1.3 Structured laser beam and atom

The development of structured laser beams such as Laguerre-Gaussian (LG) beams has enabled the coherent production of large number of identical spiral photons [44–47], which opens a new set of research in quantum optics in atomic level. LG beams are characterized by three quantum numbers: (i) the wave number k, (ii) the intertwined helical wave fronts l (an integer number) of azimuthal phasedependent that features a screw dislocation, and (iii) the radial nodes p [48, 50]. Each spiral photon in the LG beams carries lℏ of intrinsic orbital angular momentum (OAM) along the direction of propagation [51], which is arising from their nonuniform spatial intensity distribution. The characteristic shape of the intensity distribution of LG beams as well as their intrinsic orbital angular momentum can be observed as a result of this interaction. The exchange of angular momentum induces a torque and an azimuthal shift in the resonant frequency besides the usual axial Doppler shift and recoil shift which tailored the control of the motion of the center of mass of the atoms to rotate about the beam axis [52, 53]. In higher order quadrupole transition processes, the internal motion of the atom participates in orbital angular momentum exchange between structured light and atom [54–57]. However, in dipole transition, the interaction of LG beam with atom and therefore

transferring of OAM from LG beam to the external degree of freedom of the atom presents in the quantization of the center of mass motion of the atom [58].

A Rydberg atom with near classical size comparable to the wavelength of photon is big enough to feel phase differences of the helical wave front of LG beams. The LG beam has an advantage of control and narrows the resonances in electromagnetically induced transparency. The orbital angular momentum of LG beams has emerged as a popular choice for experiments on high-dimensional quantum information [59]. The dark center of blue-detuned Laguerre-Gaussian beam is emerged as an important feature to trap neutral atoms to decrease the atomic heating and decoherence rates and minimize the AC-Stark shift [60, 61]. In quantum information processing and data transmission, optical orbital angular momentum of spiral photons can provide an extra degree of entanglement [48] to carry quantum information in the different degrees of freedom by the higher dimensionality of the Hilbert space. In other words, the information carried by each photon can be increased significantly, from a qubit to a qudit, where d is the number of orthogonal basis vectors of the Hilbert space in which the photon lives.
