  In this paper, we investigate the spin and tunneling dynamics of a
spin-orbit-coupled noninteracting Bose-Einstein condensate in a periodically
driven non-Hermitian double-well potential. Under high-frequency driving, we
obtain the effective time-averaged Hamiltonian by using the standard
time-averaging method, and analytically calculate the Floquet quasienergies,
revealing that the parity-time (PT)-breaking phase transition appears even for
arbitrarily small non-Hermitian parameters when the spin-orbit coupling
strength takes half-integer value, irrespective of the values of other
parameters used. When the system is PT-symmetric with balanced gain and loss,
we find numerically and analytically that in the broken PT-symmetric regions,
there will exist the net spin current together with a vanishing atomic current,
if we drop the contribution of the exponential growth of the norm to the
current behaviors. When the system is non-PT-symmetric, though the
quasienergies are partial complex, a stable net spin current can be generated
by controlling the periodic driving field, which is accompanied by a spatial
localization of the condensate in the well with gain. The results deepen the
understanding of non-Hermitian physics and could be useful for engineering a
variety of devices for spintronics.
