Time-delayed collective dynamics in waveguide QED and bosonic quantum networks

TL;DR

This work develops a non-Markovian, time-delayed framework for quantum emitters coupled through waveguide QED environments by deriving a set of Heisenberg-Langevin equations supplemented with input-output relations. The approach yields exact results in the linear (bosonic) limit and in the single-excitation regime for saturable emitters, and it is numerically tractable for moderate system sizes and photon numbers, enabling detailed benchmarks against Wigner-Weisskopf methods. A key contribution is the demonstration of cascaded super- and subradiant emission arising from photon retardation and light-cone propagation, including a time-delayed superradiant burst and a maximal emission-rate scaling analysis for chains of emitters. The formalism offers a versatile tool for modeling quantum networks, with potential extensions to topological setups and correlated photon-state generation, and provides practical pathways for comparison with circuit QED and related platforms.

Abstract

This work introduces a theoretical framework to model the collective dynamics of quantum emitters in highly non-Markovian environments, interacting through the exchange of photons with significant retardations. The formalism consists on a set of coupled delay differential equations for the emitter's polarizations $σ^\pm_i$, supplemented by input-output relations that describe the field mediating the interactions. These equations capture the dynamics of both linear (bosonic) and nonlinear (two-level) emitter arrays. It is exact in some limits$-$e.g., bosonic emitters or generic systems with up to one collective excitation$-$and can be integrated to provide accurate results for larger numbers of photons. These equations support a study of collective spontaneous emission of emitter arrays in open waveguide-QED environments. This study uncovers an effect we term cascaded super- and sub-radiance, characterized by light-cone-limited propagation and increasingly correlated photon emission across distant emitters. The collective nature of this dynamics for two-level systems is evident both in the enhancement of collective emission rates, as well as in a superradiant burst with a faster than linear growth. While these effects should be observable in existing circuit QED devices or slight generalizations thereof, the formalism put forward in this work can be extended to model other systems such as network of quantum emitters or the generation of correlated photon states.

Figures (8)

• Figure 1: Quantum emitters (atoms, for instance) connected to a non-Markovian network (blue shades). The emitters interact by exchanging photons that travel through the common bosonic network or bath. The delays of the photons $\tau_{ij}\propto d_{ij}/c$ when travelling between any pair of emtitters $i$ and $j$, causes the network to act as a non-Markovian environment. In this work we mainly explore (a) open and (b) closed one-dimensional environments, but the formalism applies to more general bosonic quantum networks (c).
• Figure 2: Illustration of the memory functions for different systems of emitters and waveguides. (a) Single emitter in an infinite waveguide. (b) two emitters in an infinite waveguide (c) two emitters at the ends of a finite-length waveguide ,
• Figure 3: Population of the bosonic emitters as function of the dimensionless time $\gamma t$ and the emitters' label. For both simulations we have fixed $\phi_{0} = 2\pi$ and $\ket{\psi(0)} = \ket{001100}$. In the subfigure (a) we compare a numerical integration (dot symbols) with the analytical solution (solid line), considering the nearly-Markovian regime $\gamma \tau_{12} = \pi / 20$. In the subfigure (b) we repeat the same comparison, now considering the Non-Markovian regime with $\gamma \tau_{12} = 5\pi/4$, as we set $\phi_0=50\pi$.
• Figure 4: Population of the two-level emitters' excited state $P_{n}(t)$ as a function of the dimensionless time $\gamma t$ and the emitters' label. Both simulations compare the outcome of the HLS equations (dot symbols) with an exact diagonalization of the Wigner-Weisskopf ansatz (solid line), either in (a) a nearly-Markovian regime $\phi_0=2\pi$, $\gamma \tau_{12}=\pi/20$ or in (b) a deeply retarded configuration $\phi_0=50\pi$, $\gamma\tau_{12}=5\pi/4$.
• Figure 5: Population of the excited state $P(t)$ of two two-level emitters in a closed waveguide as a function of the adimensionalized time. The solid and dotted lines represent the results from the exact diagonalization and the HL equation methods, respectively. Colors blue and purple are used to denote observations on the first and second qubit, respectively. The dashed gray lines indicates the time interval $\tau_{12}$ needed for a photon to traverse the waveguide. In both simulations we have fixed $\phi_{0} = 2\pi$ and $\gamma \tau_{0} = 8 \pi$. Fig.(a) shows the qubit's excitation probabilities as a function of time, for the initial state with one excited qubit $\ket{10}$. Figs. (b-c) show the same quantities, for an initial state with two excitations $\ket{11}$.