I recently read about BEC loaded into the optical lattice p.200

Looking at a condensate released from a lattice after a time of flight typically on the order of a few milliseconds amounts to observing its momentum distribution. A harmonically trapped condensate has a Gaussian momentum distribution in the limit of small interactions, whereas in the Thomas-Fermi limit in which the interactions dominate over the kinetic energy contribution it has a parabolic density profile and expands self-similarly after being released. By contrast, a condensate in a periodic potential contains higher momentum contributions in multiples of 2 kL, their relative weights depending on the depth of the lattice. In fact, in the tight-binding limit see Sec. IV we can consider the condensate to be split up into an array of local wave functions that expand independently after the lattice has been switched off. Eventually they all overlap and form an interference pattern that in the absence of interactions is the Fourier transform of the initial condensate.

When there is no lattice potential all particles occupy the same state, but when the lattice appears is it still a single mode condensate or multi-mode one due to appearance of additional interference peaks (particle condense in more than one state)?

These were obtained after suddenly releasing the atoms from an optical lattice potential with different potential depths V0 after a time of flight of 15 ms. Values of V0 were: a, 0 Er; b, 3 Er; c, 7 Er; d, 10 Er; e, 13 Er; f, 14 Er; g, 16 Er; and h, 20 Er. [source](http://www.nature.com/nature/journal/v415/n6867/fig_tab/415039a_F2.html)


There is a single condensate only. The additional peaks appear due to the small occupation of higher-energy states, as the increasing well depth encourages localization of particles.

See also a similar question here.

  • $\begingroup$ This is exactly what I am not sure about. You start with the condensate - according to definition all particles occupy the same state - and then as you increase lattice depth you are seeing additional peaks which following your interpretation corresponds to small occupation of higher-energy states. Do all particles at this stage occupy the same state (even with admixture of higher-energy states) or some of them jump to the other energy level? If there is a single condensate why do I see the Mott insulator which is definitely not a condensate according to definition? $\endgroup$ – WoofDoggy Nov 8 '16 at 22:28
  • $\begingroup$ It seems that I have been wrong. It turns out that the change of the Wannier functions is important for the emergence of the extra peaks. I.e., as the lattice depth is increased, the Wannier functions get squeezed. I will update my answer once I feel I have a complete understanding of the problem. $\endgroup$ – ffc Nov 12 '16 at 13:43

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