Summary

The Hamiltonian is in general:

  $$\hat{\mathcal{H}} = \sum_{n=1}^{N} \hat{\mathcal{H}}(n) -... ..._{n}) \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}$$ (5)

The first term $\hat{\mathcal{H}}(n)$ denotes the Hamiltonian of a subsystem $n$ (e. g. an ion, or cluster of ions). The second term describes a bilinear interaction between different subsystems through the operators $\hat{\mathcal{I}}_{\alpha}^n$, with $\alpha = 1, 2, \dots, m$. The operators $\hat{\mathcal{H}}(n)$ and $\hat{\mathcal{I}}_{\alpha}^n$ act in the subspace $n$ of the Hilbert space, i. e. $[\hat{\mathcal{I}}_{\alpha}^n, \hat{\mathcal{I}}_{\alpha}^{n^{\prime}}] = 0$, $[\hat{\mathcal{H}}(n), \hat{\mathcal{I}}_{\alpha}^{n^{\prime}}] = 0$ and $[\hat{\mathcal{H}}(n), \hat{\mathcal{H}}(n^{\prime})] = 0$ for $n \ne n^{\prime}$.

Next we give specific examples which fill the above expression with life.