commensurate ($F=1$, full magenta line) case. For comparison the black line corresponds to the case where no lattice is present.[]{data-label="fig:momentumdistributionasfunctionofdepth"}](momentumdistributionasfunctionofdpeth.pdf){width="\linewidth"}

It is instructive to consider two distinct defect regimes of the incommensurate phase. If the deviation from the commensurate phase at $F=1$ is microscopic, such that $N=M\pm1$, the phase becomes gapless with a quadratic dispersion relation. On the other hand, for a macroscopic number of defects, such as $N=M\pm M/2$, the phase is also gapless, but with a linear dispersion relation [@Astrakharchik2017]. These regimes are, however, continuously connected which can be seen in Fig.\[fig:momentumdistributionasfunctionofratio\](a), where the momentum distribution as a function of $F$ is plotted for a constant depth $V_0 = 20 E_r$. Far from commensurate values, i.e. for a macroscopic number of defects, the multiple peaks stemming from the delocalised superfluid are the dominant contribution to the momentum distribution (see Fig. \[fig:momentumdistributionasfunctionofratio\](b)), but as $F\rightarrow 1$ a Gaussian contribution corresponding to the pinned phase momentum distribution becomes more dominant (see Fig. \[fig:momentumdistributionasfunctionofratio\](c)). In fact, for a microscopic numbers of defects, such as for $N=M\pm 1$, the momentum distribution essentially corresponds to that of the delocalised pinned phase, but with additional small peaks on top of it. Beyond unit filling 