  Geometric phases, arising from cyclic evolutions in a curved parameter space,
appear in a wealth of physical settings. Recently, and largely motivated by the
need of an experimentally realistic definition for quantum computing
applications, the quantum geometric phase was generalized to open systems. The
definition takes a kinematical approach, with an initial state that is evolved
cyclically but coupled to an environment --- leading to a correction of the
geometric phase with respect to the uncoupled case. We obtain this correction
by measuring the nonunitary evolution of the reduced density matrix of a spin
one-half coupled to an environment. In particular, we consider a bath that can
be tuned near a quantum phase transition, and demonstrate how the criticality
information imprinted in the decoherence factor translates into the geometric
phase. The experiments are done with a NMR quantum simulator, in which the
critical environment is modeled using a one-qubit system.
