Formula_24957

\[\begin{gathered}\boldsymbol{H}=\frac{1}{2} \sum_{i=1}^{N} \hbar \varepsilon_{i} \sigma_{i}^{z}+\sum_{k} \hbar \omega_{k} a_{k}^{+} a_{k}- \\-\sum_{i=1}^{N} \sum_{k}\left(g_{i k} \sigma_{i}^{+}+g_{i k}^{*} \sigma_{i}^{-}\right)\left(a_{k} e^{i \overrightarrow{\boldsymbol{k}} \vec{r}_{i}}+a_{k}^{+} e^{-i \overrightarrow{\boldsymbol{k}} \vec{r}_{i}}\right)\end{gathered}\]