# In Floquet theory, what is the physical significance of the Fourier coefficients?

Floquet theory is the study of (quasi)-periodic solutions of the time-dependent Schrödinger equation when the system is subjected to a time-periodic Hamiltonian.$\newcommand{\ket}{\left|#1\right\rangle}$

That is, suppose we have a system under the time-dependent Hamiltonian $\hat H(t)=\hat H(t+T)$. One can show that, because of this symmetry, there must exist solutions of the Schrödinger equation $i \partial_t \ket{\psi(t)} = \hat H(t)\ket{\psi(t)}$ of the form $$\ket{\Psi_\varepsilon(t)}=e^{-i\varepsilon t}\ket{\Phi(t)},$$ where $\ket{\Phi(t)}$ is periodic in time and $\varepsilon$ is known as the Floquet quasienergy. In this formalism, one then looks for periodic solutions of the equation $$\left[i\partial_t - \varepsilon -\hat H(t)\right]\ket{\Phi(t)}=0,$$ which is somewhat easier to solve (as it can be seen as an eigenvalue equation for a differential operator over a bigger domain, instead of a time-dependent Schrödinger equation as such).

Moreover, as a way to analyze this solution, because it is periodic, it is quite common to represent $\ket{\Phi(t)}$ in terms of its Fourier series: $$\ket{\Phi(t)} = \sum_{n} e^{-in\omega t} \ket{\Phi_n}.$$ What is the physical significance of the Fourier coefficients $\ket{\Phi_n }$? Are they solutions of a Schrödinger equation on their own?* Do they have a simple relation to the original Hamiltonian? Do they admit a simple physical picture (e.g. in terms of photon absorption)? What role does e.g. their spatial wavefunction $\left\langle \mathbf r \middle | \Phi_n\right\rangle$ play (or, more generally, the 'direction' of $\ket{\Phi_n}$ in Hilbert space), and what physical significance does it have?

*Upon further reflection, this is almost certainly not the case. Cleaning up §5.5.1 of these notes on Driven Quantum Systems by Peter Hänggi, the Fourier coefficients $\ket{\Phi_n}$ as defined here obey a set of coupled time-independent Schrödinger equations, $$\sum_{k=-\infty}^\infty \hat{H}_{k}\ket{\Phi_{n+k}} = (\varepsilon +n\omega)\ket{\Phi_n}$$ where the Hamiltonians and couplings, $\hat{H}_n = \frac1T \int_0^T \hat H(t) e^{in\omega t}\mathrm d t$, are the Fourier components of the original Hamiltonian. (Moreover, if $\hat H(t)$ is reasonably harmonic, e.g. with only a sinusoidal dependence on $t$, then only a few couplings with very low $k$ will be nonzero.)

Given this, it seems extremely unlikely that the dynamics of the $\ket{\Phi_n}$ can be reformulated into uncoupled TISEs for each of them, and certainly not in the general case. Furthermore, given the presence of the couplings, it's not clear at all how one can extract a physical significance for the $\ket{\Phi_n}$ from the coupled TISEs.

Also: hat tip to Kevin Tham for pointing out these notes in a comment to this question.

• Oh man. I spent months during grad school trying to answer this question for myself. I never did find a satisfactory answer. :( Oct 4, 2016 at 20:15
• I found Section 3 of this paper useful for generating an abstract formalism for Floquet states. Essentially they expand Hilbert space by taking a tensor product with kets $|n\rangle$ which correspond to the $e^{-in \omega t}$'s and re-write operators on the original space in the new space. I could never quite use this to understand the Floquet Fourier coefficients, though. Another useful paper for me was this paper, which is a lot more physics-y. In that paper... Oct 10, 2016 at 18:38
• one can see that at times, going into "Floquet space" is the same as transforming to a reference frame in which the time-dependence disappears. See: interaction picture for RWA or going into the frame of a rotating object like a rigid rotor molecule in the presence of a periodic driving of some sort. Oct 10, 2016 at 18:40
• In my dim memory, one of the useful things about Floquet theory is that it allows you to systematically find steady states where there are no stationary states (the steady states are the stationary states of the Floquet Hamiltonian), and in that case, I think the Floquet Fourier coefficients can be interpreted, but it's been too long for me to say for sure. Finally: I think Floquet theory can be used for photo-absorption in the case of strong fields: in that case, the difference between $\sqrt{n}$ and $\sqrt{n+1}$ can be neglected for photons, ... Oct 10, 2016 at 18:46
• and so $\hbar\omega$, which is the difference between the quasi-energies of the ladder of Floquet stationary states that all correspond to the same physical state can be interpreted as a resonance (and, if I remember correctly, they can be interpreted as different resonances even though the ladder of stationary states all correspond to the same physical state). That still leaves open the question of the Fourier coefficients. That's all I've got, since it's been a long time, and I was never able to sort it out completely anyway. Oct 10, 2016 at 18:48