  Monopoles are intriguing topological objects, which play a central role in
gauge theories and topological states of matter. While conventional monopoles
are found in odd-dimensional flat spaces, such as the Dirac monopole in three
dimensions and the non-Abelian Yang monopole in five dimensions, more exotic
objects were predicted to exist in even dimensions. This is the case of "tensor
monopoles", which are associated with generalized (tensor) gauge fields, and
which can be defined in four dimensional flat spaces. In this work, we
investigate the possibility of creating and measuring such a tensor monopole,
by introducing a realistic three-band model defined over a four-dimensional
parameter space. Our probing method is based on the observation that the
topological charge of this tensor monopole, which we relate to a generalized
Berry curvature, can be directly extracted from the quantum metric. We propose
a realistic three-level atomic system, where tensor monopoles could be
generated and revealed through quantum-metric measurements.
