  We investigate the regime of strong coupling of an ensemble of
two-dimensional electrons to a single-mode cavity resonator. In particular, we
realized such a regime of light-matter interaction by coupling the cyclotron
motion of a collection of electrons on the surface of liquid helium to the
microwave field in a semi-confocal Fabry-Perot resonator. For the co-rotating
component of the microwave field, the strong coupling is pronouncedly
manifested by the normal-mode splitting in the spectrum of coupled
field-particle motion. We present a complete description of this phenomenon
based on classical electrodynamics, as well as show that the full quantum
treatment of this problem results in mean-value equations of motion that are
equivalent to our classical result. For the counter-rotating component of the
microwave field, we observe a strong resonance when the microwave frequency is
close to both the cyclotron and cavity frequencies. We show that this
surprising effect, which is not expected to occur under the rotating-wave
approximation, results from the mixing between two polarization components of
the microwave field in our cavity.
