  Ultralight axion-like particles can contribute to the dark matter near the
Sun, leading to a distinct, stochastic signature in terrestrial experiments. We
search for such particles through their neutron-spin coupling by re-analyzing
approximately 40 days of data from a K-$^3$He co-magnetometer with a new
frequency-domain likelihood-based formalism that properly accounts for
stochastic effects over all axion coherence times relative to the experimental
time span. Assuming that axions make up all of the dark matter in the Sun's
vicinity, we find a median 95% upper limit on the neutron-spin coupling of $2.4
\times 10^{-10}$ GeV$^{{-1}}$ for axion masses from 0.4 to 4 feV, which is
about five orders of magnitude more stringent than previous laboratory bounds
in that mass range. Although several peaks in the experiment's magnetic power
spectrum suggest the rejection of a white-noise null hypothesis, further
analysis of their lineshapes yields no positive evidence for a dark matter
axion.
