{aligned}$$ Note that this last expression can also be used, reversing the sign, to give the force exserted on the BEC by a laser of beam profile given by ${\cal U}_p$.

At zero temperature, i.e. $\gamma=0$, and neglecting the effect of quantum fluctuations [@roberts2006force; @sykes2009drag], the impurity moves without any drag through a uniform condensate below a critical velocity, which is the low-wavelength speed of sound $c=\sqrt{\mu/m}$, as determined by the condensate linear excitation spectrum, in agreement with Landau’s criterion of superfluidity [@astrakharchik2004motion]. Above the critical speed, the impurity will create excitations, and depending on the size of the impurity these excitations range from acoustic waves (Bogoliubov excitation spectrum) to vortex dipoles and to von-Karman street of vortex pairs [@reeves2015identifying]. Previous studies focused on the theoretical investigations of the self-induced drag force and energy dissipation rate in the presence of Bogoliubov excitations emitted by a pointwise  [@astrakharchik2004motion; @roberts2006force; @sykes2009drag] or finite-size [@pinsker2017gaussian] particle, or numerical investigations of the drag force due to vortex emissions [@winiecki2000motion; @griffin2017vortex; @shukla2018particles]. The energy dissipation rate depends on whether the impurity is heavier, neutral or lighter with respect to the mass of the condensate particles [@shukla2018particles]. The dependence on the velocity of the self-induced drag force above the critical velocity changes with the spatial dimensions [@astrakharchik2004motion]. This means that the energy dissipation rate is also dependent on the spatial dimensions. If instead of a single impurity one considers many of them there will be, besides direct inter-particle interactions, additional forces between the impurities mediated by the flow, leading to a much more complex many-body dynamics even in an otherwise uniform condensate, as discussed in [@shukla2016sticking]. Here we neglect all these eff