2.3.1 Doppler and recoil shift compensation

The axial Doppler shift of the first, second, and Rydberg states by inserting Eqs. (3) and (4) in Eq. (14) can be, respectively, given as

$$\begin{aligned} \Delta\_{1D} &= -k\_1 v\_1 \cos \theta\_1 + k\_1 v\_1 \sin \theta\_1, \\ \Delta\_{2D} &= -k\_2 v\_2 \cos \theta\_2 - k\_2 v\_2 \sin \theta\_2, \\ \Delta\_{3D} &= k\_3 v\_3 + \Delta\_{LG}. \end{aligned} \tag{16}$$

It is noticeable that Δ3D contains the azimuthal Doppler shift [80], ΔLG ¼ <sup>R</sup> vΦ ⊥ , in addition to the axial Doppler shift, where v<sup>Φ</sup> is the azimuthal velocity of the atom.

Doppler broadening due to atomic motion leads to imperfect Rydberg excitation which limits the fidelity of the entanglement that is created using Rydberg interactions. By adjusting θi, it is possible to get rid of the recoil and axial Doppler shift. According to Figure 2(a), the excitation is recoil-free when

$$k\_1 \sin \theta\_1 = k\_2 \sin \theta\_2. \tag{17}$$

By substituting Eqs. (5) and (6) into Eq. (16), it is found that this flexible geometry of the excitation system results in the axial Doppler-free excitation condition in which the total detuning Δ is independent from the atomic vibrations but not rotations.
