  The linewidth of a laser plays a pivotal role in ensuring the high fidelity
of ion trap quantum processors and optical clocks. As quantum computing
endeavors scale up in qubit number, the demand for higher laser power with
ultra-narrow linewidth becomes imperative, and leveraging fiber amplifiers
emerges as a promising approach to meet these requirements. This study explores
the effectiveness of Thulium-doped fiber amplifiers (TDFAs) as a viable
solution for addressing optical qubit transitions in trapped barium ion qubits.
We demonstrate that by performing high-fidelity gates on the qubit while
introducing minimal intensity noise, TDFAs do not significantly broaden the
linewidth of the seed lasers. We employed a Voigt fitting scheme in conjunction
with a delayed self-heterodyne method to accurately measure the linewidth
independently, corroborating our findings through quadrupole spectroscopy with
trapped barium ions. Our results show linewidth values of $160 \pm 15$ Hz and
$156 \pm 16$ Hz, respectively, using these two methods, underscoring the
reliability of our measurement techniques. The slight variation between the two
methods can be attributed to factors such as amplified spontaneous emission in
the TDFA or the influence of 1/f noise within the heterodyne setup delay line.
These contribute to advancing our understanding of laser linewidth control in
the context of ion trap quantum computing as well as stretching the
availability of narrow linewidth, high-power tunable lasers beyond the C-band.
