BEC Relaxation to Dispersion Minimum

I am reading this paper, which looks at the emergence of a double well dispersion in an optical lattice through shaking as in the figure below.

I am curious as to what mechanism results in the atoms actually relaxing into the double minima. I would understand it more if that were a potential, since the atoms would want to sit where the energy is minimized. Starting from a BEC (wavepacket of $$k$$ centered ~ 0), I though that in order to move the quasimomentum $$k$$, there needed to be a force such that $$dk/dt = F$$. There is also curvature (an effective mass), but that once again requires an external

So my ultimate question is what causes the relaxation to the minima and would it be possible to model the motion of k through either a semiclassical equation of motion or a GP approach?

• Consider to spell out acronyms. – Qmechanic Feb 9 at 20:08