# Are there exact expressions for the Floquet states of a periodically-forced, undamped harmonic oscillator?

For this question I was looking for the Floquet states of a quantum harmonic oscillator driven by a non-resonant harmonic force, and I had a rather harder time finding it than the simplicity of the question would suggest. I'm sure the result exists, but it's somewhat buried under a mound of papers dealing with damped oscillators (which are of course interesting, including many real, useful applications in open quantum systems, but the undamped oscillator is also important) and many papers exploring formalisms which are not necessarily that crucial to the result. I would therefore like to ask it here so hopefully there's an easier waymarker for future askers of this question.

It seems, moreover, that these states are not as easy to get at. For instance, the paper

Squeezing in Floquet States and Quasi-energies of Harmonic Oscillator Driven by a Strong Periodic Field. M. Janati-Idrissi et al. African Journal Of Mathematical Physics 10, 21-30 (2011).

is quite recent, and it only goes to second order in the driving. This suggests that maybe these states are not actually known in exact form. In addition, it seems (Commun. Math. Phys. 215, 245 (2000)) that the Floquet quasienergy spectrum is rather different than from the static case (purely absolutely continuous instead of discrete) so there's more at play, with some disagreements in recent literature.

So: Are there known, exact expressions for the Floquet states of a periodically-forced, undamped harmonic oscillator? I'm looking for explicit expressions for a complete, orthogonal set of Floquet states, as well as a characterization of the quasienergy spectrum, and an explanation of any significant differences it has with respect to the unperturbed oscillator and the perturbative solutions.

• Stupid little question: Taking limits of damping going to zero on results for a damped oscillator does not yield anything? Or do you not know damped results either? – Danu Jan 4 '16 at 16:24
• First result from google search: its.caltech.edu/~hmabuchi/Ph125c/fho.pdf Analytical solution for any force coupled to $x$. – Meng Cheng Jan 4 '16 at 18:39
• @MengCheng Can you expand on how that paper is relevant specifically for the Floquet states that I'm actually asking about? – Emilio Pisanty Jan 4 '16 at 18:53
• The problem is exactly solved, closed form expression for evolution operator found. What else do you need to construct whatever Floquet states you are interested in? – Meng Cheng Jan 4 '16 at 19:03
• I can't comment on your post yet, so I will write here. It appears that the quasienergies and Floquet modes are given in this source here: physik.uni-augsburg.de/theo1/hanggi/Chapter_5.pdf I hope that is what you are looking for. – Kevin Tham Oct 26 '16 at 14:02