One can show (and this is part of the project) that $$ \left [ \hat{P}_+, \hat{P}_- \right ] = \sum_{k> 0} \left( \hat{a}^\dagger_k \hat{a}_k + \hat{a}^\dagger_{-{k}} \hat{a}_{-{k}} - 1 \right) = \hat{N} - \Omega. $$ Now define $$ \hat{P}_z = \frac{1}{2} ( \hat{N} -\Omega). $$ Finally you can show $$ \left [ \hat{P}_z , \hat{P}_\pm \right ] = \pm \hat{P}_\pm. $$ This means the operators \( \hat{P}_\pm, \hat{P}_z \) form a so-called \( SU(2) \) algebra, and we can use all our insights about angular momentum, even though there is no actual angular momentum involved (this is similar to project 1).

So we rewrite the Hamiltonian to make this explicit: $$ \hat{H} = -G \hat{P}_+ \hat{P}_- = -G \left( \hat{P}^2 - \hat{P}_z^2 + \hat{P}_z\right) $$