The Schrodinger equation is
$i\hbar\partial_t\psi(t)=H(t)\psi(t)$.$$i\hbar\partial_t\psi(t)=H(t)\psi(t).$$
Now, given that the situation that the Hamiltonian is periodically driven, i.e., $H(t+T)=H(t)$, then the equation can be solved by the Floquet ansatz,
$\psi(t)=e^{-i\varepsilon_n t/\hbar}u_n(t)$,$$\psi(t)=e^{-i\varepsilon_n t/\hbar}u_n(t),$$
where $\varepsilon_n$ is called the quasienergy, and the Floquet state $u_n(t)$ satisfies the periodic condition $u_n(t+T)=u_n(t)$. This can be written in a compact form by defining the Floquet Hamiltonian as
$H_F(t)=H(t)-i\partial_t\\ H_F(t)u_n(t)=\varepsilon_n u_n(t)$.\begin{align} H_F(t) & =H(t)-i\partial_t\\ H_F(t)u_n(t) & =\varepsilon_n u_n(t). \end{align}
Then if we treat time $t$ as a new degree of freedom, the $H_F$ is then defined in the composite Hilbert space $\mathcal{R}+\mathcal{T}$$\mathcal{R}\otimes\mathcal{T}$. Formally, $H_F$ is analogous to the Hamiltonian for stationary states of the "time"-independent form and all mathematical conclusion of the stationary theory can be obtained.
My question is, although we can formally define the Floquet state $u_n(t)$ and the quasienergy $\varepsilon_n$, how can we detect them? I think that only the real wave function $\psi$ describes the real physics, so how can we draw the information of the Floquet state $u_n(t)$ from $\psi$?
