Rotating-frame solution for a harmonically driven two-level system
= Rotating-frame solution for a harmonically driven two-level system
For
$$
H(t)=
\begin{pmatrix}
0&ce^{i\omega t/\hbar}\\
ce^{-i\omega t/\hbar}&0
\end{pmatrix},
$$
set $a=U(t)b$ with
$$
U(t)=\operatorname{diag}
\left(e^{i\omega t/(2\hbar)},e^{-i\omega t/(2\hbar)}\right).
$$
Then $b$ evolves under the constant Hamiltonian
$$
H_{\rm eff}=
\begin{pmatrix}
\omega/2&c\\
c&-\omega/2
\end{pmatrix},
\qquad
H_{\rm eff}^2=\left(c^2+\frac{\omega^2}{4}\right)I.
$$