Nondestructive probing of means, variances, and correlations of ultracold-atomic-system densities via qubit impurities

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 that 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.