Quasiperiodic Dynamics in Bose-Einstein Condensates

Here's another technical post...

This paper is a project dating back to my Georgia Tech days. We first submitted this for publication in June 2004 and ended up needing to submit it to another journal. (The referee for the first journal never actually rejected us, but we ended up deciding that we wouldn't be able to please him and that we should instead just submit the paper somewhere else.)

My coauthors here are all mathematicians who write almost all of their papers in theorem-proof style and this is my only true theorem-proof paper to date. (I have a couple other papers that have some small proofs, but this entire paper is in the theorem-proof style.) They are Shui-Nee Chow, Yingfei Yi, and Martijn van Noort. (Martijn has since left science to pursue other things).

This project arose because one Shui-Nee Chow noticed that the ordinary differential equations for BEC standing waves that I was studying were very similar to the forced one degree-of-freedom Hamiltonian systems he, Martin, and Yingfei were studying theoretically. They needed an example and I could use some theorems, so this became a classic you-put-your-theorems-in-my-BECs (Bose-Einstein condensates) situations. We worked together on this for a semester, submitted the paper initially when Martijn left Georgia Tech for another postdoc, and the rest is history.

Here is the abstract:

We employ KAM theory to rigorously investigate quasiperiodic dynamics in cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and superlattices. Toward this end, we apply a coherent structure ansatz to the Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation describing the spatial dynamics of the condensate. For shallow-well, intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather sets to prove that one obtains mostly quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, where the minimal amplitude depends on the experimentally adjustable BEC parameters. We show that this threshold scales with the square root of the inverse of the two-body scattering length, whereas the rotation number of tori above this threshold is proportional to the amplitude. As
a consequence, one obtains the same dynamical picture for lattices of all depths, as an increase in depth essentially affects only scaling in phase space. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers.


The work I had done before this paper (the papers on period-multiplied solutions I had written with Predrag Cvitanovic') concentrated on situations with small amplitude periodic potentials, and this one instead considered a different near-integrable situation.

One interesting thing to do to get an idea of the range my work can cover on the math--physics spectrum is to compare the phrasing of the abstracts in this paper and the other one about which I blogged below. If you want an even better idea, compare how things are phrased in the two papers. (This actually leaves a big part out of the "physics" spectrum, as I also have papers which concentrate completely on real data and others which have both real data and experiments---though I've never been the one who actually does any of the experiments. That said, I am starting to have serious thoughts of eventually having an applied math lab, as my phononic crystals collaborator at Caltech is an experimentalist but had a theorist as a Ph.D. advisor and although that theorist isn't in a math department, he's basically an applied mathematician. I know others who have done this without any experimental training, so I may well do this at some point. For BECs, it won't be possible, but for tabletop things like the chains of beads it is definitely feasible.)