The problem looks very simple. We have a time-dependent Hamiltonian:
$$H(t) = B(t)H_0$$,
where ##B(t)## is a numerical function, and matrix ##H_0## is time-indpendent.
Let us consider:
$$B(t) = \begin{cases}
1,&\text{for $0\leq t\leq t_0$}\\
A,&\text{for $t>t_0$.}
\end{cases}$$
Also, let us...