Emergent non-Markovianity in time-delayed waveguide QED

How does environmental memory, caused by finite photon propagation times, affect the validity of the Markov approximation in many-body waveguide quantum electrodynamics (wQED) as the system size increases? The authors investigate whether collective effects in arrays of emitters can render the standard Markovian treatment—which assumes an instantaneous environment—inaccurate.

Using a single-excitation model of $N$ two-level emitters coupled to a one-dimensional waveguide, the authors analyze the dynamics of collective states (both superradiant and subradiant). They hold the total array length fixed while varying the number of emitters to isolate the role of collective scaling. By comparing the exact time-delayed dynamics against the Markovian master equation, they identify the relevant system and environmental timescales that determine when memory effects become non-negligible.

• Emergent Non-Markovianity: As the number of emitters $N$ increases at a fixed array length, the system exhibits a transition from Markovian to non-Markovian dynamics. This is characterized by population revivals and spectral splitting, which are direct consequences of radiation being reabsorbed by the array.
• The Scaling Parameter $\chi$: The crossover to non-Markovian behavior is governed by the dimensionless parameter $\chi = N \Gamma \tau_\ell$, where $\Gamma$ is the single-emitter coupling rate and $\tau_\ell$ is the end-to-end propagation delay. This parameter effectively captures the ratio between the environment's memory time and the collective system's lifetime.
• State-Dependent Memory: The impact of retardation is highly state-selective. While superradiant states consistently follow the scaling defined by $\chi$, subradiant states exhibit varied behavior depending on their spatial structure. Some subradiant states are robust against these memory effects, while others are highly sensitive, highlighting that the validity of a Markovian treatment depends on the specific collective state used to encode information.