  This work demonstrates that two systematic errors, coherent betatron
oscillations (CBO) and muon losses can be reduced through application of radio
frequency (RF) electric fields, which ultimately increases the sensitivity of
the muon $g-2$ experiments. As the ensemble of polarized muons goes around a
weak focusing storage ring, their spin precesses, and when they decay through
the weak interaction, $\mu^+ \rightarrow e^+ \nu_e \bar{\nu_\mu}$, the decay
positrons are detected by electromagnetic calorimeters. In addition to the
expected exponential decay in the positron time spectrum, the weak decay
asymmetry causes a modulation in the number of positrons in a selected energy
range at the difference frequency between the spin and cyclotron frequencies,
$\omega_\text{a}$. This frequency is directly proportional to the magnetic
anomaly $a_\mu =(g-2)/2$, where $g$ is the g-factor of the muon, which is
slightly greater than 2. The detector acceptance depends on the radial position
of the muon decay, so the CBO of the muon bunch following injection into the
storage ring modulate the measured muon signal with the frequency
$\omega_\text{CBO}$. In addition, the muon populations at the edge of the beam
hit the walls of the vacuum chamber before decaying, which also affects the
signal. Thus, reduction of CBO and unwanted muon loss increases the $a_\mu$
measurement sensitivity. Numerical and experimental studies with RF electric
fields yield more than a magnitude reduction of the CBO, with muon losses
comparable to the conventional method.
