The Hamiltonian is periodic in time with some period \(T\), viz.,
\begin{equation} H(t + T) = H(t) \end{equation}
Such systems can be thought of as driven, where energy is periodically pumped in and out of the system. That is, energy is conserved modulo single photon exchanges, with \(\omega = 2\pi/T\),
\begin{equation} h/T = hf = \hbar\omega \end{equation}
The key idea underlying Floquet theory is to separate the long (stroboscopic) and short time scales, for \(n\) driving periods we have,
\begin{equation} U(t + nT) = U(t)[U(T)]^n \end{equation}
Here, the Floquet operator is defined as,
\begin{equation} U(T) := \mathcal{T} \exp{-\iota\int_{t=0}^T H(t)dt} \end{equation}
where \(\mathcal{T}\) denotes time ordering in the exponential. We define the Floquet Hamiltonian \(H_F\) as,
\begin{equation} U(T) =: e^{-\iota H_F T} \end{equation}
Thus we have a separation into short scale (through \(V(t)\)) and long scale (through \(H_F\)) dynamics as,
\begin{equation} U(t) \sim V(t) \cdot \exp{(-\iota H_F t)}
\end{equation}
where \(V(t) = V(t+T)\). The Floquet operator is unitary and admits eigenvalues on the unit circle \(\Lambda(U(T)) \subset \mathcal{U}(1)\),
\begin{equation} U(T)|\phi_{\alpha}\rangle = \lambda_{\alpha}|\phi_{\alpha}\rangle \end{equation}
with \(\lambda_{\alpha}\in\mathcal{U}(1)\). If the spectrum is gapped, we identify a cut \(\varepsilon\) such that \(e^{-\iota \varepsilon} \not\in \Lambda(U(T))\). Then we have,
\begin{equation} H_F := \frac{\iota}{T}\log_{-\varepsilon}U(T) \end{equation}
also has the same eigenstates,
\begin{equation} H_F|\phi_{\alpha}\rangle = \epsilon_{\alpha} |\phi_{\alpha}\rangle \end{equation}
but the quasienergies \(\epsilon_{\alpha}\) are only defined in
\begin{equation} \epsilon_{\alpha} \in [-\pi/T,\pi/T] \end{equation}
Time periodic Floquet modes are defined as the action of the microscopic operator on the eigenstates,
\begin{equation} |u_{\alpha}(t)\rangle := V(t)|\phi_{\alpha}\rangle = |u_{\alpha}(t+T)\rangle \end{equation}
we note that
\begin{equation} U(t)|\phi_{\alpha}\rangle = V(t)e^{-\iota H_Ft}|\phi_{\alpha}\rangle = e^{-\iota \epsilon_{\alpha}t}V(t)|\phi_{\alpha}\rangle = e^{-\iota \epsilon_{\alpha}t}|u_{\alpha}(t)\rangle \end{equation}
Thus, \begin{equation} U(t,t’) = \sum_{\alpha} e^{−\iota (t-t’)\epsilon_{\alpha}} |u_{\alpha}(t)\rangle \langle u_{\alpha}(t’)| \end{equation}
The \(u_{\alpha}(t)\) describe the micro motion, and are the same as the \(\phi_{\alpha}\) at every stroboscopic time point.