open systems—that connect to broader themes in quantum simulation, phase transitions, and non-equilibrium dynamics.
The Model
Hamiltonian and basic ingredients
The standard Dicke Hamiltonian captures N two-level systems coupled to a single bosonic mode and, up to trivial redefinitions of parameters, can be written as H = ω a†a + ω0 Jz + (g/√N) (a + a†) (J+ + J-). Here, a and a† are the annihilation and creation operators of the bosonic (cavity) mode with frequency ω, while ω0 is the energy splitting of the two-level systems. The collective spin operators Jz, J± describe the ensemble of emitters, with J = ∑n σn/2 representing the total angular momentum of the N emitters and Jx = (J+ + J-)/2, Jy = (J+ − J-)/(2i). The coupling strength g scales with the ensemble size as 1/√N to maintain a meaningful thermodynamic limit. In this form, the model emphasizes the exchange of excitations between the emitters and the field in a fully collective manner.
The same physics is sometimes expressed with equivalent parametrizations that emphasize the role of the field and the transverse coupling, for example through the relation (a + a†)Jx, or by using the Tavis–Cummings form when counter-rotating terms are neglected. Regardless of convention, the essential ingredients are a bosonic mode with frequency ω, a set of N two-level systems with splitting ω0, and a collective, symmetric coupling that links the two subsystems.
Symmetry and ground-state structure
The Dicke model possesses a discrete Z2 (parity) symmetry: the transformation a → −a and J± → −J± leaves the Hamiltonian invariant. This symmetry is dynamically realized in the ground state and can be spontaneously broken in the thermodynamic limit. When the coupling g is below a critical value gc, the ground state has essentially no photons in the mode and the atomic ensemble remains in a symmetric, unpolarized configuration. Above gc, the system undergoes a quantum phase transition to a superradiant phase with a macroscopic occupation of the photonic mode and a reorganization of the collective spin.
In the thermodynamic limit (N → ∞), mean-field and semiclassical analyses predict a sharp boundary at g = gc separating the normal and superradiant phases. The precise value of gc depends on the chosen parameterization of the Hamiltonian, but a common benchmark is that gc scales with the square root of the product of the field and emitter energies, and the transition is characterized by a nonzero expectation value for the photon field and a symmetry-broken ground state.
The no-go issue and scientific debate
A classic point of discussion is the so-called no-go theorem for a true superradiant phase transition in the minimal Dicke model when one includes the A2 (diamagnetic) term that arises in realistic light–matter couplings. In cavity quantum electrodynamics, this term can preclude a true phase transition, leading some to argue that what is observed or predicted in simple models is a fictional transition rather than a genuine thermodynamic phase change. However, researchers have shown that the no-go constraint can be circumvented in several ways: by considering multimode cavities, by tailoring the coupling geometry, or by using artificial light–matter platforms such as circuit quantum electrodynamics where the effective A2 term can be engineered differently. Consequently, a broad and active line of inquiry examines the conditions under which a Dicke-type transition can manifest in real systems, as well as the open-system, driven-dissipative dynamics that accompany it.
Extensions and related models
The Dicke model sits at the crossroads of several well-known theoretical constructs. Its collective spin description connects to [spin-boson models] and to Holstein–Primakoff-type mappings that approximate the many-body problem by bosonic degrees of freedom. Its generalizations include multimode Dicke models (where several bosonic modes couple to a single or multiple collective spins), inhomogeneous couplings (reflecting nonidentical emitters), and driven or dissipative variants that account for cavity losses and external drives. The model is closely related to the [Tavis–Cummings model], which arises when counter-rotating terms are neglected, and to broader studies in [quantum phase transitions] and [non-equilibrium quantum dynamics].
Experimental realizations and quantum simulations
Ultracold atoms in optical cavities
A landmark realization occurred with ultracold atomic gases placed inside optical cavities, where a BEC of atoms coupled collectively to a single cavity mode exhibited a Dicke-type quantum phase transition. In experiments led by Esslinger and collaborators, the system transitioned from a normal phase to a superradiant phase as the effective light–matter coupling was tuned, providing a clean, controllable platform to study critical behavior, symmetry breaking, and the interplay between coherent dynamics and dissipation. See Dicke quantum phase transition for a detailed discussion of the phenomenon in this setting.
Circuit quantum electrodynamics and solid-state platforms
Superconducting qubits in microwave resonators (circuit QED) offer another route to realize Dicke-like physics with a high degree of tunability and strong coupling. In these systems, engineered collective couplings of many qubits to a common mode allow exploration of the same basic phase-transition physics, often with different limitations and advantages compared to atomic realizations. Researchers also explore related open-system dynamics, driven scenarios, and the robustness of the phase transition under loss and noise.
Other approaches
Beyond cold atoms and superconducting circuits, researchers have pursued Dicke-type physics with trapped ions, quantum dots, and photonic platforms, highlighting the versatility of the model as a paradigm for collective light–matter interactions. These experiments illuminate how coherence, entanglement, and critical behavior emerge when many constituents share a common electromagnetic field degree of freedom.
Implications and outlook
The Dicke model remains a touchstone for understanding how collective coupling reshapes the behavior of many-body quantum systems. It provides a clear setting in which a large-N limit yields nontrivial macroscopic phenomena—superradiance, symmetry breaking, and critical fluctuations—that are accessible to experimental test and numerical simulation. By connecting foundational ideas in quantum optics with modern quantum technologies, the model informs ongoing work in quantum simulations, non-equilibrium statistical mechanics, and the design of devices that exploit collective light–matter interactions for sensing, information processing, and fundamental tests of quantum theory.