Quantum state heralding using photonic integrated circuits with free electrons

Recently, integrated photonic circuits have brought new capabilities to electron microscopy and been used to demonstrate efficient electron phase modulation and electron-photon correlations. Here, we quantitatively analyze the feasibility of high fidelity and high purity quantum state heralding using a free electron and a photonic integrated circuit with parametric coupling, and propose schemes to shape useful electron and photonic states in different application scenarios. Adopting a dissipative quantum electrodynamics treatment, we formulate a framework for the coupling of free electrons to waveguide spatial-temporal modes. To avoid multimode-coupling induced state decoherence, we show that with proper waveguide design, the interaction can be reduced to a single-mode coupling to a quasi-TM00 mode. In the single-mode coupling limit, we go beyond the conventional state ladder treatment, and show that the electron-photon energy correlations within the ladder subspace can still lead to a fundamental purity and fidelity limit on complex optical and electron state preparations through heralding schemes. We propose applications that use this underlying correlation to their advantage, but also show that the imposed limitations for general applications can be overcome by using photonic integrated circuits with an experimentally feasible interaction length, showing its promise as a platform for free-electron quantum optics.