  The Jaynes-Cummings Hamiltonian is at the core of cavity quantum
electrodynamics, and is ubiquitous in a variety of quantum technologies. The
ability to implement and control the various aspects of this Hamiltonian is
thus of paramount importance. However, conventional implementations relying on
bound-electron systems are fundamentally limited by the Coulomb potential that
bounds the electron, in addition to suffering from practical limitations such
as requiring cryogenic temperatures for operation and fabrication
inhomogeneity. In this work, we propose theoretically a new approach to realize
the Jaynes-Cummings Hamiltonian using low energy free-electrons coupled to
dielectric microcavities, and exemplify several quantum technologies made
possible by this approach. Our approach utilizes quantum recoil, which causes a
large detuning that inhibits the emission of multiple consecutive photons,
effectively transforming the free-electron into a two-level system coupled to
the cavity mode. We show that this approach can be used for generation of
single photons with unity efficiency and high fidelity. We then generalize the
concept to achieve a multiple-level quantum emitter through a suitable design
of cavity modes, allowing for deterministic photon-pair generation and even a
quantum SWAP gate between a cavity photon and a free-electron qubit. An
increase in coupling strength by a factor of \sqrt(N) can be achieved when an
entangled symmetric state of N electrons is used to drive the cavity. Tunable
by their kinetic energy and phase-matching to light waves, quantum
free-electrons are inherently versatile emitters with an engineerable emission
wavelength. As such, they pave the way towards new possibilities for quantum
interconnects between photonic platforms at disparate spectral regimes.
