Bosonic Josephson Junction Controlled by a Single Trapped Ion

We theoretically investigate the properties of a double-well bosonic Josephson junction coupled to a single trapped ion. We find that the coupling between the wells can be controlled by the internal state of the ion, which can be used for studying mesoscopic entanglement between the two systems and to measure their interaction with high precision. As a particular example we consider a single ${}^{87}\mathrm{Rb}$ atom and a small Bose-Einstein condensate controlled by a single ${}^{171}\mathrm{Yb}^{+}$ ion. We calculate interwell coupling rates reaching hundreds of Hz, while the state dependence amounts to tens of Hz for plausible values of the currently unknown $s$-wave scattering length between the atom and the ion. The analysis shows that it is possible to induce either the self-trapping or the tunneling regime, depending on the internal state of the ion. This enables the generation of large scale ion-atomic wave packet entanglement within current technology.

Figure 1

Bose-Einstein condensate trapped in a double well potential with an ion in the center. The internal state of the ion can be tuned by laser light and controls the tunneling rate $\widehat{J}$.

Figure 2

(a) Wave functions for the ground states ${\Phi }_g(z)$ (blue or dark) and ${\Phi }_e(z)$ (gray), at $q=2.06R^{*}$; the black curve shows the potential with ${\omega }_q=2\pi$ 3.9 kHz. The wave functions are multiplied by a constant for clarity. (b) Spectrum for a double well as a function of $q$ and short-range phases of $\varphi =-\pi /4$ (solid lines) and $\varphi =-\pi /3$ (dashed lines) with the ground states shown in bold gray and blue or dark. The dashed vertical line marks the separation $q=2.06R^{*}$.

Figure 3

Interwell coupling $J$ for various short-range phases with ${\omega }_q=2\pi$ 3.9 kHz and $q=2.06R^{*}$.

Figure 4

(a) Time dependence of the interwell distance $q$ for generating state dependent tunneling. (b) Probability distributions at various times. The initial state prepared at $q=2.3R^{*}$ is an equal superposition of the two ground states $\psi (z,t=0)=({\Phi }_g(z)-{\Phi }_e(z))/\sqrt{2}$. At the end of the sequence the atom has tunneled when the ion is prepared in $|\uparrow ⟩$ (top panel b) and returned to the left well after tunneling multiple times when the ion is prepared in $|\downarrow ⟩$ (lowest panel). For the calculation we assume ${\varphi }_{\uparrow }=-\pi /4$ and ${\varphi }_{\downarrow }=-\pi /3$. At the end of the sequence $>98\%$ is in the right or left well, respectively.

Figure 5

Relative population in the left well as a function of time for $|\uparrow ⟩$ (black) and $|\downarrow ⟩$ (gray) $N=20$ atoms. Obtained by integrating Eq. (4) numerically, assuming ${\varphi }_{\uparrow }=-\pi /4$ and ${\varphi }_{\downarrow }=\pi /4$.