Quantum entanglement, a key resource for quantum technologies, requires efficient generation of high-fidelity entangled states. While light-matter interactions can generate entanglement, the necessary nonlinear interactions are often weak. This research investigates a novel approach using a single two-level emitter deterministically coupled to light in a nanophotonic waveguide to overcome this limitation. The interaction between a quantum light pulse and a two-level emitter offers a path towards encoding substantial quantum complexity, leveraging the emitter's highly nonlinear operation on the incoming pulse without the need for complex preparation schemes. The core experimental challenge is enhancing the radiative coupling of the emitter to dominate over decoherence. Significant progress has been made using semiconductor quantum dots (QDs) in photonic crystal waveguides (PhC WGs) and cavities. Existing research has demonstrated phenomena like antibunching, quadrature squeezing, and two-photon correlation dynamics. However, whether this nonlinear response enables non-local quantum entanglement remained unexplored until this study. This paper aims to demonstrate that a two-level emitter can induce strong energy-time entanglement between two scattered photons, verified by violating a Bell inequality, thus offering a highly efficient and energy-saving pathway to non-Gaussian photonic operations compared to existing methods like four-wave mixing, which operate at much higher energy consumption levels and require complex pumping schemes.

Previous work on entanglement generation using quantum emitters has focused on strong excitation (Mollow) regimes in bulk samples or QD biexciton radiative cascades. These approaches differ significantly from the presented passive scattering method using weak excitation. Theoretical predictions suggested two-level nonlinear responses could induce photon-photon correlations, but their ability to generate non-local quantum entanglement was not experimentally demonstrated before this study. The work builds upon advancements in quantum nonlinear responses using various emitters (QDs, color centers, atoms, molecules), including observations of antibunching, quadrature squeezing, and two-photon correlation dynamics and photonic bound states. The development of multi-emitter systems and the inclusion of spin degrees of freedom have paved the way for photon sorters, quantum logic gates, and single-photon transistors. This research extends the understanding of many-body waveguide quantum electrodynamics towards strongly correlated light and matter.

The experiment uses a single two-level emitter (QD) deterministically coupled to a single propagating spatial mode in a PhC WG. A weak coherent input field interacts with the emitter, with coupling efficiency β determined by the ratio of radiative decay rate into the waveguide mode (γ) to the QD total decay rate (Γ). For β ≈ 1 and negligible decoherence, single-photon components are elastically reflected, while two-photon components are inelastically scattered in the forward direction. This process can be interpreted as the emitter mediating an energy exchange between two photons, creating energy-time entanglement. The two-photon scattering is investigated using a Franson interferometer with time-resolved photon correlation measurements under weak resonant excitation. The Clauser-Horn-Shimony-Holt (CHSH) Bell inequality is used to test for entanglement, with a Bell parameter S=2 representing the locality bound. Experimental imperfections (finite β, pure dephasing, incoming light strength) are considered, particularly the sensitivity of S to γ and n. The relationship between S and β is approximated by S(β) ≈ 2√2[1 − (1 − β)⁴]. The two-photon joint spectral and temporal intensity distributions illustrate the quantum correlations, showing the output entangled photon pair being broadened by the QD linewidth and exhibiting anti-correlation in the joint spectral density. The experimental setup includes a PhC WG chip, spectral filter, and Franson interferometer with two unbalanced Mach-Zehnder interferometers (UMZIs) for entanglement analysis. The UMZIs project the two-photon state into discrete time bins. The phases of the UMZIs are controlled to observe constructive/destructive interference, demonstrating energy-time entanglement. The high extinction of transmission intensity (beyond 85%) highlights the efficient radiative coupling to the PhC WG. Second-order photon correlation measurements quantify the successful two-photon component preparation, revealing pronounced photon bunching (g⁽²⁾(0) ≈ 210). A narrow bandwidth notch filter suppresses residual laser leakage. The Franson interference visibility (V = 95(4)%) and the CHSH Bell parameter (S = 2.67(16) > 2) confirm the presence and non-local nature of the entanglement. The power dependence of the S parameter is also investigated, aligning with the theoretical model. The sample is a suspended GaAs membrane with InAs QDs, mounted in a cryostat at 4 K. Phase locking of the UMZIs is achieved using a feedback loop with a photodiode, piezo-mounted mirror, and PID module.

The study experimentally demonstrates the violation of the CHSH Bell inequality through weak scattering of a single two-level emitter coupled to light in a PhC WG. A Bell parameter of S = 2.67(16) > 2 was obtained, significantly exceeding the classical limit and confirming the presence of non-local correlations. This violation is achieved using a passive scattering approach, requiring no elaborate excitation or active spin control. The high efficiency of the radiative coupling to the PhC WG is evidenced by the extinction of transmission intensity exceeding 85%. Second-order photon correlation measurements reveal pronounced photon bunching with g⁽²⁾(0) ≈ 210, signifying a substantial alteration of the input Poissonian photon distribution due to the strong nonlinear interaction with the QD. The Franson interference visibility of 95(4)% further supports the presence of entanglement. The extracted coupling efficiency is β = 92%, and Purcell enhancement is Fp = 15.9, increasing the QD decay rate to Γ/2π = 2.3 GHz. The experimental data are in good agreement with the theoretical model, considering the effects of coupling efficiency, pure dephasing, and multi-photon scattering processes. The study also demonstrates the high spectral brightness of the entanglement source, exceeding the capabilities of many parametric sources.

The successful violation of the Bell inequality definitively demonstrates the generation of energy-time entangled photons through the passive scattering of a single two-level emitter. The observed entanglement originates from the nonlinear interaction between two photons mediated by the emitter in the waveguide. This approach offers several advantages over traditional methods. First, its passive nature eliminates the need for complex and decoherence-sensitive pumping schemes, reducing energy consumption significantly. Second, it simplifies the experimental setup by avoiding the need for elaborate spin control techniques usually associated with spin-based entanglement generation. The high efficiency of the method, as indicated by the high coupling efficiency and pronounced photon bunching, promises significant improvements in scalability and practicality for future quantum technologies. The good agreement between experimental data and the theoretical model validates the underlying physical principles and allows for a thorough understanding of the entanglement generation mechanism.

This research successfully demonstrates the generation of energy-time entangled photons via the passive scattering of a single two-level emitter coupled to a PhC WG, violating the CHSH Bell inequality. The method's passive nature, high efficiency, and simplicity offer a promising alternative for on-chip entanglement generation in quantum technologies. Future research could explore high-dimensional entanglement, synthesis of more complex photonic quantum states, and the engineering of inelastic scattering processes using coupled QDs. The approach has significant potential for applications in photonic quantum computing, communication, and sensing.

While the experiment successfully demonstrates entanglement generation, the study's limitations include the influence of experimental imperfections like finite coupling efficiency, pure dephasing, and multi-photon scattering, which affect the quality of the entanglement. The theoretical model, while accurately describing the experimental data, does not explicitly include the filter's effect. Further research could explore optimizing these parameters for achieving even higher-fidelity entanglement.