in Fig. \[fig:momentumdistributionasfunctionofdepth\](b). In free space ($V_0=0$) the TG gas is delocalised in position space and therefore localises around a single, central peak in momentum space (black solid line in Fig. \[fig:momentumdistributionasfunctionofdepth\](b)), whereas for $V_0=20$ in the pinned phase ($F=1$) the gas is localised in position space and is therefore delocalised in momentum-space (magenta line). In the incommensurate phase ($F=\frac{2}{5}$) the momentum distribution can be seen to display additional back-scattering peaks at multiples of $2 k_l$, the number of which and their intensity increases with the lattice depth. The gas in this phase therefore represents a momentum delocalised superfluid that exhibits some of the distinct characteristics of both the superfluid and the pinned phase. The observed momentum distribution is similar to the momentum distributions found in supersolids [@Boninsegni2012] and it has recently been suggested that the incommensurate phase of the TG gas in deep lattices is similar to a defect-induced superfluid phase and can be utilized to investigate the Andreev-Lifschitz-Chester mechanism [@Cinti2014; @Astrakharchik2017].

![(a) Normalised momentum distribution for an incommensurate state with $F=\frac{2}{5}$ as a function of lattice depth. (b) Normalised momentum distribution in a deep lattice ($V_0 = 20 E_r$) for the incommensurate ($F=\frac{2}{5}$, red dashed line) and the 