Classical motion of a harmonically confined chain
February 6, 2026
Consider a chain of \(N\) ions in a harmonic potential. We assume that the system is in an equilibrium configuration where the ions have positions \(\vec{r}_{0,i} + \vec{r}_i\), where \(\vec{r}_i = (x_i,y_i,z_i)\). The external potential energy of the \(i\)th ion is then
\[U_i(\vec{r}_i) = \frac{1}{2}m(\omega_x^2x_i^2 + \omega_y^2y_i^2 + \omega_z^2z_i^2)\]We assume that the ions are weakly confined along the trap axis, so that \(\omega_x \ll \omega_y, \omega_z\). The potential energy of the electrostatic interaction between the \(i\)th and \(j\)th ions is
\[V_{ij}(|\vec{r}_i-\vec{r}_j|) = k_{e}e^2\frac{1}{|\vec{r}_i-\vec{r}_j|}\]So the entire potential energy for the system is
\[\sum_i U_i(\vec{r}_i) + k_ee^2\sum_{i<j}\frac{1}{|\vec{r}_i-\vec{r}_j|}\]To Taylor expand \(1/|\vec{r}_i-\vec{r}_j|\) about the equilibrium point we define the difference between the \(i\)th and \(j\)th equilibrium positions as as \(d_{ij}\) and note that this vector lies pureply along the trap axis: \(\vec{r}_{0,i} - \vec{r}_{0,j} \equiv d_{ij}\,\hat{\mathbf{x}}\). We also define the difference between deviations from equilbrium positions \(\vec{r}_{ij} = \vec{r}_i - \vec{r}_j\).
\[\frac{1}{|\vec{r}_i-\vec{r}_j|} = \left((d_{ij}\,\hat{\mathbf{x}} + \vec{r}_{ij})^2\right)^{-1/2} = \left(d_{ij}^2 + 2d_{ij}(x_i-x_j) + r_{ij}^2 \right)^{-1/2}.\]Then
\[\begin{align} \frac{1}{|\vec{r}_i-\vec{r}_j|} &= d_{ij}^{-1}(1 + 2(x_i-x_j)/d_{ij} + r_{ij}^2/d_{ij}^2)^{-1/2} \\ &= d_{ij}^{-1}(1 - (x_i-x_j)/d_{ij} - r_{ij}^2/2d_{ij}^2 + \cdots) \end{align}\]The constant term we can ignore. The linear term shifts both the \(i\)th and \(j\)th ion's equilibrium positions. And the \(r_{ij}^2\) term gives both a motional coupling term proportional to \(x_ix_j\) as well as quadratic terms proportional to \(x_i^2\) and \(x_j^2\) which shift the \(x\) motional mode frequencies for ions \(i\) and \(j\).