A significantly stable mode of the ultracold atomic wave packet in amplitude modulated parabolic optical lattices

We show that a conspicuous wave packet of ultracold noninteracting Bosonic atoms emerges in a 1-dimensional parabolic optical lattice as in the setup of the Aarhus experiment [P. L. Pedersen ${\it et}$ ${\it al.}$, Phys. Rev. A ${\bf 88}$, 023620 (2013)], given the lattice height is harmonically modulated with a particular amplitude at a resonant frequency. We show that this wave packet, coined "${\it 4bandPWP}$" here, executes stable time-wise periodic motion for infinitely long time. We apply the Floquet theory to analyze the parameter dependence of ${\it 4bandPWP}$ in detail. Our analysis shows that it consists mainly of two principal Floquet eigenstates of the periodically driven Hamiltonian. The informative Husimi representation yields temporal slices of the phase space of ${\it 4bandPWP}$, visually identifying moments where the inter-band transitions take place. The provided data should aid the experiment in locating ${\it 4bandPWP}$.