Dynamics of a Ground-State Cooled Ion Colliding with Ultracold Atoms

Ultracold atom-ion mixtures are gaining increasing interest due to their potential applications in ultracold and state-controlled chemistry, quantum computing, and many-body physics. Here, we studied the dynamics of a single ground-state cooled ion during few, to many, Langevin (spiraling) collisions with ultracold atoms. We measured the ion’s energy distribution and observed a clear deviation from the Maxwell-Boltzmann distribution, characterized by an exponential tail, to a power-law distribution best described by a Tsallis function. Unlike previous experiments, the energy scale of atom-ion interactions is not determined by either the atomic cloud temperature or the ion’s trap residual excess-micromotion energy. Instead, it is determined by the force the atom exerts on the ion during a collision which is then amplified by the trap dynamics. This effect is intrinsic to ion Paul traps and sets the lower bound of atom-ion steady-state interaction energy in these systems. Despite the fact that our system is eventually driven out of the ultracold regime, we are capable of studying quantum effects by limiting the interaction to the first collision when the ion is initialized in the ground state of the trap.

Figure 1

(a) Experimental system. Our apparatus is composed of two vacuum chambers. In the top chamber, we collected atoms and cooled them to a temperature of several $\mu \mathrm{K}$ in a ${\mathrm{CO}}_2$ dipole trap. We then transferred the atoms to the bottom chamber using a 1D moving optical lattice (1064 nm YAG laser) where they were retrapped in a crossed dipole trap and overlapped with a single ground-state cooled ion. (b) ${}^{88}{\mathrm{Sr}}^{+}$ ion energy levels.

Figure 2

Carrier Rabi spectroscopy for few atom-ion Langevin collisions. (a)–(f) Each graph corresponds to a different interaction time [0, 0.5, 2, 3.5, 5, 6.5 ms for graphs (a)–(f), respectively]. We scanned the shelving pulse time $t_R$ and measured the shelving probability $P_D$. Each data point corresponds to 170 experiments. Error bars are binomial distribution standard deviation. We fitted the data using Eq. (2) together with the Tsallis distribution [Eq. (1)]. We extracted the distribution free parameters ($T$ and $n$) using maximum likelihood estimation [34]. The resulting curve is shown by solid lines. Dashed lines show the fit of our data to a thermal distribution. (g) The ion’s temperature [$T_{\mathrm{ion}}=Tn/(n-2)$] (filled colored circles) is plotted as function of the interaction time (top x-axis) and the average number of Langevin collisions (bottom x-axis) and it increases linearly with a rate of $296(37)\mu \mathrm{K}/\mathrm{ms}$, which is equivalent to $100(13)\mu \mathrm{K}$ per collision. Error bars are $1\sigma$ standard deviation. Shaded area represents linear fit confidence bounds ($1\sigma$). Open circles are the results of a simulation that takes into account the reaction of the polarization potential on the ion’s position. Black dots are simulation results taking into account only hard-sphere collisions. (h) Ion’s power-law parameter $n$. The ion’s energy distribution starts with $n\gg 1$, consistent with a Maxwell-Boltzman distribution, and converges to $n=4.0(2)$ after $\sim 10$ collisions. For $n>10$, thermal and Tsallis distributions are almost indistinguishable as can be seen in (a)–(c). The gray shaded area represents the non-normalizable region of the distribution ($n<3$). The dashed lines represent the threshold ($n=4$) for mean energy divergence.

Figure 3

Ion’s steady-state temperature after hundreds of atom-ion collisions. (a) DRC signal (pink dots, $150\mu \mathrm{s}$ moving average) for $150\mathrm{mK}·k_B$ EMM experiment and a fit for power-law (black solid line) and thermal (blue dashed line) energy distributions. (b) Ion’s temperature as a function of the dc electric field at the trap center (bottom $x$ axis; top $x$ axis shows the differential voltage on the compensation electrode). Solid blue line is a parabolic fit. In the main figure, the two opposing-sign dc voltage configurations are averaged to show the ion temperature versus the EMM kinetic energy. Error bars account for both fit confidence intervals ($1\sigma$) and the statistics of averaging over the two opposing EMM points. The solid blue line is a linear fit. Shaded area represents fit confidence bounds ($1\sigma$). The black points are the results of a simulation that takes into account only the effect of hard-sphere collisions.

Figure 4

Ion’s heating dynamics. Ion’s temperature for $100\mathrm{mK}·k_B$ EMM energy (purple diamonds) measured using DRC. Solid purple line is an exponential fit. Error bars are recooling fit confidence intervals ($1\sigma$). Shaded area represents fit confidence bounds ($1\sigma$). We compared this result to the heating rate measured using Rabi spectroscopy [Fig. 2; blue circles in this figure] and the steady-state measured using DRC (Fig. 3; not shown in this figure) for EMM energy less than $0.5\mathrm{mK}·k_B$. Solid blue line is an exponential fit. For comparison, we show a hard-sphere collisions simulation (black dots) with EMM energy ($12\mathrm{mK}·k_B$), which produces the same steady-state temperature, but, however, disagrees with the heating dynamics measured.