= Solution
Differentiate $U_I=U_0^\dagger U$. The free propagator satisfies $\dot U_0^\dagger=(i/\hbar)U_0^\dagger H_0$, so the <Schrödinger equation> gives
$$
\begin{aligned}
i\hbar\dot U_I
&=-U_0^\dagger H_0U+U_0^\dagger(H_0+fH_1)U\\
&=fU_0^\dagger H_1U_0U_I.
\end{aligned}
$$
Thus the <interaction picture> removes the drift and gives $\boxed{i\hbar\dot U_I=f(t)H_I(t)U_I,\quad H_I=U_0^\dagger H_1U_0}$, with $U_I(0)=I$. The transformed control <Quantum Hamiltonian> is time-dependent even though the original control matrix is constant.
Back to article page
