Measurement of the low-energy ${\mathrm{Na}}^{+}-\mathrm{Na}$ total collision rate in an ion-neutral hybrid trap

We present measurements of the total elastic and resonant charge-exchange ion-atom collision rate coefficient $k_{\mathrm{ia}}$ of cold sodium (Na) with optically dark low-energy ${\mathrm{Na}}^{+}$ ions in a hybrid ion-neutral trap. To determine $k_{\mathrm{ia}}$, we measured the trap loading and loss rates from both a Na magneto-optical trap (MOT) and a linear radio-frequency quadrupole Paul trap. We found the total rate coefficient to be $7.4\pm 1.9\times {10}^{-8}{\mathrm{cm}}^3/\mathrm{s}$ for the type-I Na MOT immersed within an $\approx 140-\mathrm{K}$ ion cloud and $1.10\pm 0.25\times {10}^{-7}{\mathrm{cm}}^3/\mathrm{s}$ for the type-II Na MOT within an $\approx 1070-\mathrm{K}$ ion cloud. Our measurements show excellent agreement with previously reported theoretical fully quantal ab initio calculations. In the process of determining the total rate coefficient, we demonstrate that a MOT can be used to probe an optically dark ion cloud's spatial distribution within a hybrid trap.

Figure 1

Schematic of the hybrid trap apparatus within the vacuum chamber (top view, not to scale). The Na magneto-optical trap (MOT) is created in the center of the segmented linear Paul trap (LPT) with six 589-nm laser beams. MOT fluorescence measurements can be taken with our CMOS camera or with our photomultiplier tube (PMT). The MOT can be translated within the LPT using the electromagnet shim coils. A pair of anti-Helmholtz coils and a third shim coil sit outside the vacuum chamber and are not shown in the figure. The LPT is loaded with ${\mathrm{Na}}^{+}$ ions by photoionization (PI) of excited Na($3\mathrm{P}$) MOT atoms with a 405-nm laser beam collinear with one MOT beam. Ions are destructively detected by extracting the ions toward the Channeltron electron multiplier (CEM). The mesh is both used for extraction ion optics and to create a more uniform gain across the CEM cone [40].

Figure 2

CMOS camera image, without false coloring, of the smaller, denser, and colder type-I MOT (a) and the larger, warmer type-II MOT (b). As depicted in Fig. 1, the camera is looking down the axial (long) dimension of the segmented LPT. The inner edges of the LPT's end-segment electrodes can be seen in the corners of the image. The images are saturated to be more visually striking, but care is taken to avoid saturation when taking data.

Figure 3

Energy-level diagram that shows the hyperfine structure of the D2 Na line. We indicate the cooling laser (C) and the repumping laser (R) transitions for the type-I and -II MOTs. Each detuning from resonance (${\delta }_{23}$ and ${\delta }_{22}$) was chosen for maximum MOT fluorescence intensity. For the type-I MOT, the cooling laser is the carrier signal from our electro-optical modulator (EOM). For the type-II MOT, the cooling laser is the EOM sideband signal.

Figure 4

We tested for deviation from the expected exponential behavior of the CEM gain as a function of the voltage applied to the CEM detection cone $V_c$ at a fixed ion input current. We find that the output ion signal (which is proportional to the gain) looks linear when plotted with a log-linear plot scaling, indicating that we are not saturating the CEM. The uncertainty in the power supply voltage $V_c$ and the ion signal is approximately the size of the plot markers. In the experiments presented here, we operate at a $V_c=1250$ V.

Figure 5

Fluorescence from a type-II MOT as it loads, with the corresponding fits to Eq. (7). Curve (a) shows the raw PMT data (light blue) and fit (royal blue) of an isolated MOT loaded with a total loss rate ${\gamma }_t={\gamma }_b$. Curve (b) shows the raw PMT data (gray) and fit (green) of a MOT loaded with PI ($I_{\mathrm{pi}}\approx$ 80 mW/${\mathrm{cm}}^2$), making ${\gamma }_t={\gamma }_b+{\gamma }_{\mathrm{pi}}$. Curve (c) shows the raw PMT data (magenta) and fit (black) of a MOT loaded with the same PI intensity as curve (b), but the MOT is also immersed in a saturated LPT ion cloud, making ${\gamma }_t={\gamma }_b+{\gamma }_{\mathrm{pi}}+{\gamma }_{\mathrm{ia}}$.

Figure 6

Plot of the total MOT loss rate in the presence of PI (${\gamma }_t={\gamma }_b+{\gamma }_{\mathrm{pi}}$) as a function of the total peak PI intensity and corresponding linear fits. Curve (a) shows type-I MOT data and curve (b) shows type-II MOT data. The $y$ intercepts (at $I_{\mathrm{pi}}=0$) are ${\gamma }_b$ and the slopes are proportional to ${\sigma }_{\mathrm{pi}}{f}_e$. The statistical uncertainty in the rates is smaller than the plot markers. The uncertainty in the intensity is primarily due to power fluctuations and the precision of the beam waist measurement.

Figure 7

Plot of the ion-atom loss rate ${\gamma }_{\mathrm{ia}}$ as a function of increasing rf voltage amplitude $V_{\mathrm{rf}}$. Because the saturated LPT density increases quadratically with $V_{\mathrm{rf}}$, the ion-atom loss rate also appears to increase quadratically. Although, the data could also be interpreted as being linearly propostional to $V_{\mathrm{rf}}$.

Figure 8

Plot of the ion-atom loss rate as a function of the saturated LPT steady-state ion population and corresponding linear fit to Eq. (3). Curve (a) shows type-II MOT data and curve (b) shows type-I MOT data. The type-II MOT has larger loss rates and fewer steady-state ions at saturation because it produces a hotter lower density ion cloud that has a slightly larger $k_{\mathrm{ia}}$. The uncertainty in the measurements is discussed in the text.

Figure 9

CEM measured LPT loading (from the type-I MOT) as a function of time and corresponding two-parameter fits to the solution to rate Eq. (14). Each curve corresponds to a different PI intensity: Curve (a) is measured with $I_{\mathrm{pi}}\approx 670$ mW/${\mathrm{cm}}^2$, curve (b) is with $I_{\mathrm{pi}}\approx 108$ mW/${\mathrm{cm}}^2$, curve (c) is with $I_{\mathrm{pi}}\approx 42$ mW/${\mathrm{cm}}^2$, and curve (d) is with $I_{\mathrm{pi}}\approx 15$ mW/${\mathrm{cm}}^2$. The uncertainties are smaller than the size of the plot markers.

Figure 10

CEM measured LPT loading rate as a function of $I_{\mathrm{pi}}$ and corresponding fit to Eq. (14). Curve (a) shows the LPT loaded from the type-II MOT and curve (b) shows the LPT loaded from the type-I MOT. The fit to Eq. (14) is only a single parameter $y$-scaling constant, whose value is the CEM calibration. All other parameters are independently determined from MOT fluorescence measurements.

Figure 11

Plot of the ion-atom loss rate ${\gamma }_{\mathrm{ia}}$ as a function of PI intensity for the type-I MOT (black) squares and the type-II MOT (red) circles. We find that ${\gamma }_{\mathrm{ia}}$ measurements of the MOT fluorescence are not equivalent to the LPT loss rate measurements seen in Fig. 12. Furthermore, we show fits to the type-I (black dashed line) and type-II (solid red line) MOT data obtained as the solution of Ref. [17]'s intensity-loss rate equation.

Figure 12

Plot of the LPT loss rate $\lambda$ as a function of $I_{\mathrm{pi}}$. Each value is determined from fits to loading curves like the ones seen in Fig. 9. Curve (a) is the $\lambda$ fitting value when the LPT is loaded with the type-II MOT and curve (b) is the $\lambda$ fitting value when the LPT is loaded with the type-I MOT.

Figure 13

Experimental LPT ion population decay as a function of time on a log-log scale. The LPT is saturated with the type-I MOT and $I_{\mathrm{pi}}\approx 670$ mW/${\mathrm{cm}}^2$. Loading from the type-II MOT gives qualitatively identical behavior. The decay is initially nonexistent and is then followed by a sudden rapid decay (after 0.1 s) and then a slower decay (after 1.1 s). We have fit the decay curve to three solutions: the red dashed curve is the solution to Eq. (10) with $L_I=0$, the blue dotted-dashed curve is the solution to Eq. (16) with $L_I=0$, and the solid green curve is for a double-exponential decay. The uncertainty in the data is smaller than the plot markers.

Figure 14

Ion trap simulations (red solid curve) and numerical diffusing Gaussian model (black dashed curve) of 500 ions decaying from an idealized ion trap. The simulation and model calculations show good quantitative agreement with each other and good qualitative agreement with the experimental data in Fig. 13.

Figure 15

MOT ion-atom loss rate ${\gamma }_{\mathrm{ia}}$ normalized by the steady-state ion population ${\tilde{N}}_I$ as a function of the center position of the type-I MOT (as measured with the CMOS camera) relative to the geometric center of the LPT [$x_{0,1}$ in Eq. (4)]. The data are fit to Eq. (3) and the fitted ion cloud radius is $r_{I,1}=1.6\pm 0.1$ mm.