Superradiant decay in non-Markovian Waveguide Quantum Electrodynamics

TL;DR

The paper investigates how finite photon propagation times in a 1D waveguide QED setup modify the canonical Dicke superradiant decay. Using a tensor-network approach (collision-based matrix product states), it resolves non-Markovian dynamics and reveals that the traditional superradiant burst fragments into a train of correlated photon pulses, with excitation partially trapped in bound states that emerge at long times. The study also shows that non-Markovian delays generate emitter–emitter entanglement and, for highly radiant states like the symmetric Dicke state, can transiently enhance decay beyond Markovian predictions. The results illuminate a rich non-Markovian regime with potential realizations in circuit QED and matter-wave platforms, and point to further explorations of driven steady states and scattering of input pulses.

Abstract

An array of initially excited emitters coupled to a one-dimensional waveguide exhibits superradiant decay under the Born-Markov approximation, manifested as a coherent burst of photons in the output field. In this work, we employ tensor-network methods to investigate its non-Markovian dynamics induced by finite time delays in photon exchange among the emitters. We find that the superradiant burst breaks into a structured train of correlated photons, each intensity peak corresponding to a specific photon number. We quantify the emitter-photon and emitter-emitter entanglement generated during this process and show that the latter emerges in the long-time limit, as part of the excitation becomes trapped within the emitters' singlet subspace. We finally consider the decay of the system's most radiant state, the symmetric Dicke state, and show that time delay can lead to decay rates exceeding those predicted by the Markovian approximation.

Figures (8)

• Figure 1: Panel (a). Schematic representation of an array of $N$ two-level quantum emitters, separated by equal distances $d$, initially prepared in their excited state $\otimes_{j}|{e_j}\rangle$. The emitted photons can propagate in a waveguide, with the isolated-emitter decay rate $\gamma$ determining the strength of the coupling between each emitter and the field. The propagation time of a photon between two neighboring emitters is $\tau=d/v_g$ with $v_g$ being the photon group velocity. In the Markovian regime, where $\eta=\gamma \tau \ll 1/N$, the output field is characterized by a superradiant burst of photons. However, as $\eta$ increases, the emission turns into a train of correlated photon pulses, with each intensity peak corresponding to a specific photon number. Panel (b) Graphical representation of the time-delayed phase-locking process, that progressively involves larger portions of the atomic array, at the basis of the phenomenology of superradiance in the non-Markovian regime.
• Figure 2: Panels (a)--(b). Time evolution of the emitter excitation number $N_{\mathrm{exc}}$, defined in Eq. \ref{['eq_pmean']} and normalized to the number $N$ of emitters, from the $N$-excitation initial state \ref{['eq:Nexc']}. The quantity is plotted in panel (a) for different numbers of emitters with fixed $\eta=0.3$, and in panel (b) for different values of $\eta$ with fixed $N=6$, where the non-Markovianity parameter $\eta=\gamma\tau$ represents the ratio of the propagation delay to the single-emitter lifetime. In both panels, the black dashed lines represent for comparison the behavior in the Markovian limit, which is obtained by solving the emitters' master equation \ref{['eq:ME_Dicke']}. The inset in panel (a) displays the corresponding dynamics on a logarithmic scale while the inset in panel (b) shows the instantaneous decay rate \ref{['eq:rate']}, compared with the decay rate of an individual emitter (dashed line) given by $\gamma$. Panels (c)--(d). Spatial profile of the field energy density along the waveguide in the long-time limit $t\approx 9\gamma^{-1}$ for (c) $N=4$ with $\eta=0,0.4,0.8$, and at $t=16\gamma^{-1}$ for (d) $N=6$ and $\eta=0.6$. The vertical dashed lines indicate the positions of the emitters. The inset in panel (d) provides a magnified view of the energy profile of the field trapped between the emitters.
• Figure 3: Panels (a)--(b). Spatial profiles of the second- and third-order auto-correlation functions, $G_a^{(2)}$ and $G_a^{(3)}$, as defined in Eq. \ref{['eq_correlation']}, together with the corresponding $m$-th power of the intensity, are shown for $N = 4$ emitters and time delay $\eta = 0.5$. All profiles correspond to a snapshot at time $t_0=8\tau$ of the left part of the waveguide between the left edge and the first emitter. Panels (c)--(d). Spatial profiles of the corresponding normalized auto-correlation functions $g_a^{(m)}$. In gray we show the correlations computed at zero time delay, $\eta=0$, corresponding to the Markovian regime.
• Figure 4: Time evolution of the entanglement generated from the initial state \ref{['eq:Nexc']} for $N = 4$ and different values of $\eta=\gamma\tau$. Panel (a) shows the atom–field entanglement entropy $S_{\mathrm{e-f}}$; panel (b) the entanglement entropy $S_{\mathrm{in-out}}$ between the trapping region and the outer parts of the waveguide; panel (c) the logarithmic negativity $N_L$ computed for half of the emitter chain. The gray box indicates the time window during which the maximum of the output intensity occurs for the corresponding values of $\eta$.
• Figure 5: Populations $P^{(N_{\mathrm{exc}})}$ of the emitters' Hilbert space sectors characterized by different excitation numbers, for $N=4$ qubits at $\eta=0.4$ [panel (a)], and $\eta=0.8$ [panel (b)]. In panel (c), we report as a function of $\eta$ the populations of the $N_{\mathrm{exc}}=1$ and $N_{\mathrm{exc}}=2$ excitations subspaces, and of the two states in Eqs. \ref{['eq:singl1']}-\ref{['eq:singl2']}, which span the singlet subspace.