How to observe Floquet state?

Question

The Schrodinger equation is

$$i\hbar\partial_t\psi(t)=H(t)\psi(t).$$

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,

$$\psi(t)=e^{-i\varepsilon_n t/\hbar}u_n(t),$$

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

\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 $\mathcal{R}\otimes\mathcal{T}$. 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 $\psi$?

Answer 1

They have been observed in the following experiment:

Observation of Floquet-Bloch States on the Surface of a Topological Insulator. YH Wang, H Steinberg, P Jarillo-Herrero and N. Gedik. Science 342, 453 (2013), arXiv:1310.7563.

In this paper, there is a periodic laser pulse hitting the sample which gives a time-periodic Hamiltonian. They then used ARPES (angle-resolved photoemission) to see the band structure repeated in energy steps, $\epsilon_n=\hbar \omega_n$. Without the time-dependent laser pulse, however, the "replica bands" are not visible. Of course the experiment is not perfect, which manifests itself in the fact that there are not infinitely many "replica bands" but at least this is strong evidence that under perfect experimental conditions, a perfect Floquet state could exist.