  High-fidelity operation of quantum computers requires precise knowledge of
the physical system through characterization. For motion-mediated entanglement
generation in trapped ions, it is crucial to have precise knowledge of the
motional-mode parameters such as the mode frequencies and the Lamb-Dicke
parameters. Unfortunately, the state-of-the-art mode-characterization schemes
do not easily render the mode parameters in a sufficiently scalable and
accurate fashion, due to the unwanted excitation of adjacent modes in the
frequency space when targeting a single mode, an effect known as the
\textit{cross-mode coupling}. Here, we develop an alternative scheme that
leverages the degrees of freedom in pulse design for the characterization
experiment such that the effects of the cross-mode coupling is actively
silenced. Further, we devise stabilization methods to accurately characterize
the Lamb-Dicke parameters even when the mode frequencies are not precisely
known due to experimental drifts or characterization inaccuracies. We
extensively benchmark our scheme in simulations of a three-ion chain and
discuss the parameter regimes in which the shaped pulses significantly
outperform the traditional square pulses.
