\hat{a}_i\hat{b}^+_j\hat{b}_j\hat{b}_j=\langle \hat{a}_i\hat{b}^+_j \rangle \hat{b}_j\hat{b}_j+\langle \hat{b}_j\hat{b}_j
 \rangle \hat{a}_i\hat{b}^+_j+2\langle \hat{a}_i\hat{b}_j\rangle\hat{b}^+_j\hat{b}_j+2\langle \hat{b}^+_j\hat{b}_j\rangle
 \hat{a}_i\hat{b}_j