Long-Lived Mesoscopic Entanglement outside the Lamb-Dicke Regime

We create entangled states of the spin and motion of a single ${}^{40}\mathrm{Ca}^{+}$ ion in a linear ion trap. We theoretically study and experimentally observe the behavior outside the Lamb-Dicke regime, where the trajectory in phase space is modified and the motional coherent states become squeezed. We directly observe the modification of the return time of the trajectory, and infer the squeezing. The mesoscopic entanglement is observed up to $\Delta \alpha =5.1$ with coherence time $170\mu \mathrm{s}$ and mean phonon excitation $\overline{n}=16$.

Figure 1

Trajectories in phase space $\alpha (\tau )$ (dotted and dashed lines) and motional Wigner function (contour lines at 0,1,2 standard deviations) for the parameter values ${\omega }_0/2\pi =536\mathrm{kHz}$, $\eta =0.244$, $\Omega /2\pi =93\mathrm{kHz}$, $\delta /2\pi =3.4\mathrm{kHz}$ and varying forcing time $\tau$. The trajectories are shown for the two spin states for ${\Phi }_w=1.41$, up to the time $2\pi /\delta =286\mu \mathrm{s}$. ${\Phi }_w=|{\varphi }_{\uparrow }-{\varphi }_{\downarrow }|$ is the phase angle between the forces on the two spin states. The return time is $t_r=192\mu \mathrm{s}$. For clarity, the Wigner function is shown for just one of the trajectories at three example times, $\tau =0$, 96, and $286\mu \mathrm{s}$. The squeezing is $\simeq 3$ at $t_r/2$. In the experiments, superpositions of motion along both members of such pairs of trajectories are created.

Figure 2

Observed fringe amplitudes as a function of $\tau$, normalized to the value at $\tau =0$. Each point is obtained from a sinusoidal fit to a scan over $\varphi$. Dashed horizontal lines separate data sets taken at different trap frequencies. Dashed vertical lines are drawn at $\tau =2\pi /\delta$.

Figure 3

Observed phase offset of the fringes as a function of $\tau$, successive data sets are offset by 20 rad.