Multi-Species Open Dicke Model

• Multi-Species Open Dicke Model is a driven-dissipative light–matter system where multiple spin ensembles interact with a common lossy cavity, leading to non-conserved excitation dynamics.
• Redfield treatments and nonlinear mean-field analyses reveal key features such as non-reciprocal couplings, exceptional points, and distinct dynamical phase transitions.
• Experimental realizations in cavity-QED setups demonstrate synchronization, transient chaos, and dissipative quantum scarring across varied parameter regimes.

Searching arXiv for the cited multi-species/open Dicke model papers to ground the article in current literature. The multi-species open Dicke model denotes a class of driven-dissipative light–matter systems in which several collective spin ensembles couple to a common lossy cavity mode, so that excitation number is not conserved and the cavity mediates both coherent and dissipative inter-species couplings. In the recent literature, this includes the two-component non-reciprocal Dicke model, its multi-species spin-only Redfield reduction, and the open coupled-top Dicke model realized by coupling a two-species Bose–Josephson junction to a lossy cavity (Chiacchio et al., 2023, Jachinowski et al., 10 Jul 2025, Mondal et al., 21 May 2026).

1. Model class and physical realizations

A common starting point is the Hepp-Lieb-Dicke setting with multiple spin species coupled to a single cavity mode. In the two-component formulation, one considers two collective spin-$\tfrac12$ ensembles of equal size $N$, with collective operators

$$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$

and Hamiltonian

$$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$

A standard parametrization is $g_1=\lambda e^{+i\phi}$, $g_2=\lambda e^{-i\phi}$, which makes the relative phase structure explicit (Chiacchio et al., 2023).

The multi-species generalization analyzed in the spin-only Redfield treatment considers $N_m$ spin-$\tfrac12$ atoms of species $m=-M,\dots,M$ coupled collectively to a single cavity mode $a$, with the cavity itself coupled to a broad photonic bath. In that formulation, at most two populated species $N$0 are assumed for explicit calculations, while the generalization to more species is stated to be straightforward. A key structural feature is that the phases $N$1 cannot in general be gauged away when more than one species is present (Jachinowski et al., 10 Jul 2025).

The open coupled-top Dicke model is a distinct two-species realization in which the matter sector consists of two collective spins of length $N$2, built from two bosonic species in a double well with $N$3 atoms in each species. Its Hamiltonian contains the cavity energy $N$4, tunneling $N$5, an antiferromagnetic interaction $N$6, and a light–matter coupling $N$7 between the cavity quadrature and $N$8. The paper identifies this model as effectively realizable by coupling a two-species Bose–Josephson junction to a lossy cavity (Mondal et al., 21 May 2026).

Variant	Degrees of freedom	Distinctive focus
Two-component non-reciprocal Dicke model	Two collective spin ensembles plus cavity	Non-reciprocal interactions, $N$9-type symmetry, dynamical phase (Chiacchio et al., 2023)
Multi-species nonreciprocal Dicke model	Multiple spin species plus cavity and bath; spin-only reduction	Redfield spin-only dynamics, limit cycles, exceptional point, finite-size Liouvillian analysis (Jachinowski et al., 10 Jul 2025)
Open coupled-top Dicke model	Two collective spins of length $$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$0 plus lossy cavity	Synchronization, transient chaos, dissipative quantum scarring, chaos-assisted tunneling (Mondal et al., 21 May 2026)

2. Open-system generators and reduced descriptions

The open Dicke setting is defined by a Lindblad master equation in which cavity photon loss is the primary dissipative channel. For the two-component model,

$$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$1

with

$$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$2

Here $$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$3 is the cavity decay rate and $$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$4 is an optional spin-decay rate (Chiacchio et al., 2023). In the coupled-top model, photon decay is described by the single jump operator $$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$5, and the Liouvillian is written as

$$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$6

The same work also gives the equivalent sum-over-jumps representation of the Lindbladian (Mondal et al., 21 May 2026).

A central methodological development is the replacement of naive adiabatic elimination by a Redfield treatment. In the multi-species nonreciprocal model, one first derives a Redfield equation for the joint spin+cavity density operator $$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$7, retaining nonsecular terms. The paper states that retaining these nonsecular terms is crucial to correctly capture the Dicke-transition physics. Tracing out both the cavity and the extra-cavity bath then yields an effective spin-only Redfield Liouvillian whose coefficients are controlled by a zero-temperature Lorentzian spectral density,

$$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$8

with memory time $$J_\alpha^{(i)}=\frac12\sum_{j=1}^N \sigma^\alpha_{j,(i)},\qquad \alpha=x,y,z,$$9 (Jachinowski et al., 10 Jul 2025).

In the fast-cavity limit, $$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$0, the spin-only Redfield dynamics reduces to a Lindblad form with effective couplings parameterized by

$$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$1

This separation between Redfield and adiabatic-elimination descriptions is not merely technical: the multi-species analysis states that adiabatic elimination misses $$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$2 and $$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$3 terms and can therefore produce quantitatively, and sometimes qualitatively, different predictions, especially at finite $$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$4 or small $$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$5 (Jachinowski et al., 10 Jul 2025).

3. Mediated interactions, nonreciprocity, and symmetry structure

The defining interaction mechanism is cavity-mediated coupling between distinct spin species. In the two-component non-reciprocal Dicke model, cavity loss together with complex couplings $$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$6 generates effective asymmetrical inter-species couplings. In the adiabatic limit, these are written as

$$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$7

and, equivalently,

$$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$8

Non-reciprocity is therefore identified by $$H=\omega_c a^\dagger a+\sum_{i=1}^2 \omega_i J_z^{(i)}+\frac{1}{\sqrt N}\sum_{i=1}^2\bigl[g_i a+g_i^* a^\dagger\bigr]J_x^{(i)}.$$9 (Chiacchio et al., 2023).

The spin-only multi-species formulation makes the same point at the Liouvillian level. Both the induced Hamiltonian and the dissipator carry phase factors $g_1=\lambda e^{+i\phi}$0. Writing the couplings in a $g_1=\lambda e^{+i\phi}$1 pseudospin basis reveals off-diagonal terms mixing $g_1=\lambda e^{+i\phi}$2 with $g_1=\lambda e^{+i\phi}$3 whose coefficients are proportional to $g_1=\lambda e^{+i\phi}$4. These terms are odd under exchange $g_1=\lambda e^{+i\phi}$5 and hence break reciprocity (Jachinowski et al., 10 Jul 2025).

The symmetry structure is correspondingly richer than in the single-species open Dicke model. The multi-species Redfield analysis identifies superradiant parity

$g_1=\lambda e^{+i\phi}$6

as a weak $g_1=\lambda e^{+i\phi}$7 symmetry commuting with the adjoint Lindbladian. For two species with equal $g_1=\lambda e^{+i\phi}$8 and $g_1=\lambda e^{+i\phi}$9, it also defines a bipartite swap $g_2=\lambda e^{-i\phi}$0 exchanging $g_2=\lambda e^{-i\phi}$1, together with time-reversal operations $g_2=\lambda e^{-i\phi}$2 or $g_2=\lambda e^{-i\phi}$3. The combination $g_2=\lambda e^{-i\phi}$4 is a weak symmetry of the full and fast-cavity Lindbladians for $g_2=\lambda e^{-i\phi}$5; setting $g_2=\lambda e^{-i\phi}$6 restores $g_2=\lambda e^{-i\phi}$7 symmetry of the fast-cavity model, while $g_2=\lambda e^{-i\phi}$8 explicitly breaks it (Jachinowski et al., 10 Jul 2025).

In the earlier two-component analysis, the nonlinear mean-field equations are invariant for $g_2=\lambda e^{-i\phi}$9 and $N_m$0 under the combined transformation consisting of species exchange $N_m$1 and $N_m$2. That work further states that the associated non-reciprocal phase transition does not necessitate the presence of any underlying broken symmetry or exceptional points in the spectrum, both believed to be essential requirements for non-reciprocal phase transitions (Chiacchio et al., 2023). This directly addresses a common misconception imported from simpler non-Hermitian settings.

4. Mean-field fixed points, instabilities, and phase diagrams

At the semiclassical level, the two-component model takes $N_m$3 and factorizes operator products in terms of $N_m$4 and spin polarizations $N_m$5. The normal phase is

$N_m$6

and its stability is determined by the eigenvalues of an $N_m$7 fluctuation matrix $N_m$8. The normal state loses stability when the leading eigenvalue satisfies $N_m$9. For $\tfrac12$0 and $\tfrac12$1, the static superradiant threshold is

$\tfrac12$2

while the dynamical threshold is given by

$\tfrac12$3

The resulting phase diagram in the $\tfrac12$4 plane contains a normal phase, static superradiant phases, and a dynamical non-stationary phase consisting of stable limit cycles (Chiacchio et al., 2023).

The multi-species Redfield treatment recasts the same problem in terms of nonlinear Bloch equations for $\tfrac12$5. The normal fixed point is $\tfrac12$6, $\tfrac12$7, and linearization yields a $\tfrac12$8 Jacobian. For two species with equal $\tfrac12$9, the superradiant pitchfork thresholds $m=-M,\dots,M$0 are obtained analytically; in the fast-cavity limit they simplify in terms of

$m=-M,\dots,M$1

A Hopf bifurcation occurs when a complex-conjugate pair crosses into $m=-M,\dots,M$2, producing a self-sustained oscillation with

$m=-M,\dots,M$3

The mean-field phase diagram includes NS, SR, and DS regions, together with coexistence pockets near $m=-M,\dots,M$4 between $m=-M,\dots,M$5-broken cycles. Along $m=-M,\dots,M$6, the work identifies a codimension-two exceptional point where Floquet multipliers coalesce on the unit circle (Jachinowski et al., 10 Jul 2025).

The coupled-top realization has a distinct but related fixed-point structure. Its classical equations for $m=-M,\dots,M$7 and $m=-M,\dots,M$8 admit four families of fixed points: NP$m=-M,\dots,M$9, NP$a$0, FSR$a$1, and FSR$a$2. The bifurcation from NP$a$3 to NP$a$4 occurs at

$a$5

and the Dicke transition from NP$a$6 to FSR$a$7 is

$a$8

These boundaries define three dynamical regimes in the $a$9 plane: Region I, with regular decay into NP$N$00 and fast synchronization; Region II, with transient chaos, eventual decay to NP$N$01, and slow synchronization; and Region III, with superradiant dynamics around FSR$N$02 and restored coherent oscillations (Mondal et al., 21 May 2026).

5. Dissipation-free subspaces and spontaneous synchronization

A distinctive feature of the open coupled-top Dicke model is the emergence of a decoherence-free subspace despite cavity loss. Defining symmetric and antisymmetric spin combinations,

$N$03

the lossy cavity is not populated when two conditions hold: the cavity amplitude vanishes, $N$04, and the spins satisfy

$N$05

The corresponding projector onto the decoherence-free subspace commutes with the Lindblad jump operator (Mondal et al., 21 May 2026).

Under the master-equation dynamics, any component of an initial state that overlaps this subspace survives, while orthogonal components decay. The constraints $N$06 and $N$07 imply $N$08 and $N$09 in the Schwinger-boson description. The dissipative evolution therefore projects the system onto a configuration in which the two species are perfectly antisynchronized in $N$10 and have opposite phase. In the classical $N$11 limit, this appears as spontaneous synchronization once the photon number $N$12 tends to zero (Mondal et al., 21 May 2026).

This mechanism is significant because the synchronization is not imposed by an external locking term; it is induced by photon loss through projection onto a dissipation-free sector. A plausible implication is that, within this realization, synchronization and dissipation are not competing tendencies but can instead be co-generated by the same Lindbladian structure. The phase diagram corroborates this interpretation by placing fast synchronization in Region I, slow synchronization after transient chaos in Region II, and restored coherent oscillations in the superradiant Region III (Mondal et al., 21 May 2026).

6. Dissipative quantum scarring, tunneling, finite-size signatures, and experimental scales

The coupled-top model uses non-Hermitian fixed points rather than closed-system eigenstates as the organizing structures for quantum scarring. In Region II, two distinct dissipative scarring phenomena are identified. The first is the NP$N$13-protected scar, obtained by preparing the coherent state $N$14 peaked at the unstable NP$N$15 saddle and monitoring the mixed-state survival probability

$N$16

The survival probability oscillates periodically with period $N$17 set by the homoclinic orbit of NP$N$18, and remains large without decay even for $N$19. Husimi plots show stretching along the homoclinic loop followed by recombination near the saddle at each revival, leading to the designation “dissipation-protected scar” (Mondal et al., 21 May 2026).

The second is the FSR$N$20 superradiant scar. Preparing one unstable excited superradiant branch $N$21, one defines

$N$22

In the closed system, $N$23 display slow oscillations with tunneling period $N$24, where $N$25 is the small level splitting. In the open system,

$N$26

so the memory of the unstable FSR$N$27 saddle survives parametrically longer than generic states. For small $N$28, Husimi distributions display back-and-forth tunneling between the two symmetry-related branches before eventual diffusion (Mondal et al., 21 May 2026).

The same work links this behavior to chaos-assisted macroscopic quantum tunneling. In the isolated limit, the splitting between the two FSR$N$29 coherent states obeys the WKB form

$N$30

with $N$31. Once the classical dynamics enters the chaotic regime of Region II, small chaotic layers appear around the barrier and enhance tunneling, so that $N$32 becomes a nonmonotonic function of $N$33 and control parameters (Mondal et al., 21 May 2026).

Beyond mean field, the multi-species Redfield treatment uses permutation symmetry to reduce the Liouvillian into blocks labeled by the conserved total spins $N$34. Exact diagonalization in this reduced basis yields finite-size signatures of the mean-field transitions. In the superradiant phase, the steady-state spin-averaged Wigner function $N$35 develops two peaks whose locations match the mean-field values of $N$36; as $N$37, the peaks of one species wash out, reflecting dephasing from nonreciprocity. The Liouvillian gap $N$38 changes character across the phase diagram and closes at $N$39 as $N$40, signaling the onset of the dynamical state (Jachinowski et al., 10 Jul 2025).

The coupled-top study also specifies experimentally relevant scales for cavity-QED implementations: atom numbers $N$41–$N$42 per species, corresponding to $N$43–$N$44; tunneling $N$45–$N$46 kHz; collisional interaction $N$47–$N$48 kHz; cavity frequency $N$49 kHz; cavity decay $N$50–$N$51 kHz; and light–matter coupling $N$52–$N$53 kHz. Within this regime, the work states that one can traverse from normal through synchronized to superradiant phases, observe transient chaos and synchronization at single-trajectory level, and probe both dissipative scars and chaos-enhanced macroscopic tunneling by varying $N$54 (Mondal et al., 21 May 2026).