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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 $\psi$?

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  • $\begingroup$ those wavefunctions are representing the same state $\endgroup$
    – Phoenix87
    Jan 18, 2015 at 12:10
  • $\begingroup$ Looks like you could extract the quasi-energies by a form of Ramsey interferometry. That is, you measure the evolution of a coherence between two Floquet eigenstates over a range of waiting times. You would probably have to use only waiting times that were a multiple of the period $T$ in order to eliminate the contributions from the time dependence of the $u_n(t)$. Therefore the coherence time must be much larger than $T$. But if $T$ is an optical period this is hardly a limitation. $\endgroup$ Jan 18, 2015 at 12:14
  • $\begingroup$ Thanks for your suggestion, and I will learn the Ramsey interferometry technique. $\endgroup$
    – pchenweis
    Feb 10, 2015 at 7:29

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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.

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  • $\begingroup$ I have noted the paper you provide. However, here my question is how to detect the Floquet wave function rather than the quasienergy. $\endgroup$
    – pchenweis
    Feb 10, 2015 at 7:31
  • $\begingroup$ How does one ever measure a wavefunction? I suppose one could measure density, but to measure a wavefunction outright is impossible! $\endgroup$
    – Xcheckr
    Feb 10, 2015 at 13:18
  • $\begingroup$ You are correct. To detect the wave function, I actually mean to detect the probability and the geometric phase. $\endgroup$
    – pchenweis
    Feb 14, 2015 at 13:07

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