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

Abstract

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 QED system, directly comparing and contrasting two different quantum theoretical simulation methods: (i) an input-output scattering approach that uses frequency-dependent scattering matrices, and (ii) a matrix product states (MPS) approach, which uses quantum noise operators in time bins and a tensor network technique to solve the time-dependent waveguide function for the entire system. Beginning with pulsed excitation using one-photon and two-photon Fock state pulses, we first show how to compute time-dependent observables with the scattering matrix approach, in terms of frequency integrals that encode the pulse spectrum, including how to extract the 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 bird-like 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 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 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 clear 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.

Figures (4)

• Figure 1: Schematic of the light-matter system of interest, which includes a TLS coupled to an infinite (i.e., open) waveguide, where $\gamma_R$ and $\gamma_L$ correspond to the right/left coupling, and the emitter is excited by a quantum pulse (Fock state) containing one or a few photons with a pulse envelope shape $f(t)$. Here, $n_{\rm pulse}$ represents the flux of the input pulse, and $n_R$ and $n_L$ represent the right/left photon fluxes after the interaction with the TLS. For a chiral emitter, then $\gamma_L=0$.
• Figure 2: Two-times first-order quantum correlation functions of a TLS symmetrically coupled to a waveguide and driven by a quantum pulse with a Gaussian envelope. Figures (a,b) show the right output of the first-order correlation function with a single-photon pulse, $G^{(1)}_{1,\rm RR}$, calculated using MPS in (a), and the scattering matrix theory in (b). Figures (c,d) show the same first-order correlation function, but in the case of having a pulse containing 2 photons, $G^{(1)}_{2,\rm RR}$, where again both methods are compared, with MPS results in (c) and S-matrix results in (d).
• Figure 3: Population dynamics and second-order two-times correlation functions of a 2-photon Gaussian pulse with $\gamma \sigma_t=1$ and centered at $\gamma t_c=3$ interacting with a TLS symmetrically coupled to a waveguide in (a,b,c) and right-chirally coupled to a waveguide in (c,d,e). (a,d) Population dynamics, including the TLS population $n_{\rm TLS}$, input pulse flux $n_{\rm pulse}$, photon flux transmitted to the right $n_{\rm R}$ and reflected to the left $n_{\rm L}$. Solid curves correspond to MPS calculations, and circles correspond to results using S-matrix theory. (b,c) Right channel second-order correlation function, $G^{(2)}_{2,\rm RR} (t,t+\tau)$, comparing MPS results in (b) and S-matrix ones in (c). (e,f) Second-order correlation function in the chiral coupling solution, $G^{(2)}_{2}(t,t+\tau)$, again comparing MPS results in (e) and S-matrix ones in (f).
• Figure 5: Population dynamics of an $n$-photon Gaussian Fock-state pulse with $\gamma \sigma_t=1$ and centered at $\gamma t_c=3$, interacting with a TLS symmetrically coupled (a,b,c) and chirally coupled (d,e) to a waveguide. (a,d) TLS population dynamics. (b,e) Transmitted photon flux, and (c) reflected photon flux. The input pulse is shown in yellow for reference (renormalized to a maximum of 0.4).