Twisted-Light–Ion Interaction: The Role of Longitudinal Fields

The propagation of light beams is well described using the paraxial approximation, where field components along the propagation direction are usually neglected. For strongly inhomogeneous or shaped light fields, however, this approximation may fail, leading to intriguing variations of the light-matter interaction. This is the case of twisted light having opposite orbital and spin angular momenta. We compare experimental data for the excitation of a quadrupole transition in a single trapped ${}^{40}{\mathrm{Ca}}^{+}$ ion from Schmiegelow et al. [Nat. Commun. 7, 12998 (2016)] with a complete model where longitudinal components of the electric field are taken into account. Our model matches the experimental data and excludes by 11 standard deviations the approximation of a complete transverse field. This demonstrates the relevance of all field components for the interaction of twisted light with matter.

Figure 1

Illustration of how different beams may or may not have a longitudinal field component. (a),(b) Transverse cut of Laguerre-Gaussian beams with orbital angular momentum $\ell =1$ and different spin or circular polarization $\sigma =\pm 1$. In the top rows (a) and (c) the spin and orbital angular momentum are parallel; in the bottom rows (b) and (d) they are antiparallel. The arrows indicate the projection of the field direction on this plane. (c),(d) The beam as it propagates after a lens. At the focus position the resulting component at the center of the beam is drawn in black. When the spin and orbital angular momentum are parallel there is no longitudinal component. Conversely, when they are antiparallel, there is a strong longitudinal component. The interference giving rise to a longitudinal component can be understood as well from a modal decomposition of the beam.

Figure 2

(a) Energy levels for the Zeeman split ${}^{40}{\mathrm{Ca}}^{+}$ quadrupole $S_{1/2}\text{−}{D}_{5/2}$ transition. The transitions are labeled from $a$ to $j$ and shown in different colors indicating the photon orbital angular momentum content. Note that there are two different configurations to drive the $\mathrm{\Delta }m=0$ transitions, either with $\ell =+1$ or $-1$. (b) Relative transition strengths expressed as ratios of Rabi frequencies $\mathrm{\Omega }$ for different transitions. Data for the $\ell =1$ transitions (light blue) and for the $\ell =-1$ transitions (dark purple) are plotted together with theory predictions. Dark orange bars show the prediction with the model presented in this Letter where longitudinal fields are taken into account. Light orange bars show the prediction where the longitudinal field is neglected. (c) Normalization is done to a chosen transition ${\mathrm{\Omega }}_a$ and to account for the difference in interaction strength between transitions driven with Gaussian or vortex beams with ${\mathrm{\Omega }}_{\mathrm{norm}}={\mathrm{\Omega }}_a({\delta }_{|\ell |,1}+{\delta }_{\ell ,0}k{w}_0)$. All measured transition strengths and predictions are labeled as in (a).