Buffer-Gas Cooling of a Single Ion in a Multipole Radio Frequency Trap Beyond the Critical Mass Ratio

We theoretically investigate the dynamics of a trapped ion immersed in a spatially localized buffer gas. For a homogeneous buffer gas, the ion’s energy distribution reaches a stable equilibrium only if the mass of the buffer gas atoms is below a critical value. This limitation can be overcome by using multipole traps in combination with a spatially confined buffer gas. Using a generalized model for elastic collisions of the ion with the buffer-gas atoms, the ion’s energy distribution is numerically determined for arbitrary buffer-gas distributions and trap parameters. Three regimes characterized by the respective analytic form of the ion’s equilibrium energy distribution are found. Final ion temperatures down to the millikelvin regime can be achieved by adiabatically decreasing the spatial extension of the buffer gas and the effective ion trap depth (forced sympathetic cooling).

Figure 1

Schematic of an ion trajectory (blue) in a linear octupole trap (pole order $n=4$) colliding with a buffer-gas atom (red). The velocities ${\overrightarrow{v}}_i$ and ${\overrightarrow{u}}_i$ are associated with the ion’s micro and macromotion, respectively, while the atom’s velocity is indicated by ${\overrightarrow{v}}_a$. In the lower graph, the effective ponderomotive potential $V_{\text{eff}}$ of the ion trap is shown together with the spatial distribution of the buffer-gas atoms (spatial extension $2{\sigma }_a$). In the right panels, typical ion trajectories are shown for a collision near the trap center, yielding a decrease of the ion’s energy (upper panel), and near the classical turning point, resulting in an increase of the ion’s energy (lower panel).

Figure 2

Normalized equilibrium energy distributions for different mass ratios $\xi$ in a Paul trap (pole order $n=2$). The buffer-gas cloud distribution is given by a Gaussian of size ${\sigma }_a=R_0/100$ and temperature of $T_a=200\mu \mathrm{K}$. Also shown is the energy distribution in the Boltzmann regime (red curve) and the energy distribution for $\xi =34$ (purple curve), according to the expressions in Table 1. The inset compares the exponents $\kappa$ of the power law in the energy distribution (as defined in Table 1), for different models. The condition $\kappa =-2$ separates the regimes of stable from unstable ion motion. Results found by Zipkes et al. have been corrected as ${\kappa }_{\text{corr}}=\kappa -1$ (see Supplemental Material [33]).

Figure 3

Forced sympathetic cooling. Shown are four energy distributions in a Paul trap for different buffer-gas sizes. The buffer gas has a temperature of $T_a=4\mathrm{K}$ and $\xi =10$. The two curves correspond to the analytic expressions found in Table 1 for ${\sigma }_a=0.01$ and 0.64. The inset shows the final temperature of the ion as a function of the buffer-gas size. The analytic expression corresponds to Eq. (5); the numeric result was obtained by fitting a modified Boltzmann distribution to the numeric data.