  The strongest interaction between microscopic spins in magnetic materials is
the exchange interaction $J_\text{ex}$. Therefore, ultrafast control of
$J_\text{ex}$ holds the promise to control spins on ultimately fast timescales.
We demonstrate that time-periodic modulation of the electronic structure by
electric fields can be used to reversibly control $J_\text{ex}$ on ultrafast
timescales in extended antiferromagnetic Mott insulators. In the regime of weak
driving strength, we find that $J_\text{ex}$ can be enhanced and reduced for
frequencies below and above the Mott gap, respectively. Moreover, for strong
driving strength, even the sign of $J_\text{ex}$ can be reversed and we show
that this causes time reversal of the associated quantum spin dynamics. These
results suggest wide applications, not only to control magnetism in condensed
matter systems, for example, via the excitation of spin resonances, but also to
assess fundamental questions concerning the reversibility of the quantum
many-body dynamics in cold atom systems.
