Coherent states of a harmonic oscillator on a sphere in the motion of a trapped ion

The preparation of coherent states of the harmonic oscillator on a sphere is considered. Based on the nonlinear coherent states of the atomic center-of-mass motion of a trapped ion, one may control the laser-ion interaction in such a way that sphere coherent states are obtained as motional dark states of the system. The curvature can be controlled by the Rabi frequencies of the driving lasers. The nonclassical properties of the resulting quantum states are analyzed. By variation of the curvature of the sphere, the states can be tuned continuously between spin-squeezed states and motional Fock states.

Figure 1

Expansion coefficients $c_n$ of a sphere coherent state, cf. Eq. (16), versus $\lambda$, for $N=3$ and $\mu =0.25$; $c_0$, dashed line; $c_1$, dotted line; $c_2$, dashed-dotted line; $c_3$, solid line.

Figure 2

The normalized Rabi frequencies ${\Omega }_1∕{\Omega }_0$ (solid curve) and ${\Omega }_2∕{\Omega }_0$ (dotted curve) as functions of the curvature $\lambda$, for $N=2$ and $\mu =0.25$.

Figure 3

The normalized Rabi frequencies ${\Omega }_1∕{\Omega }_0$ (solid curve), ${\Omega }_2∕{\Omega }_0$ (dotted curve), and ${\Omega }_3∕{\Omega }_0$ (dashed curve) as functions of the curvature $\lambda$, for $N=3$ and $\mu =0.25$.

Figure 4

$\mid G(k,\phi )\mid$ as a function of $k$, for $\phi =0$, $N=2$, $\mu =0.25$, and various values of $\lambda$. $\lambda =0.1$, dashed-dotted line; $\lambda =0.5$, dashed line; $\lambda =1$, dotted line; $\lambda =2$, solid line. The thick solid curve shows $G_{\mathrm{gr}}\left(k\right)$.

Figure 5

$\mid G(k,\phi )\mid$ as function of $k$, for $\phi =0$, $N=2$, $\mu =0.25$, and various values of $\lambda$. $\lambda =2$, solid line; $\lambda =5$, dashed-dotted line; $\lambda =10$, dashed line, $\lambda =25$, dotted line. The thick solid curve shows $G_{\mathrm{gr}}\left(k\right)$.

Figure 6

$\mid G(k,\phi )\mid$ as function of $k$, for $\phi =0$, $N=3$, $\lambda =0.5$, and various values of $\mu$. $\mu =0.2$, dashed-dotted line; $\mu =1$, dashed line; $\mu =3$, dotted line; $\mu =10$, solid line. The thick solid curve shows $G_{\mathrm{gr}}\left(k\right)$.