  We investigated coherent betatron oscillations of a deuteron beam in the
storage ring COSY, excited by a detuned radio-frequency Wien filter. These beam
oscillations were detected by conventional beam position monitors, read out
with lock-in amplifiers. The response of the stored beam to the detuned Wien
filter was modelled using the ring lattice and time-dependent 3D field maps of
the radio-frequency Wien filter. The influence of uncertain system parameters
related to manufacturing tolerances and electronics was investigated using the
polynomial chaos expansion. With the currently available apparatus, we show
that oscillation amplitudes down to $\SI{1}{\micro \meter}$ can be detected.
Future measurements of the electric dipole moment of protons will, however,
require control of the relative position of counter-propagating beams in the
sub-picometer range. Since the stored beam can be considered as a rarefied gas
of uncorrelated particles, we moreover demonstrate that the amplitudes of the
zero-point betatron oscillations of individual particles are within a factor of
10 of the Heisenberg uncertainty limit. As a consequence of this, we conclude
that quantum mechanics does not preclude the control of the beam centroids to
sub-picometer accuracy. The smallest Lorentz force exerted on a single particle
that we have been able to determine is $\SI{10}{aN}$.
