Excitation transfer and many-body dark states in waveguide quantum electrodynamics

TL;DR

This paper addresses excitation transport in one-dimensional waveguide QED with infinite-range dissipative interactions by constructing a symmetry-based set of $M$-excitation dark states for two emitter ensembles in a mirror configuration. It derives analytic projections for the time evolution of these dark states, enabling a compact, exact description of transport and storage dynamics with reduced computational cost. The authors show that the steady-state transfer is optimized at a pumped fraction $(N_p/N)_{\mathrm{Max}} \approx 0.55$ for large systems, reflecting a balance between dark-state capacity and collective decay. They further assess robustness to positional disorder, nonradiative decay, and dephasing, finding that moderate imperfections preserve efficient transfer and reveal how symmetry-breaking channels can support long-lived subradiant states, with implications for dissipative many-body dynamics in integrated WQED platforms.

Abstract

In one-dimensional waveguide quantum electrodynamics systems, quantum emitters interact through infinite-range, dispersive, and dissipative dipole-dipole interactions mediated by guided photonic modes. These interactions give rise to long-range periodic behavior and rich many-body physics absent in free space. In this work, we construct a set of symmetrized multi-excitation dark states and derive analytic expressions for their time-evolution projections. This framework captures the essential dynamics of excitation transport and storage while significantly reducing computational complexity compared to full quantum simulations. Our analysis reveals a fundamental bound on energy redistribution governed by the structure of dark states and collective dissipation, and discovers that optimal excitation transfer between emitter ensembles converges toward an initial pumped fraction of $N_\text{p}/N \approx 0.55$ for large system sizes. We further examine the robustness of this mechanism under realistic imperfections, including positional disorder, nonradiative decay, and dephasing. These results highlight the role of many-body dark states in enabling efficient and controllable energy transfer, offering new insights into dissipative many-body dynamics in integrated quantum platforms.

Figures (5)

• Figure 1: Schematic representation of the system setup. Two 1D emitter arrays, consisting of $N_{\text{p}}$ and $N_{\text{np}}$ emitters, interact via an infinite-range DDI mediated by a 1D waveguide mode. Each emitter is a two-level system with ground state $\ket{g}$, excited state $\ket{e}$, and transition frequency $\omega_0$. The emitters are evenly spaced with a separation of $d$. Without loss of generality, we consider the emitters in the left ensemble (of size $N_{\text{p}}$) are initially pumped to the excited state $\ket{e}$, while those in the right ensemble (of size $N_{\text{np}}$) remain in the ground state $\ket{g}$. In addition to the collective emission into the waveguide (rate $\gamma$), each emitter undergoes nonradiative decay at a rate $\gamma_{\text{nr}}$.
• Figure 2: Full master equation simulation of the system. The time evolution of the mean excitation number in each emitter ensemble (red for the pumped ensemble and blue for the un-pumped ensemble), along with the projection of the density matrix onto $M$-excitation dark states. The dashed lines in (a) and (b) represent the expectation values contributed from only the dark-state manifold. In (c) and (d), we separate the projection onto the dark states with different excitation number $M$. In all panels, the total number of emitters is $N = 10$. However, the pumped ensemble size is $N_\text{p} = 4$ in (a) and (c), while it is $N_\text{p} = 6$ in (b) and (d).
• Figure 3: Steady-state excitation transfer. (a) Heatmap of the excitation transfer ratio $T$ as a function of total emitter number $N$ and number of initially pumped emitters $N_\text{p}$. A clear transition occurs near the critical point $N_\text{p} = N/2$, where the dark-states no longer localize significant excitations in the pumped ensemble, resulting in enhanced excitation transfer. (b) The maximal transfer ratio $T_\text{Max}$ as a function of total emitter number $N$. The peak value increases slowly with system size and shows no sign of saturation. (c) The ratio $(N_\text{p}/N)_\text{Max}$ at which maximal transfer occurs, plotted against $N$. This optimal pumped fraction converges toward approximately $0.55$ for large system sizes, slightly above the half-pumped threshold.
• Figure 4: Time evolution under system imperfections. (a–c) Time evolution of the mean excitation number $\langle n_e \rangle$ in the pumped (blue) and un-pumped (red) ensembles for varying levels of (a) positional disorder, (b) nonradiative decay, and (c) dephasing, modeled respectively by $\epsilon = 0.001$ (dashed), $0.01$ (dash-dotted), and $0.1$ (dotted), where $\epsilon$ characterizes either the spatial deviation or the decay/dephasing rate relative to the collective decay rate $\gamma$. (d–f) Corresponding dark-state projections onto $|\psi_D^{(3,1)}\rangle$, $|\psi_D^{(3,2)}\rangle$, and $|\psi_D^{(3,3)}\rangle$ under each imperfection scenario. The yellow curves show the sum of projections onto all $|\psi_D^{(3,M)}\rangle$ dark states, including the zero-excitation ground state, thereby representing the total population within the ideal dark-state manifold (constructed according to Eq. (\ref{['DS_expression']})). Notably, even when the total dark-state projection decreases, the excitation transfer remains effective—indicating that excitations are redistributed into other subradiant states emerging from symmetry breaking. The curves for positional disorder are averaged over 200 random positional disorder sampled from a Gaussian distribution.
• Figure 5: Decay rates for different emitter spacings. The dissipative interaction in Eq. (\ref{['master_eq']}), where $\gamma_{m,n} = \gamma \cos(k_0 x_{m,n})$, can be diagonalized into a set of $N$ jump operators $\mathcal{J}_i$ with corresponding decay rates $\Gamma_i$, which are the eigenvalues of the dissipative interaction matrix $\gamma_{m,n}$clemens2003collectivemasson2022universality. The figure shows the decay rates for $N=10$ under different emitter spacings $d$.