Collective oscillations of a stored deuteron beam close to the quantum limit

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 $\SI{10}{aN}$.