A similar effect has been reported in optical lattices near the edge of the Brillouin zone
[22,23,35]. These self-trapping effects in lattices have been attributed to several different phenomena. One, based on a Josephson effect with suppressed tunneling between neighboring sites
[23,32], was predicted from a variational framework
[36]. Another explanation is that the sharp boundary is a “gap soliton”
[37], though this explanation is disputed by Ref.
[32] on the basis that solitons should remain stable whereas the latter observed self-trapping only for a finite period of time. Finally, self-trapping has been explained in terms of the Peierls-Nabarro energy barrier
[38]. In all of these cases, the self-trapping occurs where the effective mass becomes negative, but the interpretation of the self-trapping effect in optical lattices is complicated by the presence of spatial modulations in the potential.
The beauty of engineering dispersions with SOC BECs is that lattice complications are removed. The success of the single-band model in reproducing the experiment demonstrates clearly that the self-trapping results from the effective dispersion relationship. A single-band model using the dispersion of the lowest band in an optical lattice is able to explain the previous observation of self-trapping in lattice systems, clearly demonstrating the importance of the band structure and deemphasizing the role of the underlying lattice geometry of coupled wells. This result is confirmed by using a tight-binding approximation to map the optical lattice of Ref.
[32] to a single band model, which reproduces their results. In our simulations, although the boundary appears to be very stable, it is “leaky.” This can be seen from Fig.
3 where the boundary maintains its shape, but permits a small number of fast moving particles to escape. Similarly, in optical lattice systems such fast moving particles are responsible for the continued increase in the width of the cloud seen by Ref.
[32] even though the boundary remains stopped. This may resolve the apparent discrepancy between Refs.
[32] and
[37] as a quasistable but leaky gap soliton.
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