2.3 The interaction of a four-level atom with three excitation laser fields

Let us begin with the four-level atom introduced in Figure 1. In general, the state of the atom for the coupled atom-laser system can be written in terms of the eigenstates of Hatom, the unperturbed Hamiltonian for the atomic system in the absence of the excitation laser, as

$$
\langle \boldsymbol{\mu}(\mathbf{r}, t) \rangle = c\_0(t) \boldsymbol{\mu}\_0(\mathbf{r}) e^{-i \frac{E\_0}{\hbar} t} |\mathbf{0}\rangle + \sum\_i c\_i(t) \boldsymbol{\mu}\_i(\mathbf{r}) e^{-i \frac{E\_i}{\hbar} t} |i\rangle,\tag{8}
$$

where c<sup>i</sup> (i = 0, 1, 2, 3) is the probability amplitude of ith state. The Hamiltonian for the system can be taken as

$$
\stackrel{\frown}{H} = \stackrel{\frown}{H}\_{\text{atom}} + \stackrel{\frown}{H}\_{\text{intb}} \tag{9}
$$

\_ \_ \_ where Hint ¼ Hd þ Hq, where EðR⊥; Φ; ZÞ ¼ E1ðR⊥; Φ; ZÞ þ E1ðR⊥; Φ; ZÞ þ E3lpðR⊥; Φ; ZÞ is the total

\_ \_ electric field of excitation lasers, and where Hd and Hq are the coupling to the electric dipole and quadrupole moment, respectively [79]. The first two excitations proceed via dipole transition employing the first two Gaussian laser beams, while the Rydberg excitation takes place through the quadrupole transition using the polarized LG beam.

To study the dynamics of the population of the different levels of the atom, we need to know c<sup>i</sup> (t) by solving the Schrodinger equation:

$$i\hbar\frac{\partial}{\partial t}|\varphi(\mathbf{r},t)\rangle = \hat{H}\,'|\varphi(\mathbf{r},t)\rangle,\tag{10}$$

where

$$
\stackrel{\frown}{H}{}' = \stackrel{\frown}{U}\stackrel{\frown}{H}\stackrel{\frown}{U}^{-1} + -i\hbar \stackrel{\frown}{U} \frac{\partial}{\partial t} \stackrel{\frown}{U}^{-1}.\tag{11}
$$

In matrix notation, the Schrodinger equation after rotating wave approximation can be given by

Localized Excitation of Single Atom to a Rydberg State with Structured Laser Beam for Quantum… DOI: http://dx.doi.org/10.5772/intechopen.82319

$$\frac{\partial}{\partial t} \begin{pmatrix} c\_0(t) \\ c\_1(t)e^{j\Delta\_1 t} \\ c\_2(t)e^{j(\Delta\_2 + \Delta\_1)t} \\ c\_3(t)e^{j\Delta t} \end{pmatrix} = -i \begin{pmatrix} 0 & \frac{\Omega\_1}{2} & 0 & 0 \\ \frac{\Omega\_1}{2} & \Delta\_1 & \frac{\Omega\_2}{2} & 0 \\ 0 & \frac{\Omega\_2}{2} & \Delta\_1 + \Delta\_2 & \frac{\Omega\_3}{2} \\ 0 & 0 & \frac{\Omega\_3}{2} & \Delta \end{pmatrix} \begin{pmatrix} c\_0(t) \\ c\_1(t)e^{j\Delta\_1 t} \\ c\_2(t)e^{j(\Delta\_2 + \Delta\_1)t} \\ c\_3(t)e^{j\Delta t} \end{pmatrix}. \tag{12}$$

Here

$$\mathbf{D}\_{\mathbf{i}\_{12}}\left(\boldsymbol{R}\_{\perp}\right) = \left(\frac{2e^2 I\_{\mathbf{i},2}}{\hbar^2 c \varepsilon\_0}\right)^{\frac{1}{2}} \mathbf{r}\_{\mathbf{0},\mathbf{l}\rightarrow\mathbf{1},2} \cdot \mathbf{e}\_{\mathbf{l},2}\left(\boldsymbol{\theta}\_{\mathbf{l},2}, \boldsymbol{\varphi}\_{\mathbf{l},2}\right) f\_{G\mathbf{l},2}\left(\boldsymbol{R}\_{\perp}\right) \tag{13}$$

and

$$\mathfrak{Q}\_{3}(R\_{\perp}) = \left(\frac{2e^{2}I\_{3}}{\hbar^{2}c\epsilon\_{0}}\right)^{\frac{1}{2}} \frac{2r\_{\perp}\mathbf{r}\_{2\to 3}}{\sqrt{\pi}w\_{03}} \cdot \mathbf{e}\_{3}(\theta\_{3},\rho\_{3})\varepsilon^{i\rho} \frac{d}{dR\_{\perp}}f\_{LG3}(R\_{\perp})\tag{14}$$

are the first and second dipole and the last quadrupole Rydberg excitation Rabi frequencies, respectively, where <sup>r</sup><sup>i</sup>�1!<sup>i</sup> <sup>¼</sup> <sup>h</sup><sup>i</sup> � <sup>1</sup>jψ<sup>∗</sup> ð Þr rψið Þ<sup>r</sup> j i<sup>i</sup> stand for the atomic <sup>i</sup>�<sup>1</sup> dipole momentums between states i�1 and i, with respective eigenfunctions ψi-1(r) Ei�<sup>1</sup> and <sup>ψ</sup>i(r); <sup>Δ</sup><sup>i</sup> <sup>¼</sup> <sup>ω</sup><sup>i</sup> � Ei� <sup>þ</sup> <sup>Δ</sup>iD is the detuning of state <sup>i</sup> affected by residual <sup>ℏ</sup> Doppler shifts:

$$
\Delta\_{i\mathcal{D}} = \Theta\_i. \mathbf{v}\_i \tag{15}
$$

due to the random very small displacements and vibrations of the atom (see Eqs. (3) and (4)), even though the atom is cooled and trapped at the center of the LG beam, and <sup>Δ</sup> <sup>¼</sup> <sup>∑</sup><sup>3</sup> <sup>1</sup>Δ<sup>i</sup> <sup>¼</sup> <sup>E</sup><sup>3</sup> � <sup>E</sup><sup>0</sup> � <sup>∑</sup><sup>i</sup> <sup>ω</sup><sup>i</sup> stands for the total detuning to the <sup>i</sup><sup>¼</sup> Rydberg state.

According to Eq. (14), the quadrupole Rabi frequency of the Rydberg excitation step in addition to the polarization depends on the orbital angular momentum of the LG beam. The phase factor ei<sup>φ</sup> changes the parity symmetry and transfers one unit of optical orbital angular momentum to the internal motion of atom. The quadratic radial, r⊥r<sup>2</sup>!3, in Eq. (14) is the consequence of the transversal variation of the LGbeam intensity.
