Quantum dynamics of few-photon pulsed waveguide QED with a single artificial atom: Frequency-dependent scattering theory and time-dependent matrix product states

We present a quantum dynamical study of pulsed few-photon scattering from a single artificial atom, consisting of a two-level system (TLS) or qubit, in a waveguide-quantum electrodynamics system, directly comparing and contrasting two different quantum theoretical simulation methods: (1) an input-output scattering approach that uses frequency-dependent scattering matrices, and (2) a matrix product states (MPS) approach, which uses quantum noise operators in discretized time bins and a tensor network technique to directly solve the time-dependent observables for the entire quantum waveguide-qubit system. Beginning with pulsed quantum-field excitation, using one-photon and two-photon Fock-state pulses, we first show how to compute time-dependent observables with a scattering matrix approach, in terms of frequency integrals that encode the pulse spectrum, including how to extract population dynamics of the excited quantum emitter, as well as the linear and nonlinear contributions. We present solutions for both symmetric and chiral TLS coupling. We then show how to compute the qubit and field observables in a more direct way using MPS, and obtain the characteristic birdlike shape for the two-photon correlation function at two times, which has been observed in recent experiments. We compare and contrast both of these methods, for one-photon and two-photon excitation pulses, and show excellent agreement. We also present a study of the linear and nonlinear contributions, which can easily be calculated using scattering theory, and which show the important role of pulse duration. Finally, we demonstrate the clear advantages of MPS by easily going to higher N -photon excitations, and show selected example population dynamics of up to eight-photon Fock-state pulses, manifesting in nonlinear population oscillations during the pulse interaction, similar to classical Rabi oscillations, but with quantum input fields that have a vanishing electric field expectation value.