We investigated coherent betatron oscillations of a deuteron beam in the storage ring cooler synchrotron and storage ring, excited by a detuned rf Wien filter (WF). The beam oscillations were detected by conventional beam position monitors. With the currently available apparatus, we show that oscillation amplitudes down to 1 μm can be detected. The interpretation of the response of the stored beam to the detuned rf WF is based on simulations of the beam evolution in the lattice of the ring and realistic time-dependent 3D field maps of the WF. 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 here the stored beam can be considered as a rarefied gas of uncorrelated particles, we moreover demonstrate that the amplitudes of the zero-point (ground state) betatron oscillations of individual particles are only a factor of about 10 larger than 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 10 aN
