  The pseudogap regime of the underdoped cuprates arguably remains one of the
most enigmatic phenomena of correlated quantum matter. Recent theoretical ideas
suggest that a pair density wave (PDW) or a "fractionalized PDW" could be a key
ingredient for the understanding of the pseudogap physics. These ideas are to
be contrasted to the scenario where charge density wave order and
superconductivity coexist at low temperatures. In this paper, we present a few
tests to compare the two scenarios in a Josephson junction setup. For a PDW
scenario, we observe a beat-like structure of AC Josephson current. The
additional frequencies for the AC Josephson current appear at the half-odd
integer multiple of the standard Josephson frequency. We can extract the
modulation wavevector of the PDW state by studying the average Josephson
current. Furthermore, the usual sharp Shapiro steps break down. In contrast,
these signatures are absent for the simple coexistence of orders. Any detection
of such signatures in a similar experimental setup will strongly support the
PDW scenario for the pseudogap phase.
