Dicke Lattice Model Overview

• The Dicke lattice model is a spatial extension of Dicke superradiance, integrating cavity–spin units with photon hopping to create structured light–matter interactions.
• It exhibits diverse ordered phases—including patterned superradiant, superfluid, and Mott-insulating regimes—driven by the interplay of local coupling, hopping, detuning, and dissipation.
• Experimental implementations in cavity arrays and optical lattices demonstrate multistability and finite-momentum instabilities, offering insights into nonequilibrium quantum dynamics.

The Dicke lattice model denotes a class of spatial extensions of Dicke superradiance in which collective light–matter coupling is embedded in a lattice or otherwise spatially structured setting. In its standard form, each site of a cavity or resonator array hosts a bosonic mode coupled to a collective spin, while photon hopping couples neighboring sites; related usages include bosonic optical lattices whose density-wave degree of freedom couples to a single cavity mode, as well as momentum-space constructions built from timed Dicke states (Wei et al., 1 Jul 2026, Klinder et al., 2015, Wang et al., 2014). Across these variants, the defining feature is that superradiant ordering no longer occurs in a spatially featureless environment: it competes with hopping, detuning, on-site interactions, boundary conditions, and dissipation, producing normal, patterned superradiant, superfluid, and Mott-insulating regimes.

1. Definition and range of usages

In the cavity-array formulation, the Dicke lattice model is a spatial extension of the Dicke model to an array of coupled cavity–spin units. Each lattice site $i$ contains a single bosonic cavity mode $c_i$ of frequency $\omega_c$ and an ensemble of $N_a$ identical two-level atoms or spins described by a collective spin $S_i$ of frequency $\omega_a$, with neighboring cavities coupled by photon hopping $\xi$ (Wei et al., 1 Jul 2026). The competition between local Dicke coupling and inter-site hopping organizes superradiant order into lattice-symmetry sectors distinguished by momentum and sign pattern.

A second usage arises in ultracold atoms in optical cavities. There, a Bose–Einstein condensate loaded into an optical lattice inside a high-finesse cavity realizes what was termed a Dicke–Hubbard setting: a lattice of bosonic sites whose collective density-wave operator couples to a single radiation mode, generating an effective infinite-range interaction across the lattice (Klinder et al., 2015). In that setting, the Dicke degree of freedom is not a local two-level atom per site, but a coarse-grained checkerboard density-wave mode embedded in a Bose–Hubbard system.

Further related constructions broaden the term’s scope. The long-range Dicke–Ising model consists of a single global bosonic mode coupled uniformly to a spin lattice with algebraically decaying antiferromagnetic Ising interactions; it realizes a “Dicke lattice” in the sense of a spatially structured spin sector, but not an array of hopping photon modes (Koziol et al., 4 Mar 2025). The “superradiance lattice” goes further and realizes a tight-binding lattice in momentum space from timed Dicke states coupled by electromagnetically induced transparency (EIT) fields (Wang et al., 2014). Taken together, these works show that “Dicke lattice model” is a family resemblance term rather than a single Hamiltonian.

2. Hamiltonian structure and symmetry

The standard closed Dicke lattice Hamiltonian on a periodic ring is

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

with

$$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$

Its dissipative extension is governed by the Lindblad equation

$$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$

with cavity loss $c_i$0 taken as the dominant dissipative channel in the analyses under discussion (Wei et al., 1 Jul 2026).

Photon hopping reduces the independent local $c_i$1 symmetries of uncoupled sites to a single global $c_i$2 and imposes discrete translation symmetry. Superradiant phases are therefore classified by irreducible representations of the translation group, equivalently by lattice momenta $c_i$3, and on finite rings by the sitewise sign structure of $c_i$4, where $c_i$5 (Wei et al., 1 Jul 2026). In Dicke–Hubbard lattices with counter-rotating terms, the relevant conserved symmetry is likewise a global $c_i$6 parity,

$c_i$7

which is spontaneously broken when the photonic order parameter $c_i$8 becomes nonzero (Lu et al., 2016).

In the cavity-coupled Bose–Hubbard realization, the matter sector is

$c_i$9

while the cavity field and light–matter coupling are

$\omega_c$0

with

$\omega_c$1

After adiabatic elimination of the cavity in the stationary weakly dissipative regime,

$\omega_c$2

and the atoms experience the effective interaction

$\omega_c$3

For red detuning $\omega_c$4, the induced infinite-range term lowers the energy when $\omega_c$5 grows, favoring checkerboard density-wave order and superradiance (Klinder et al., 2015).

3. Ordered phases, order parameters, and critical behavior

For dissipative cavity arrays, the normal phase is defined by $\omega_c$6 and $\omega_c$7, while a superradiant phase has finite $\omega_c$8 and $\omega_c$9 (Wei et al., 1 Jul 2026). On a four-site ring, the principal superradiant configuration classes are the uniform pattern $N_a$0, the staggered pattern $N_a$1, the stripe-like pattern $N_a$2, and a low-symmetry three-minus pattern $N_a$3, with degeneracies $N_a$4, respectively. The first three were labeled HSRP, ISRP1, and ISRP2; the fourth, obtained numerically, was labeled ISRP3 (Wei et al., 1 Jul 2026).

The linear instability of the normal phase in the dissipative model occurs at

$N_a$5

where $N_a$6 is the photon normal-mode frequency. In the closed model, minimizing the mean-field energy yields

$N_a$7

For $N_a$8, the equilibrium ground state is uniform; for $N_a$9, it is staggered (Wei et al., 1 Jul 2026). The same work further found that, in the dissipative problem, different nonequilibrium branches can belong to different universality classes: NP$S_i$0HSRP and NP$S_i$1ISRP1 have $S_i$2, whereas NP$S_i$3ISRP2 has $S_i$4. By contrast, the closed Dicke lattice shares the equilibrium Dicke exponent $S_i$5 across distinct configurations (Wei et al., 1 Jul 2026).

In the optical-lattice realization, three phases were observed in the ground-state phase diagram. The homogeneous superfluid has $S_i$6, $S_i$7, and $S_i$8; the self-organized superradiant superfluid has $S_i$9, $\omega_a$0, and $\omega_a$1; and the self-organized superradiant Mott insulator has $\omega_a$2, $\omega_a$3, and $\omega_a$4 (Klinder et al., 2015). Here $\omega_a$5 measures matter-wave coherence, $\omega_a$6 the coherent intracavity field, and $\omega_a$7 the checkerboard density-wave order. The cavity-induced term $\omega_a$8 with $\omega_a$9 reshapes the Bose–Hubbard phase structure by suppressing number fluctuations on the “wrong” sublattice and stabilizing a Mott insulator inside a self-organized density wave (Klinder et al., 2015).

4. Dissipation, boundaries, and multistability

A defining nonequilibrium feature of the dissipative Dicke lattice is multistability. Photon hopping organizes the candidate superradiant steady states by lattice symmetry, and dissipation can stabilize several symmetry-distinguished configurations simultaneously (Wei et al., 1 Jul 2026). For the dissipative four-site ring, the complete phase diagram contains a stable normal region, regions with only one stable superradiant phase, twofold coexistence regions, threefold coexistence regions, and a fourfold coexistence region in which HSRP, ISRP1, ISRP2, and ISRP3 are all dynamically stable. A representative parameter set exhibiting fourfold coexistence is $\xi$0, $\xi$1, $\xi$2, and $\xi$3 (Wei et al., 1 Jul 2026). Which branch is reached depends on initial conditions because several attracting fixed points can coexist.

Open boundary conditions qualitatively alter this picture. For the dissipative Dicke lattice with finite $\xi$4, open boundaries are implemented by setting the end-to-end hopping $\xi$5, whereas periodic boundaries correspond to $\xi$6. Under open boundary conditions, the photonic normal-mode dispersion changes from

$\xi$7

to

$\xi$8

The normal-to-superradiant threshold remains

$\xi$9

but the phase structure becomes strongly boundary-sensitive (Wei et al., 14 Aug 2025).

For $$H=\sum_{i=1}^{N} H_i^{\rm Dicke}-\xi\sum_{i=1}^{N}(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i),$$0, open boundaries generate a “zoo of superradiant phases” not present in the corresponding infinite system: O1 with $$H=\sum_{i=1}^{N} H_i^{\rm Dicke}-\xi\sum_{i=1}^{N}(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i),$$1 and equal signs on all sites, O2 with $$H=\sum_{i=1}^{N} H_i^{\rm Dicke}-\xi\sum_{i=1}^{N}(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i),$$2 and $$H=\sum_{i=1}^{N} H_i^{\rm Dicke}-\xi\sum_{i=1}^{N}(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i),$$3, O3 with all three amplitudes unequal and nonzero, and O4 with equal edge amplitudes opposite in sign to the center (Wei et al., 14 Aug 2025). The same study showed that a homogeneous superradiant steady state is absent under open boundary conditions for any finite $$H=\sum_{i=1}^{N} H_i^{\rm Dicke}-\xi\sum_{i=1}^{N}(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i),$$4, because the edge and bulk steady-state constraints cannot be satisfied by one uniform complex $$H=\sum_{i=1}^{N} H_i^{\rm Dicke}-\xi\sum_{i=1}^{N}(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i),$$5. This produces monostable, bistable, and tristable regions that are absent or greatly simplified under periodic boundaries (Wei et al., 14 Aug 2025).

A related dissipative effect is finite-momentum instability in hybrid microwave-cavity arrays. There the mode-resolved threshold is

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

so $$H=\sum_{i=1}^{N} H_i^{\rm Dicke}-\xi\sum_{i=1}^{N}(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i),$$7. For $$H=\sum_{i=1}^{N} H_i^{\rm Dicke}-\xi\sum_{i=1}^{N}(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i),$$8, the instability is at $$H=\sum_{i=1}^{N} H_i^{\rm Dicke}-\xi\sum_{i=1}^{N}(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i),$$9, giving a homogeneous superradiant transition. For $$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$0, the transition occurs at finite momentum, with

$$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$1

This finite-$$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$2 selection is a distinct non-equilibrium signature of the driven open Dicke lattice model (Zou et al., 2014).

5. Implementations and direct observations

A direct realization of Dicke–Hubbard physics was achieved with a cigar-shaped $$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$3 Bose–Einstein condensate of $$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$4 atoms in $$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$5, overlapped with a high-finesse cavity mode along $$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$6 and an optical lattice inside the cavity (Klinder et al., 2015). The cavity had waist $$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$7, finesse $$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$8, Purcell factor $$H_i^{\rm Dicke}=\omega_c c_i^\dagger c_i+\omega_a S_i^z+\frac{2g}{\sqrt{N_a}}(c_i+c_i^\dagger)S_i^x.$$9, and $$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$0. Two standing waves at $$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$1 were used: a transverse pump along $$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$2 and an external lattice along $$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$3 at fixed depth $$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$4, with

$$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$5

For $$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$6, the dispersive shift was $$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$7, placing the system in a strong cooperative-coupling regime (Klinder et al., 2015).

Superradiance was detected through the intracavity photon number $$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$8 obtained from leaked photons, while matter-wave coherence and density-wave order were monitored by $$\frac{d\rho}{dt}=-i[H,\rho]+\kappa\sum_i D[c_i]\rho, \qquad D[O]\rho=2O\rho O^\dagger-O^\dagger O\rho-\rho O^\dagger O,$$9 time-of-flight absorption images. Bragg peaks at $c_i$00 arose from the pump lattice, and peaks at $c_i$01 from the cavity–pump interference lattice (Klinder et al., 2015). The HSF$c_i$02SSF transition appeared as the onset of nonzero $c_i$03 in the negative-$c_i$04 half-plane. The SSF$c_i$05SMI transition was identified by a sudden increase in the zero-momentum width $c_i$06, together with continued growth of $c_i$07 and kinks in $c_i$08 versus $c_i$09 that were interpreted as reduced number fluctuations (Klinder et al., 2015).

A complementary implementation was proposed for arrays of superconducting microwave cavities coupled to ensembles of nitrogen-vacancy centers in diamond. By cavity-assisted Raman transitions between two NV spin states, the scheme realizes a generalized Dicke interaction containing both Jaynes–Cummings and anti–Jaynes–Cummings terms, with tunable effective detunings and couplings (Zou et al., 2014). The effective lattice Hamiltonian is

$c_i$10

with photon loss at rate $c_i$11 (Zou et al., 2014). The paper quoted realistic values $c_i$12, bare collective coupling $c_i$13–$c_i$14, Raman coupling $c_i$15 for $c_i$16 and $c_i$17 up to $c_i$18, cavity decay $c_i$19–$c_i$20, inhomogeneous broadening $c_i$21, and arrays of $c_i$22–$c_i$23 cavities as feasible (Zou et al., 2014).

Finite cavity or resonator arrays with engineered boundaries are also experimentally relevant in circuit QED and quantum optics. Proposed observables include site-resolved cavity fields $c_i$24, photon numbers $c_i$25, spin polarizations $c_i$26, emitted-light phase patterns, and multistability under parameter sweeps (Wei et al., 14 Aug 2025).

6. Related models, conceptual extensions, and open directions

The Dicke lattice idea has been extended in several directions that change the balance between locality, interaction range, and the role of counter-rotating terms. In the Dicke–Hubbard lattice with full counter-rotating interaction,

$c_i$27

mean-field plus extended coherent-state calculations found a localization–delocalization transition marked by a finite photonic order parameter $c_i$28, spontaneous parity breaking, complete suppression of Mott lobes, and monotonic enhancement of $c_i$29 with increasing $c_i$30 (Lu et al., 2016). In the large-$c_i$31 limit, the critical line is

$c_i$32

and the model contrasts sharply with the rotating-wave Dicke–Hubbard or Tavis–Cummings lattice, where lobe physics survives (Lu et al., 2016).

A different extension couples a single bosonic mode to a long-range Ising lattice. In the long-range Dicke–Ising model on square and triangular lattices, the $c_i$33 limit exhibits devil’s staircase magnetization plateaux, while finite light–matter coupling produces both uniform and magnetically ordered superradiant phases with finite photon density (Koziol et al., 4 Mar 2025). Examples include a three-sublattice $c_i$34–$c_i$35–$c_i$36 superradiant phase on the square lattice and a superradiant Wigner crystal with a four-site unit cell, the $c_i$37–$c_i$38 SR phase, on the triangular lattice (Koziol et al., 4 Mar 2025). There, the transition from normal to superradiant phases is second order with Dicke universality when magnetic order is preserved and first order when it changes (Koziol et al., 4 Mar 2025).

The superradiance lattice is conceptually more distant but clarifies how Dicke collectivity can itself define a lattice. Timed Dicke states of three-level atoms form momentum-space sites, while a standing-wave EIT coupling induces nearest-neighbor hopping between them. In one dimension, detuning between the two standing-wave components produces an effective uniform force in momentum space, enabling Bloch oscillations, Wannier–Stark ladders, Bloch band collapse, and dynamic localization (Wang et al., 2014). In two dimensions, three coupling beams generate a honeycomb superradiance lattice with graphene-like Dirac physics (Wang et al., 2014). This construction differs from the conventional real-space Dicke lattice of coupled cavities, but it preserves the central idea that collective light–matter states can be arranged into lattice-like structures with tunable connectivity (Wang et al., 2014).

Several open directions follow directly from the surveyed literature. Multimode cavities could realize richer Dicke lattices with competing patterns and frustrated long-range interactions; higher dimensions and different geometries may host additional superradiant insulating and supersolid phases; and theoretical treatments incorporating full cavity dynamics, such as Keldysh or Langevin approaches, are expected to refine the description of hysteresis and bistability beyond static cavity elimination (Klinder et al., 2015). In finite arrays, boundary engineering already emerges as a control parameter of the same status as hopping and dissipation (Wei et al., 14 Aug 2025). These developments indicate that the Dicke lattice model is best understood not as one fixed Hamiltonian, but as a broad framework for studying how collective radiation reorganizes many-body order once spatial structure is made dynamical.