Generation and stabilization of entangled coherent states for the vibrational modes of a trapped ion

We propose a scheme for preparation of entangled coherent states for the motion of an ion in a two-dimensional anisotropic trap. In the scheme, the ion is driven by four laser beams along different directions in the ion trap plane, resulting in carrier excitation and couplings between the internal and external degrees of freedom. When the total quantum number of the vibrational modes initially has a definite parity, the competition between the unitary dynamics and spontaneous emission will force the system to evolve to a steady state, where the vibrational modes are in a two-mode cat state. We show that the method can be extended to realization of entangled coherent states for three vibrational modes of an ion in a three-dimensional anisotropic trap.

Figure 1

(a) Fidelity of the generated vibrational state to the expected state as a function of $\lambda t$ with the initial state $\left|g\right.〉{\left|0\right.〉}_a{\left|0\right.〉}_b$. The generated vibrational state is calculated by numerically solving Eq. (14), with the parameters ${\eta }_x=0.15$, ${\eta }_y=0.1$, $\varepsilon =16\lambda$, $\mathrm{\Gamma }=100\lambda$, and ${\gamma }_a={\gamma }_b=0.0005\lambda$. The corresponding expected state is ${\mathcal{N}}_{+}({\left|\alpha \right.〉}_a{\left|\alpha \right.〉}_b+{\left|-\alpha \right.〉}_a{\left|-\alpha \right.〉}_b)$ with $\alpha =2$. The black-solid and red-dashed lines represent the results with and without consideration of the dominant higher-order terms of the Hamiltonian, respectively. (b) Fidelity of the generated vibrational state to the expected state as a function of $\lambda t$ with the initial state $\left|g\right.〉\left({\left|1\right.〉}_a{\left|0\right.〉}_b+{\left|0\right.〉}_a{\left|1\right.〉}_b\right)/\sqrt{2}$. The corresponding expected state is ${\mathcal{N}}_{-}({\left|\alpha \right.〉}_a{\left|\alpha \right.〉}_b-{\left|-\alpha \right.〉}_a{\left|-\alpha \right.〉}_b)$ with $\alpha =2$. The unit of this figure is $\lambda$.

Figure 2

(a) Plane cut of the two-mode Wigner function $W(\beta ,\chi )$ along the $\mathrm{Im}(\beta$)-$\mathrm{Im}(\chi$) axes for the ideal cat state ${\mathcal{N}}_{+}{({\left|\alpha \right.〉}_a{\left|\alpha \right.〉}_b+|-\alpha \rangle }_a{|-\alpha \rangle }_b)$ with $\alpha =2$. (b) Plane cut of $W(\beta ,\chi )$ along the $\mathrm{Im}(\beta$)-$\mathrm{Im}(\chi$) axes obtained by numerically solving Eq. (8) with the initial state $\left|g\right.〉{\left|0\right.〉}_a{\left|0\right.〉}_b$. The parameters are the same as those in Fig. 1, and the dominant higher-order terms in the Hamiltonian are included. The unit of this figure is $\lambda$.

Figure 3

(a) Plane cut of $W(\beta ,\chi )$ along the $\mathrm{Im}(\beta$)-$\mathrm{Im}(\chi$) axes for the ideal cat state ${\mathcal{N}}_{-}{({\left|\alpha \right.〉}_a{\left|\alpha \right.〉}_b-|-\alpha \rangle }_a{|-\alpha \rangle }_b)$ with $\alpha =2$. (b) Plane cut of $W(\beta ,\chi )$ along the $\mathrm{Im}(\beta$)-$\mathrm{Im}(\chi$) axes for the vibrational modes obtained by simulating the master equation with the initial state $\left|g\right.〉\left({\left|1\right.〉}_a{\left|0\right.〉}_b+{\left|0\right.〉}_a{\left|1\right.〉}_b\right)/\sqrt{2}$. The parameters are the same as those in Fig. 1, and the dominant higher-order terms in the Hamiltonian are included. The unit of this figure is $\lambda$.