  We report measurements of dynamical tunneling rates of a Bose-Einstein
condensate across a barrier in classical phase space. The atoms are initially
prepared in quantum states that extend over a classically regular island
region. We focus on the specific system of quantum accelerator modes of the
kicked rotor in the presence of gravity. Our experimental data is supported by
numerical simulations taking into account imperfections mainly from spontaneous
emission. Furthermore, we predict experimentally accessible parameter ranges
over which direct tunneling could be readily observed if spontaneous emission
was further suppressed. Altogether, we provide a proof-of-principle for the
experimental accessibility of dynamical tunneling rates in periodically driven
systems.
