  We study the dynamics of entanglement for a one-dimensional spin chain with a
nearest neighbor time dependent Heisenberg coupling J(t) between the spins in
presence of a time dependent external magnetic field h(t) at zero and finite
temperatures. We consider different forms of time dependence for the coupling
and magnetic field; exponential, hyperbolic and periodic. We examined the
system size effect on the entanglement asymptotic value. It was found that for
a small system size the entanglement starts to fluctuate within a short period
of time after applying the time dependent coupling. The period of time
increases as the system size increases and disappears completely as the size
goes to infinity. We also found that when J(t) is periodic the entanglement
shows a periodic behavior with the same period, which disappears upon applying
periodic magnetic field with the same frequency. Solving the particular case
where J(t) and h(t) are proportional exactly, we showed that the asymptotic
value of entanglement depends only on the initial conditions regardless of the
form of J(t) and h(t) applied at t > 0.
