  An array of ions in a linear radio-frequency (RF) Paul trap is a good
candidate for investigating structural phase transitions, such as the
linear-to-zigzag (LZ) transition, due to the convenient control provided by
modification of the trap confinement strength. In such studies, the trap
secular frequencies are a key factor that limits the stability of the critical
point (CP). In this paper, we implement secular-frequency stabilization,
including active feedback stabilization of the RF voltage near the trap
electrodes, and achieve a stability of better than 5~ppm over 200~s for both
transverse and axial potentials. To evaluate the combined long-term stability
of the trap potential in both directions, we measure the zigzag (ZZ) mode
frequency near the CP, where the effect of instability in both trap directions
is substantially amplified. The short-term noise within a limited spectral
window is also suppressed by this scheme, as inferred from decoherence rates of
the ZZ mode measured at different frequencies with Ramsey fringes. We also
identify that the limitation of the secular frequency stability is mainly from
the imperfect temperature insensitivity of voltage dividers and RF detectors,
and as a result improvement of lab temperature stability is of great help for
stabilizing the trap frequency.
