We introduce a picture to describe and interpret waveguide-QED problems in the non-Markovian regime of long photonic retardation times resulting in delayed coherent feedback. The framework is based on an intuitive spatial decomposition of the waveguide into blocks. Among these, the block directly coupled to the atoms embodies an effective lossy multimode cavity leaking into the rest of the waveguide, in turn embodying an effective white-noise bath. The dynamics can be approximated by retaining only a finite number of cavity modes, which grows with the time delay. This description captures the atomic as well as the field’s dynamics, even with many excitations, in both emission and scattering processes. As an application, we show that the recently identified non-Markovian steady states can be understood by retaining very few or even only one cavity mode.
We introduce a picture to describe and interpret waveguide-QED problems in the non-Markovian regime of long photonic retardation times resulting in delayed coherent feedback. The framework is based on an intuitive spatial decomposition of the waveguide into blocks. Among these, the block directly coupled to the atoms embodies an effective lossy multimode cavity leaking into the rest of the waveguide, in turn embodying an effective white-noise bath. The dynamics can be approximated by retaining only a finite number of cavity modes, which grows with the time delay. This description captures the atomic as well as the field’s dynamics, even with many excitations, in both emission and scattering processes. As an application, we show that the recently identified non-Markovian steady states can be understood by retaining very few or even only one cavity mode.
We present two different methods for modeling non-Markovian quantum light-matter interactions in waveguide QED systems, using matrix product states (MPSs) and a space-discretized waveguide (SDW) model.After describing the general theory and implementation of both approaches, we compare and contrast these methods directly on three topical problems of interest in waveguide-QED, including (i) a two-level system (TLS) coupled to an infinite (one-dimensional) waveguide, (ii) a TLS coupled to a terminated waveguide with a time-delayed coherent feedback, and (iii) two spatially separated TLSs coupled within an infinite waveguide.Both approaches are shown to efficiently describe multiphoton nonlinear dynamics in highly non-Markovian regimes, and we highlight the advantages and disadvantages of these methods for modeling waveguide QED interactions, including their implementation in PYTHON, computational run times, and ease of conceptual understanding.We explore both vacuum dynamics as well as regimes of strong optical pumping, where a weak excitation approximation cannot be applied.The MPS approach scales better when modeling multiphoton dynamics and long delay times and explicitly includes non-Markovian memory effects.In contrast, the SDW model accounts for non-Markovian effects through space discretization and solves Markovian equations of motion, yet rigorously includes the effects of retardation.The SDW model, based on an extension of recent collisional pictures in quantum optics, is solved through quantum trajectory techniques and can more easily add in additional dissipation processes, including off-chip decay and TLS pure dephasing.The impact of these processes is shown directly on feedback-induced population trapping and TLS entanglement between spatially separated TLSs.
The emergence of integrated (or solid state) photonic systems, including quantum dots, waveguides, and cavities, has provided a base to harness quantum optic phenomena for use in current and future quantum technologies. These systems also provide opportunities for exploring fundamentally new regimes in quantum optics. In order to fully realize the potential of these systems, many figures of merit need to be improved, for example by increasing the stability and coherent lifetimes of these systems. Following the notable improvements using measurement-based feedback, time-delayed coherent optical feedback has been proposed as one such method of stabilizing and improving these systems. Furthermore, modelling coherent feedback itself presents an interesting fundamental problem due to its non-Markovian nature. In this thesis, we use quantum trajectory (QT) theory to derive two models for simulating cavity quantum electrodynamic (cavity-QED) systems with time-delayed coherent feedback. First, an explanation of QT theory is given and the related time discretized waveguide (TDW) model is derived. Next we present a model for feedback using the frequency modes of the waveguide and results are presented in the ``one photon in the loop" approximation. We demonstrate how the photon lifetime can be improved in typical cavity-QED with coherent feedback and we explore some nonlinear effects. We then expand the system to allow for two photons in the feedback loop which requires us to use the TDW model. Lastly, this new approach is used to model two two-level systems coupled via a waveguide and connected to a feedback loop. These findings have implications on improving quantum optic systems by creating a new degree of control, as well as for modelling non-Markovian features in quantum optics.
Spontaneous emission of a quantum emitter, such as an excited atom, is a fundamental process in quantum electrodynamics (QED), typically associated with exponential decay to the ground state accompanied by irreversible photon emission. This simple Markovian picture, however, is profoundly modified in the presence of time-delayed feedback, structured continua, or cooperative emission, as occurs when an emitter radiates in front of a mirror, when several emitters radiate collectively, or in the case of a giant atom. In such regimes, strong non-Markovian dynamics arise from photon reabsorption and interference effects, leading to pronounced deviations from exponential decay. Here we demonstrate the emergence of exceptional points (EPs) in these highly non-Markovian waveguide-QED environments, i.e., non-Markovian EPs. These EPs appear directly in the relaxation dynamics as sharp transitions to oscillatory behavior, manifested by the appearance of real zeros in the excited-state amplitude. We analyze in detail the spontaneous emission of giant atoms with two or more coupling points, highlighting the mechanisms leading to non-Markovian EPs, and show that similar phenomena arise in other waveguide-QED settings, such as the collective spontaneous emission of spatially separated point-like emitters. Our results reveal waveguide-QED systems as experimentally accessible platforms for realizing and exploring non-Markovian EP physics.
We analyze the non-Markovian spontaneous-emission dynamics of a two-level test atom placed in a cavity formed by two atomic arrays in a waveguide quantum electrodynamics (QED) setup. We demonstrate a crossover from single-mode to multimode strong-coupling cavity QED as the cavity length $\ensuremath{\sim}d$ becomes comparable to the coherence length associated with collective spontaneous emission $\ensuremath{\sim}v/(N\ensuremath{\gamma})$. The resulting non-Markovian dynamics of the test atom and the emergent spectral density of the field are analyzed as a function of various tunable atomic array parameters: the number of atoms, length of the atomic cavity, and resonance frequency of the atoms forming the atomic mirrors. Our results show limitations to cooperatively enhanced light-matter coupling in the presence of time-delayed feedback. We further illustrate that the non-Markovian system dynamics can be efficiently approximated in terms of a few modes of the emergent spectral density of the field.
In the multimode regime of cavity quantum electrodynamics (cQED), light-matter coupling can be strong enough such as to allows a quantum emitter to couple with multiple resonant field modes. This regime leaves distinctive signatures in the behavior of both the field and the emitter, which are already accessible by modern experimental techniques. To model such effects in lossy cavities, effective discrete-mode approaches have been widely employed, despite their accuracy beyond two modes remaining rarely benchmarked. In this work, we demonstrate the necessity of employing multiple modes to describe the multimode signatures by analyzing the dynamics of an atom in a one-dimensional lossy resonator. The atom's dynamics is simulated using the formalism developed in Weitzel et al. (2026) (arXiv:2512.18478), where a resonance quantization prescription is provided. We investigate its dependency with respect to two different parameters: the light-matter coupling strength and the atom's position. In the former case, we find that the minimal number of modes necessary to capture the essential features of the atom's dynamics increases with the coupling strength. In the latter, we further observe that the multimode regime leads to a strong position-dependency of the atom's dynamics. In all cases, we discuss the physical interpretation of our results and further discuss about how the employed formalism can be used to recover physically relevant quantities.
We study the quantum dynamics of multiple two-level atoms (qubits) in a waveguide quantum electrodynamics system, with a focus on modified superradiance effects between two or four atoms with finite delay times. Using a numerically exact matrix product approach, we explore both Markovian and non-Markovian regimes, and highlight the significant influence of time-delayed feedback effects and the clear breakdown of assuming instantaneous coupling dynamics. We first show a system composed of two spatially separated qubits, prepared in a doubly excited state (both fully excited), and provide a comprehensive study of how delayed feedback influences the collective system decay rates, as well as the quantum correlations between waveguide photons, atoms, and between atom and photons. The system is then extended to include two additional qubits located next to the initial ones (four qubits in total), and we demonstrate, by manipulating the initial excitations and the time-delay effects, how long-term quantum correlations and light-matter entangled states can be established.