Atom-ion scattering and doped ultracold samples

Scattering of charged particles has a long history. However, the interaction of ions and neutral particles in the ultracold regime is a nascent sub-field in AMO physics. In our early studies of Na colliding with Na⁺, we found that the cross sections of elastic and charge exchange processes are very large due to the very long-range C ₄/R⁴ polarization interaction potential between an ion and neutral atom (for molecular species, additional angular dependence may exist). In systems such as Na+Na⁺, both the ion and its parent approach each other along two potential curves (a gerade and ungerade curve), and the scattering cross sections are given by

,

where η_ℓ ^g/u is the phase shift for the ℓ-th partial wave corresponding to the gerade and ungerade symmetry, respectively. The total and the charge exchange cross sections are very large, as shown in the figure beside: the elastic cross sections are many orders of magnitude larger than the corresponding cross sections between neutral Na+Na. The charge exchange is referred to as resonant charge transfer, and both cross sections follow simple behavior when many partial waves contribute to the process:

,

where μ is the reduced mass of the system. Both approximations, shown in the figure, agree well with the full calculations. Using such results, we computed the transport properties (diffusion and mobility) of an ion in its parent gas. We found that the mobility is rather constant for a large range of temperatures, but increases at low temperatures.

An interesting question arises: how does the charge diffuse when the temperature T becomes extremely low ? From the charge transfer cross section σ_{c h} = πρ_ch ², we can define a charge transfer length scale ρ_ch. As E ∼ k_B T decreases, σ_ch ∼ T ^-1/2 leads to ρ_ch scaling as T ^-1/4 while the de Broglie's wavelength of the ion/atom scales as T ^-1/2.

As we sketch in the figure (a) beside, as the temperature is lowered, the charge exchange radius increases faster than the de Broglie wavelength (shown as the gaussian envelope), so that there is a non-zero probability P(x) of having the atom-ion pair within the charge exchange radius. By integrating over all possible location x, we obtain the total probability of being within the charge exchange radius. At that point, while the ion and atom are basically immobile, charge exchange can take place very efficiently by having the electron (or equivalently the hole) hopping between atom and ion.

This hopping conductivity should be very efficient and dominate the transport properties at very low temperatures. We proposed a simple experiment to detect such behavior. This is sketched beside: after preparing an ion at a given location in a cold sample (e.g. using two crossed laser beams), a small electric field is applied and the ion is detected. Depending on the temperature, two time scales should occur: at higher temperature where the charge is moving with the "heavy" ion, it would take more time to accelerate the charge to the detector, while at lower temperatures, the "light" hole would hop easily from atom to atom before resting on a last ion reaching the detector, leading to a shorter time scale.

We also investigate other neutral-charged mixtures. For example, we are interested in charge exchange processes when the ion and atom are isotopes of the same element. In that case, because of the slight shift in energy levels (arising mainly from the mass difference) the processes are not resonant, and the charge transfer can be identified since different isotopes have slightly transition frequencies/wavelengths. Alkaline-earth (group II) elements are particularly interesting, since both the ion and the neutral atom can be optically detected. We have been studying one of the simplest one: Be. We are introducing other interactions, such as the hyperfine and Zeeman interactions; as in the case of scattering in neutral samples, we expect to witness Feshbach resonances strongly affecting the collision and transport properties in mixed samples.

Another type of mixtures involving an ion in a cold sample of distinguishable neutral atoms is being studied experimentally by several groups. We initiated this effort using Ca⁺ ions immersed in an ultracold sample of Na atoms. Because charge transfer occurs by a radiative process for ground state collisions (i.e. a photon must be emitted to transfer the system from Na+Ca⁺ to Na⁺+Ca), the rate is small (i.e. the process not very fast), which allows cooling of Ca⁺ by elastic collisions with ultracold Na atoms. Several experimental groups are exploring similar systems (e.g., Rb with Ba⁺ or Yb⁺ or Ca⁺). One of the motivation is our prediction of the formation of large mesoscopic molecular ions when an ion is immersed into a BEC. As shown in the figure beside, the ion provides an attractive polarization potential, and when two atoms (yellow discs) scatter in its presence, one can be captured into a bound level with a rate W_cap, while the other atom takes the extra momentum q (red discs). In a BEC, this creates an excitation, or a quasi-particle (phonon) emission. We predicted large capture rates, and the existence of large metastable mesoscopic molecular ion "cluster" in steady-state (containing hundreds of atoms). (see Degenerate quantum gases, BEC and BCS for more details).