Correlation Localization in Waveguide QED with Delayed Interactions

Technical Summary: Correlation Localization in Waveguide QED with Delayed Interactions

Problem Statement
Waveguide quantum electrodynamics (wQED) systems typically exhibit long-range atom-atom interactions mediated by guided electromagnetic fields. In the standard Markovian regime, where photon propagation time is negligible compared to atomic timescales, these interactions are treated as instantaneous, leading to memoryless dynamics described by Lindblad master equations. However, as experimental platforms achieve larger interatomic separations, photon propagation delays become significant, inducing non-Markovian dynamics where the Born approximation fails. While previous studies have explored how these delays alter collective decay and excitation propagation, the transport of correlations—crucial for entanglement distribution and coherence preservation—remains largely unexplored in this context. Specifically, the behavior of correlation transport in the presence of delay-induced non-Markovianity, particularly within collective systems initialized with a single excitation, requires further investigation.

Methodology
The authors investigate an array of $N$ identical two-level atoms coupled to a one-dimensional waveguide under the Bragg condition, where interatomic distances are set such that the propagation phase is a multiple of $2\pi$. The study focuses on the relaxation dynamics starting from a single excited atom (the central atom) with all others in the ground state.

To model the system, the authors employ the Weisskopf-Wigner theory of spontaneous emission without invoking the Born-Markov approximation. They derive a set of delay differential equations for the atomic excitation coefficients $\beta_j(t)$, accounting for the finite photon travel time $\tau_{j,j'} = |x_j - x_{j'}|/v$ between atoms. The interaction Hamiltonian includes both left- and right-propagating modes with symmetric decay rates. Due to the intractability of analytically inverting the transcendental characteristic equations arising from the delay terms in the Laplace domain, the authors utilize numerical methods to solve the equations of motion directly in the time domain.

The analysis focuses on:

1. Excitation Dynamics: Tracking the probability of excitation for the central atom and neighboring atoms as they evolve toward a steady state.
2. Correlation Functions: Quantifying atom-atom correlations using the spin-spin correlation function $g(j) = \langle \sigma_0^z \sigma_j^z \rangle - \langle \sigma_0^z \rangle \langle \sigma_j^z \rangle$, where index 0 denotes the initially excited central atom.
3. Disorder Effects: Introducing spatial disorder via Gaussian random deviations in interatomic spacing to observe its impact on localization.

Key Results
The study reveals a distinct transition from delocalized collective behavior to localized dynamics as photon propagation delays increase:

• Markovian Regime ($\gamma\tau = 0$): In the absence of delay, the excitation spreads evenly across the array. The steady-state excitation probability of the central atom increases with system size, asymptotically approaching unity for large $N$, reflecting collective excitation sharing.
• Non-Markovian Regime (Finite $\tau$): When delays are introduced, the steady-state excitation probability of the central atom decreases and rapidly saturates as $N$ increases. This indicates that a fraction of the excitation remains localized near the initially excited atom, effectively decoupling the central atom from distant neighbors. The dynamics become governed by nearest-neighbor interactions rather than the full ensemble.
• Correlation Length Scaling: The atom-atom correlation function decays exponentially with distance from the central atom, $g(j) \propto e^{-|j|/\xi}$. The correlation length $\xi$ decreases as the delay $\tau$ increases. Crucially, the authors find that $\xi$ scales with the delay time according to a power law: $\xi \propto \tau^{-1/2}$. This scaling is derived analytically by substituting an evanescently localized ansatz into the equations of motion, linking the localization length to the interplay between the interaction delay and the temporal coherence length of emitted photons ($\gamma^{-1}$).
• Disorder Effects: The introduction of spatial disorder further reduces the correlation length. While weak disorder follows a power-law scaling similar to the delay-induced case, it eventually leads to a saturation of the correlation length at a finite minimum value (slightly below a single lattice site). This establishes a disorder-imposed localization limit. The study finds that delay-induced non-Markovian effects confine correlations more strongly than spatial randomness alone, though even weak disorder sets a lower bound on the correlation length.

Significance and Claims
The paper claims to elucidate a novel non-Markovian transport phenomenon where delayed interactions induce the localization of atomic correlations in waveguide QED systems. The primary significance lies in demonstrating that retardation effects fundamentally modify excitation transport by suppressing long-range coherence and confining correlations to finite spatial regions, even in the single-excitation regime.

The authors highlight that this localization phenomenon is a steady-state effect arising from the specific combination of the Bragg condition and perfect emitter-waveguide coupling. They note that while deviations from these ideal conditions would introduce decay channels preventing a true steady state, the transient dynamics would still exhibit characteristic localization effects.

Conceptually, the work draws parallels between the Markovian limit and the Kuramoto model of coupled oscillators, while the non-Markovian regime with delays resembles the Kac-Ising model with finite-range interactions. The authors suggest that these findings offer new scenarios for studying localization phenomena and controlling correlation confinement in wQED, potentially aiding in the design of quantum devices that exploit non-Markovian effects to isolate excitations. The study does not propose specific experimental implementations or future applications beyond these theoretical insights and the potential for controlling correlation transport.