Micromotion in trapped atom-ion systems

We examine the validity of the harmonic approximation, where the radio-frequency ion trap is treated as a harmonic trap, in the problem regarding the controlled collision of a trapped atom and a single trapped ion. This is equivalent to studying the effect of the micromotion since this motion must be neglected for the trapped ion to be considered as a harmonic oscillator. By applying the transformation of Cook and Shankland we find that the micromotion can be represented by two periodically oscillating operators. In order to investigate the effect of the micromotion on the dynamics of a trapped atom-ion system, we calculate (i) the coupling strengths of the micromotion operators by numerical integration and (ii) the quasienergies of the system by applying the Floquet formalism, a useful framework for studying periodic systems. It turns out that the micromotion is not negligible when the distance between the atom and the ion traps is shorter than a characteristic distance. Within this range the energy diagram of the system changes remarkably when the micromotion is taken into account, which leads to undesirable consequences for applications that are based on an adiabatic process of the trapped atom-ion system. We suggest a simple scheme for bypassing the micromotion effect in order to successfully implement a quantum controlled phase gate proposed previously and create an atom-ion macromolecule. The methods presented here are not restricted to trapped atom-ion systems and can be readily applied to studying the micromotion effect in any system involving a single trapped ion.

Figure 1

A trapped atom-ion system. O is the center of the atom trap which is chosen as the coordinate origin; ${\vec{r}}_1$ and ${\vec{r}}_2$ are the position vectors of the atom and ion, respectively; and $\vec{d}$ is the position vector of the center of the ion trap.

Figure 2

(a) Eigenenergies and unperturbed quasienergies. The solid lines indicate the two eigenenergy levels, the dash ones show the zone structure of the unperturbed quasienergies. (b) A comparison of the exact quasienergies, represented by the dotted curves, with the unperturbed quasienergies. A typical avoided crossing caused by the oscillating field is marked by AC.

Figure 3

The unperturbed eigenergies of (a) the vibrational levels and (b) the low-lying molecular levels. The energy gaps between adjacent molecular levels are very large compared with those between the vibrational levels which appear as a continuum in the bottom plot.

Figure 4

A plot of (a) the eigenenergies $E_n$ together with $E_n-\omega$ shows the $\omega$ resonances and (b) the eigenenergies $E_n$ together with $E_n-2\omega$ shows the $2\omega$ resonances. These resonances appear as crossings.

Figure 5

Coupling strengths for (a) $V_1$ and (b) $V_2$ at $d=9l_i$ (diamonds), $d=5.5l_i$ (circles), $d=5.3l_i$ (squares), and $d=5.1l_i$ (crosses). The coupling strengths between the ground state and (c) an intermediate excited state $n=5$ and (d) a very high excited state $n=100$ of $V_1$ (diamonds) and $V_2$ (circles) are also plotted as functions of the trap distance.

Figure 6

(a) Quasienergies of the trapped atom-ion system plotted in the first zone $[-\hslash \omega /2,\hslash \omega /2]$. While the vibrational energy levels are affected by the micromotion at small trap distances, the energy curves for the low-lying molecular states stay in good agreement with the results obtained by the harmonic approximation. (b) An enlarged plot of the ground level Floquet state. The micromotion-induced avoided crossings appear and the energy curve begins to disintegrate around the characteristics distance $d_{\mathrm{mm}}\simeq 5.7l_i$.

Figure 7

Quasienergy levels obtained when only the first term $V_1$ of the micromotion is included, the energy curve of the ground level is similar to the result obtained by the harmonic approximation except that the new avoided crossings appear at what was the $\left(2\omega \right)$ resonances in the approximate energy diagram shown in Fig. 4.

Figure 8

A plot of the coupling strength $|\langle 0|V_{3,4}|5\rangle |$ as a function of trap distance.