Chaos to Synchronization and Dissipative Quantum Scarring in Open Coupled top-Dicke model in a Lossy Cavity

Abstract: We present a variant of the Dicke model, termed as the open coupled-top Dicke model, which enables the exploration of rich non-equilibrium phenomena, particularly the fate of quantum scars in an open environment. This model can effectively be realized by coupling a two-species Bose-Josephson junction to a lossy cavity. Photon loss induces spontaneous synchronization via projection onto a dissipation-free subspace, along with transient chaos followed by restoration of synchronization and coherence. We identify two distinct scarring phenomena in the presence of dissipation. One remains protected, exhibiting persistent revivals, while the scar associated with the superradiant phase displays a dissipation-induced slow decay of the survival probability. Remarkably, for sufficiently small spin magnitude, the chaos-assisted macroscopic quantum tunneling is linked to the latter type of scarring. The results can be readily tested in ongoing cavity QED experiments and have broader applicability in other platforms.

• The paper demonstrates that engineered photon loss via Lindblad dynamics drives the transition from chaos to synchronization in the open coupled‐top Dicke model.
• The paper reveals persistent dissipation-protected quantum scars and chaos-assisted macroscopic tunneling, highlighting distinct survival probabilities and localization phenomena.
• The paper analyzes fixed-point configurations and dynamical regimes, establishing mechanisms for robust quantum state stabilization in cavity QED architectures.

Chaos, Synchronization, and Dissipative Quantum Scarring in the Open Coupled-Top Dicke Model

Model Definition and Physical Realization

The paper introduces the open coupled-top Dicke (OCTD) model, a hybrid quantum system comprising two antiferromagnetically interacting spin-$S$ particles coupled to a single photonic cavity mode with frequency $\omega_c$ and photon loss rate $\kappa$. This construction generalizes the canonical Dicke model by including spin-spin interactions and species degrees of freedom. Experimentally, the model is realized via a two-component Bose-Josephson junction embedded in a lossy optical cavity, effectively mapped to large collective spins using Schwinger boson mapping. Crucially, photon loss is modeled via Lindblad dynamics, resulting in non-unitary evolution and mixed states in the long-time limit (2605.22953).

Figure 1: Schematic of the OCTD setup, phase diagram in the $\lambda$-$V$ plane for fixed $\kappa$, fixed-point configurations, and decorrelator dynamics characterizing transient chaos.

Fixed Points, Dynamical Classes, and Phase Structure

The classical equations of motion admit fixed points corresponding to normal and superradiant phases:

• Normal Phase (NP$_1$, NP$_2$): Both characterized by vanishing photon number ($n^*=0$), with NP$_1$ exhibiting parallel spins and NP$\omega_c$0 an antiferromagnetic configuration determined by $\omega_c$1. For $\omega_c$2 (with $\omega_c$3 in the isolated case), NP$\omega_c$4 experiences a bifurcation, leading to symmetry breaking.
• Ferromagnetic Superradiant (FSR$\omega_c$5): Nonzero photon occupation with parallel spins, exhibiting symmetry-broken branches.
• Superradiant Excited State (FSR$\omega_c$6): Phase occurs as an excited state in the isolated system; in open systems, it becomes an unstable fixed point, serving as the locus for dissipative quantum scarring.

Spin-exchange symmetry allows reduction to symmetric and antisymmetric dynamical classes. The antisymmetric class yields a decoherence-free subspace (DFS) with vanishing photon field, structuring regular periodic spin dynamics, and underpinning synchronization. Stability analysis reveals partial attractors: only a subset of modes exhibits decay, while the remainder sustains oscillatory motion at frequencies set by the imaginary parts of the stability matrix eigenvalues.

Dissipation-Induced Synchronization and Transient Chaos

Photon loss dynamically projects the system onto the DFS, enforcing synchronization between spins: the symmetric variables $\omega_c$7 and $\omega_c$8 decay to zero independent of initial conditions. Two parameter regions emerge:

• Region I (Regular Dynamics): Fast decay and synchronization; quantum trajectories manifest coherent oscillations and suppressed phase fluctuations.
• Region II (Chaotic Regime and Transient Chaos): Initially chaotic mixing sustains phase fluctuations and decorrelator growth, followed by slow projection onto the DFS and eventual synchronization, as quantified by the decay of $\omega_c$9.

  Figure 2: Synchronization and transient chaos: dynamics of photon number, collective phase, and population imbalance for individual quantum trajectories in both regular and chaotic regimes.

The photon sector vanishes at late times, decoupling the spin evolution, which then reduces to integrable LMG-type dynamics.

Figure 3: Classical trajectories of photon number and spin observables demonstrate rapid synchronization in region I, but delayed convergence in region II due to transient chaos.

Dissipative Quantum Scarring: Survival and Tunneling Phenomena

The study identifies two categories of dissipative scarring, both originating from unstable fixed points:

• Dissipation-Protected Scar (NP$\kappa$0): Dynamics initialized at the unstable NP$\kappa$1 yields persistent survival probability oscillations with negligible decay, contrasting generic chaotic initial states which decorrelate rapidly. The Husimi distributions show recurrent localization near the saddle, consistent with classical homoclinic orbits.
• Scar of Unstable Superradiant Phase (FSR$\kappa$2): For initial states near FSR$\kappa$3, survival probability decays slowly under dissipation, remaining localized in phase space for extended times before diffusing. For small $\kappa$4, survival probability exhibits oscillatory tunneling ("macroscopic quantum tunneling," MQT) between two symmetry-broken branches, supported by complementary Husimi density accumulation.

  Figure 4: Dissipative quantum scarring from unstable NP$\kappa$5 and FSR$\kappa$6: time evolution of survival probabilities and Husimi phase-space distributions.

The chaos-assisted nature of MQT is established: tunneling is suppressed in the stable regime but enhanced under instability—chaotic mixing facilitates transitions between branches.

Figure 5: Quantum trajectory ensemble data demonstrating synchronization and phase fluctuation suppression across dynamical regimes.

Additional results highlight that synchronization persists even without explicit spin-spin coupling ($\kappa$7), showing robustness of the DFS under dissipation.

Figure 6: Coherent oscillatory evolution of collective phase and population imbalance in the regular regime, robust under quantum trajectory averaging.

Practical and Theoretical Implications

The OCTD model provides a testbed for probing the effects of engineered dissipation on non-equilibrium quantum phenomena, including quantum scars and information scrambling. Dissipation-induced synchronization offers mechanisms for stabilizing many-body coherence, relevant for large-spin qudit encoding and superradiant cat state generation in quantum computation and error correction. Controlled scrambling and chaos-suppression channels could be harnessed for quantum information processing and cavity QED architectures.

The persistence of certain quantum scars despite mixed steady states in the Lindblad framework challenges extant notions about eigenstate-based scarring, motivating further exploration of dynamical scarring in open systems, and raising questions about the universality of decoherence-free subspaces in many-body settings.

Figure 7: Superradiant phase stationary photon distributions and emergent self-organization in region III, reflecting non-unique attractors sensitive to initial conditions.

Conclusion

This work elucidates the interplay between chaos, dissipation, and synchronization in a generic many-body quantum system. The identification of dissipation-protected quantum scars and chaos-assisted tunneling in open quantum environments establishes novel dynamical mechanisms, with experimental relevance for cavity-coupled bosonic systems and broader implications for quantum control, information, and ergodicity-breaking. Future directions include characterization of scar stability in more complicated open systems and applications of synchronization for robust quantum state engineering (2605.22953).