Abstract
We present two different methods for modelling 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 model multi-photon nonlinear dynamics in highly non-Markovian regimes, and we highlight the advantages and disadvantages of these methods for modelling 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 modelling multi-photon 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.
Highlights
Waveguide quantum electrodynamics (QED) deals with quasi one-dimensional (1d) systems that couple atoms and photons through waveguide geometries, where the atoms and two-level systems (TLSs) are coupled to a continuum of quantized field modes [1,2,3,4,5,6,7,8,9,10]
We have presented two different models for solving quantum nonlinear light-matter in waveguide-QED systems, using matrix product states (MPSs) and a space-discretized waveguide (SDW) model
Both approaches are shown to efficiently describe the complicated non-Markovian cases of a time-delayed coherent feedback and two spatially separated TLSs. We applied these models to study three different topical systems in waveguide quantum circuits, including a TLS coupled to an infinite waveguide, a TLS coupled to a semi-infinite waveguide, and two spatially separated TLS coupled to an infinite waveguide
Summary
Waveguide quantum electrodynamics (QED) deals with quasi one-dimensional (1d) systems that couple atoms and photons through waveguide geometries, where the atoms and two-level systems (TLSs) are coupled to a continuum of quantized field modes [1,2,3,4,5,6,7,8,9,10] Such systems can result in a significant enhancement of the spontaneous emission rate when coupled to slow light waveguide modes, with very little off-chip decay [11,12,13].
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