Controlled collisions of a single atom and an ion guided by movable trapping potentials

We consider a system composed of a trapped atom and a trapped ion. The ion charge induces in the atom an electric dipole moment, which attracts it with an $r^{-4}$ dependence at large distances. In the regime considered here, the characteristic range of the atom-ion interaction is comparable or larger than the characteristic size of the trapping potential, which excludes the application of the contact pseudopotential. The short-range part of the interaction is described in the framework of quantum-defect theory, by introducing some short-range parameters, which can be related to the $s$-wave scattering length. When the separation between traps is changed we observe trap-induced shape resonances between molecular bound states and vibrational states of the external trapping potential. Our analysis is extended to quasi-one-dimensional geometries, when the scattering exhibit confinement-induced resonances, similar to the ones studied before for short-range interactions. For quasi-one-dimensional systems we investigate the effects of coupling between the center of mass and relative motion, which occurs for different trapping frequencies of atom and ion traps. Finally, we show how the two types of resonances can be employed for quantum state control and spectroscopy of atom-ion molecules.

Figure 1

(Color online) Schematic drawing of a controlled collision between a single atom and a single ion, whose center-of-mass wave packets are guided by a time-dependent atom trap and ion trap, respectively. Labels distinguish different phases of the process: (a) initially particles prepared in the motional ground state; (b) collision (overlap of the wave packets); (c) excitation of the motional states in the traps; and (d) particles in some excited states after the collision.

Figure 2

(Color online) (a) Schematic drawing of the considered setup consisting of a trapped atom and ion. (b) and (c) Long and short-range regimes, respectively, in the controlled collisions between a trapped atom and ion.

Figure 3

(Color online) Contour plot of the total potential $U\left(\mathbf{r}\right)=V\left(\mathbf{r}\right)+\frac{1}{2}\mu {\omega }^2{\left(\mathbf{r}-\mathbf{d}\right)}^2$ for the relative motion in the atom-ion system. The figure shows the case of spherically symmetric trapping potentials with the same trapping frequencies, separated by $\mathbf{d}=\left(0,0,2R^{\ast }\right)$, and for $R^{\ast }=2l$ $\left[l=\sqrt{\hslash ∕\left(\mu \omega \right)}\right]$.

Figure 4

(Color online) Potential for the atom-ion relative motion (upper panel) and the relative wave function (lower panel) for the quasi-one-dimensional system with $l_{\perp }=0.09R^{\ast }$ and without trapping in the axial direction.

Figure 5

(Color online) Even and odd short-range phases ${\phi }_e$ and ${\phi }_o$ calculated for $l_{\perp }=R^{\ast }$. Numerical results (red solid lines) are compared with predictions of the model replacing $r^{-4}$ with the energy-dependent pseudopotential.

Figure 6

(Color online) (a) Even short-range phase ${\phi }_e$ and (b) odd short-range phase ${\phi }_o$ calculated numerically for $l_{\perp }=0.1R^{\ast }$, versus 3D short-range phase $\phi$.

Figure 7

(Color online) 3D short-range phase $\phi$ versus energy of bound states in the presence of the transverse confinement with $l_{\perp }=0.1R^{\ast }$. The bound states in the combined harmonic and $r^{-4}$ potentials (solid lines) are compared with the bound states of pure $r^{-4}$ interaction (dashed lines).

Figure 8

(Color online) Energy spectrum of the relative motion for an atom and an ion in 1D traps versus the distance $d$ between traps, calculated for ${\phi }_e=-\pi ∕4$, ${\phi }_o=\pi ∕4$, and $R^{\ast }=3.48l$ (see text for details). The dashed curve shows the line $\frac{1}{2}\mu {\omega }^2{d}^2$, giving the approximate shift of the bound state in the trapping potential (see Sec. 3C for derivation).

Figure 9

(Color online) Eigenstates of the relative motion for an atom and an ion in 1D traps at different separations $d$ between the traps, calculated for ${\phi }_e=-\pi ∕4$, ${\phi }_o=\pi ∕4$, and $R^{\ast }=3.48l$. Presented eigenstates correspond to labeled points in Fig. 8. The horizontal lines present the corresponding eigenenergies and the thick black line shows the potential energy.

Figure 10

(Color online) Energy spectrum of the relative motion for trapped atom and ion in 1D traps versus distance $d$ between traps, calculated for ${\phi }_e={\phi }_o=-\pi ∕4$ and $R^{\ast }=3.48l$ (see text for details).

Figure 11

(Color online) Eigenstates of the relative motion for trapped atom and ion in 1D at trap separation $d=0.1R^{\ast }$, calculated for ${\phi }_e={\phi }_o=-\pi ∕4$ and $R^{\ast }=3.48l$. The horizontal lines present the corresponding eigenenergies and the thick black line shows the potential energy.

Figure 12

(Color online) Energy spectrum of the relative motion for an atom and an ion confined in spherically symmetric traps versus the distance $d$ between traps, calculated for $\phi =-\pi ∕4$ and $R^{\ast }=3.48l_r$ (see text for details). The blue dashed line shows the shift of the bound states with $d$, given approximately by $\frac{1}{2}\mu {\omega }^2{d}^2$.

Figure 13

(Color online) Schematic drawing of the avoided crossing between vibrational and molecular states in the system of a trapped atom and an ion. An adiabatic change of the distance $d$ between traps induces a transition from the vibrational to the molecular states, while a diabatic process leaves the particles in the vibrational states.

Figure 14

(Color online) Energy splitting at the avoided crossing between the vibrational ground state and the molecular state in 1D and 3D systems. Semiclassical results (solid lines) are compared to the numerical values extracted from the adiabatic energy spectra for different combinations of ${\phi }_e$ and ${\phi }_o$. 3D results present avoided crossings widths for $s$-wave bound states.

Figure 15

(Color online) Energy splitting at the avoided crossing between the vibrational ground state and molecular states with different angular momenta for a 3D spherically symmetric trap. Semiclassical results obtained by means of the instanton technique (solid lines) are compared to the numerical values extracted from the adiabatic energy spectra. Semiclassical calculations stop at the distance $d=1.1R^{\ast }$, where the potential barrier disappears.

Figure 16

Energy spectrum for atom and ion confined in harmonic traps with ${\omega }_i=5.5{\omega }_a$ as a function of the distance $d$. Calculations are performed for ${\phi }_e=-\pi ∕4$, ${\phi }_o=\pi ∕4$, and $l_a=0.9R^{\ast }$ (see text for details).