e MI-phase as $v$ is varied ($F=0$, $L=N=7$).[]{data-label="fig1"}](fig1.eps){width="8cm"}

We proceed with the multi-particle case. A natural extension of the tight-binding model (\[1\]), which accounts for the repulsive interaction of the atoms, is given by the Bose-Hubbard model [@Fish89], $$H=-\frac{J}{2}\left(\sum_{l=1}^L \hat{a}^\dag_{l+1}\hat{a}_l
  +h.c.\right)
  +\frac{W}{2}\sum_{l=1}^L \hat{n}_l(\hat{n_l}-1)$$ $$\label{2}
+2\pi F\sum_{l=1}^L l\hat{n}_l \;.$$ In Eq. (\[2\]), $\hat{a}_l^\dag$ and $\hat{a}_l$ are the bosonic creation and annihilation operators, $\hat{n}_l= \hat{a}_l^\dag\hat{a}_l$ is the occupation number operator of the $l$th lattice site, and the parameter $W$ is proportional to the integral over the Wannier function raised to the fourth power. Since the Bose-Hubbard Hamiltonian conserves the total number of atoms $N$, the wave function of the system can be represented in the form $|\Psi\rangle=\sum_{\bf n} c_{\bf n}|{\bf n}\rangle$, where the vector ${\bf n}$, consisting of $L$ integer numbers $n_l$ ($\sum_l n_l=N$), labels the $N$-particle bosonic wave function constructed from $N$ Wannier functions. (In what follows, if not stated otherwise, $|\Psi\rangle$ refers to the ground state of the system.) As known, in the thermodynamic limit, and for $F=0$, the system (\[2\]) shows a quantum phase transition from a superfluid (SF) to a Mott insulator (MI) phase as the ratio $J/W$ is varied (see [@Sach01] and references therein). It is interesting to note that an indication of this transition can already be observed in a system of few atoms [@Jaks98]. As an example, Fig. \[fig1\] shows the diagonal elements of the one-particle density matrix, $$\label{4}
\rho(k,k')=\langle\Psi|\hat{\Phi}^\dag(k)\hat{\Phi}(k')
 |\Psi\rangle \;,\quad 
\hat{\Phi}(k)=\sum_{l=1}^L \hat{a}_l\phi_l(k) \;,$$ for $N=L=7$, $5\le v\le 35$, and $W=0.1\int dk \phi_l^4(k)$ (here, $\phi_l(k)$ are the Wannier states in the momentum representation, and $k=p/(2\pi\hbar/d)$ is the dimensionless momentum). Physically, this qua