  The detection of microwave fields at single-photon power levels is a much
sought-after technology, with practical applications in nanoelectronics and
quantum information science. Here we demonstrate a simple yet powerful
criticality-enhanced method of microwave photon detection by operating a
magnetic-field tunable Kerr Josephson parametric amplifier near a first-order
quantum phase transition. We obtain a 73% efficiency and a dark-count rate of
167 kHz, corresponding to a responsivity of $1.3 \times
10^{17}~\mathrm{W}^{-1}$ and noise-equivalent power of 3.28 zW/$\sqrt{\rm Hz}$.
We verify the single-photon operation by extracting the Poissonian statistics
of a coherent probe signal.
