  We develop a framework to solve a large class of linearly driven
non-Hermitian quantum systems. Such a class of models in the Hermitian scenario
is commonly known as multi-state Landau-Zener models. The non-hermiticity is
due to the anti-Hermitian couplings between the diabatic levels. We find that
there exists a new conservation law, unique to this class of models, that
describes the simultaneous growth of the unnormalized wavefunctions. These
models have practical applications in Bose-Einstein condensates, and they can
describe the dynamics of multi-species bosonic systems. The conservation law
relates to a pair-production mechanism that explains the dissociation of
diatomic molecules into atoms. We provide a general framework for both solvable
and semiclassically solvable non-Hermitian Landau-Zener models. Our findings
will open new avenues for a number of diverse emergent phenomena in explicitly
time-dependent non-Hermitian quantum systems.

Quantum Physics;Atomic Physics