2.3.3 Transition selection rules

To coherently excite a single atom to a Rydberg state for quantum information processing, it is crucial to determine the strength of the radial and angular momentum couplings between the ground and Rydberg states. Moreover, the quantum

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

qudit encoding is based on the spin and orbital angular momentum exchange in the interaction of atom with excitation laser beams, which is expressed in angular momentum coupling term. The internal atomic wave function in spherical polar coordinate is written as

$$
\Psi\_{nlm}(\mathbf{r}) = u\_{nl}(r)Y\_{lm}(\theta,\varphi),
\tag{21}
$$

where unl(r) is the radial part of the electron wave function, which for a Rydberg state can be approximated using quantum defect theory [82], and n, l, and m are quantum numbers, which characterize the atom states. Substituting Eq. (21) into Eqs. (13) and (14) and considering the polarization orientation with respect to the quantization axis, the Rabi frequency of each transition can be given by

� �<sup>1</sup> <sup>2</sup> ðπð π <sup>8</sup> <sup>2</sup> <sup>π</sup>I<sup>1</sup> <sup>Ω</sup>1ð0Þ ¼ eRn0l0!n1l<sup>1</sup> ð Þ<sup>r</sup> <sup>∑</sup>σs1<sup>σ</sup>ðθ1; <sup>φ</sup>1<sup>Þ</sup> sin <sup>θ</sup>dθdφY<sup>m</sup><sup>0</sup> <sup>∗</sup>Y1 <sup>σ</sup>Y<sup>m</sup><sup>1</sup> <sup>l</sup><sup>0</sup> <sup>l</sup><sup>1</sup> 3ℏ<sup>2</sup> ε0c 0 0 (22) � �<sup>1</sup> <sup>2</sup> <sup>2</sup>I<sup>1</sup> <sup>¼</sup> eRn0l0!n1l<sup>1</sup> ð Þ<sup>r</sup> <sup>β</sup>1dp <sup>ℏ</sup><sup>2</sup> ε0c � �<sup>1</sup> <sup>2</sup> ðπð π <sup>8</sup> <sup>2</sup> <sup>π</sup>I<sup>2</sup> <sup>Ω</sup><sup>2</sup> <sup>0</sup> eRn1l1!n2l<sup>2</sup> ð Þ∑σs2<sup>σ</sup>ð Þ sin <sup>θ</sup>dθdφY<sup>m</sup> l1 1 1Y<sup>m</sup> l2 <sup>2</sup> <sup>ð</sup> Þ ¼ <sup>r</sup> <sup>θ</sup>2; <sup>φ</sup><sup>2</sup> <sup>∗</sup>Y<sup>σ</sup> 3ℏ<sup>2</sup> ε0c 0 0 (23) � �<sup>1</sup> <sup>2</sup> <sup>2</sup>I<sup>2</sup> <sup>¼</sup> eRn1l1!n2l<sup>2</sup> ð Þ<sup>r</sup> <sup>β</sup>2dp <sup>ℏ</sup><sup>2</sup> ε0c � �<sup>1</sup>rffiffiffiffiffi <sup>2</sup> ðπð π <sup>2</sup> <sup>n</sup>3l<sup>3</sup> <sup>8</sup>πI<sup>3</sup> <sup>32</sup> eRn2l2! ð Þ<sup>r</sup> <sup>Ω</sup>3ð0Þ ¼ <sup>∑</sup>σs3<sup>σ</sup>ðθ3; <sup>φ</sup>3<sup>Þ</sup> sin <sup>θ</sup>dθdφY<sup>m</sup><sup>2</sup> ∗Y<sup>σ</sup> 1Y1 1 Y<sup>m</sup><sup>3</sup> <sup>l</sup><sup>2</sup> <sup>l</sup><sup>3</sup> 3ℏ<sup>2</sup> ε0c 3 w<sup>3</sup> 0 0 � �<sup>1</sup> <sup>16</sup>I<sup>3</sup> <sup>2</sup> eRn2l2!n3l<sup>3</sup> ð Þ<sup>r</sup> <sup>¼</sup> <sup>β</sup>3qp <sup>π</sup>ℏ<sup>2</sup> ε0c w<sup>03</sup> (24)

where Rni�1, li�<sup>1</sup> ! ni, li represent the overlap integral between the radial wave functions of the electron and the dipole-quadrupole moment and βli� are the <sup>1</sup>mi�1!limi angular coupling expressed in terms of Clebsch-Gordan coefficients [77]. βli�<sup>1</sup>mi�1!limi contributes in precise quadrupole Rydberg excitation with elemental parameters θ<sup>i</sup> and Φi. Different combinations of θ<sup>i</sup> and Φ<sup>i</sup> provide all possible transitions accessible, which is applicable in precision measurements [83]. In case of θ<sup>3</sup> = 0 as shown in Figure 2(a), for left and right circularly polarized LG beam with p = 0 and ℓ = 1, the angular momentum transferred to the internal motion of the Rydberg atom via quadrupole transition is |Δm| = 2 and 0, respectively. Therefore, there is a possibility for the Rydberg atom to gain two units of angular momentum due to the quadrupole LG excitation: one from the polarization and the other from optical orbital angular momentum of LG beam. Additionally, the radial overlap integral of the quadrupole transition Rn2l2!n3l<sup>3</sup> ð Þr is considerable for high-lying Rydberg state with respect to w03, which increases the quadrupole Rabi frequency. Moreover, the w03-dependence of the Rydberg excitation Rabi frequency reflects the fact that the electric quadrupole transitions for an LG beam scales with w03 (compared to the plane wave, which scales with wave number k). Therefore, a relevant focusing with respect to the diffraction limit in addition to sufficient LGbeam power enhances the probability of the effective quadrupole excitation.
