  We study the quantum motion of an impurity atom immersed in a Bose Einstein
condensate in arbitrary dimension. The Bogoliubov excitations of the Bose
Einstein condensate act as a bosonic bath for the impurity. We present a
detailed derivation of the $d$-dimensional Langevin equations that describe the
quantum dynamics of the system, and of the associated generalized tensor that
describes the spectral density in the full generality. When the impurity is not
trapped, we calculate the mean square displacement, showing that the motion is
super diffusive. We obtain also explicit expressions for the super diffusive
coefficient in the small and large temperature limits. We find that, in the
latter case, the maximal value of this coefficient is the same in all
dimensions. We study also the behaviour of the average energy and compare the
results for various dimensions. In the trapped case, we study squeezing and
find that the stronger position squeezing can be obtained in lower dimensions.
We quantify the non-Markovianity of the particle's motion, and find that it
increases with dimensionality.
