Ion-trap analog of particle creation in cosmology

We consider the transversal modes of ions in a linear radio-frequency trap where we control the time-dependent axial confinement to show that we can excite quanta of motion via a two-mode squeezing process. This effect is analogous to phenomena predicted to occur in the early universe, in general out of reach for experimental investigation. As a substantial advantage of this proposal in comparison to previous ones we propose to exploit the radial and axial modes simultaneously to permit experimental access of these effects based on state-of-the-art technology. In addition, we propose to create and explore entanglement between the two ions.

Figure 1

Pictorial representation of the main mechanism. (a) Two uncoupled pendula have independent ground states. (b) For two pendula coupled by a string, the ground state is an entangled state. (c) After removing the coupling suddenly, the entangled state remains such that each pendulum separately is in an excited state. (b) represents the initial quantum vacuum state. The sudden removal of the spring then corresponds to the tearing-apart of waves (e.g., due to the cosmic expansion), which then leads to the creation of entangled pairs of particles.

Figure 2

Two ions with coordinates ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ in the potential energy landscape of a nonisotropic harmonic trap with time-dependent axial trap frequency ${\omega }_{\text{ax}}\left(t\right)$ and radial frequency ${\omega }_{\text{rad}}$. The $z$ direction is not shown. Classical motion along the axial $x$ direction can amplify quantum fluctuations in the rocking mode $\delta {\widehat{y}}_{-}$ along the $y$ direction.

Figure 3

Example of a time-dependent axial confinement characterized by ${\omega }_{\text{ax}}\left(t\right)$ that leads to the classical motion illustrated in Fig. 4.

Figure 4

Numerically calculated trajectory $\mathrm{\Delta }x$ for two ions confined in the axial potential with frequency ${\omega }_{\text{ax}}\left(t\right)$ presented in Fig. 3. After the collision the ions oscillate relative to their initial equilibrium positions. The dashed red line shows the critical distance at which ${\mathrm{\Omega }}_{-}=0$, where the ion chain becomes instable.

Figure 5

Bogoliubov coefficient ${\beta }_{-}$ obtained (crosses) numerically or (asterisks) based on Eq. (23) for collisions induced by an axial potential as presented in Fig. 3, where the peak confinement ${\omega }_{\text{ax}}^{\text{max}}$ is varied.