Motional entanglement with trapped ions and a nanomechanical resonator

We study the entangling power of a nanoelectromechanical system (NEMS) simultaneously interacting with two separately trapped ions. To highlight this entangling capability, we consider a special regime where the ion-ion coupling does not generate entanglement in the system and any resulting entanglement will be the result of the NEMS acting as an entangling device. We study the dynamical behavior of the bipartite NEMS-induced ion-ion entanglement as well as the tripartite entanglement of the whole system ($\text{ions}+\text{NEMS}$). We found some quite remarkable phenomena in this hybrid system. For instance, the two trapped ions initially uncorrelated and prepared in coherent states can become entangled by interacting with a nanoelectromechanical resonator (also prepared in a coherent state) as soon as the ion-NEMS coupling achieves a certain value, and this can be controlled by external voltage gate on the NEMS device. We also show that, dynamically, the tripartite entanglement presents a more pronounced robustness against the destructive effects of dissipation when compared to the bipartite content.

Figure 1

Sketch of the physical setup considered in this paper. Two trapped ions (traps not shown) interact with the flexural motion of a doubly clamped nanobeam which is charged by a bias gate not shown in the picture.

Figure 2

Logarithmic negativity for the ions when the system is initially prepared in a product of coherent states. Two coupling constants between the ions and the NEMS were considered: $\kappa =1$ (solid) and $\kappa =3$ (dashed). Also, $\omega =0.5$ has been considered.

Figure 3

Logarithmic negativity for the ions when the NEMS is prepared in a thermal state characterized by $\alpha$ and the ions are initially prepared in a product of coherent states. Thin lines correspond to zero temperature ($\alpha =1$) and thick lines correspond to a finite temperature ($\alpha =5$). For the NEMS-ions coupling we used $\kappa =1$ (solid) and $\kappa =3$ (dashed). Also, $\omega =0.5$ has been considered.

Figure 4

Logarithmic negativity for the ions as a function of time $t$ and $\alpha$, which is an increasing function of the temperature of the NEMS (see main text). The ions were initially prepared in a product of coherent states. The other parameters are $\omega =0.5$ and $\kappa =3$.

Figure 5

Open-system dynamics for bi- and tripartite entanglements in the system consisting of one NEMS coupled to two trapped ions. Thin lines refer to $\kappa =1$ and thick lines to $\kappa =1.5$. The other system parameters are $\zeta =0.01$ and $\omega =0.5$. The mean thermal occupation number in the reservoir is $\overline{N}=4.5$. Dotted lines correspond to ${\tau }_{012}$, dashed lines to $N_{01}$, and continuous lines to $N_{12}$.

Figure 6

Open-system dynamics for bi- and tripartite entanglements in the system for two different NEMS's decay constants. Thin lines refer to $\zeta =0.01$ and thick lines to $\zeta =0.02$. The other system parameters are $\kappa =1$ and $\omega =0.5$. The mean thermal occupation number in the reservoir is $\overline{N}=4.5$. All lines have the same meanings as in Fig. 5.

Figure 7

Open-system dynamics for bi- and tripartite entanglements in the system for two different temperatures. Thin lines refer to $\overline{N}=4.5$ and thick lines to $\overline{N}=15$. The other system parameters are $k=1$, $\omega =0.5$, and $\zeta =0.01$. All lines have the same meanings as in Fig. 5.

Figure 8

First local maximum of the logarithmic negativity (circles) and the period of the oscillation (squares) as a function of $\kappa$ for the same situation of Fig. 2. The continuous line is the numerical fit performed with (B2), (B3), and parameters (B4). The relative errors of these fits are smaller than $5\%$.

Figure 9

First local maximum of the logarithmic negativity as a function of $\alpha$. We consider here the same situation of Fig. 3, with $\kappa =1$ (circles), $\kappa =3$ (squares), and $\kappa =10$ (diamonds). We also plot a numerical fit for both set of points, the relative errors of these fits are smaller than $5\%$ for all points and $\omega =0.5$.

Figure 10

Behavior of entanglement as a function of $\overline{N}$ for the fixed instant of time $t=10$. We consider here $\kappa =1$, $\zeta =0.01$, and $\omega =0.5$. In the inset we plot the local maxima of entanglement as function of time for $\overline{N}=10$. We show the tripartite entanglement's measure (circles), the negativity of NEMS-ion subsystem (squares), and the bipartite ion-ion negativity (diamonds). We also plot a numerical fit for all sets of points.

Figure 11

Behavior of entanglements measure as a function of $\zeta$ for a fixed instant of time for $t=10$. We consider here $\kappa =1$, $\overline{N}=4.5$, and $\omega =0.5$. All lines and symbols have the same meanings as in Fig. 10.

Figure 12

Time in which the entanglement measures attain the maximum as a function of $\kappa$. We consider here $\overline{N}=4.5$, $\zeta =0.1$, and $\omega =0.5$. The relative error of these fits are smaller than $5\%$. All lines and symbols have the same meanings as in Fig. 10.