Dynamics of a single trapped ion immersed in a buffer gas

We provide a comprehensive theoretical framework for describing the dynamics of a single trapped ion interacting with a neutral buffer gas, thus extending our previous studies on buffer-gas cooling of ions beyond the critical mass ratio [B. Höltkemeier et al., Phys. Rev. Lett. 116, 233003 (2016)]. By transforming the collisional processes into a frame, where the ion's micromotion is assigned to the buffer-gas atoms, our model allows us to investigate the influence of nonhomogeneous buffer-gas configurations as well as higher multipole orders of the radio-frequency trap in great detail. Depending on the neutral-to-ion mass ratio, three regimes of sympathetic cooling are identified which are characterized by the form of the ion's energy distribution in equilibrium. We provide analytic expressions and numerical simulations of the ion's energy distribution, spatial profile, and cooling rates for these different regimes. Based on these findings, a method for actively decreasing the ion's energy by reducing the spatial expansion of the buffer gas arises (forced sympathetic cooling).

Figure 1

Schematic of a linear rf trap with $n=4$. The atoms are indicated in red; the ion is shown in blue. The eight rf electrodes are indicated by the gray cylinders. Such rods are commonly used to replace the ideal hyperbolic electrodes [20]. The two insets on the right illustrate two elastic collisions, one close to the trap center and the other close to the ion's turning point $r_{\mathrm{turn}}$. The trap radius $r_0$ is indicated by the arrow.

Figure 2

Probability distributions of a relative energy change $\mathrm{\Delta }{E}_i/E_i$ in an elastic ion-neutral collision for an ion at energy $E_i$ as a function of the normalized collision radius $r/r_{\max }$. The buffer gas is assumed to be at rest (temperature $T_a=0$). The left graphs show the distribution for a Paul trap (pole order $n=2$), and the right graphs show the distribution for an octupole trap ($n=4$). The bottom, middle, and top graphs correspond to mass ratios of $\xi =0.25$, 1, and 4, respectively. The color code provides the normalized probability function $p_r(\mathrm{\Delta }{E}_i/E_i)$ of the energy change at a given radius $r$. Indicated by the red dashed line is the critical cooling radius [Eq. (10)] given by the condition that average energy transfer $〈\mathrm{\Delta }{E}_i/E_i〉$ becomes zero.

Figure 3

The ion's spatial probability distribution. (a) The ion's minimum and maximum radius for different angular momenta and multipole orders. The angular momentum has been normalized using the maximum turning point $r_{\mathrm{turn},\max }$, given by the condition $V_{\mathrm{eff}}\left(r_{\mathrm{turn},\max }\right)=E_{\mathrm{r}}$, resulting in $L_{\mathrm{norm}}=|(\vec{r}/r_{\mathrm{turn},\max })\times [{\vec{v}}_{\mathrm{d},\mathrm{r}}/v_{\mathrm{d},\mathrm{r}}(r=0)]|$. The inset shows an exemplary trajectory (red), with $r_{\mathrm{turn}}$ and $r_{\min }$ being indicated by the dashed circles. (b) The radial probability distribution as a function of the angular momentum. In analogy to a one-dimensional pendulum, the probability distribution diverges at the radial turning points as the radial velocity $\dot{r}$ becomes zero.

Figure 4

Normalized equilibrium energy distributions for different thermalicity parameters $\zeta$ in a Paul trap ($n=2$). 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 ($\zeta =0.96$) and an energy distribution in the localization regime for $\zeta =0.07$, according to Eqs. (19) and (24).

Figure 5

The ion's energy distribution in the Boltzmann regime. The difference between the energy distribution of the ion and the one of the buffer gas (dashed line) is caused by the effective potential, confining the ion in the trap. Depending on the multipole order, the influence of the effective potential varies. The inset shows the ion's mean energy as a function of the trap order. The mean energy is largest for an ion inside a Paul trap and converges towards the mean energy of the buffer gas for large multipole orders.

Figure 6

Energy distributions in the power-law regime. (a) Five distributions corresponding to five different mass ratios. With increasing mass ratio the energy distribution exhibits a growing power law towards higher energies. The solid lines correspond to Tsallis functions [Eq. (21)] which were fitted to the numerical data. (b) Mass-ratio dependence of the power-law exponent $\kappa$ for four different multipole orders. For a fixed mass ratio, the power-law exponent is always largest for an ion inside the Paul trap and monotonically decreases with increasing multipole order. The solid lines correspond to Eq. (22), and the dashed line shows the analytic result of Chen et al. [11] for a Paul trap, which shows reasonable agreement with our numeric results.

Figure 7

Spatial distribution in the power-law regime. Shown are five distributions for an ion inside a Paul trap, corresponding to different mass ratios. For small mass ratios, the spatial distribution corresponds to the one in the Boltzmann regime [Eq. (20)]. For larger mass ratios, the distributions exhibit a power law towards large radii, much like the energy distributions. The inset shows the mass-ratio dependence of the power-law exponent for different multipole orders. The solid lines are a guide to the eye.

Figure 8

Thermalization rates in different rf traps. Shown is the average energy loss per collision for different mass ratios and multipole orders. The gray curve shows the average energy loss in the absence of the rf trap, which is equivalent to a trap with an infinite multipole order. The mass ratio where the average energy loss turns into an energy gain corresponds to the critical mass ratio. The inset shows two exemplary energy traces. The average energy loss was extracted from the exponential decay at the beginning of the trace, where the ion's energy exceeds that of the buffer gas.

Figure 9

The ion's energy distribution for a localized buffer gas in a Paul trap. The buffer gas has a Gaussian density distribution with standard deviation ${\sigma }_a=r_0/10$ and an atom-to-ion mass ratio of $\xi =3$. The five graphs correspond to five different multipole stability parameters $\eta$ [Eq. (2)].

Figure 10

Energy distributions in the localization regime. (a) The ion's energy distribution for different mass ratios in a Paul trap. The energy scale $E_{\mathrm{a},\mathrm{tot}}$ was kept constant at 1 eV. (b) Power-law exponents for different multipole orders as indicated by the dashed lines in the main graph. Shown are the power-law exponents as a function of the mass ratio. The solid curves correspond to the analytic expression given in Eq. (25), which shows good agreement with the numerical data for large mass ratios. For small mass ratios, however, the exponent is better described by Eq. (22), as indicated by the close-up of the area marked in gray shown in the inset. Shown are the numerical data for $n=2,6$ and the corresponding analytic expressions from Eq. (25) (solid lines) and Eq. (22) (dashed lines).

Figure 11

Forced sympathetic cooling. Shown are five energy distributions in an octupole trap $(n=4)$ for different buffer-gas sizes. The buffer gas has a temperature of $T_a=4\mathrm{K}$ and $\xi =30$. The two curves correspond to the analytic expressions found in the text. The inset shows the ion's mean energy and the effective buffer-gas energy [Eq. (23)] as a function of the buffer-gas size for three different trap orders. Both energies are in good agreement.