Fractional-period excitations in continuum periodic systems

This paper just appeared in Physical Review A. My coauthors and I originally sent a shorter version of this paper to PRL. In the words of one of them, it kicked off the goalpost. One of the referees indicated to publish it 'as is' and the other asked for us to address some concerns. We attempted to do this; he/she thanked us for the obviously large amount of effort it took to do this. (This referee did his/her job. We doubted a couple of points at first, but he/she was right on every single point. I have to give credit where credit is due.) In our opinion, we satisfied everything, but the referee wasn't convinced and suggested we add more details in appendices to turn it into a PRA article. A new referee also suggested PRA because, in his/her view, no paper without any experiments should ever be published in PRL. WTF? (This person needs to check out the journal's masthead and quite a few of the past articles that have been published in it.)

We could have brought that up with the editors, but it probably would have been a long fight anyway (and the chance of success didn't seem that great because this would have involved at least one more review) because of the other referee, so we decided it wasn't worth the trouble. So, continuing the soccer terminology, after the goal post, instead of taking another shot at the goal, we got nailed against a fence and have a PRA instead of a PRL. (The original plan was to have the PRL and then submit an archival paper to a SIAM journal, but we instead just have a physicsy version of the archival paper in PRA.) I'm happy for the PRA, but there's a bit of an issue of what might have been. (To provide context, by the way, this was moved over from PRL to PRA before either of my two PRLs got accepted, so at the time we were dealing with this, this had been my best shot ever at getting a PRL. I.e., it matters much less now than it seemed to at the time.)

OK, so now that I've resuscitated a dead rant (braaaaaaaains...) that is no longer relevant (if it ever was overly relevant in the first place), let's acknowledge my coauthors and discuss the science a bit.

My main coauthors on this paper were Hector Nistazakis, Panos Kevrekidis, and Dimtri Frantzeskakis, who are all members of the Greek BEC mafia. Alex Nicolin was a theoretical consultant and a familiar person (Jit Kee Chin '01) was an experimental consultant. Some experiments were performed in an afternoon or two in the Ketterle lab, but they didn't work out and the equipment had to be used for experiments in the group's official agenda, so the project ended up only having theoretical and computational components. (We were hoping to try to observe our stuff experimentally, but it didn't work. It can work in theory, and I hope to see that happen someday.)

Here is the abstract:

We investigate the generation of fractional-period states in continuum periodic systems. As an example, we consider a Bose-Einstein condensate confined in an optical-lattice potential. We show that when the potential is turned on nonadiabatically, the system explores a number of transient states whose periodicity is a fraction of that of the lattice. We illustrate the origin of fractional-period states analytically by treating them as resonant states of a parametrically forced Duffing oscillator and discuss their transient nature and potential observability.


You'll notice that Bose-Einstein condensates (BECs) aren't actually mentioned in the title of the paper. (Note: I've provided a brief explanation of BECs in a prior post, so I'll just let you google that if interested.) The reason is that this is a very general phenomenon that can occur in continuum systems (modeled by partial differential equations) which also have some sort of periodicity in the a dependent variable that turns a continuous translational symmetry into a discrete one. BECs, which can be modeled by a nonlinear Schrödinger (NLS) equation known as the Gross-Pitaevskii (GP) equation, provided our focus example.

In the presence of a periodic potential (which is an optical lattice for BECs), a natural thing to do is to look for solutions of the same period (using Bloch theory, for example). For BECs, one can construct such solutions using either the GP equation or with a discrete model (such as the Bose-Hubbard model) in which the lattice lengthscale is imposed on the model via the discretization. One can also construct solutions whose periodicity is a multiple of the lattice period, which I did analytically using Hamiltonian perturbation theory (and KAM considerations) in previous papers. Alex saw these things numerically using a discrete NLS equation in a paper that came out at the same time. I think my eventual-PRE and his eventual-PRA may even have been posted on the arxiv on consecutive days. (We saw each other's papers and started talking to each other.)

About a year after my work was published, the Chu group at Stanford constructed period-doubled states experimentally. (His group was motivated by Alex's paper, as that came out of the Pethick-Smith BEC group, and my collaborator and I were completely unknown in the community, as we come instead from the nonlinear science community.) I was really excited when that paper was posted on the arXiv, especially given that when I spoked about my work at the 2004 March Meeting, there seemed to be some skepticism in the audience as to whether such states could actually be observed. (One of my big messages for that paper is that although Bloch theory gives canonical solutions whose lengthscale matches the one imposed by the lattice that from a dynamical systems perspective, other types of states were also natural even though people trained in atomic physics might not expect it.)

Fractional-period states can be constructed similarly. (I used multiple-scale perturbation theory for the analytics.) If one ignores the harmonic trap, one can construct them. With the harmonic trap, they arise as potentially long-lived transient solutions. (We studied the situation with the trap numerically and that without the trap both analytically and numerically.) A discrete NLS cannot possibly pick them up a priori because the lattice lengthscale is imposed in the modeling. It's not always clear when the GP is a better description and when the Bose-Hubbard model (or another discrete NLS equation) is a better description for the macroscopic dynamics of BECs, so it's really nice to show a solution that one can have and the other can't. Obviously, observing this stuff experimentally will really be nice, but I don't think it will be easy and most of the BEC labs have moved on to things like fermions, so we'll see if this ever comes to pass.