  We propose a method by which the quantization of the Hall conductance can be
directly measured in the transport of a one-dimensional atomic gas. Our
approach builds on two main ingredients: (1) a constriction optical potential,
which generates a mesoscopic channel connected to two reservoirs, and (2) a
time-periodic modulation of the channel, specifically designed to generate
motion along an additional synthetic dimension. This fictitious dimension is
spanned by the harmonic-oscillator modes associated with the tightly-confined
channel, and hence, the corresponding "lattice sites" are intimately related to
the energy of the system. We analyze the quantum transport properties of this
hybrid two-dimensional system, highlighting the appealing features offered by
the synthetic dimension. In particular, we demonstrate how the energetic nature
of the synthetic dimension, combined with the quasi-energy spectrum of the
periodically-driven channel, allows for the direct and unambiguous observation
of the quantized Hall effect in a two-reservoir geometry. Our work illustrates
how topological properties of matter can be accessed in a minimal
one-dimensional setup, with direct and practical experimental consequences.,
