Period doubling, two-color lattices, and the growth of swallowtails in Bose-Einstein condensates

Abstract

The band structure of a Bose-Einstein condensate is studied for lattice traps of sinusoidal, Jacobi elliptic, and Kronig-Penney form, all in the context of the nonlinear Schrödinger equation. It is demonstrated that the physical properties of the system are independent of the type of lattice. The Kronig-Penney potential, which admits a full exact solution in closed analytical form, is then used to understand the swallowtails, or loops, that form in the band structure. The appearance of swallowtails is explained by adiabatically tuning a second lattice with half the period. Such a two-color lattice, which can be easily realized in experiments, has intriguing physical properties. For instance, swallowtails appear even for weak nonlinearity, which is the experimental regime. We determine the stability properties of this system and relate them to current experiments.

Loop-structure stability of a double-well-lattice Bose-Einstein condensate

“…Similar looped band structures also appear for BEC on a double-well optical lattice [22,25]. Unlike the conventional looped structure described above, however, we find that a significantly large loop is induced for any interaction strength, and a large energy separation from the excited band is possible by suitably tuning the lattice depth and the lattice staggering.…”

“…3. This loop structure could also be understood from the period-doubled solution [22]. It is known that this tight-binding model without staggering ( = 0) admits period-p solutions, where p is any positive integer.…”

“…2(a)], in which case our unit cell is twice the lattice's natural unit cell. It is known that an extra solution exists at the band edge [22] and a staggering splits the upper and lower bands and hence a loop is induced [ Fig. 2(b)].…”

“…In Ref. [22], a one-dimensional (1D) Kronig-Penney potential was used to demonstrate period doubling in a double-well lattice in one dimension. The special form of the potential allowed analytic solutions to be obtained.…”

Multi-barrier resonant tunnelling for the one-dimensional nonlinear Schrödinger Equation

“…These results strongly motivate a study of related effects in the context of resonant transmission through structures consisting of more than one well (or more than two barriers, respectively) where one expects the occurrence of both bistability and symmetry breaking. The limiting case of resonant transport in infinitely extended periodic structures has also been of recent interest (see, e. g. [29][30][31]). The occurrence of looped Bloch bands is one of the major effects of nonlinearity in these systems.…”