  A pair-density wave state has been suggested to exist in underdoped cuprate
superconductors, with some supporting experimental evidence emerging over the
past few years from scanning tunneling spectroscopy. Several studies have also
linked the observed quantum oscillations in these systems to a reconstruction
of the Fermi surface by a pair-density wave. Here, we show, using semiclassical
analysis and numerical calculations, that a Fermi pocket created by first-order
scattering from a pair-density wave cannot induce such oscillations. In
contrast, pockets resulting from second-order scattering can cause
oscillations. We consider the effects of a finite pair-density wave correlation
length on the signal, and demonstrate that it is only weakly sensitive to
disorder in the form of $\pi$-phase slips. Finally, we discuss our results in
the context of the cuprates and show that a bidirectional pair-density wave may
produce observed oscillation frequencies.
