  We report details on the axion dark matter search experiment that uses the
innovative technologies of a High-Temperature Superconducting (HTS) magnet and
a Josephson Parametric Converter (JPC). An 18 T HTS solenoid magnet is
developed for this experiment. The JPC is used as the first stage amplifier to
achieve a near quantum-limited low-noise condition. The first dark matter axion
search was performed with the 18 T axion haloscope. The scan frequency range is
from 4.7789 GHz to 4.8094 GHz (30.5 MHz range). No significant signal
consistent with Galactic dark matter axion is observed. Our results set the
best limit of the axion-photon-photon coupling ($g_{a\gamma\gamma}$) in the
axion mass range of 19.764 to 19.890 $\mu$eV. Using the Bayesian method, the
upper bounds of $g_{a\gamma\gamma}$ are set at
0.98$\times|g_{a\gamma\gamma}^{\text{KSVZ}}|$
(1.11$\times|g_{a\gamma\gamma}^{\text{KSVZ}}|$) in the mass ranges of 19.764 to
19.771 $\mu$eV (19.863 to 19.890 $\mu$eV), and at 1.76
$\times|g_{a\gamma\gamma}^{\text{KSVZ}}|$ in the mass ranges of 19.772 to
19.863 $\mu$eV with 90\% confidence level, respectively. We report design,
construction, operation, and data analysis of the 18 T axion haloscope
experiment.
