Reply to “Comment on ‘Nonlinear band structure in Bose-Einstein condensates: Nonlinear Schrödinger equation with a Kronig-Penney potential’ ”

In response to Danshita and Tsuchiya’s comment on our work on nonlinear band theory, we show that the size of the region of the swallowtail with a negative effective mass is inversely proportional to the interaction strength, i.e., for large interaction strengths, the region becomes negligibly small. We explain why the appearance of swallowtails is not related to superfluidity, but instead to a more universal nonlinear feature valid for both signs of the underlying atomic interactions: period doubling.

Figure 1

Width of the region of the lower portion of the swallowtail that has a negative effective mass decreases rapidly (solid curve) with increasing interaction strength, while the width of the positive mass region increases (dashed curve). The strength of the potential is given by $V_0=E_0$. Note that the lines are guides to the eye.

Figure 2

(Color online) Band structure of a two-color lattice with strong nonlinearity, $gn=10V_0$, as presented in Ref. 5. The quasimomentum is given in units of $q_0\equiv \pi ∕d$. (a) Shown are period-doubled solutions for a one-color lattice of period $d$ (thin black curve shows trivial solutions; thick blue curve shows nontrivial solutions). (b),(c) Bloch-wave solutions with period $2d$ (solid curve shows lowest band; dashed curve shows second band). (b) A two-color lattice in which the energy of every other lattice site has been lowered slightly. Note that the central swallowtail is derived from the nontrivial period-doubled solution of (a). (c) A one-color lattice with double the period of (a), $2d$.