Dissipative Dicke Lattice Model

• The dissipative Dicke lattice model is an open quantum many-body system featuring collective light–matter interactions with photon hopping and finite cavity loss.
• It employs NV ensembles and superconducting microwave resonators to engineer effective Dicke interactions, enabling tunable superradiant and finite‐k phases.
• Dissipation drives pattern formation and multistability, inducing nonequilibrium instabilities and phase boundaries absent in closed systems.

The dissipative Dicke lattice model is an open quantum many-body model in which each site of a cavity array hosts a collective light–matter degree of freedom, while photons hop between neighboring cavities and leak out at a finite rate. In its canonical form, the model combines ultrastrong on-site Dicke interactions, including both rotating and counter-rotating terms, with inter-cavity hopping and Markovian cavity loss. In the formulation developed for hybrid quantum system arrays of nitrogen-vacancy (NV) ensembles coupled to superconducting microwave resonators, the model supports normal, uniform superradiant, finite-$k$ superradiant, and unstable nonequilibrium regimes, with several phase-boundary features that are induced by dissipation and have no direct equilibrium counterpart (Zou et al., 2014).

1. Model definition and effective Hamiltonian

In the hybrid implementation, an ensemble of NV centers in diamond is placed on top of a superconducting stripline microwave cavity. The NV defect has an electronic spin $S=1$ ground-state triplet $|m_s\rangle = |0\rangle, |\pm 1\rangle$, with zero-field splitting $D \approx 2.88\,{\rm GHz}$. A static magnetic bias field sets the Zeeman splitting $\delta_B$ between $|\pm 1\rangle$, while two classical microwave tones at frequencies $\omega_1 \approx D+\delta_B$ and $\omega_2 \approx D-\delta_B$ generate cavity-assisted Raman transitions between $|-1\rangle$ and $|+1\rangle$, allowing adiabatic elimination of $S=1$0 and producing an effective Dicke interaction (Zou et al., 2014).

For a single cavity, the resulting effective Hamiltonian is

$S=1$1

where $S=1$2 act in the $S=1$3 subspace. The effective parameters are

$S=1$4

$S=1$5

When $S=1$6, the interaction reduces to the full Dicke form $S=1$7.

For a homogeneous ensemble of $S=1$8 spins, one defines collective operators

$S=1$9

with collective coupling $|m_s\rangle = |0\rangle, |\pm 1\rangle$0. Extending to an array of $|m_s\rangle = |0\rangle, |\pm 1\rangle$1 coupled cavities yields the Dicke lattice Hamiltonian

$|m_s\rangle = |0\rangle, |\pm 1\rangle$2

with cavity dispersion

$|m_s\rangle = |0\rangle, |\pm 1\rangle$3

This form makes explicit that the lattice generalization is not a perturbative correction to the single-mode Dicke model, but a momentum-resolved open Dicke system whose instability structure depends on both hopping and dissipation.

2. Open-system dynamics and order parameters

The dominant dissipation channel in the NV-resonator realization is photon loss from each microwave cavity at rate $|m_s\rangle = |0\rangle, |\pm 1\rangle$4, with $|m_s\rangle = |0\rangle, |\pm 1\rangle$5 appearing in the Lindblad equation as half the energy decay rate. Spin relaxation is neglected because $|m_s\rangle = |0\rangle, |\pm 1\rangle$6 at cryogenic temperatures, while spin dephasing enters primarily through static inhomogeneous broadening $|m_s\rangle = |0\rangle, |\pm 1\rangle$7 rather than an explicit Lindblad term (Zou et al., 2014).

For the lattice, the master equation is

$|m_s\rangle = |0\rangle, |\pm 1\rangle$8

This driven-open structure is essential: the steady states are determined by the competition between coherent collective pair creation processes and cavity dissipation, not by minimization of a closed-system free energy.

The normal phase is characterized by

$|m_s\rangle = |0\rangle, |\pm 1\rangle$9

with stability determined by the linearized fluctuation spectrum. Superradiant phases are diagnosed by finite cavity expectation values and transverse spin coherence. A convenient photonic order parameter is

$D \approx 2.88\,{\rm GHz}$0

The spin observables most commonly used are $D \approx 2.88\,{\rm GHz}$1 and $D \approx 2.88\,{\rm GHz}$2. Across the superradiant transition, $D \approx 2.88\,{\rm GHz}$3 increases from $D \approx 2.88\,{\rm GHz}$4, while transverse coherence develops.

A common simplification is to regard the dissipative Dicke lattice as a closed Dicke lattice with a phenomenological linewidth. The open-system formulation shows that this is incomplete: the linear stability problem is intrinsically non-Hermitian, and the ordering wavevector can be selected by dissipation itself rather than by equilibrium mode softening.

3. Nonequilibrium phases and dissipation-induced instabilities

For a single cavity with homogeneous couplings and no Stark-shift correction to the threshold, the open Dicke critical coupling is

$D \approx 2.88\,{\rm GHz}$5

Above threshold, the semiclassical steady state has finite photon amplitude and transverse spin order (Zou et al., 2014).

In the lattice, the instability condition becomes mode dependent:

$D \approx 2.88\,{\rm GHz}$6

This produces several distinct regimes.

A uniform superradiant transition occurs at $D \approx 2.88\,{\rm GHz}$7 when $D \approx 2.88\,{\rm GHz}$8, with

$D \approx 2.88\,{\rm GHz}$9

In that phase, all cavities acquire the same coherent field.

More distinctively, there is a dissipation-induced finite-$\delta_B$0 transition in the parameter window

$\delta_B$1

for which the first unstable mode has finite wavevector

$\delta_B$2

and the critical coupling becomes

$\delta_B$3

The steady state then exhibits spatial modulation,

$\delta_B$4

with a random offset $\delta_B$5 from run to run. The original analysis identifies this as a genuine nonequilibrium, dissipation-induced pattern-forming phase absent in the equilibrium Dicke lattice (Zou et al., 2014).

A further regime appears when $\delta_B$6 and $\delta_B$7. Then some photonic modes satisfy $\delta_B$8, and the normal phase is unstable for arbitrarily small $\delta_B$9. The physical mechanism is the counter-rotating $|\pm 1\rangle$0 pair-creation term: it injects energy into the spin sector while photons dissipate, leading to large-amplitude spin oscillations with small photon amplitude rather than a simple stationary superradiant state.

These results delimit a central conceptual point of the dissipative Dicke lattice model: loss does not merely suppress ordering. It can select the ordering wavevector, create pattern-forming phases, and generate instability windows that are not inherited from the closed model.

4. Inhomogeneous broadening and ultrastrong effective coupling

A defining feature of the NV implementation is strong static inhomogeneous broadening arising from strain, spin–spin dipolar couplings, and hyperfine interactions, notably from $|\pm 1\rangle$1 and $|\pm 1\rangle$2. The spin-frequency distribution $|\pm 1\rangle$3 has full width at half maximum $|\pm 1\rangle$4, with typical $|\pm 1\rangle$5 in dense samples. To treat this, spins are grouped into sub-ensembles and then described by a spectral density

$|\pm 1\rangle$6

The normal-phase instability condition becomes the principal-value equation

$|\pm 1\rangle$7

For a Lorentzian distribution,

$|\pm 1\rangle$8

A key figure of merit is the collective cooperativity

$|\pm 1\rangle$9

The non-equilibrium superradiant transition can be observed if $\omega_1 \approx D+\delta_B$0, even when $\omega_1 \approx D+\delta_B$1 (Zou et al., 2014).

The same analysis reports that for q-Gaussian-like distributions, with $\omega_1 \approx D+\delta_B$2 observed experimentally, $\omega_1 \approx D+\delta_B$3 is reduced relative to the Lorentzian case and can even lie below the homogeneous critical coupling. This does not imply that broadening enhances coherence in a generic sense; rather, it indicates that the detailed line shape, not only the linewidth, enters the principal-value stability condition.

The ultrastrong-coupling aspect is also effective rather than microscopic. The actual magnetic single-spin coupling is small, $\omega_1 \approx D+\delta_B$4, but Raman engineering makes the effective couplings $\omega_1 \approx D+\delta_B$5 tunable, and the collective scale $\omega_1 \approx D+\delta_B$6 can become comparable to the effective detunings $\omega_1 \approx D+\delta_B$7 and $\omega_1 \approx D+\delta_B$8, which are in the MHz range. Retaining the counter-rotating sector is therefore not optional: it is essential to the full Dicke interaction and to the dissipative finite-$\omega_1 \approx D+\delta_B$9 and unstable regimes.

5. Boundary conditions, finite-size phases, and multistability

Later work showed that finite-size dissipative Dicke lattices are highly sensitive to boundary conditions. For a one-dimensional chain of $\omega_2 \approx D-\delta_B$0 coupled cavities with cavity loss $\omega_2 \approx D-\delta_B$1, periodic boundary conditions (PBC) use a boundary link $\omega_2 \approx D-\delta_B$2, while open boundary conditions (OBC) set $\omega_2 \approx D-\delta_B$3. The corresponding photonic dispersions are

$\omega_2 \approx D-\delta_B$4

and the normal-phase threshold is

$\omega_2 \approx D-\delta_B$5

For finite $\omega_2 \approx D-\delta_B$6, OBC qualitatively reorganize the phase structure: a homogeneous superradiant steady state is absent for any finite $\omega_2 \approx D-\delta_B$7, because the steady-state constraints at the edges and in the bulk are incompatible unless the uniform field vanishes, while a “zoo” of inhomogeneous superradiant phases, together with extended bistable and tristable regions, appears (Wei et al., 14 Aug 2025).

For $\omega_2 \approx D-\delta_B$8, the OBC phase diagram includes patterns such as the edge-localized state $\omega_2 \approx D-\delta_B$9, with $|-1\rangle$0 and $|-1\rangle$1, as well as configurations with $|-1\rangle$2 but different sign structures. The normal-phase stability window also differs between PBC and OBC. For $|-1\rangle$3,

$|-1\rangle$4

As $|-1\rangle$5, these differences shrink, and the threshold becomes boundary independent.

A complementary symmetry-based analysis on periodic rings organizes superradiant phases by the sign pattern of $|-1\rangle$6, modulo cyclic translations and a global sign flip. For a dissipative four-site ring, the representative classes are $|-1\rangle$7, $|-1\rangle$8, $|-1\rangle$9, and $|+1\rangle$0, corresponding respectively to a homogeneous superradiant phase and several inequivalent inhomogeneous phases. The complete nonequilibrium phase diagram contains regions with up to four simultaneously stable superradiant phases. Using the discrete truncated Wigner approximation, the same work found two nonequilibrium universality classes: the homogeneous and staggered-like phases have $|+1\rangle$1, while the dimer-like phase has $|+1\rangle$2 (Wei et al., 1 Jul 2026).

Taken together, these results show that finite-size and boundary effects are not minor corrections. In dissipative Dicke lattices, they can eliminate homogeneous superradiance, generate edge-localized order, and restructure multistability. A common misconception is that translationally invariant infinite-lattice results automatically describe experimentally achievable small arrays; the finite-size analyses show that this need not hold.

6. Experimental realization, observables, and conceptual significance

The NV-resonator proposal specifies an experimentally grounded parameter regime. Single-spin magnetic couplings are $|+1\rangle$3, while collective coupling $|+1\rangle$4 has been reported experimentally for a single module. With Raman engineering, achievable values include $|+1\rangle$5, $|+1\rangle$6 up to $|+1\rangle$7, $|+1\rangle$8, cavity loss $|+1\rangle$9, spin broadening $S=1$00, effective detunings $S=1$01, and $S=1$02 spins per cavity. Arrays of one- and two-dimensional superconducting cavities with nearly identical $S=1$03 and controllable hopping $S=1$04 are described as standard (Zou et al., 2014).

The primary observables are photonic. Cavity output fields provide $S=1$05, photon number, and emission spectra, while the structure factor $S=1$06 distinguishes uniform order from finite-$S=1$07 order by peaks at $S=1$08 or $S=1$09. Site-resolved or mode-resolved measurements can reveal the cosine modulation of the finite-$S=1$10 phase, and a random $S=1$11 across experimental runs signals spontaneous breaking of translation symmetry. Spin observables include $S=1$12 and $S=1$13, which can be inferred through cavity-mediated spectroscopy, dispersive shifts, or NV ensemble probes such as ODMR.

The finite-$S=1$14 window

$S=1$15

is especially significant experimentally because it provides a clear signature of dissipation: the transition occurs at

$S=1$16

a feature explicitly identified as absent in the equilibrium Dicke lattice. Likewise, the unstable regime at $S=1$17 implies that failure to observe a simple stationary superradiant pattern does not necessarily mean that the system remains in the normal phase; it may instead reflect a dissipative instability with large-amplitude spin dynamics and small field amplitudes.

The broader significance of the dissipative Dicke lattice model lies in its role as a controlled setting for nonequilibrium pattern formation, symmetry breaking, and multistability in collective light–matter systems. The foundational NV-based formulation established that cavity dissipation qualitatively reshapes the phase diagram, while subsequent finite-size and configuration-based studies showed that boundaries and lattice symmetry can further organize or proliferate stationary phases. The combined picture is that the dissipative Dicke lattice is not simply a lossy version of the Dicke model on a graph, but a distinct nonequilibrium many-body system whose phase structure is set jointly by ultrastrong collective coupling, photon dispersion, loss, and geometry (Zou et al., 2014).