\hat{a}^+_i\hat{a}_i=\hat{a}^+_i\langle \hat{a}_i \rangle+\langle\hat{a}^+_i\rangle \hat{a}_i-\langle\hat{a}^+_i\hat{a}_i\rangle+(\hat{a}^+_i-\langle \hat{a}^+_i\rangle)(\hat{a}_i-\langle \hat{a}_i\rangle )