Pulsed two-photon scattering from a single atom in a waveguide with delay-modified temporal correlations

Abstract

Quantum nonlinearity is an essential ingredient for many quantum technologies, but often the nonlinearity is too weak to be exploited at the few-photon level. However, few photons interacting strongly with single quantum emitters in a waveguide environment can impact a significant nonlinear response, opening up a wide range of photon-photon correlations. Using a waveguide-QED system containing a single atom (treated as a two-level system) chirally coupled to a waveguide, we theoretically investigate two-photon nonlinearities with delay-controlled temporal correlations. We use both matrix product states (MPS) and a frequency-dependent scattering theory approach to analyze the exact population dynamics, as well as the first-order and second-order photon correlation functions in transmission of the system, when pumped by a two-photon Fock-state pulse with a bimodal temporal pulse envelope. The two-photon Fock-state pulses are considered to be either two single photons localized to each peak of the pulse, or both photons delocalized (but correlated) between the two peaks. We consider the regimes of a short, moderate, and (relatively) long distance between the two pulse peaks, comparing the important differences in the temporal correlations with the two types of two-photon pulses. We demonstrate the strikingly different nonlinear features and quantum correlations that occur for uncorrelated and correlated two-photon pairs in experimentally accessible regimes.

Figures (8)

• Figure 1: Schematic of the TLS chirally coupled (photon scattering is only in the forward/right direction) to the waveguide and an incident Fock pulse with a pulse envelope comprised of two spatial peaks, specified by $f_1(t)$ and $f_2(t)$ respectively. The spatial separation distance between these peaks, $d$, is given by $d=v_g t_b$, where $v_g$ is the group velocity of the waveguide mode.
• Figure 2: Population dynamics for two subsequent single photon top-hat pulses of length $\gamma t_p=1$ incident a chiral TLS-waveguide system with break times between the pulse centers of $\gamma^{-1}$ (a), $2\gamma^{-1}$ (c), and $11\gamma^{-1}$ (e) simulated with MPS. Results on the right side (b,d,f) are the same but for a single two photon pulse with the same temporal pulse envelope separated into two segments of length $\gamma^{-1}$. The blue shaded regions are to emphasize the pulsed area of the graph and the observables are defined by $n_{\rm TLS}=\braket{\sigma^+\sigma^-}(t)$, $n_{\rm in}$ being the input pulse flux, $n_T=\braket{\Delta B_R^\dag(t) \Delta B_R(t)}/(\Delta t)^2$, and $n_{TT} = G_{TT}^{(2)}(t,0)$.
• Figure 3: Two time point second-order correlation function in transmission, $G_{TT}^{(2)}(t,\tau)$ of the same simulations as \ref{['fig:mpsChiralTophatPops']} calculated with MPS. Plots are between two independent time points, $t$ and $t+\tau$.
• Figure 4: Population dynamics for two subsequent single photon Gaussian pulses of standard deviation $\gamma\sigma_t=1$ and the first pulse centered at $\gamma t_c=4$ incident a chiral TLS-waveguide system with break times ($t_b$) between the pulse centers of $\gamma^{-1}$ (a), $2\gamma^{-1}$ (c), and $8\gamma^{-1}$ (e) calculated using the scattering theory. Results on the right side (b,d,f) are the same but for a single two photon pulse into the TLS. The blue shaded regions are to emphasize the pulsed area of the graph and the observables are defined by $n_{\rm TLS}=\braket{\sigma^+\sigma^-}(t)$, $n_{\rm in}$ being the input pulse flux, $n_T=\braket{a_{\rm out}^\dag a_{\rm out}}(t)$, and $n_{TT} = G_{TT}^{(2)}(t,0)$.
• Figure 5: Second-order correlation function $G_{TT}^{(2)}(t,\tau)$ of the same simulations as \ref{['fig:fanChiralGaussiansPops']} and calculated using the scattering theory method.