Manipulation of single neutral atoms in optical lattices

We analyze a scheme to manipulate quantum states of neutral atoms at individual sites of optical lattices using focused laser beams. Spatial distributions of focused laser intensities induce position-dependent energy shifts of hyperfine states, which, combined with microwave radiation, allow selective manipulation of quantum states of individual target atoms. We show that various errors in the manipulation process are suppressed below ${10}^{-4}$ with properly chosen microwave pulse sequences and laser parameters. A similar idea is also applied to measure quantum states of single atoms in optical lattices.

Figure 1

(Color online) The scheme for manipulating single atoms in optical lattices using focused lasers and microwave radiation. (a) ${}^{87}\mathrm{Rb}$ hyperfine structure. ${\Delta }_0$ is the detuning of the focused laser from the transition $5^2{S}_{1∕2}\to 6^2{P}_{3∕2}$. (b) Position-dependent hyperfine splitting induced by the spatial distribution of the focused laser intensity. Microwave radiation couples two hyperfine states of atoms. (c) Optical potentials for atoms around the focused laser. The length unit is the wavelength $\lambda =850\mathrm{nm}$ of the optical lattice. The energy unit is the recoil energy $E_r$. Top: a 2D plot for atoms at state $\mid 0⟩$. Bottom: a 1D plot along $y=0$. Solid and dashed lines for states $\mid 0⟩$ and $\mid 1⟩$, respectively.

Figure 2

(Color online) (a) Time evolution of $\Delta \mid c_1\left(t\right){\mid }^2$ during the microwave pulse. ${\Omega }_0∕{\omega }_0=1.81$ is the Rabi amplitude for a pulse whose area is slightly larger than $\pi$. Atoms $A$ (dashed dotted) and $B$ (solid) with initial state $(c_0,c_1)=(\sqrt{\frac{1}{2}},i\sqrt{\frac{1}{2}})$; atoms $A$ (dashed) and $B$ (dotted) with initial state $(1,0)$. (b) $\Delta \mid c_1\left(t_f\right){\mid }^2$ is the population variation of atom $A$ after the microwave pulse. Initial states $(1,0)$ (dashed dotted); $(\sqrt{\frac{1}{2}},i\sqrt{\frac{1}{2}})$ (dotted), $(\sqrt{\frac{2}{3}},\sqrt{\frac{1}{4}}+i\sqrt{\frac{1}{12}})$ (dashed), and $(\sqrt{\frac{2}{3}},\sqrt{\frac{1}{4}}-i\sqrt{\frac{1}{12}})$ (solid).

Figure 3

Manipulation error $ϵ=\mid \mid c_1\left(t_f\right){\mid }^2-\mid c_1(-t_f){\mid }^2\mid$ for the nearest neighboring atom $B$. $ϵ$ is smaller for other atoms because they have larger detuning $\delta \left(r\right)$. $-7\le {\omega }_{0}t\le 7$ in the numerical integration of Eq. (2). Initial states are same as those in Fig. 2b.