Abstract
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. Employing static and dynamical polaron Ans\"atze 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 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 by the coupling, due to significant frequency renormalization. This constitutes another dynamical consequence of the USC regime.
Highlights
In most scenarios, photons are weakly coupled to matter, and they travel fast relative to the matter dimensions
Employing theoretical tools based on the polaron transformation, we have addressed the problem of two distant emitters interacting with a one-dimensional Ohmic waveguide beyond RWA
We have performed numerical simulations to compare those results with the ones obtained from the proposed discrete model and explore the full dynamical behavior for finite separation distance between the emitters
Summary
Photons are weakly coupled to matter, and they travel fast relative to the matter dimensions. This has been recently studied outside of the USC regime showing that even in a simple system of two separated two-level emitters, the collective dynamics can be significantly affected by non-Markovian interference caused by radiation-delayed feedback between them [32,33,34,35,36,37,38] In this case retardation effects become important when the atomic lifetime γ −1 ∼ x/vg, for a separation distance x. In this work we present a first step in generalizing the effect of distance to the USC coupling regime, combining both sources of non-Markovianity This is done by considering an Ohmic environment with nonflat spectral density, and by taking into account delay-memory effects due to finite distance separation between emitters. The paper ends with the conclusions, while several technical aspects are assigned to the Appendixes
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