  Long-lived ionic quantum states known as metastable electronic states in
highly-charged ions (HCIs) are of great interest in fundamental physics.
Especially, it generates transitions with very narrow natural linewidth which
is a promising candidate for use in the next generation HCI atomic clocks to
reach an accuracy below $10^{-19}$. A recent experiment reported in
[Nature,581(7806) 2020], used Penning trap mass spectrometry to measure the
energy of an extremely long-lived metastable electronic state, thus opening
doors to search for HCI clock transitions. Building upon prior research, this
study introduces an experimental proposal with the goal of measuring lifetimes
of the metastable states beyond seconds. Our approach employs a sequential
pulse-and-phase measurement scheme, allowing for direct observations of the
decay processes from metastable electronic states through single-ion mass
spectrometry in a Penning trap. This measurement poses a significant challenge
to conventional techniques like fluorescence detection. To demonstrate the
effectiveness of this method, we conducted a comprehensive simulation under
real experimental conditions, yielding promising results in a specific
scenario. Two suitable candidates are proposed for testing this method, and the
state-of-the-art MCDHF theory are employed for accurate energy levels and
transition rate calculations. Some future prospects in the experimental
determinations of a wide range of energy and lifetimes of long-lived metastable
electronic states, probing hyperfine and magnetic quenching effects on
high-order forbidden transitions and search for highly quality HCI clock
transitions are discussed.
