Improving efficiency of sympathetic cooling in atom-ion and atom-atom confined collisions

We propose a way for sympathetic cooling of ions in an electromagnetic Paul trap: it implies the use for this purpose of cold buffer atoms in the region of atom-ion confinement-induced resonance (CIR). The problem is that the unavoidable micromotion of the ion and the long-range nature of its interaction with the environment of colder atoms in a hybrid atomic-ion trap prevent its sympathetic cooling. We show that the destructive effect of ion micromotion on its sympathetic cooling can, however, be suppressed in the vicinity of the atom-ion CIR. Here, the resonant blocking of the complete approach of an atom with an ion during a collision also blocks the enhancement of its micromotion. We investigate the effect of sympathetic cooling around CIRs in atom-ion and atom-atom confined collisions within the quasiclassical-quantum approach using the $\mathrm{Li}-{\mathrm{Yb}}^{+}$ and Li-Yb confined systems as an example. In this approach, the Schrödinger equation for a cold light atom is integrated simultaneously with the classical Hamilton equations for a hotter heavy ion or atom during collision. We have found the region near the atom-ion CIR where the sympathetic cooling of the ion by cold atoms is possible in a hybrid atom-ion trap. We also show that it is possible to improve the efficiency of sympathetic cooling in atomic traps by using atomic CIRs.

Figure 1

Schematic representation of the atom-ion system confined in a hybrid trap. Here, the ion is situated in the cloud of cold atoms confined by an optical atomic trap inside the electromagnetic Paul trap. The time-dependent rf field confines the ion transversally, whereas longitudinally a static confinement is formed by the dc field. The dimensions of the confinement region of the ion are determined by the frequencies of the Paul trap ${\omega }_i$ and ${\mathrm{\Omega }}_{\text{rf}}$. The atomic waveguide along the longitudinal axis, $z$, of the linear Paul trap confines the atoms in the transverse $x,y$ directions. The width of the atomic trap $a_{\perp }=\sqrt{\hslash /\left(m_a{\omega }_{\perp }\right)}$ is determined by the frequency ${\omega }_{\perp }$ of the harmonic approximation for the trap shape. Inside the hybrid trap occur paired atom-ion collisions.

Figure 2

The calculated atom scattering amplitude $f^{+}\left(t\right)$, transmission coefficient $T\left(t\right)$, coupling constant $g_{1\text{D}}\left(t\right)$, and the ion deviation from the center of the trap $r_i\left(t\right)$ for the ion being initially at rest (i.e., with zero energy before the collision with the atom) for four different values of the ratio $a_{\perp }/a_s$. This parameter fixes the coupling constant $g_{1\text{D}}$ (13) for four different cases: resonant atom-ion repulsion near the atom-ion CIR (left panel: $a_{\perp }/a_s=1.56$ at $b^2=0.179R^{*2},c^2=0.1R^{*2}$ in Eq. (7)), strong atom-ion repulsion (center panel: $a_{\perp }/a_s=2.64$ at $b^2=c^2=0.1R^{*2}$), very weak atom-ion attraction (right panel: $a_{\perp }/a_s=-260$ at $b^2=0.01R^{*2},c^2=0.06R^{*2}$), and strong attraction (inset in bottom right plot: $a_{\perp }/a_s=1.45$ and $g_{1\text{D}}=-68$ at $b^2=0.189R^{*2},c^2=0.1^{*2}$).

Figure 3

Schematic representation of the spectrum of the atom-ion system confined in a hybrid atom-ion trap as a function of the ratio $a_{\perp }/a_s$. The pair $(n,I)$ indicates the atom quantum number $n$ and the set of ion quantum numbers $I$. The point of the cross of the energy curve of the first excited state with respect to the atomic motion $(2,I)$ with the threshold of the entrance channel of the system $(0,I)$ defines the position of the CIR on the $a_{\perp }/a_s$ axis.

Figure 4

The calculated mean ion energy $\langle E_i^{\left(\text{out}\right)}\rangle$ after atom-ion collision, effective coupling constant $g_{1\text{D}}$, transmission coefficient $T$, and the molecule formation probability $P_{\text{mol}}$ as a function of the ratio $a_{\perp }/a_s$. The black circles connected by the solid black lines indicate the results of calculations performed for the “head-on collisions” with the initial conditions for the ion (18). The open circles are connected by the solid red lines related to “not head-on collisions” obtained with the initial conditions for the ion (19).

Figure 5

The calculated time evolution of the ion momentum ${\mathbf{p}}_i\left(t\right)$ and its kinetic energy $E_i\left(t\right)$ during the collision with the atom. The left column of graphs (graphs (a)) presents the result of the calculation with the potential of atomic-ion interaction $V_{ai}$ fixing the ratio $a_{\perp }/a_s=1.697$ slightly to the right of the CIR and with the initial condition (19) specifying the 3D motion of the ion in the Paul trap before the collision. The results presented in the central and right columns of the graphs (graphs (b) and (c)) were obtained with the initial condition (18) specifying the 2D motion of the ion in the Paul trap before the collision. The results shown in the central graphs (graphs (b)) were obtained with the potential $V_{ai}$ giving the ratio $a_{\perp }/a_s=1.576$ in the resonance region slightly to the right of the CIR. The right graphs (graphs (c)) illustrate the dynamics of the ion outside the resonance region for $a_{\perp }/a_s=1.796$.

Figure 6

The calculated in the time-independent secular approximation (22) mean ion energy $\langle E_i^{\left(\text{out}\right)}\rangle$ after Li-${\mathrm{Yb}}^{+}$ collision, effective coupling constant $g_{1\text{D}}$, transmission coefficient $T$, and the molecular ion formation probability $P_{\text{mol}}$ as a function of the ratio $a_{\perp }/a_s$.

Figure 7

The results of calculations with the potential (23) simulating Li-Yb collision in the time-independent secular approximation (22): the mean energy $\langle E_i^{\left(\text{out}\right)}\rangle$ of the Yb atom after collision, effective coupling constant $g_{1\text{D}}$, transmission coefficient $T$, and the molecule formation probability $P_{\text{mol}}$ as a function of the ratio $a_{\perp }/a_s$.