Detecting Vacuum Entanglement in a Linear Ion Trap

We propose and study a method for detecting ground-state entanglement in a chain of trapped ions. We show that the entanglement between single ions or groups of ions can be swapped to the internal levels of two ions by sending laser pulses that couple the internal and motional degrees of freedom. This allows us to entangle two ions without actually performing gate operations. A proof of principle of the effect can be realized with two trapped ions and is feasible with current technology.

Figure 1

Scheme for detecting ground-state entanglement: the entanglement between two groups of ions is swapped to the internal level of ions $A$ and $B$ by sending separate laser pulses that induce an interaction between the internal levels and the position of each ion.

Figure 2

Histogram of the final density matrix of the internal levels of ions $A$ and $B$. The entanglement of formation of this state accounts for 97% of the computed initial ground-state motional entanglement.

Figure 3

Logarithmic negativity between two groups consisting of 1, 3, and 5 ions, as a function of their separation in a chain of 20 ions.

Figure 4

(a) Classical propagation of a perturbation originating at the center of the chain. (b) The commutation relation between the displacement operators of the $n$th ion and the central ion in a chain of 80 trapped ions, at different time slices.

Figure 5

The ratio $\eta$ for ions in a chain of $N=20$ ions as a function of the detuning $\delta$. (a) The ion probes are located at $l=6,15$ and $T=0.8$ (in units wherein ${\nu }_0=1$). (b) $l=10,11$ and $T=0.05$. The range $\eta >1$ signifies entanglement.