Persistent subradiant correlations in a random driven Dicke model

1. Problem Statement

The paper addresses the stability of collective quantum phenomena in open systems subject to disorder. Specifically, it investigates a driven-dissipative ensemble of $N$ two-level atoms (a Dicke model) coupled to a single photonic mode.

• The Challenge: In the absence of an external drive, strong fluctuations in atomic resonant frequencies (inhomogeneous broadening/disorder) typically destroy collective effects like superradiance and subradiance.
• The Question: Can collective subradiant states and correlations survive in the presence of significant disorder when the system is subjected to a strong external drive? Furthermore, how do these states manifest in the Liouvillian spectrum (the generator of open-system dynamics)?
• Context: While "dissipative time crystals" (non-equilibrium phases with persistent oscillations) have been studied, they often require the thermodynamic limit ($N \to \infty$) or specific symmetries. The authors seek to understand if robust, long-living correlations exist in finite-size systems with disorder.

2. Methodology

The authors employ a combination of analytical theory, group representation theory, and numerical diagonalization.

• Theoretical Model:
  • The system is described by a random Dicke model with a Liouvillian superoperator $\mathcal{L}$:
    $$\mathcal{L}\rho = -i[H, \rho] + \frac{\gamma}{2} (2\hat{J}_-\rho\hat{J}_+ - \rho\hat{J}_+\hat{J}_- - \hat{J}_+\hat{J}_-\rho)$$
  • Hamiltonian ($H$): Includes a collective drive term ($2\Omega\hat{J}_x$) and a disorder term ($\delta H = \sum \omega_n \sigma_n^z$), where $\omega_n$ are random resonant frequencies.
  • Dissipation: Modeled by collective decay via the jump operator $\hat{J}_- = \sum \sigma_n$.
• Spectral Analysis:
  • The dynamics are analyzed via the eigenvalues $\lambda_i$ of $\mathcal{L}$, where $\rho(t) = \sum c_i \rho_i e^{\lambda_i t}$.
  • Key Metric: The real part, $\text{Re}(\lambda_i)$, determines the decay rate (lifetime $\tau \sim -1/\text{Re}(\lambda_i)$). Subradiant states correspond to eigenvalues with small $|\text{Re}(\lambda_i)|$. The imaginary part, $\text{Im}(\lambda_i)$, indicates oscillatory behavior.
• Symmetry Analysis:
  • The authors utilize Group Representation Theory to analyze the degeneracy and symmetry of the eigenstates. They examine how the system's symmetry (Permutation group $S_N$, Dihedral group $D_N$, or Mirror group $C_s$) dictates the number of dark states and oscillating modes.
  • They extend the model to include nearest-neighbor dipole-dipole interactions ($\delta H_{NN}$) to study the interplay between drive, disorder, and coherent coupling.

3. Key Contributions

1. Discovery of Disorder-Immune Subradiance: The paper demonstrates that while disorder destroys collective subradiance in the weak-drive regime, a strong external drive ($\Omega \gg \delta\omega$) can recover these states.
2. Mechanism of Recovery (Dynamical Decoupling): The authors explain that a strong drive acts as a dynamical decoupling mechanism. It aligns the effective magnetic field along the drive axis ($x$-axis), suppressing transitions between different total angular momentum manifolds ($j$) caused by disorder. This restores the "darkness" of specific states against the collective decay channel.
3. Finite-Size Robustness: Unlike many time-crystal phases that require $N \to \infty$, these subradiant correlations persist in finite-size systems.
4. Oscillating Correlations: The study reveals that while strong drive suppresses disorder, it does not fully suppress dipole-dipole interactions. These residual interactions break the degeneracy of the subradiant states, leading to non-trivial, long-living oscillations in the correlation functions.
5. Symmetry-Based Counting: The authors provide a rigorous formula based on group theory to predict the number of long-living states: $\sum_i |\dim D_i|^2$, where $D_i$ are irreducible representations of the system's symmetry group.

4. Results

• Phase Diagram: Numerical simulations for $N=4$ show a phase transition in the Liouvillian spectrum.
  • Weak Drive ($\Omega \lesssim \gamma$): Strong disorder leads to rapid decay; no long-living correlations exist.
  • Strong Drive ($\Omega \gg \delta\omega$): Long-living correlations emerge regardless of the disorder strength. The smallest non-zero decay rate becomes parametrically small ($|\text{Re}(\lambda)| \ll \gamma$).
• Bloch Sphere Visualization:
  • In the non-driven case, disorder mixes angular momentum manifolds, destroying dark states (except the ground state).
  • In the driven case, the system evolves into "mixed dark states" distributed along orbits spinning around the $x$-axis. These orbits are invariant to the jump operator $\hat{J}_-$, preserving subradiance.
• Oscillations vs. Decay:
  • As drive strength increases, the lifetime of correlations increases (decay rate decreases), but the oscillation frequency (Im $\lambda$) is suppressed.
  • However, when dipole-dipole interactions are included, the symmetry is reduced (e.g., from $S_N$ to $D_N$ or $C_s$). This lifts the degeneracy of the eigenvalues, creating distinct non-zero imaginary parts. Consequently, the system exhibits persistent oscillations in time-dependent correlation functions $\langle \sigma_n^\dagger \sigma_m \rangle$.
• Quantitative Agreement: The number of oscillating frequencies predicted by group theory (Table I) matches the exact numerical diagonalization of the Liouvillian. For $N=4$ with periodic boundaries ($D_4$), 2 oscillating frequencies are found; with open boundaries ($C_s$), 4 are found.

5. Significance

• Fundamental Physics: The work bridges the gap between disorder, driving, and collective dissipation. It proves that strong driving can effectively "shield" a quantum system from inhomogeneous broadening, a counter-intuitive result given that disorder usually destroys coherence.
• Time Crystals and Non-Equilibrium Phases: The identified states represent a new class of non-equilibrium phases. They differ from standard dissipative time crystals by existing in finite systems and being robust to disorder, offering a more practical route to observing time-crystalline behavior.
• Experimental Relevance: The findings are applicable to various platforms, including:
  • Atomic ensembles in optical cavities.
  • Solid-state nuclear spin ensembles.
  • Superconducting qubit arrays coupled to waveguides (Waveguide QED).
• Control of Quantum Dynamics: The study provides a theoretical framework for engineering long-lived quantum correlations and controlling oscillation frequencies via symmetry breaking (e.g., boundary conditions) and drive strength, which is crucial for quantum memory and sensing applications.

In summary, Leppenen and Poddubny demonstrate that strong driving can resurrect collective subradiant correlations in disordered atomic ensembles, transforming a system that would otherwise decay rapidly into one exhibiting long-lived, potentially oscillating, quantum dynamics.