  The merger of a neutron star (NS) binary may result in the formation of a
long-lived, or indefinitely stable, millisecond magnetar remnant surrounded by
a low-mass ejecta shell. A portion of the magnetar's prodigious rotational
energy is deposited behind the ejecta in a pulsar wind nebula, powering
luminous optical/X-ray emission for hours to days following the merger. Ions in
the pulsar wind may also be accelerated to ultra-high energies, providing a
coincident source of high energy cosmic rays and neutrinos. At early times, the
cosmic rays experience strong synchrotron losses; however, after a day or so,
pion production through photomeson interaction with thermal photons in the
nebula comes to dominate, leading to efficient production of high-energy
neutrinos. After roughly a week, the density of background photons decreases
sufficiently for cosmic rays to escape the source without secondary production.
These competing effects result in a neutrino light curve that peaks on a few
day timescale near an energy of $\sim10^{18}$ eV. This signal may be detectable
for individual mergers out to $\sim$ 10 (100) Mpc by current (next-generation)
neutrino telescopes, providing clear evidence for a long-lived NS remnant, the
presence of which may otherwise be challenging to identify from the
gravitational waves alone. Under the optimistic assumption that a sizable
fraction of NS mergers produce long-lived magnetars, the cumulative
cosmological neutrino background is estimated to be $\sim 10^{-9}-10^{-8}\,\rm
GeV\,cm^{-2}\,s^{-1}\,sr^{-1}$ for a NS merger rate of $10^{-7}\,\rm
Mpc^{-3}\,yr^{-1}$, overlapping with IceCube's current sensitivity and within
the reach of next-generation neutrino telescopes.
