Experimental Reconstruction of the Few-Photon Nonlinear Scattering Matrix from a Single Quantum Dot in a Nanophotonic Waveguide

Coherent photon-emitter interfaces offer a way to mediate efficient nonlinear photon-photon interactions, much needed for quantum information processing. Here we experimentally study the case of a two-level emitter, a quantum dot, coupled to a single optical mode in a nanophotonic waveguide. We carry out few-photon transport experiments and record the statistics of the light to reconstruct the scattering matrix elements of one- and two-photon components. This provides direct insight to the complex nonlinear photon interaction that contains rich many-body physics.

Figure 1

(a),(b) Illustration of single- and two-photon scattering processes for a TLE in a waveguide. In the former case, either elastic reflection or transmission may occur. In the latter case, the two photons may exchange energy via the interaction with the TLE, leading to different scattering processes. (c) Experimental setup to extract the few-photon scattering matrices of the system from intensity $I_t$ and photon-correlation measurements $g_{tt}^{(2)}$, $g_{rr}^{(2)}$, and $g_{tr}^{(2)}$ between different scattering channels. The QD is embedded in a photonic-crystal waveguide (PhCW) and excited by a cw laser source. Beam splitters (BS), single-photon detectors (SPDs), and an electronic time tagger are used to record second-order photon correlations. Polarizing optical elements, such as two linear polarizers (LPs), a half- and a quarter-wave plate (HWP and QWP, respectively) are used for extinction of the laser light and collection of the reflected light from the QD.

Figure 2

(a) Measured (light blue) and fitted (dark blue) transmission intensity $I_t(\omega )$ as a function of the detuning of the excitation laser from the QD resonance. Inset: time resolved dynamics of the QD (in logarithmic scale) and exponentially decaying fitting function, convolved with the instrument response of the detector, to characterize the radiative decay rate. (b) Measured (light blue) and fitted (dark blue) second-order correlation function $g_{tt}^{(2)}(\tau )$ in transmission, with time delay $\tau$, obtained on resonance ($\omega \approx {\omega }_0$).

Figure 3

(a) Experimentally reconstructed (solid line) modulus and phase of the complex single-photon transmission (blue) and reflection (red) coefficients and comparison to theory (dashed line). (b) Experimentally acquired second-order correlation functions for the three different configurations: $g_{tt}^{(2)}$ (blue), $g_{rr}^{(2)}$ (red), and $g_{rt}^{(2)}$ (green). The measurements are well fitted by the theoretical model including imperfections (dashed lines).

Figure 4

(a) Real part of the two-photon correlated coefficient $\mathrm{Re}[\mathcal{T}]$ reconstructed from experimental data (blue) and comparison to theory (black). The theoretical imaginary part $\mathrm{Im}[\mathcal{T}]$ is also included (dashed pink). (b) Theoretical prediction of the real (solid black) and imaginary (dashed pink) parts of $\mathcal{T}$ in the absence of external experimental imperfections.