  Typically, energy levels change without bifurcating in response to a change
of a control parameter. Bifurcations can lead to loops or swallowtails in the
energy spectrum. The simplest quantum Hamiltonian that supports swallowtails is
a non-linear $2 \times 2$ Hamiltonian with non-zero off-diagonal elements and
diagonal elements that depend on the population difference of the two states.
This work implements such a Hamiltonian experimentally using ultracold atoms in
a moving one-dimensional optical lattice. Self-trapping and non-exponential
tunneling probabilities, a hallmark signature of band structures that support
swallowtails, are observed. The good agreement between theory and experiment
validates the optical lattice system as a powerful platform to study, e.g.,
Josephson junction physics and superfluidity in ring-shaped geometries.
