$q$ even for incommensurate particle densities. A detailed explanation of the behavior in the different regimes shown in Fig. \[fig:currentasfunctionofspeed\] can be found in Appendix A.

![(a) Single-particle energy spectrum for PBC (odd number of particles) and (b) A-PBC (even number of particles), plotted as a function of the driving momentum $q$ for $M=4$ and $V_0=0.06 E_r$.[]{data-label="fig:oddevenspectrum"}](oddevenspectrum.pdf){width="\linewidth"}

Many-body phases in the driven system
-------------------------------------

As we are interested in using the driven lattice to probe the phases of the system, we restrict our simulations to small values of the rotation speed, namely $q = 2.9 \pi/ L$, which is off resonant so that the rotation does not excite the particles into higher bands. This therefore allows for a clear distinction between the dynamics of the superfluid and insulating phases in the system. In order to understand the nature of the bosonic phases, we investigate the coherence and the momentum distribution in addition to the average momentum.

### Time-averaged Dynamics: {#subsubsec:timeaveraged}

In Fig. \[fig:dynamicphasediagram\] we show the time-averaged average momentum $\langle K \rangle/N$ and the time averaged coherence $\langle \lambda_0 \rangle$ as a function of the filling ratio, $F$, and the lattice depth, $V_0$. For $\langle \lambda_0\rangle$ one can see qualitatively 