the same behavior as for the static coherence shown in Fig. \[fig:staticphasediagram\](a), although for $\langle \lambda_0 \rangle$ after the quench an odd-even effect is present near commensurability in shallow lattices (see the inset in Fig. \[fig:dynamicphasediagram\](b)). The same odd-even effect is present for $\langle K \rangle/N$, as discussed in section \[subsubsec:rotationspeed\], and for which a detailed explanation of the origin can be found in Appendix A. The minimum in the coherence obtained for $F=1$ indicates the presence of the pinned phase and the maximum in $\langle K \rangle/N$ at $F=1$ (and $F=2$) can therefore be attributed to this state, which restricts tunneling between the sites resulting in the gas being moved along with the lattice. For incommensurate values of $F$, the relatively large degree of coherence indicates the presence of a superfluid phase (as does the small value of $\langle K \rangle/N$), where particles can freely tunnel through the lattice as it rotates through the gas.

![(a) Time-averaged average momentum and (b) time-averaged coherence as a function of the lattice depth $V_0$ and the filling ratio $F$. The insets shows the same near $F=1$ for a lattice depth of $V_0=0.1 E_r$. These plots are based on calculations for $M=50$ and 