A general framework for interactions between electron beams and quantum optical systems

We provide a theoretical framework to describe the dynamics of a free-electron beam interacting with quantized bound systems in arbitrary electromagnetic environments. This expands the quantum optics toolbox to incorporate free-electron beams for applications in highly tunable quantum control, imaging, and spectroscopy at the nanoscale. The framework recovers previously studied results and shows that electromagnetic environments can amplify the intrinsically weak coupling between a free-electron and a bound electron to reach previously inaccessible interaction regimes. We leverage this enhanced coupling for experimentally feasible protocols in coherent qubit control and towards the nondestructive readout and projective control of the electron beam’s quantum-number statistics. Our framework is broadly applicable to microwave-frequency qubits, optical nanophotonics, cavity quantum electrodynamics, and emerging platforms at the interface of electron microscopy and quantum information.

💡 Research Summary

This paper introduces a unified theoretical framework for describing the dynamics of a free‑electron beam interacting with a quantized bound system—typically modeled as a two‑level spin qubit—within an arbitrary electromagnetic environment. By representing the electron, the qubit, and the surrounding field as three distinct quantum subsystems (a unidirectional infinite ladder, a discrete TLS, and a bidirectional “quantum rotor”), the authors derive an effective Hamiltonian using the dyadic Green’s function of the environment and a Markovian master‑equation approach that adiabatically eliminates the photonic degrees of freedom.

In free space, the interaction reduces to a simple magnetic dipole coupling whose dimensionless strength ϕ₀ scales inversely with the impact parameter r⊥ and involves a modified Bessel function K₁. Practical constraints on r⊥ (typically >10 µm to avoid beam damage) limit ϕ₀ to 10⁻⁶–10⁻¹⁰, rendering single‑qubit control infeasible. To overcome this limitation, the authors embed the qubit and the electron trajectory inside a microwave cavity operated in the dispersive regime (detuning Δ≫cavity linewidth γ). The cavity mediates an interaction Hamiltonian H_int = ħ g g*el Δ σ b†{ω₀/v}+h.c., where g_sp, g_d, and g_el denote the spin‑magnetic, dipole‑electric, and electron‑electric coupling rates, respectively, and g_Q = g_el T quantifies the electron‑cavity coupling over the interaction time T. Because g_Q does not decay with distance, it can reach |g_Q|≈1, leading to an enhanced overall coupling ϕ_cav = |g_Q||g|/Δ that can be two to three orders of magnitude larger than ϕ₀.

The resulting scattering matrix S = exp