  When very high magnetic fields suppress the superconductivity in underdoped
cuprates, an exceptional new electronic phase appears. It supports remarkable
and unexplained quantum oscillations and exhibits an unidentified density wave
(DW) state. Although generally referred to as a "charge" density wave (CDW)
because of the observed charge density modulations, theory indicates that this
could actually be the far more elusive electron-pair density wave state (PDW).
To search for evidence of a field-induced PDW in cuprates, we visualize the
modulations in the density of electronic states $N(\bf{r})$ within the halo
surrounding Bi$_2$Sr$_2$CaCu$_2$O$_8$ vortex cores. This reveals multiple
signatures of a field-induced PDW, including two sets of $N(\bf{r})$
modulations occurring at wavevectors $\bf{Q}_P$ and $2\bf{Q}_P$, both having
predominantly $s$-symmetry form factors, the amplitude of the latter decaying
twice as rapidly as the former, along with induced energy-gap modulations at
$\bf{Q}_P$ . Such a microscopic phenomenology is in detailed agreement with
theory for a field-induced primary PDW that generates secondary CDWs within the
vortex halo. These data indicate that the fundamental state generated by
increasing magnetic fields from the underdoped cuprate superconducting phase is
actually a PDW with approximately eight CuO$_2$ unit-cell periodicity ($\lambda
= 8a_0$) and predominantly $d$-symmetry form factor.
