  We describe the design of the dark matter experiment Haloscope At Yale
Sensitive To Axion Cold Dark Matter (HAYSTAC), and report the results of a
haloscope search for dark matter axions. We exclude axion models with
axion-photon couplings $g_{a\gamma\gamma} \underset{\sim}{{>}}2\times10^{-14}$
GeV$^{-1}$ over the range $23.55<m_{a}<24.0$ $\mu$eV. This sensitivity is a
factor of 2.7 above KSVZ model coupling, averaged over the given mass range.
Phase I achieved a noise temperature a factor of 2 over the standard quantum
limit. Phase II, now entering commissioning, incorporates a squeezed-vacuum
state receiver to evade the quantum limit, which will deliver a factor of
$\sim2$ improvement in scanning rate over the current single Josephson
parametric amplifier (JPA) receiver. The sensitivity of the HAYSTAC Phase II
receiver will lead to precise constraints on two photon coupling strengths and
represents the most sensitive axion cavity detector probing $m_a>20~\mu$eV to
date.
