within the model. We note that a similar multi-peak structure for incommensurate systems is also obtained when a super-lattice is imposed on the BHM, as this also breaks the spatial symmetry and introduces oscillations in the correlation function [@Rigol2006].

![Auto-correlation functions as a function of $r$ for different filling factors and for a lattice depth of (a) $V_0 =5 E_r$ (except the red line) and (b) $V_0=20 E_r$. The dashed lines correspond to a $r^{-\frac{1}{2}}$ decay and are displayed for $V_0 =0$ and $F=\frac{2}{5},\frac{7}{5}$.[]{data-label="fig:correlationfunctions"}](correlationfunctions.pdf){width="\linewidth"}

Probing phases with a driven lattice {#sec:driven}
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Sudden perturbations are a powerful method for studying many-body effects in quantum systems, and for TG gases notable examples that have been investigated in recent years include the orthogonality catastrophe and decay [@Campo2011; @Goold2011; @Cetina2015; @Parish2016; @Demler2016; @Demler2018], persistent currents and probing superfluidity [@Citro2009; @Hallwood2010; @Schenke2011; @Schenke2012; @Schloss2016; @Minguzzi2017; @Minguzzi2017b; @Minguzzi2018]. Such exploration of the non-equilibrium dynamics can be used to further characterise the different phases we have examined in the static model, and quenches can be used to detect phase transitions due to the creation of non-trivial dynamics induced by large quantum fluctuations at criticality [@Silva2008; @Karrasch2013; @Campbell2016; @Heyl2017; @Fogarty2017].

Here we quench by assuming that at 