Non-Markovian time evolution of quantum systems is a challenging problem, often mitigated by employing numerical methods or making simplifying assumptions. In this work, we address this problem in waveguide quantum electrodynamics (QED) by developing a diagrammatic approach, which performs fully analytical non-Markovian time evolution of single-photon states. By revisiting Fermi's two-atom problem, we tackle the impeding question of whether the rotating-wave approximation violates causality in single-photon waveguide QED. Afterward, we introduce and prove the no upper half-plane poles (no-UHP) theorem, which connects the poles of scattering parameters to the causality principle. Finally, we visualize the time-delayed coherent quantum feedback mediated by the field and discuss the Markovian limit for microscopically separated qubits where short-distance causality violations occur and the emergence of collective decay rates in this limit. Our diagrammatic approach is a method to perform exact and analytical non-Markovian time evolution of multiemitter systems in waveguide QED.
Non-Markovian time evolution of quantum systems is a challenging problem, often mitigated by employing numerical methods or making simplifying assumptions. In this work, we address this problem in waveguide quantum electrodynamics (QED) by developing a diagrammatic approach, which performs fully analytical non-Markovian time evolution of single-photon states. By revisiting Fermi's two-atom problem, we tackle the impeding question of whether the rotating-wave approximation violates causality in single-photon waveguide QED. Afterward, we introduce and prove the no upper half-plane poles (no-UHP) theorem, which connects the poles of scattering parameters to the causality principle. Finally, we visualize the time-delayed coherent quantum feedback mediated by the field and discuss the Markovian limit for microscopically separated qubits where short-distance causality violations occur and the emergence of collective decay rates in this limit. Our diagrammatic approach is a method to perform exact and analytical non-Markovian time evolution of multiemitter systems in waveguide QED.
We develop a formalism for modelling exact time dynamics in waveguide quantum electrodynamics (QED) using the real-space approach. The formalism does not assume any specific configuration of emitters and allows the study of Markovian dynamics fully analytically and non-Markovian dynamics semi-analytically with a simple numerical integration step. We use the formalism to study subradiance, superradiance and bound states in continuum. We discuss new phenomena such as subdivision of collective decay rates into symmetric and anti-symmetric subsets and non-Markovian superradiance effects that can lead to collective decay stronger than Dicke superradiance. We also discuss possible applications such as pulse-shaping and coherent absorption. We thus broaden the range of applicability of real-space approaches beyond steady-state photon transport.
We give an exposure to diagrammatic techniques in waveguide quantum electrodynamics (QED) systems. A particular emphasis is placed on the systems with delayed coherent quantum feedback. Specifically, we show that the N-photon scattering matrices in single-qubit waveguide QED systems, within the rotating wave approximation, admit for a parametrization in terms of ($N\ensuremath{-}1$)-photon effective vertex functions and provide a detailed derivation of a closed hierarchy of generalized Bethe-Salpeter equations satisfied by these vertex functions. The advantage of this method is that the above-mentioned integral equations hold independently of the number of radiation channels, their bandwidth, the dispersion of the modes they are supporting, and the structure of the radiation-qubit coupling interaction, thus enabling one to study multiphoton scattering problems beyond the Born-Markov approximation. Further, we generalize the diagrammatic techniques to the systems containing more than a single emitter by presenting an exact set of equations governing the generic two- and three-photon scattering operators. The above-described theoretical machinery is then showcased on the example of a three-photon scattering on a giant acoustic atom, recently studied experimentally [G. Andersson et al., Nat. Phys. 15, 1123 (2019)].
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.
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.
Starting from the paradigmatic spin-boson model (SBM), we investigate the static and dynamical properties of a system of two distant two-level emitters coupled to a one-dimensional Ohmic waveguide beyond the rotating wave approximation (RWA). Employing static and dynamical polaron ansatze we study the effects of finite separation distance on the behavior of the photon-mediated Ising-like interaction, qubit frequency renormalization, ground state magnetization, and entanglement entropy of the two qubit system. Based on previous works we derive an effective approximate Hamiltonian for the two-impurity SBM that preserves the excitation-number and thus facilitates the analytical treatment. In particular, it allows us to introduce non-Markovianity arising from delay-feedback effects in two distant emitters in the so-called ultrastrong coupling (USC) regime. We test our results with numerical simulations performed over a discretized circuit-QED model, finding perfect agreement with previous results, and showing interesting new dynamical effects arising in ultrastrong waveguide-QED with distant emitters. In particular, we revisit the Fermi two-atom problem showing that, in the USC regime, initial correlations yield two different evolutions for symmetric and antisymmetric states even before the emitters become causally connected. Finally, we demonstrate that the collective dynamics, e.g., superradiance or subradiance, are affected not only by the distance between emitters, but also on the coupling, due to significant frequency renormalization. This constitutes another dynamical consequence of the USC regime.
Quantum systems are invariably coupled to a surrounding environment. A common theoretical method to simplify the problem is to trace over the environment under a Markov approximation, which assumes that the environment holds no memory over the timescales that the system evolves under. In interacting many-body systems, it is often not trivial to extract which of the multiple collective timescales are relevant. We consider a time-delayed waveguide QED setup with a single excitation. By fixing the maximum propagation time and increasing the number of emitters, we show that, even for small delay times, collective effects are sufficient to cause non-Markovianity. The impact of memory is intrinsically state-dependent. For superradiant states, the relevant system timescale is the superradiant lifetime while the relevant bath timescale is the end-to-end delay time. For subradiant states, both timescales depend on the structure of the specific state. Our results demonstrate the importance of prudently making Markov approximations in quantum many-body systems, and highlight potential pitfalls to avoid in scaling up quantum devices to large system sizes.