Emergent spin order and steady-state superradiance in one-dimensional baths

TL;DR

The paper studies how emergent spin order and steady-state superradiance arise for ensembles of atoms coupled to one-dimensional electromagnetic baths, going beyond the Dicke limit by incorporating multimode competition and propagation. It compares a ring cavity, which supports two competing bright channels, with a bidirectional waveguide that includes coherent exchange, showing distinct orders: dynamical symmetry breaking between left/right emission in the ring, and phase-separated left/right order with edge locking in the waveguide. Using master-equation formalisms, superspin reductions, mean-field, and Truncated Wigner approximations, the authors establish $R_{ ext{max}}\propto N^2$ across models and derive (analytic or numeric) thresholds for synchronization, along with spectral analyses that indicate potential linewidth narrowing in some geometries. The results illuminate how propagation and mode competition shape steady-state order in 1D reservoirs and identify regimes where robust, ultranarrow, steady-state emission could be realized beyond the Dicke paradigm, with implications for scalable quantum light sources and quantum synchronization phenomena.

Abstract

Spontaneous collective decay in driven atomic ensembles can generate coherence far from equilibrium, as illustrated by superradiant lasers where decay into a single-mode cavity synchronizes atomic phases into a macroscopic dipole and yields superradiant emission of light with an ultranarrow spectrum. Whether similar ordering persists in multimode reservoirs with propagation and competing collective decay channels remains an open question. We address this problem by analyzing atoms coupled to one-dimensional electromagnetic baths through two models. The first is a ring cavity supporting two bright collective decay channels, and the second is a bidirectional waveguide where, in addition to competition between channels, propagation induces coherent dipole-dipole interactions. For suitable incoherent pumping strengths, the dynamics enters a synchronization window in which collective decay overcomes disordering processes, leading to spontaneous steady-state phase ordering and superradiant emission. We extract the thresholds marking the onset of synchronization and show that the maximum intensity scales quadratically in both models. The resulting order is not described by a single macroscopic dipole: in the ring cavity spontaneous chirality emerges at the level of individual trajectories, while the waveguide develops a local chirality with different orders dominating opposite ends of the atomic array. The analysis of the emitted light spectrum reveals a linewidth that seems to narrow with increased system size in the ring cavity, while narrowing in the waveguide remains inconclusive within accessible numerics. These results clarify how competition and propagation shape emergent order in one-dimensional reservoirs and identify regimes where steady-state superradiance may arise beyond the Dicke limit.

Figures (7)

• Figure 1: Schematics of the spin models for a single-mode cavity, ring cavity and waveguide, and their corresponding spin orders. In all settings, $N$ two-level atoms with parasitic decay $\Gamma'$ and incoherent pumping $w$ are coupled to a shared electromagnetic environment, into which they emit photons at rate $\Gamma_\text{1D}$. In a single-mode cavity (a), atoms provide global feedback to one another, leading to a synchronized steady state with a single phase. In a ring cavity (b), atoms provide competing directional feedback, driving the ensemble to synchronize to emit all the light into either the right- or the left-propagating mode. In a bidirectional waveguide (c), two types of feedback are also present. However, feedback is position dependent, leading to a phase-separated steady state.
• Figure 2: Order in the single-mode cavity. Wigner function for the electromagnetic field emitted by $N=1024$ atoms coupled to a single-mode cavity, for $\Gamma'=20\Gamma_{\rm 1D}$ and $w =N\Gamma_{\rm 1D}/2$. White squares indicate mean-field results.
• Figure 3: Order in the ring cavity. Mean-field trajectories and Wigner function in (a) the space of right- and left-field amplitudes, and (b) in the complex plane of the right field. Time is encoded in the line transparency (fainter for ealier times), and each trajectory is colored according to which field dominates in the steady state. In (b), the Wigner function in regions far from the center have been multiplied by a factor of 10 for visualization purposes. Panel (c) shows the magnetization $\mathcal{M}$ for a single realization of the phase dynamics, where the right field prevails. Parameters: $N=1024$, $\Gamma' = 0$, $w = N\Gamma_{\rm 1D}/4$.
• Figure 4: Order in a bidirectional waveguide. (a) Mean-field magnetization $\mathcal{M}$ for a single realization of the phase dynamics with parameters $(N,\Gamma',w,kd) = (1024,0,N\Gamma_{\rm 1D}/8,2\pi/3)$. (b) Schematics of the correlation phase matrix $\arg[C_{nm}]$ together with representative plots. Labels (I)–(V) denote regions with distinct behavior, as discussed in the main text. The phase correlations $\arg[C_{nm}]$ along the horizontal cuts shaded in black in the schematics, $\arg[C_{1m}] \simeq kd(1 - m)$ and $\arg[C_{Nm}] \simeq -kd(N - m)$, are shown in the upper and lower panels on the right. These correlation phases align with the values predicted by the left and right ordered states as $m$ moves deeper into the corresponding domain. Correlation phases along the diagonals $(n,n+d)$ are plotted as a function of the rescaled and shifted coordinate $n'$ defined in the main text (middle right panel). Plots in (b) are done with parameters $(N,\Gamma',w,kd) = (200,2\Gamma_{\rm 1D},N\Gamma_{\rm 1D}/4,2\pi/3)$. TWA simulation results are shown in colors and ansatz predictions in black.
• Figure 5: Thresholds for superradiant light emission and scaling of maximum emission rate. (a) Emission rate and (b) excited-state population in the steady state as a function of pumping strength. Insets: zoom on the first threshold. (c) Maximum intensity vs atom number. In all plots, solid lines show the analytical expressions [from Ref. Meiser10 for the single-mode cavity and from Eqs. \ref{['ThresholdsRing']} for the ring cavity], points show the numerical results with TWA, and the dashed line in (c) indicates the estimate of $R_{\text{max}}$ for the waveguide described in the main text. Parameters: $(N,\Gamma',kd )=(1024,2\Gamma_{\rm 1D},2\pi/3)$.