Open Coupled-Top Dicke Model

• The open coupled-top Dicke model is a driven-dissipative system featuring two interacting collective spins coupled to a lossy cavity mode.
• It employs a Hamiltonian with cavity energy, transverse spin fields, and inter-spin coupling to capture synchronization and transient chaos.
• The analysis uncovers fixed points, phase transitions, and distinct dissipative scarring phenomena with implications for macroscopic quantum tunneling.

The open coupled-top Dicke model is a driven-dissipative variant of Dicke physics in which two interacting collective spins of equal magnitude $S$ are coupled to a single lossy cavity mode. It was introduced to study nonequilibrium many-body phenomena in a setting that combines direct matter-matter interaction, collective spin-photon coupling, and photon leakage, and it is proposed to be effectively realizable by coupling a two-species Bose-Josephson junction to a lossy cavity (Mondal et al., 21 May 2026).

1. Conceptual definition and scope

In the standard Dicke model, the matter sector is a single collective spin coupled to one bosonic mode. The open coupled-top Dicke model replaces that single collective spin by two interacting large spins, or “tops,” each of fixed magnitude $S$. The resulting matter sector is therefore not only collectively coupled to the cavity, but also internally structured by a direct interaction term proportional to $\hat S_{1z}\hat S_{2z}$ (Mondal et al., 21 May 2026).

The model is called “coupled-top Dicke” because each large spin can be viewed semiclassically as a rigid top on a Bloch sphere, while the cavity coupling is Dicke-like. Relative to the standard Dicke model, the defining differences are the two-spin structure, the direct matter-matter interaction $\frac{V}{S}\hat S_{1z}\hat S_{2z}$, and the exchange symmetry between the two spins, which produces dynamically distinct symmetric and antisymmetric sectors. For $V=0$, it reduces to a generalized Dicke model for a binary atomic mixture (Mondal et al., 21 May 2026).

A common source of confusion is the phrase “coupled-top.” In this context it does not denote a mere reinterpretation of a one-spin Dicke model in angular-momentum language; it denotes a model with two interacting collective spins. That distinguishes it from earlier open Dicke realizations in which only one collective pseudospin is present.

2. Hamiltonian structure and physical realization

The Hamiltonian contains four ingredients: the cavity energy $\omega_c \hat a^\dagger \hat a$, a transverse spin field $-J(\hat S_{1x}+\hat S_{2x})$, an antiferromagnetic inter-spin coupling $\frac{V}{S}\hat S_{1z}\hat S_{2z}$, and a Dicke-type spin-cavity coupling through $$\frac{\lambda}{\sqrt{2S}}(\hat S_{1z}+\hat S_{2z})(\hat a+\hat a^\dagger)$$. Energies are measured in units of $J$, and time in units of $S$0 (Mondal et al., 21 May 2026).

Openness enters through photon leakage from the cavity. The density matrix obeys the Lindblad master equation

$S$1

so dissipation acts directly only in the photonic channel, but indirectly reshapes the spin dynamics through the cavity coupling. The jump operator is the cavity annihilation operator $S$2 (Mondal et al., 21 May 2026).

The proposed microscopic realization uses a two-species Bose-Josephson junction. In that construction, each species occupies a Bose-Hubbard dimer with tunneling amplitude $S$3, interspecies interaction $S$4, and cavity coupling to the population imbalance. Under the Schwinger-boson mapping,

$S$5

each species becomes a collective spin of size $S$6. The imbalance variable

$S$7

is then the natural semiclassical coordinate for each top (Mondal et al., 21 May 2026).

The analysis assumes equal populations in both species, a collective-spin description with fixed $S$8, the semiclassical limit $S$9 for mean-field analysis, the factorization approximation $\hat S_{1z}\hat S_{2z}$0, and Markovian cavity loss.

3. Semiclassical equations, invariant sectors, and synchronization

In the large-$\hat S_{1z}\hat S_{2z}$1 limit, the cavity and spins are described by

$\hat S_{1z}\hat S_{2z}$2

with $\hat S_{1z}\hat S_{2z}$3. The semiclassical equations of motion contain cavity damping, collective feedback through $\hat S_{1z}\hat S_{2z}$4, and nonlinear spin precession generated by both the cavity field and the inter-spin coupling $\hat S_{1z}\hat S_{2z}$5 (Mondal et al., 21 May 2026).

Because the equations are invariant under exchange $\hat S_{1z}\hat S_{2z}$6, it is natural to introduce symmetric and antisymmetric variables

$\hat S_{1z}\hat S_{2z}$7

Two invariant dynamical classes are then identified. The antisymmetric class is

$\hat S_{1z}\hat S_{2z}$8

and the symmetric class is

$\hat S_{1z}\hat S_{2z}$9

The antisymmetric class is the dissipation-free or dark subspace. It has $\frac{V}{S}\hat S_{1z}\hat S_{2z}$0, but the paper stresses that $\frac{V}{S}\hat S_{1z}\hat S_{2z}$1 alone is not sufficient; the full class-I constraint is required, otherwise photons can be regenerated dynamically (Mondal et al., 21 May 2026).

Within the dark class, the surviving variables $\frac{V}{S}\hat S_{1z}\hat S_{2z}$2 follow an effective Hamiltonian dynamics equivalent to the Lipkin-Meshkov-Glick model. Photon loss therefore does not simply damp all motion uniformly. Instead, it can project trajectories onto a sector in which the cavity field vanishes while coherent collective spin dynamics persist.

This projection underlies the model’s synchronization mechanism. Cavity loss damps $\frac{V}{S}\hat S_{1z}\hat S_{2z}$3, suppresses the cavity-mediated drive, and attracts trajectories toward the antisymmetric manifold. On that manifold the synchronization conditions are

$\frac{V}{S}\hat S_{1z}\hat S_{2z}$4

Synchronization is therefore not externally imposed; it emerges through dissipative projection onto a dark sector (Mondal et al., 21 May 2026).

4. Fixed points, phase structure, and transient chaos

The model supports two normal phases and two superradiant branches. The normal fixed points satisfy $\frac{V}{S}\hat S_{1z}\hat S_{2z}$5, while the superradiant branches have $\frac{V}{S}\hat S_{1z}\hat S_{2z}$6 (Mondal et al., 21 May 2026).

Structure	Defining property	Character
$\frac{V}{S}\hat S_{1z}\hat S_{2z}$7	$\frac{V}{S}\hat S_{1z}\hat S_{2z}$8	Normal
$\frac{V}{S}\hat S_{1z}\hat S_{2z}$9	$V=0$0	Antiferromagnetic normal
$V=0$1	$V=0$2	Superradiant
$V=0$3	Second superradiant branch of isolated model	Unstable for $V=0$4

More explicitly, the antiferromagnetic normal phase is

$V=0$5

while the principal superradiant phase has

$V=0$6

Its field quadratures satisfy

$V=0$7

The $V=0$8 signs label the two symmetry-broken branches (Mondal et al., 21 May 2026).

For the isolated system, there is also an excited superradiant branch $V=0$9 with

$\omega_c \hat a^\dagger \hat a$0

stable in part of the isolated model’s phase diagram. When $\omega_c \hat a^\dagger \hat a$1, that branch becomes unstable, and this instability is central to the scarring dynamics discussed below.

An important dynamical feature is that these fixed points are partial attractors rather than full attractors: linearization shows that only two eigenvalues have negative real parts, while the rest have zero real parts. Long-time motion therefore need not collapse to a fixed point; it can settle onto reduced-phase-space orbits (Mondal et al., 21 May 2026).

The phase diagram in the $\omega_c \hat a^\dagger \hat a$2 plane separates three regimes. Region I is a normal regular regime with asymptotic photon decay and fast synchronization. Region II is a normal regime with transient chaos: the isolated system is chaotic for $\omega_c \hat a^\dagger \hat a$3, but under dissipation the chaos becomes transient and is followed by synchronization and restored regularity. Region III is a superradiant regime with stable $\omega_c \hat a^\dagger \hat a$4, nonzero long-time photon number, and periodic orbits around the superradiant fixed point (Mondal et al., 21 May 2026).

For the isolated coupled-top sector at $\omega_c \hat a^\dagger \hat a$5, $\omega_c \hat a^\dagger \hat a$6 bifurcates into the two $\omega_c \hat a^\dagger \hat a$7 branches at

$\omega_c \hat a^\dagger \hat a$8

in units $\omega_c \hat a^\dagger \hat a$9. More generally, the onset of chaos in the isolated model occurs for

$-J(\hat S_{1x}+\hat S_{2x})$0

With photon loss, chaos is no longer asymptotic in the normal region. It is diagnosed by early-time exponential growth and later decay of the photon-field decorrelator, together with time-dependent Lyapunov behavior, which identifies it as transient chaos rather than a persistent chaotic attractor.

5. Dissipative quantum scarring and macroscopic quantum tunneling

In this model, scarring is defined dynamically rather than spectrally, because dissipation drives the system into mixed states. The principal diagnostic is the survival probability

$-J(\hat S_{1x}+\hat S_{2x})$1

supplemented by branch overlaps

$-J(\hat S_{1x}+\hat S_{2x})$2

and Husimi distributions

$-J(\hat S_{1x}+\hat S_{2x})$3

for reduced spin sectors (Mondal et al., 21 May 2026).

The paper identifies two distinct dissipative scarring mechanisms. The first is a dissipation-protected scar associated with the unstable normal fixed point $-J(\hat S_{1x}+\hat S_{2x})$4. When the initial state is centered on that saddle, $-J(\hat S_{1x}+\hat S_{2x})$5 shows persistent periodic oscillations with negligible decay. The mechanism is geometric: $-J(\hat S_{1x}+\hat S_{2x})$6 lies inside the antisymmetric dark manifold, so the wavepacket spreads along a homoclinic orbit surrounding the stable $-J(\hat S_{1x}+\hat S_{2x})$7 points and repeatedly refocuses near $-J(\hat S_{1x}+\hat S_{2x})$8 without being erased by cavity loss (Mondal et al., 21 May 2026).

The second scar is tied to the unstable superradiant branch $-J(\hat S_{1x}+\hat S_{2x})$9. Starting from a coherent state on one branch, the survival probability decays, but much more slowly than for a generic chaotic initial state. The Husimi function remains localized around the unstable branch for long times before diffusing away. This scar is therefore metastable rather than protected: dissipation does not remove it immediately, but induces a slow leakage and phase-space diffusion (Mondal et al., 21 May 2026).

The distinction between the two scar types is sharp. The unstable-$\frac{V}{S}\hat S_{1z}\hat S_{2z}$0 scar is protected because it belongs to the dissipation-free sector. The unstable-$\frac{V}{S}\hat S_{1z}\hat S_{2z}$1 scar is not dissipation-free; it survives only as a long-lived transient structure with slow decay.

For sufficiently small $\frac{V}{S}\hat S_{1z}\hat S_{2z}$2, the unstable-$\frac{V}{S}\hat S_{1z}\hat S_{2z}$3 regime exhibits chaos-assisted macroscopic quantum tunneling. In that case $\frac{V}{S}\hat S_{1z}\hat S_{2z}$4 and $\frac{V}{S}\hat S_{1z}\hat S_{2z}$5 show complementary oscillations with decaying envelopes, and Husimi distributions alternate between the two unstable symmetry-broken branches. The interpretation given is tunneling between the two $\frac{V}{S}\hat S_{1z}\hat S_{2z}$6 branches, enhanced by mixing with surrounding chaotic states (Mondal et al., 21 May 2026).

6. Relation to broader open Dicke literature

The open coupled-top Dicke model sits within a broader family of open Dicke systems, but it is not reducible to any one of them. Earlier open Dicke experiments realized a single large collective pseudospin coupled to a lossy cavity mode and established nonequilibrium hysteresis, dynamical instability, and Kibble–Zurek-type scaling near the Hepp–Lieb–Dicke transition (Klinder et al., 2014). In that sense, the open coupled-top model inherits the driven-dissipative Dicke setting, but changes the matter sector qualitatively by introducing two interacting tops rather than one.

Other generalized open Dicke models showed that even one collective top coupled to a cavity can display Hopf bifurcations, oscillatory superradiance, period-doubling cascades, and chaotic attractors when co-rotating and counter-rotating couplings are independently tunable (Stitely et al., 2020). The coupled-top model extends that nonlinear phenomenology by adding exchange symmetry, an antiferromagnetic normal phase, a dark antisymmetric manifold, and explicit synchronization dynamics.

A different but closely related “multi-top” perspective appears in the open Dicke model with weak local spin-nonconserving dissipation, where fixed-total-spin sectors $\frac{V}{S}\hat S_{1z}\hat S_{2z}$7 are treated as distinct collective-spin sectors and weak local dephasing or decay couples adjacent sectors. In that formulation, each $\frac{V}{S}\hat S_{1z}\hat S_{2z}$8-subspace functions as a different collective spin or “top,” and the open Dicke problem becomes an effective inter-top dynamics in $\frac{V}{S}\hat S_{1z}\hat S_{2z}$9-space (Tong et al., 19 May 2025). That construction is conceptually adjacent to, but distinct from, the open coupled-top Dicke model: the former reorganizes one Dicke ensemble into multiple total-spin sectors, whereas the latter begins with two explicit interacting collective spins.

Two misconceptions are therefore best avoided. First, the open coupled-top Dicke model is not merely a standard open Dicke model rewritten in semiclassical language; its defining matter content is two interacting tops. Second, photon loss does not simply wash out coherent many-body structure in this model. A central result is the opposite: photon loss can project the dynamics onto a dark antisymmetric manifold, induce spontaneous synchronization, convert sustained chaos into transient chaos, and protect one class of scarred revivals while only slowly degrading another (Mondal et al., 21 May 2026).