We consider the problem of modeling a single qubit in contact with a one-dimensional waveguide beyond the standard perturbative and Markovian approximations. Using the recently developed input-output hierarchical equations of motion (io-HEOM), we investigate multiple examples of such waveguides, characterized by different spectral densities. Our examples highlight that the io-HEOM method can accurately capture non-Markovianity in waveguide QED from two distinct origins. The first source of non-Markovianity is spatially non-local coupling between the qubit and the waveguide. By examining two examples with non-local coupling, we show how the coupling function affects the steady-state bound photons, and demonstrate the release of these photons when the qubit energy is quenched. The second source of non-Markovianity is non-linear dispersion. We illustrate this scenario using the example of a cavity array with point-like coupling, where the non-linear dispersion leads to persistent oscillations due to Van Hove singularities in the spectral density.
We consider the problem of modeling a single qubit in contact with a one-dimensional waveguide beyond the standard perturbative and Markovian approximations. Using the recently developed input-output hierarchical equations of motion (io-HEOM), we investigate multiple examples of such waveguides, characterized by different spectral densities. Our examples highlight that the io-HEOM method can accurately capture non-Markovianity in waveguide QED from two distinct origins. The first source of non-Markovianity is spatially non-local coupling between the qubit and the waveguide. By examining two examples with non-local coupling, we show how the coupling function affects the steady-state bound photons, and demonstrate the release of these photons when the qubit energy is quenched. The second source of non-Markovianity is non-linear dispersion. We illustrate this scenario using the example of a cavity array with point-like coupling, where the non-linear dispersion leads to persistent oscillations due to Van Hove singularities in the spectral density.
Embedding non-Markovian open quantum dynamics into an enlarged Markovian space offers a powerful route to nonperturbative simulations, where the dynamics of the extended space can be governed by multiple distinct Markovian equations. We show that these distinct embeddings arise from different unravelings of Gaussian bath self-energies, generating a family of deterministic, time-local equations for the extended system. Using the Brownian-oscillator spectral density as an illustrative example, we clarify the relationships among existing approaches, including the Hierarchical Equations of Motion (HEOM) and the Lindblad--pseudomode formalism, and demonstrate how this framework enables numerically stable and efficient simulations. This work provides both a transparent theoretical foundation for embedding techniques and a flexible platform for developing new methods to simulate non-Markovian quantum dynamics.
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 derive an extended version of the hierarchical equations of motion (HEOM) to compute output physical properties of a bosonic environment, which is allowed to be initially prepared at an earlier time in a non-Gaussian input state and then non-perturbatively interact with a quantum system with a linear environmental operator. While spectral assumptions analogous to the ones used in the regular HEOM are imposed to compute dynamical output bath observables, they are not required to model input states or output observables at a fixed time, in this case leading to time-dependent contributions to the equations. In the Markovian limit, we use this formalism to derive an input-output Lindblad equation which can be used to extend the applicability of the regular version. For a given desired input state and output observable, the range of the indexes extending the regular HEOM is, by construction, bounded. Overall, the aim of this formalism is to take advantage of the efficiency and generality of the HEOM framework to model non-Gaussian input states and the dynamics of environmental observables in bosonic, non-Markovian open quantum systems.
We study two-level systems (2LS) coupled at different points to a one-dimensional waveguide in which one end is open and the other is either open (infinite waveguide) or closed by a mirror (semi-infinite). Upon injection of two photons (corresponding to weak coherent driving), the resonance fluorescence and photon correlations are shaped by the effective qubit transition frequencies and decay rates, which are substantially modified by interference effects. In contrast to the well-known result in an infinite waveguide, photons reflected by a single 2LS coupled to a semi-infinite waveguide are initially bunched, a result that can be simply explained by stimulated emission. As the number of 2LS increases (up to 10 are considered here), rapid oscillations build up in the correlations that persist for a very long time. For instance, when the incoming photons are slightly detuned, the transmitted photons in the infinite waveguide are highly antibunched. On the other hand, upon resonant driving, incoherently reflected photons are mostly distributed within the photonic band gap and several sharp side peaks. These features can be explained by considering the poles of the single-particle Green function in the Markovian regime combined with the time delay. Our calculation is not restricted to the Markovian regime, and we obtain several fully non-Markovian results. We show that a single 2LS in a semi-infinite waveguide can not be decoupled by placing it at the node of the photonic field, in contrast to recent results in the Markovian regime. Our results illustrate the complexities that ensue when several qubits are strongly coupled to a bus (the waveguide) as might happen in quantum information processing.
We study two-level systems (2LS) coupled at different points to a one-dimensional waveguide in which one end is open and the other is either open (infinite waveguide) or closed by a mirror (semi-infinite). Upon injection of two photons (corresponding to weak coherent driving), the resonance fluorescence and photon correlations are shaped by the effective qubit transition frequencies and decay rates, which are substantially modified by interference effects. In contrast to the well-known result in an infinite waveguide, photons reflected by a single 2LS coupled to a semi-infinite waveguide are initially bunched, a result that can be simply explained by stimulated emission. As the number of 2LS increases (up to 10 are considered here), rapid oscillations build up in the correlations that persist for a very long time. For instance, when the incoming photons are slightly detuned, the transmitted photons in the infinite waveguide are highly antibunched. On the other hand, upon resonant driving, incoherently reflected photons are mostly distributed within the photonic band gap and several sharp side peaks. These features can be explained by considering the poles of the single particle Green function in the Markovian regime combined with the time delay. Our calculation is not restricted to the Markovian regime, and we obtain several fully non-Markovian results. We show that a single 2LS in a semi-infinite waveguide can not be decoupled by placing it at the node of the photonic field, in contrast to recent results in the Markovian regime. Our results illustrate the complexities that ensue when several qubits are strongly coupled to a bus (the waveguide) as might happen in quantum information processing.
We study the non-Markovian dynamics of a giant artificial atom coupled to a one-dimensional acoustic waveguide beyond the rotating-wave and weak-coupling approximations. By combining an optimized ESPRIT-based decomposition of the bath correlation function with the hierarchical equations of motion (HEOM), we achieve numerically exact simulations in regimes with long memory times, finite temperature, and strong system-bath coupling. Benchmarking against analytical results reveals the breakdown of perturbative non-Markovian approaches such as Redfield theory even at weak coupling in the presence of delay-induced memory. We further show that non-Markovian features, including excitation revivals, remain robust at finite temperature and can be enhanced by increasing the system-bath coupling strength. Our approach provides a versatile framework for studying non-Markovian quantum dynamics in structured environments relevant to giant-atom platforms.