Theoretical method for the generation of a dark two-mode squeezed state of a trapped ion

Here we show how to generate a dark two-mode squeezed state of a trapped ion, employing a three-level ion in a V configuration with a strong decay of the excited states. The degree of squeezing can be manipulated by choosing the intensity of the driving fields. Our scheme is robust against the usual dissipation mechanism and could be implemented with present-day technology. The validity of the approximations employed in this work was tested by numerical calculations, which agreed completely with the analytical solutions.

Figure 1

Atomic levels of the trapped ion. The ground state $|0⟩$ is coupled to the excited states $|1⟩$ and $|2⟩$ through laser fields.

Figure 2

(Color online) The time evolution of the mean number of quanta, $⟨a^{†}a⟩$, of the vibrational mode $x$ for $\Gamma =10\lambda$, ${\overline{n}}_{th}=0.5$, $r=1$, and three values of the decay rate of the vibrational modes: $\gamma =0.001\lambda$ (solid line), $\gamma =0.01\lambda$ (dotted line), and $\gamma =0.1\lambda$ (dashed-dotted line). The dashed line (straight line) represents the expected value.

Figure 3

(Color online) The time evolution of the total variance $⟨{\left(\Delta \widehat{u}\right)}^2+{\left(\Delta \widehat{v}\right)}^2⟩$ for the same parameters used in Fig. 2.