Loading a linear Paul trap to saturation from a magneto-optical trap

We present experimental measurements of the steady-state ion number in a linear Paul trap (LPT) as a function of the ion-loading rate. These measurements, taken with (a) constant Paul trap stability parameter $q$, (b) constant radio-frequency (rf) amplitude, or (c) constant rf frequency, show nonlinear behavior. At the loading rates achieved in this experiment, a plot of the steady-state ion number as a function of loading rate has two regions: a monotonic rise (region I) followed by a plateau (region II). Also described are simulations and analytical theory which match the experimental results. Region I is caused by rf heating and is fundamentally due to the time dependence of the rf Paul-trap forces. We show that the time-independent pseudopotential, frequently used in the analytical investigation of trapping experiments, cannot explain region I, but explains the plateau in region II and can be used to predict the steady-state ion number in that region. An important feature of our experimental LPT is the existence of a radial cutoff ${\widehat{R}}_{\mathrm{cut}}$ that limits the ion capacity of our LPT and features prominently in the analytical and numerical analysis of our LPT-loading results. We explain the dynamical origin of ${\widehat{R}}_{\mathrm{cut}}$ and relate it to the chaos border of the fractal of non-escaping trajectories in our LPT. We also present an improved model of LPT ion-loading as a function of time.

Figure 1

Fits of ion loading as a function of time based on (2) overshoot the rise, undershoot the knee, and overshoot the plateau. This is the case regardless of whether the ion number is measured via fluorescence at low loading rates (top; reproduced from Ref. [20] with permission from The Royal Society of Chemistry) or measured using a channel electron multiplier at high loading rates from a hybrid trap by our group (bottom). Where the error bars (showing statistical errors) are not seen, they are smaller than the corresponding plot symbols.

Figure 2

A diagram of the hybrid apparatus used in these experiments. The anti-Helmholtz coils, with current directions shown by arrows, are located outside of the vacuum chamber (not shown).

Figure 3

Ion signal as a function of time for short loading times, which minimizes the effects from losses. The ion signal is proportional to the number of trapped ions; the slope is the loading rate from the MOT. Where the error bars (showing statistical errors) are not seen, they are smaller than the corresponding plot symbols.

Figure 4

Two-way measurement of the ion loading rate as a function of the intensity of the photoionizing laser for the purpose of finding the calibration factor between the ion signal and the number of ions. Measurements using the CEM (black squares) are compared to measurements of the atom loss rate from the MOT in the presence of the photoionization laser (solid red line) multiplied by a fitted scaling factor, which is the reciprocal of the CEM calibration factor. Where the error bars (showing statistical errors) are not seen, they are smaller than the corresponding plot symbols.

Figure 5

Loading curves taken at constant $q=0.3$ and plotted on a lin-log scale. The trap settings were $V_{\mathrm{rf}}=13$ V and $f=450$ kHz for the black squares, $V_{\mathrm{rf}}=16$ V and $f=500$ kHz for the red circles, and $V_{\mathrm{rf}}=19.5$ V and $f=550$ kHz for the blue triangles. Where the error bars (showing statistical errors) are not seen, they are smaller than the corresponding plot symbols.

Figure 6

Loading curves taken at constant $V_{\mathrm{rf}}=16$ V and plotted on a lin-log scale. The trap settings were $q=0.22$ and $f=580$ kHz for the black squares, $q=0.26$ and $f=535$ kHz for the red circles, $q=0.30$ and $f=500$ kHz for the upright blue triangles, and $q=0.37$ and $f=450$ kHz for the inverted turquoise triangles. The plots are in order of their trap depths. Where the error bars (showing statistical errors) are not seen, they are smaller than the corresponding plot symbols.

Figure 7

Loading curves taken at constant $f=500$ kHz and plotted on a lin-log scale. The trap settings were $q=0.22$ and $V_{\mathrm{rf}}=12$ V for the black squares, $q=0.26$ and $V_{\mathrm{rf}}=14$ V for the red circles, $q=0.30$ and $V_{\mathrm{rf}}=16$ V for the upright blue triangles, and $q=0.37$ and $V_{\mathrm{rf}}=20$ V for the inverted turquoise triangles. The plots are in order of their trap depths. Where the error bars (showing statistical errors) are not seen, they are smaller than the corresponding plot symbols.

Figure 8

Simulation results for the model LPT performed at constant $q=0.3$ for three different combinations of rf voltage and frequencies. $f=450\mathrm{kHz}, V=13\mathrm{V}$: open, blue circles; $f=500\mathrm{kHz}, V=16\mathrm{V}$: filled, green circles; $f=550\mathrm{kHz}, V=19.5\mathrm{V}$: red asterisks. Scale parameter: $\sigma =1/40$. These simulation results may be compared with the three corresponding LPT experiments shown in Fig. 5. The heavy solid line (purple) is the analytical result for $N_s\left(\lambda \right)$ in region IV [22]. The lengths of the error bars, equal to $2\mathrm{\Delta }{N}_s$, characterize the statistical fluctuations of the ion number in the saturated regime.

Figure 9

Simulations of loading curves at three scaling factors, from top to bottom: $\sigma =1/10$, small blue triangles; $\sigma =1/8$, large black triangles; and $\sigma =1/5$, red circles. Also shown are a number of experimental loading curves (squares) at $q=0.30, V_{\mathrm{rf}}=13$ V, and $f=450$ kHz. The scatter of the experimental data points gives an idea of the statistical and systematic variations in our experimental loading data. For these simulations, ${\widehat{R}}_{\mathrm{cut}}=4.5\mathrm{mm}$ was chosen, which, as a result of many simulations, akin to those shown in this figure, turned out to yield the best agreement with the experimental results. Where the experimental error bars (showing statistical errors) are not seen, they are smaller than the corresponding plot symbols.

Figure 10

The data points from Fig. 5 shown with fit using $N_s=A{\lambda }^{0.281}$. The trap settings were $V_{\mathrm{rf}}=13$ V and $f=450$ kHz for the bottom curve, $V_{\mathrm{rf}}=16$ V and $f=500$ kHz for the middle curve, and $V_{\mathrm{rf}}=19.5$ V and $f=550$ kHz for the top curve. We see that the power-law form of the fit function, as predicted by our analytical model, fits the data in region I very well. Where the error bars (showing statistical errors) are not seen, they are smaller than the corresponding plot symbols.

Figure 11

Simulation of the LPT: Comparison between the time-dependent model with rf switched on (asterisks) and the pseudopotential model with only the pseudopotential present (filled circles). We simulated the case $V_{\mathrm{rf}}=16\mathrm{V}, f=450\mathrm{kHz}$, with scaling factor $\sigma =1/40$. The lengths of the error bars, equal to $2\mathrm{\Delta }{N}_s$, characterize the statistical fluctuations of the ion number in the saturated regime. Where error bars are not seen, they are smaller than the plot symbols.

Figure 12

Ion number vs time fitted to the model (34) [thick blue (upper) line], the traditional model (2) (thin green line), that model with ${\ell }_2=0$ (dot-dashed purple line), and that model with ${\ell }_1=0$ (dotted red line). The traditional model displays the familiar overshoot-undershoot-overshoot pattern, but the model (34) fits both the knee and the plateau. Where the error bars (showing statistical errors) are not seen, they are smaller than the corresponding plot symbols.

Figure 13

Color-coded fractal of escape times. Denoting by $L$ the number of rf cycles it takes for an initial condition $(x,z)$ to reach $\left|x\right|\ge r_0=9.5\mathrm{mm}$, the colors code for $L<3$ (cyan or lightest gray), $3\le L<6$ (green or light gray), $6\le L<9$ (red or gray), $9\le L<1000$ (blue or dark gray), and $L>1000$ (black). The black area corresponds to initial conditions that lead to ion trajectories that never escape. The black area is bounded in the $x$ direction by ${\widehat{R}}_{\mathrm{cut}}\approx 5\mathrm{mm}$. The points with $x=0$, protruding from the fractal, are also shown in black, since they correspond to on-axis equilibrium points that, too, never escape.