non-interacting particles with coherence $\lambda_i=1$ for the $N$ lowest lying orbitals.

Even though the coherence can be used to identify different regimes of the system, it is not easy to measure it in a direct way. However, it is closely linked to the momentum distribution, which can be readily observed through time-of-flight experiments and calculated from the Fourier transform of $\rho_1(x,y)$ as $$n(k) = \int dx \, dy \rho_1(x,y) e^{-i k(x-y)}.$$ This can be recast in terms of the Fourier transform of the eigenstates of the RSPDM, $\widetilde{\phi_i}(k)$, using the same occupation numbers $$n(k) = \sum_{i} \lambda_i \widetilde{\phi_i}^*(k) \widetilde{\phi_i}(k).
\label{eq:mom}$$ It is therefore clear that the momentum distribution depends directly on the distinct characteristics of the coherence and long-range order of the system.

Coherence and momentum distribution
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![(a) Coherence $\lambda_0$ and (b) height of the central peak of the momentum distribution $n(k=0)$ as a function of the filling ratio $F$ and the lattice depth $V_0$ (in units of $E_r$). The number of lattice wells is fixed at $M=50$. The insets shows the coherence and the zero momentum peak as a function of $F$ for $V_0=20 E_r$.[]{data-label="fig:staticphasediagram"}](staticphasediagram.pdf){width="\linewidth"}

The lowest eigenvalue of the RSPDM, $\lambda_0$, and the peak-value of the momentum 