Mechanical Properties of Ion Crystals¶
Potential¶
In a system of trapped ions, the ions experience the following potentials:
\[
\Phi = \Phi_{\mathrm{Coulomb}} + \Phi_{\mathrm{Trap}}
\]
- \(\Phi_{\mathrm{Coulomb}} \equiv\) Coulomb potential.
- \(\Phi_{\mathrm{Trap}} \equiv\) Trapping potential.
The trapping potential can be generated with:
- Combination of static (DC) and dynamic (RF) electric fields
- Optical fields (e.g. optical cavity, optical tweezers)
Equilibrium Position¶
Calculating the equilibrium position of the ions corresponds to:
\[
\{\mathbf{r}_{i}^*\}_i^N = \mathrm{argmin}_{\{\mathbf{r}_{i}\}_i^N} \Phi\left(\{\mathbf{r}_{i}\}_i^N\right)
\]
- \(\{\mathbf{r}_{i}\}_i^N \equiv\) Position of the ions.
- \(\{\mathbf{r}_{i}^*\}_i^N \equiv\) Equilibrium position of the ions.
Vibrational Modes¶
Calculating the vibrational (phonon) modes corresponds to:
\[
\begin{aligned}
A &= \mathrm{Hess}[\Phi]\left(\{\mathbf{r}_{i}^*\}_i^N\right) \\
A &= B^* D B
\end{aligned}
\]
- \(B \equiv\) Eigenvector matrix for the ion crystal.
- \(D \equiv\) Diagonal matrix containing eigenvalues for the ion crystal.