Energy Scaling of Cold Atom-Atom-Ion Three-Body Recombination

We study three-body recombination of ${\mathrm{Ba}}^{+}+\mathrm{Rb}+\mathrm{Rb}$ in the mK regime where a single ${{}^{138}\mathrm{Ba}}^{+}$ ion in a Paul trap is immersed into a cloud of ultracold ${}^{87}\mathrm{Rb}$ atoms. We measure the energy dependence of the three-body rate coefficient $k_3$ and compare the results to the theoretical prediction, $k_3\propto E_{\mathrm{col}}^{-3/4}$, where $E_{\mathrm{col}}$ is the collision energy. We find agreement if we assume that the nonthermal ion energy distribution is determined by at least two different micromotion induced energy scales. Furthermore, using classical trajectory calculations we predict how the median binding energy of the formed molecules scales with the collision energy. Our studies give new insights into the kinetics of an ion immersed in an ultracold atom cloud and yield important prospects for atom-ion experiments targeting the $s$-wave regime.

Figure 1

Double-logarithmic plot of the measured loss rate $\mathrm{\Gamma }$ for ${\mathrm{Ba}}^{+}$ as a function of the tuned eMM energy $E_{\mathrm{fMM}}$. Red circles are the experimental data; the curve represents a fit of Eq. (1) (see text for details). The corresponding values of ${\overline{k}}_3$ are indicated on the right-hand side. The inset is the Logarithmic plot of the decay curve of the ${\mathrm{Ba}}^{+}$ ion. $p$ is the probability to recover ${\mathrm{Ba}}^{+}$ after interacting with Rb. The straight line is an exponential fit to the data. The sketch shows the ion orbit in the atom cloud. With increasing ion energy its orbit becomes comparable to the atom cloud size.

Figure 2

(a) A typical trajectory at a collision energy of $100\mu \mathrm{K}\times k_{\mathrm{B}}$ associated with the three-body collision ${\mathrm{Ba}}^{+}+\mathrm{Rb}+\mathrm{Rb}$ that leads to the formation of ${\mathrm{BaRb}}^{+}$. We show the distances $r_{ij}$ between the particles as indicated in the sketch. (b) Double log plot of $k_3$ obtained with CTC for ${\mathrm{Ba}}^{+}+\mathrm{Rb}+\mathrm{Rb}$ as a function of the collision energy $E_{\mathrm{col}}$ (circles). The straight line shows the analytically predicted power-law dependence $k_3\propto E_{\mathrm{col}}^{-3/4}$.

Figure 3

(a) Calculated ion energy distributions $P(E_{\mathrm{col}},E_1^S)$, each with a single energy scale $E_1^S$. An energy of $E_1^S=E_{\mathrm{fMM}}=1\mathrm{mK}$ (20 mK) was used for the green (red) distribution. Choosing $E_1^S=E_{\varphi \mathrm{MM}}=1\mathrm{mK}$ produces the blue distribution, which has a different shape compared to both previous distributions. (b) Comparison of the experimental (full circles) ${\overline{k}}_3$ data as a function of $E_{\mathrm{fMM}}$ with the full calculation (blue line). The red line is the same calculation but multiplied by 1.1.

Figure 4

(a) Logarithmically binned histogram of the binding energies at collision energies of $100\mu \mathrm{K}\times k_{\mathrm{B}}$ (blue) and $10\mathrm{mK}\times k_{\mathrm{B}}$ (red). The second histogram is magnified by a factor of 2. (b) Double-logarithmic plot of the typical binding energy of the formed molecule as a function of the collision energy. The dashed line represents a power law fit.