Role of higher-order electric field corrections in the quadrupole interaction of twisted light with matter

Recent studies on twisted-light-matter interactions have highlighted the importance of the longitudinal electric field component, which had previously been overlooked. Together with the transverse component, it corresponds to the lowest-order terms in a perturbative expansion over the paraxial parameter. In this work, we investigate the influence of higher-order correction terms in the perturbative expansion on the quadrupole excitation of a trapped atom. This article begins by resolving discrepancies in the literature regarding the calculation of transverse fields at second and higher even orders using the appropriate gauge. A key finding of our study reveals that including a higher-order term in the perturbative expansion is not always sufficient to enhance accuracy in describing the interaction at a given level; instead, two successive orders may need to be considered together. This is particularly evident in quadrupole transitions involving no change in the magnetic quantum number (Δm = 0), where longitudinal corrections play a more significant role than their transverse counterparts, despite the latter being lower order. This behavior is attributed to the sensitivity of quadrupole interactions to field gradients and the connection between odd- and even-order fields via transverse gradients.