We propose an innovative approach to W-state generation and verification based on the self-imaging of light in coupled waveguide arrays. Provided is also a formal proof of the W-state entanglement.
Abstract W states are maximally entangled states with excellent robustness to loss. They have been studied in ion, photon and waveguide-mode bases, however the entanglement verification has remained challenging. Here, we theoretically study W-state generation by the quantum walk of a single photon through a linearly coupled waveguide array (WGA) that supports self-imaging of light. We design W states in symmetric and asymmetric arrays with even and odd number of waveguides. Bipartite entanglement is formally proven using the von Neumann entropy of the reduced density matrix. We further use the self-imaging to construct an entanglement witness based on the exclusion principle. A key to verification is the interferometer which extends the length of a W-state generator to the revival length where the photon is fully recombined into a single waveguide mode. In addition, W-state coherence is proven by numerically demonstrating the far-field interference. The sensitivities of the proposed W-state generation and verification protocols to the fabrication tolerances are numerically evaluated in glass, silicon nitride and silicon-on-insulator WGAs, indicating the feasibility of their realisation by high-precision e-beam lithography and laser writing.
Unlike discrete photonic circuits, which manipulate photons step-by-step using a series of optical elements, arrays of coupled waveguides enable photons to interfere continuously across the entire structure. When composed of a nonlinear material, such arrays can also directly generate quantum states of light within the circuit. To clarify the similarities and distinctions between these two approaches of quantum walks, we conduct here a systematic comparison between linear waveguide arrays, injected with photons produced externally, and nonlinear arrays, where photon pairs are continuously generated via parametric down-conversion. We experimentally validate these predictions using III-V semiconductor nonlinear waveguide lattices with varied geometries, enabling us to tune the depth of the quantum walks over an order of magnitude and reveal the gradual emergence of non-classicality in the output state. Finally, we demonstrate an inverse-design approach to engineer \textit{aperiodic} waveguide arrays, whose optimized coupling profiles generate maximally entangled states such as the biphoton W-state. These results highlight the potential of continuously-coupled photonic systems to harness high-dimensional entanglement within compact architectures.
We study the possibility of generating continuous-variable (CV) entanglement on an integrated chip through nonlinear optical waveguides coupled in a circular array. We show the simultaneous generation and manipulation of multimode CV entangled states of light in χ (2) nonlinear circular arrays (NCAs), a feat unattainable with linear optical waveguides and bulk optical elements. These weakly coupled integrated waveguide arrays are compact and can be fabricated using the femtosecond laser writing technique. This technique offers interferometric stability, scalability, and less sensitivity to decoherence. Our study investigates the potential of NCA to generate bipartite and tripartite CV entanglement. We present analytical results and numerical simulations based on coupled-mode equations for the evolution of field operators. Our findings suggest promising applications of compact on-chip sources in various photonic quantum technologies, such as communication, computing, and networking.