  The remarkable progress in the field of laser spectroscopy induced by the
invention of the frequency-comb laser has enabled many new high-precision tests
of fundamental theory and searches for new physics. Extending frequency-comb
based spectroscopy techniques to the vacuum (VUV) and extreme ultraviolet (XUV)
spectral range would enable measurements in e.g. heavier hydrogen-like systems
and open up new possibilities for tests of quantum electrodynamics and
measurements of fundamental constants. The main approaches rely on
high-harmonic generation (HHG), which is known to induce spurious phase shifts
from plasma formation. After our initial report (Physical Review Letters 123,
143001 (2019)), we give a detailed account of how the Ramsey-comb technique is
used to probe the plasma dynamics with high precision, and enables accurate
spectroscopy in the VUV. A series of Ramsey fringes is recorded to track the
phase evolution of a superposition state in xenon atoms, excited by two
up-converted frequency-comb pulses. Phase shifts of up to 1 rad induced by HHG
were observed at ns timescales and with mrad-level accuracy at $110$ nm. Such
phase shifts could be reduced to a negligible level, enabling us to measure the
$5p^6 \rightarrow 5p^5 8s~^2[3/2]_1$ transition frequency in $^{132}Xe$ at 110
nm (seventh harmonic) with sub-MHz accuracy. The obtained value is $10^4$ times
more precise than the previous determination and the fractional accuracy of
$2.3 \times 10^{-10}$ is $3.6$ times better than the previous best
spectroscopic measurement using HHG. The isotope shifts between $^{132}Xe$ and
two other isotopes were determined with an accuracy of $420$ kHz. The method
can be readily extended to achieve kHz-level accuracy, e.g. to measure the
$1S-2S$ transition in $He^+$. Therefore, the Ramsey-comb method shows great
promise for high-precision spectroscopy of targets requiring VUV and XUV
wavelengths.
