“…Still, a sample initially prepared in a thermal state and thereby without quantum correlation can exhibit a gain of the PMD maximum or PSD up to the number M of internal states (or ultimately M 2 if initial correlations exist in the initial state, see SM [19]). The direct and fundamental consequence of this analysis, holding for any kind of free particles or particles in time-dependent trapping potential is that cooling mechanisms based on coherent field momentum transfer without spontaneous emission (such as adiabatic passages, bichromatic, π-pulses [5,6,8,10,11,28]) have a limited efficiency and could only lead to a position-momentum PSD gain of M . This is still of interest for studies that need more particles in a same phase space area regardless of the internal distribution (for laser manipulation, detection, collisional studies, ...).…”