2.2 Excitation laser beams

The four-level atom is driven by three laser fields, the Gaussian first and second excitation laser field with respective frequency ω<sup>1</sup> and ω<sup>2</sup> and the LG-Rydberg excitation laser field with frequency ω3. Neglecting the focusing and the radial complexity and the mode curvature of the excitation laser beams, the electric field of each one-photon transition in the cylindrical-polar coordinates is given by

$$\mathbf{E}\_{\mathsf{L},2}(R\_{\perp}, \Phi, Z) = \mathbf{e}\_{\mathsf{L},2} E\_{0\mathsf{L},2} f\_{G\mathsf{L},2}(R\_{\perp}) e^{i\left(\Theta\_{\mathsf{L},2}(\mathsf{R})\right)},\tag{1}$$

$$\mathbf{E}\_{\ell p}(\mathsf{R}\_{\perp}, \Phi, Z) = \mathbf{e}\_3 E\_{03} \xi\_{\ell p} \left(\frac{\sqrt{2}}{w\_{03}}\right)^{|\ell|} f\_{LG3}(\mathsf{R}\_{\perp}) \mathbf{e}^{i(\Theta\_{\mathbb{R}}(\mathbf{R}))},\tag{2}$$

where

$$f\_{G1,2}(R\_\perp) = e^{-\frac{\hbar^2\_\perp}{w^2\_{\text{01},2}}},\tag{3}$$

$$f\_{LG3}(R\_\perp) = R\_\perp^{|\ell|} e^{-\frac{R\_\perp^2}{w\_{03}^2}} L\_p^{|\ell|} \left(\frac{2R\_\perp^2}{w\_{03}^2}\right),\tag{4}$$

$$
\Theta\_{1,2}(\mathbf{R}) = \mathbf{k} \cdot \mathbf{R}, \tag{5}
$$

and

$$
\Theta\_3(\mathbf{R}) = \ell \rho + \mathbf{k}\_3 \cdot \mathbf{R}.\tag{6}
$$

Here ei, w0i, and k<sup>i</sup> are the polarization, beam waist at z = 0, and wave vector of the ith excitation laser field, respectively. The electric field amplitudes E0i are connected to the laser intensity I<sup>i</sup> via E0i = √(2Ii/cε0), where c is the speed of light and ε<sup>0</sup> is the vacuum permittivity. In Eq. (2), ℓ corresponds to the optical orbital angular momentum, the mode index p represents the number of radial nodes of LG qffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>p! j j beam, <sup>ξ</sup><sup>ℓ</sup><sup>p</sup> <sup>¼</sup> is the normalization constant, and <sup>L</sup> <sup>ℓ</sup> is the Laguerre poly- <sup>π</sup>ðpþj j <sup>ℓ</sup> <sup>Þ</sup> <sup>p</sup> nomial. The last excitation LG beam propagating along the z-direction in the plane of the focus of the beam appears as rings with radius R<sup>l</sup> that is scaled linearly to the optical angular momentum and proportional to the beam waist [78]. The polarization e<sup>i</sup> determines the particular transition conditions happened in ith laser beam. The appearance of the polarization vector e<sup>i</sup> depends explicitly upon the choice of

coordinate orientation. In the basis of eσ, where σ = {1, �1, 0} corresponds to the right circular, left circular, and linear polarization vector attached to the quantization axis z in xyz frame, the e<sup>i</sup> are given by

$$\mathbf{e}\_{i}(\theta\_{i},\rho\_{i}) = \sum\_{\sigma} s\_{i\sigma}(\theta\_{i},\rho\_{i})\mathbf{e}\_{\sigma} \tag{7}$$

where siσðθi; φiÞ is the relation coefficient and φ<sup>i</sup> is the azimuthal angle of respective polarization according to Figure 2(b).

The ith laser excitation beam carries linear momentum ℏki, and spin angular momentum �ℏ per photon relates to the excitation laser polarization, if circularly polarized, while the last LG beam with an azimuthal phase dependence exp (ilϕ) in addition to linear and spin angular momentum carries orbital angular momentum that can be many times greater than the spin [49]. The combination of energy selectivity, associated with the laser light frequency, and sublevel selectivity, associated with polarization as well as angular momentum of light beam, provides good controls and manipulation over the qudits in quantum information processing.
