Open Dicke Lattice Model

• The open Dicke lattice model is a dissipative many-body system where lattices of coupled cavities interact locally via Dicke-type spin couplings.
• Photon hopping introduces a band structure that drives momentum-dependent superradiant thresholds and spatially modulated ordering.
• Dissipation and boundary conditions generate unique phenomena such as multistability, finite-momentum instabilities, and distinct nonequilibrium universality classes.

Searching arXiv for papers on the open Dicke lattice model and closely related open Dicke criticality. The open Dicke lattice model denotes a class of driven or dissipative many-body light–matter systems in which cavity or resonator modes are arranged on a lattice, each site is locally coupled to a collective spin or atomic ensemble by a Dicke-type interaction, and openness enters through environmental coupling such as cavity loss. In the most direct lattice realizations, photon hopping endows the photonic sector with a dispersion or band structure, while the on-site light–matter coupling retains the full Dicke form with rotating and counter-rotating contributions (Zou et al., 2014, Wei et al., 14 Aug 2025, Wei et al., 1 Jul 2026). The topic sits at the intersection of nonequilibrium quantum optics, cavity QED, circuit QED, and dissipative many-body physics, and it is distinguished from the single-mode open Dicke model by the presence of spatial structure, boundary conditions, and momentum- or configuration-selective instabilities (Zou et al., 2014, Wei et al., 14 Aug 2025).

1. Definition and model classes

A standard open Dicke lattice consists of an array of coupled resonators, each locally coupled to a collective spin degree of freedom. One explicit formulation is the Dicke lattice Hamiltonian

$$H_{\rm DLM}=\Delta_c \sum_{\ell=1}^{N_L} a^\dagger_{\ell} a_{\ell} -t\sum_{\ell=1}^{N_L-1}(a^\dagger_\ell a_{\ell+1}+a_\ell a^\dag_{\ell+1}) + \sum_{\ell=1}^{N_L} \Delta_s J^z_\ell + \sum_{\ell=1}^{N_L} \frac{G}{\sqrt{\mathcal N}}(J^+_{\ell}+J^-_\ell)(a_\ell+a_\ell^\dag)$$

with local cavity operators $a_\ell$, collective spin operators $J_\ell^{\pm,z}$, hopping $t$, cavity detuning $\Delta_c$, spin detuning $\Delta_s$, and collective coupling $G$ (Zou et al., 2014). A closely related finite-size formulation uses

$$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$

with local Dicke blocks

$$H_{i}^{\rm Dicke} = \omega_c c_i^{\dagger} c_i + \omega_{a} S_i^z + \frac{2g}{\sqrt{N_a}} ( c_i + c_i^\dagger ) S_i^x$$

and boundary parameter $\lambda=\xi$ for periodic boundary conditions and $a_\ell$0 for open boundary conditions (Wei et al., 14 Aug 2025).

A periodic one-dimensional ring formulation is

$a_\ell$1

with $a_\ell$2 and

$a_\ell$3

This form underlies a configuration-based analysis of nonequilibrium superradiant phases in finite rings (Wei et al., 1 Jul 2026).

These models are “open” because the dynamics is not unitary. In the most common treatments, photon loss is included through a Lindblad master equation,

$a_\ell$4

or its site-indexed variants (Wei et al., 14 Aug 2025, Wei et al., 1 Jul 2026, Zou et al., 2014). By contrast, several influential works on single-mode open Dicke systems treat non-Markovian baths or additional matter dissipation; these are not lattice models, but they supply mechanisms that can be imported into lattice settings (Scarlatella et al., 2016, Nagy et al., 2015).

2. Canonical ingredients: hopping, symmetry, and photonic structure

The defining lattice ingredient is photon hopping. In the superconducting-cavity/NV-center implementation, hopping produces the photon band

$a_\ell$5

and after Holstein–Primakoff reduction the quadratic normal-phase Hamiltonian becomes

$a_\ell$6

(Zou et al., 2014). In finite lattices, the photonic normal-mode frequencies depend on boundary conditions: $a_\ell$7

$a_\ell$8

with positivity condition $a_\ell$9 (Wei et al., 14 Aug 2025). Periodic lattices therefore support plane-wave modes, whereas open chains support standing-wave modes (Wei et al., 14 Aug 2025).

The Dicke coupling preserves a discrete parity structure. In the single-site model discussed in the non-Markovian setting, the model has a $J_\ell^{\pm,z}$0 parity symmetry $J_\ell^{\pm,z}$1 (Scarlatella et al., 2016). In the lattice setting, this local Dicke symmetry coexists with lattice translation or boundary symmetries, and the ordered phases are best classified by the spatial pattern of the local cavity order parameters. In the four-site ring, the real part of the local cavity field,

$J_\ell^{\pm,z}$2

serves as the local order parameter, and the possible superradiant configurations are organized into sign-pattern classes such as $J_\ell^{\pm,z}$3, $J_\ell^{\pm,z}$4, $J_\ell^{\pm,z}$5, and $J_\ell^{\pm,z}$6 (Wei et al., 1 Jul 2026). This classification reflects the reduction of independent local parities at $J_\ell^{\pm,z}$7 to a global $J_\ell^{\pm,z}$8 parity once hopping is switched on (Wei et al., 1 Jul 2026).

3. Dissipation, steady states, and superradiant thresholds

Open Dicke lattice criticality is formulated as a stability problem for nonequilibrium steady states rather than as energy minimization. In the homogeneous single-cavity open Dicke model with cavity loss, the dissipative threshold is shifted from the closed-system value to

$J_\ell^{\pm,z}$9

(Zou et al., 2014). In the lattice, each momentum mode $t$0 has its own threshold,

$t$1

and the actual instability is determined by

$t$2

(Zou et al., 2014). In finite lattices with site-resolved mean-field equations, the normal-phase threshold is

$t$3

(Wei et al., 14 Aug 2025), while for the four-site periodic ring the normal phase loses stability at

$t$4

The isolated-site threshold remains

$t$5

when $t$6 (Wei et al., 1 Jul 2026).

The normal phase is characterized by vanishing cavity fields and fully polarized spins. In the three-site boundary-sensitive model,

$t$7

(Wei et al., 14 Aug 2025). In the four-site ring,

$t$8

(Wei et al., 1 Jul 2026). The superradiant phase is identified by nonzero coherent cavity amplitudes on at least one site, $t$9 (Wei et al., 14 Aug 2025).

A central nonequilibrium point is that the instability is controlled by the complex spectrum of damped modes. In the cavity-array implementation, the transition occurs when one eigenvalue of the linearized dynamical matrix acquires positive real part (Zou et al., 2014). This differs from closed equilibrium reasoning based only on softening of a Hermitian excitation. A closely related lesson emerges in the experimental single-mode open Dicke model: the phase boundary is not determined solely by the vanishing of the excitation frequency, because one must also track the damping or growth rate $\Delta_c$0 (Bolian et al., 17 Feb 2025). This suggests that lattice critical manifolds should generally be located by the full complex spectrum of momentum-resolved modes rather than by real-frequency softening alone.

4. Spatial ordering, finite-momentum phases, multistability, and boundaries

One of the defining features of the open Dicke lattice is the possibility of ordered phases beyond homogeneous superradiance. In the cavity-array model, if $\Delta_c$1, the first unstable mode is at $\Delta_c$2, giving a homogeneous superradiant phase with $\Delta_c$3 (Zou et al., 2014). In the parameter window $\Delta_c$4, however, the instability occurs at a finite wavevector $\Delta_c$5 satisfying

$\Delta_c$6

with threshold

$\Delta_c$7

The corresponding ordered state is spatially modulated,

$\Delta_c$8

and spontaneously breaks translation symmetry (Zou et al., 2014). For larger losses $\Delta_c$9, the preferred instability is at $\Delta_s$0, corresponding to an antiferromagnetic or staggered pattern (Zou et al., 2014).

Finite systems display an additional layer of structure. For the dissipative four-site ring, the complete phase diagram contains a normal phase and four superradiant phase types: a homogeneous superradiant phase and three inhomogeneous superradiant phases (Wei et al., 1 Jul 2026). The homogeneous phase corresponds to

$\Delta_s$1

the alternating phase to

$\Delta_s$2

and the domain-wall-like phase to

$\Delta_s$3

(Wei et al., 1 Jul 2026). The paper further identifies coexistence regions with up to four stable superradiant phases, so multistability is interpreted as simultaneous dynamical stabilization of several symmetry-allowed configuration classes rather than as unstructured nonlinear complexity (Wei et al., 1 Jul 2026).

Open boundary conditions alter the phase structure even more dramatically. For a three-site dissipative Dicke lattice, open boundaries support a “zoo of superradiant phases with broken translational symmetry,” including monostable, bistable, and tristable regions (Wei et al., 14 Aug 2025). Representative open-boundary patterns are

$\Delta_s$4

$\Delta_s$5

and

$\Delta_s$6

(Wei et al., 14 Aug 2025). A central analytic result is that, for any finite $\Delta_s$7, a spatially homogeneous steady-state solution is impossible under open boundary conditions because edge and bulk sites satisfy incompatible balance equations: $\Delta_s$8 whereas

$\Delta_s$9

This incompatibility forbids a fully homogeneous superradiant steady state under open boundary conditions for finite $G$0 (Wei et al., 14 Aug 2025).

These results establish that the open Dicke lattice is not simply the periodic Dicke lattice with edges attached. Boundaries, standing-wave mode structure, and dissipative selection reshape the available stationary phases (Wei et al., 14 Aug 2025).

5. Universality, fluctuations, and the role of dissipation

Dissipation in Dicke systems does not merely shift thresholds; it can also alter fluctuation criticality and universality. In the open four-site ring, discrete truncated Wigner calculations show that different superradiant configurations can belong to different nonequilibrium universality classes. The photon fluctuation scaling

$G$1

yields $G$2 for the normal-to-homogeneous and normal-to-ISRP1 transitions, but $G$3 for the normal-to-ISRP2 transition (Wei et al., 1 Jul 2026). By contrast, in the corresponding closed lattice the lowest excitation gap scales with exponent $G$4, and the equilibrium universality class is shared by different superradiant configurations (Wei et al., 1 Jul 2026).

Single-mode open Dicke studies clarify the microscopic reasons dissipation can matter so strongly. In the Markovian open Dicke model, the soft mode is complex, and criticality is governed by the vanishing damping gap rather than only by the real-frequency softening (Konya et al., 2012, Nagy et al., 2011). In the driven open Dicke model with a sub-Ohmic matter bath, the critical photon-number exponent is $G$5 without spin-bath coupling and becomes less than $G$6 when the spin-like mode couples to a sub-Ohmic reservoir, decreasing monotonically as the bath exponent $G$7 is reduced below $G$8 (Nagy et al., 2015). In a distinct non-Markovian construction, coupling the cavity displacement to a zero-temperature power-law bath yields a retarded hybridization function

$G$9

a bath-renormalized photon frequency

$$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$0

and a large-$$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$1 critical coupling

$$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$2

so that dissipation can promote rather than suppress superradiance (Scarlatella et al., 2016). These are single-mode results, not lattice results, but they isolate mechanisms—frequency-dependent self-energies, bath-controlled infrared behavior, and dissipative universality changes—that are directly relevant when generalized to momentum-dependent lattice propagators (Scarlatella et al., 2016, Nagy et al., 2015).

A further single-mode result of broad relevance is that local Markovian spin decay can enhance collective quantum correlations in the open Dicke model. In the large-$$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$3 fluctuation treatment with cavity loss $$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$4 and local spin decay $$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$5, the critical coupling is

$$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$6

and local dissipation can lead to enhancement of logarithmic negativity near the critical line and to a superradiant phase with nonzero spin-boson entanglement (Boneberg et al., 2021). This suggests that local dissipation in lattice settings may also modify fluctuation structure in ways that are not reducible to simple decoherence.

6. Implementations, methods, and scope of the field

A concrete implementation of the Dicke lattice was proposed in arrays of superconducting microwave cavities coupled to NV-center ensembles in diamond. The microscopic model uses cavity-assisted Raman transitions to engineer an effective Dicke interaction, with cavity loss providing openness (Zou et al., 2014). The starting Hamiltonian contains a cavity mode, NV-center ground-state triplet levels $$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$7, and two microwave drives; after adiabatic elimination of the intermediate state, the effective two-level model becomes

$$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$8

(Zou et al., 2014). In the homogeneous limit $$H= \sum_{i=1}^{N} H_i^{\rm Dicke} - \sum_{i=1}^{N-1} \xi \left( c_{i}^{\dagger} c_{i+1} + c_{i} c_{i+1}^{\dagger} \right) - \lambda \left( c_{1}^{\dagger} c_{N} + c_{1} c_{N}^{\dagger} \right),$$9, this reduces to an effective Dicke form and can be extended to a cavity array with hopping (Zou et al., 2014). The same work emphasizes robustness of the superradiant transition to substantial inhomogeneous broadening, encapsulated in the generalized instability condition

$$H_{i}^{\rm Dicke} = \omega_c c_i^{\dagger} c_i + \omega_{a} S_i^z + \frac{2g}{\sqrt{N_a}} ( c_i + c_i^\dagger ) S_i^x$$0

and, for a Lorentzian spin distribution, the broadened critical coupling

$$H_{i}^{\rm Dicke} = \omega_c c_i^{\dagger} c_i + \omega_{a} S_i^z + \frac{2g}{\sqrt{N_a}} ( c_i + c_i^\dagger ) S_i^x$$1

(Zou et al., 2014).

Methodologically, the field combines mean-field theory, Holstein–Primakoff bosonization, linear stability analysis, Routh–Hurwitz criteria, Keldysh functional integrals, discrete truncated Wigner approximation, and, in single-mode problems, Liouvillian spectral analysis (Wei et al., 14 Aug 2025, Wei et al., 1 Jul 2026, Scarlatella et al., 2016, Nagy et al., 2015, Villaseñor et al., 2023). The Liouvillian-spectral work on the single-mode open Dicke model is not a lattice study, but it is notable for showing that complex spectral windows can display 2D Poisson statistics in regular regimes and GinUE statistics in chaotic ones, together with an eigenstate-based convergence criterion for bosonic Liouvillians (Villaseñor et al., 2023). This suggests a possible route for future open Dicke lattice studies of nonequilibrium chaos, though the spatial case is substantially more difficult because of hopping, larger Hilbert spaces, and momentum structure.

A persistent source of confusion is the use of “open Dicke” for models that are not lattices. Several important papers treat a single cavity mode coupled collectively to matter and then add photon loss, matter baths, or local spin dissipation (Scarlatella et al., 2016, Nagy et al., 2015, Boneberg et al., 2021). These are not Dicke lattice models, because they lack photon hopping, site-resolved cavity fields, and lattice momentum structure. Their relevance lies instead in identifying local or mode-resolved mechanisms—bath-induced cavity softening, dissipative critical exponents, local-dissipation-induced sector selection, or open-system correlation structure—that can serve as building blocks for genuine open Dicke lattices (Scarlatella et al., 2016, Nagy et al., 2015, Tong et al., 19 May 2025, Boneberg et al., 2021).

In current usage, then, the open Dicke lattice model is best understood not as a single universal Hamiltonian but as a family of dissipative lattice light–matter theories defined by three ingredients: local Dicke coupling, spatially extended photonic dynamics, and explicit openness. Across implementations and formulations, the field’s most characteristic results are the emergence of finite-momentum and boundary-induced superradiant phases, the organization of steady states into symmetry-related configuration classes, and the fact that dissipation can create qualitatively new nonequilibrium phase structure rather than merely perturb the equilibrium Dicke transition (Zou et al., 2014, Wei et al., 14 Aug 2025, Wei et al., 1 Jul 2026).