  We report on the design, realization and experimental investigation by
spatially resolved monochromated electron energy loss spectroscopy (EELS) of
high quality factor cavities with modal volumes smaller than $\lambda^3$, with
$\lambda$ the free-space wavelength of light. The cavities are based on a slot
defect in a 2D photonic crystal slab made up of silicon. They are optimized for
high coupling of electrons accelerated to 100 kV, to quasi-Transverse
Electrical modes polarized along the slot direction. We studied the cavities in
two geometries. The first geometry, for which the cavities have been designed,
corresponds to an electron beam travelling along the slot direction. The second
consists in the electron beam travelling perpendicular to the slab. In both
cases, a large series of modes is identified. The dielectric slot modes
energies are measured to be in the 0.8- 0.85 eV range, as per design, and
surrounded by two bands of dielectric and air modes of the photonic structure.
The dielectric even slot modes, to which the cavity mode belongs, are highly
coupled to the electrons with up to 3.2$\%$ probability of creating a slot
photon per incident electron. Although the experimental spectral resolution
(around 30 meV) alone does not allow to disentangle cavity photons from other
slot photons, the remarkable agreement between the experiments and FDTD
simulations permits us to deduce that amongst the photons created in the slot,
around 30$\%$ are stored in the cavity mode. A systematic study of the energy
and coupling strength as a function of the photonic band gap parameters permits
to foresee increase of coupling strength by fine-tuning phase matching. Our
work demonstrates free electron coupling to high quality factor cavities with
low mode density, sub-$\lambda^3$ modal volume, making it an excellent
candidate for applications such as quantum nano-optics with free electrons.
