I am currently reading the paper Boson localization and the superfluid-insulator transition by Fisher, Weichman, and Fisher. Equation 2.1 defines the Hamiltonian of a Bose Lattice gas as
$$ H = -\sum_{i}(-J_0 + \mu + \delta \mu_i)\hat{N}_i + 0.5 V \sum_i \hat{N}_i (\hat{N}_i - 1) - 0.5 \sum_{i,j} J_{ij}(\hat{\Phi}_i^\dagger \hat{\Phi}_j + \text{H.C} ) $$
where $\hat{\Phi}_j$ is a boson operator at site j.
Equation 2.3 defines the Josephson junction array Hamiltonian as
$$ H = 0.5V \sum_{i}\hat{n}_i^2 - \sum_{i,j} J_{ij} \cos(\hat{\phi}_i - \hat{\phi}_j) $$.
Based on the comments under equation 2.3 I see how to the get the cosine term. The paper mentions to write $\hat{\Phi}_j = |\Phi_j| \exp(i\phi_j)$. However, how do I get the other term and prove equality? I tried reading the reference paper (Quantum Critical Phenomena in Charged Superconductors) but I am still stuck. Please let me know.