What is a floquet (optical) sideband?

[taishizhiqiu]

In literature, I read:"(exposed in a beam of light) in floquet theory, the quasi-static eigenvalue spectrum at finite driving field A shows copies of the original bands shifted by integer multiples of $\Omega$, the so-called Floquet sidebands"

I have read something about floquet theory and found out that quasienergy is uncertain by integer multiples of $\hbar \Omega$. However, I cannot understand why quasienergy band should be the shape of original band.
Can anyone help me?

 

[eljose79]

Hello,

Thank you for your question. In floquet theory, the quasienergy spectrum is determined by the time-periodic Hamiltonian of a system. This Hamiltonian can be written as H(t) = H0 + V(t), where H0 is the unperturbed Hamiltonian and V(t) is the time-periodic perturbation. When the system is exposed to a driving field A, the perturbation V(t) takes the form of eAcos(ωt), where ω is the frequency of the driving field and A is the amplitude.

Now, in order to understand why the quasienergy band has the shape of the original band, we need to consider the time evolution operator U(t). This operator describes how the state of the system evolves over time and can be written as U(t) = e^(-iH(t)t). Using this operator, we can calculate the quasienergy spectrum by diagonalizing U(t). This leads to the following expression for the quasienergy spectrum:

E(q) = E0 + nℏω, where E0 is the unperturbed energy of the system and n is an integer.

From this expression, we can see that the quasienergy spectrum is shifted by integer multiples of ℏω, which is the energy of a photon with frequency ω. This is why the quasienergy band has the shape of the original band, as it is simply shifted by these integer multiples.

I hope this explanation helps to clarify the relationship between the quasienergy spectrum and the original band in floquet theory. If you have any further questions, please let me know. Thank you.