  Ultracold bosonic atoms trapped in a two-leg ladder pierced by a magnetic
field provide a minimal and quasi-one-dimensional instance to study the
interplay between orbital magnetism and interactions. Using time-dependent
matrix-product-states simulations, we investigate the properties of the
so-called "Meissner" and "vortex" phases which appear in such system, focusing
on experimentally accessible observables. We discuss how to experimentally
monitor the phase transition, and show that the response to a modulation of the
density imbalance between the two legs of the ladder is qualitatively different
in the two phases. We argue that this technique can be used as a tool for
many-body spectroscopy, allowing to quantitatively measure the spin gap in the
Meissner phase. We finally discuss its experimental implementation
