  We propose experimentally feasible means for non-destructive thermometry of
homogeneous Bose Einstein condensates in different spatial dimensions
($d\in\{1,2,3\}$). Our impurity based protocol suggests that the fundamental
error bound on thermometry at the sub nano Kelvin domain depends highly on the
dimension, in that the higher the dimension the better the precision.
Furthermore, sub-optimal thermometry of the condensates by using measurements
that are experimentally feasible is explored. We specifically focus on
measuring position and momentum of the impurity that belong to the family of
Gaussian measurements. We show that, generally, experimentally feasible
measurements are far from optimal, except in 1D, where position measurements
are indeed optimal. This makes realistic experiments perform very well at few
nano Kelvin temperatures for all dimensions, and at sub nano Kelvin
temperatures in the one dimensional scenario. These results take a significant
step towards experimental realisation of probe-based quantum thermometry of
Bose Einstein condensates, as it deals with them in one, two and three
dimensions and uses feasible measurements applicable in current experimental
setups.
