  We show how impurity atoms can measure moments of ultracold atomic gas
densities, using the example of bosons in a one-dimensional lattice. This
builds on a body of work regarding the probing of systems by measuring the
dephasing of an immersed qubit. We show this dephasing is captured by a
function resembling characteristic functions of probability theory, of which
the derivatives at short times reveal moments of the system operator to which
the qubit couples. For a qubit formed by an impurity atom, in a system of
ultracold atoms, this operator can be the density of the system at the location
of the impurity, and thus, means, variances, and correlations of the atomic
densities are accessible.
