2.3.2 Adiabatic approximation and effective two-level transition

With the presence of decoherency, the evolution of the atomic states in memoryless environment will be expressed as density matrix formalism [81]. However, storing quantum information for long periods needs a decoherence-free quantum memory. In order to have coherent excitation, the spontaneous photon scattering should be limited by far detuning of laser excitation frequency from the respective excited state. In the limit of very large intermediate detuning such that, |Δi| ≫ Ω<sup>i</sup> and |Δi| ≫ |Δ|, the population of intermediate states is very low, and the system can behave as a two-level system with an effective coupling between ground and Rydberg states [77]. Supposing that the atom is initially in the ground state |0>, the time dependence of the Rydberg state population by the GGLG-beam excitation scheme can be obtained from Eq. (12) as

$$\left|c\_{3}\right|^{2} = \frac{\left\|\Delta\_{\text{eff}}\right\|^{2}}{\left\|\Delta\_{\text{eff}}\right\|^{2} + \left\|\Delta\_{\text{eff}}\right\|^{2}}\sin^{2}\left(\sqrt{\left\|\Delta\_{\text{eff}}\right\|^{2} + \left\|\Delta\_{\text{eff}}\right\|^{2}}t\right),\tag{18}$$

where

$$
\Delta\_{\rm eff}(R\_{\perp}) = \Delta + \frac{\Omega\_3^2(R\_{\perp})}{4\Delta\_3} + \frac{\Omega\_1^2(R\_{\perp})}{4\Delta\_1} + \Delta\_{LG} \tag{19}
$$

and

$$
\Omega\_{\rm eff}(R\_\perp) = \frac{\Omega\_1(R\_\perp)\Omega\_2(R\_\perp)\Omega\_3(R\_\perp)}{4\Delta\_1\Delta\_3} \tag{20}
$$

are the effective detuning and resonant Rabi frequency characterizing effective quadrupole coupling between ground and Rydberg states due to the dipoledipole-quadrupole transition under the adiabatic approximation.

The transverse profile of three-photon GGLG effective quadrupole Rabi frequency has a narrow central peak compared to the one-photon quadrupole LG Rabi frequency without any sidelope. Then, according to Eq. (18), the nonzero excitation probability is limited to the position of a single atom which is localized at the center of the trap leading to enhancement of the high-accuracy single-atom excitation.
