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arXiv:1707.04776 · Physical Review Letters

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

G. F. Quinteiro, Ferdinand Schmidt‐Kaler, Christian Tomás Schmiegelow

Published Dec 20, 2017• 91 citations
arXivHTMLPDF

Abstract

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}Ca^{+} ion from Schmiegelow et al. [Nat. Commun. 7, 12998 (2016)NCAOBW2041-172310.1038/ncomms12998] 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.

DOI
10.1103/physrevlett.119.253203

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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}Ca^{+} ion from Schmiegelow et al. [Nat. Commun. 7, 12998 (2016)NCAOBW2041-172310.1038/ncomms12998] 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.
Experimental evidence and theory on the head-on excitation of atoms by paraxial Laguerre–Gaussian beams revealed that the longitudinal component of the field has to be taken into account. Optical vortices are in fact a large family of fields, Laguerre–Gaussian being only one particular case. Here, we extend the previous study to a broader set of vortex fields. We demonstrate that, in general, paraxial optical vortices that have opposite orbital and spin angular momenta exhibit a longitudinal component that cannot be disregarded in the light–matter interaction.
It is well established that the longitudinal component of paraxial optical vortices has a key role in specific atomic quadrupole transitions near the beam axis when the spin and orbital angular momenta are antiparallel. By deriving analytical expressions for the position-dependent selection rules, this work shows that the significant role of the longitudinal fields is not limited to this case, but rather is a more general feature for any paraxial light including, for example, optical vortices with parallel spin and orbital angular momenta as well as Guassian beams. Numerically, the transition strengths induced by the weaker longitudinal component can be twice as high as those by the stronger transverse component. We also show that there are transitions that can be induced exclusively by the longitudinal component for light carrying two quanta of orbital angular momentum.
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 (Δ m = 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.
We study the interaction of a light beams carrying angular momentum with a single, trapped and well localized ion. We provide a detailed calculation of selection rules and excitation probabilities for quadrupole transitions. The results show the dependencies on the angular momentum and polarization of the laser beam as well as the direction of the quantization magnetic field. In order to observe optimally the specific effects, focusing the angular momentum beam close to the diffraction limit is required. We discuss a protocol for examining experimentally the effects on the S1/2_{1/2}1/2​ to D5/2_{5/2}5/2​ transition using a 40^{40}40Ca+^++ ion. Various applications and advantages are expected when using light carrying angular momentum: In quantum information processing, where qubit states of ion crystals are controlled, parasitic light shifts could be avoided as the ion is excited in the dark zone of the beam at zero electric field amplitude. Such interactions also open the door to high dimensional entanglement between light and matter. In spectroscopy one might access transitions which have escaped excitation so far due to vanishing transition dipole moments.
Twisted light has received a lot of attention in recent years. And there are justified reasons for this. The properties of twisted light are unusual and differ fundamentally from the plane-wave properties. The phase fronts of these light beams have a helical structure, giving them the name twisted. This helical phase structure originates from the well-defined orbital angular momentum (OAM) that they carry along their propagation direction and in addition to the spin angular momentum (SAM). Moreover, they have a characteristic intensity profile, which consists of concentric rings with a central minimum, and a phase singularity in the beam center. This thesis is based on two main parts, each dealing with a different aspect of twisted light. These investigations are carried out using Bessel beams, one type of twisted light. Twisted light beams are cylindrically symmetrical. Therefore, they are defined on the basis of circularly polarized light. This fact leads to the first question: Is it possible to generate twisted light in other polarization states? In this work, we derive how this can be achieved by using the superposition principle and how this can be used to construct linearly, radially, or azimuthally polarized Bessel beams. The focus in the second part is on excitations of atoms by twisted Bessel beams. In particular, we analyze how the use of twisted light can modify individual atomic multipole transitions. The obtained results show that for an efficient modification a precise localization of the target atom is required. In the last chapter, both aspects are combined. We analyze transitions between magnetic hyperfine states in a single trapped atom (or ion), which is exposed to an additional external magnetic field. The presented calculations for these multipole excitations are based on a formalism that accounts for the alignment of the applied external magnetic field with respect to the light propagation direction, and additionally different polarization states of the radiation field are considered. The obtained results indicate that twisted light could be of particular interest for precision experiments. One possible application is the use in single-ion clocks. The work shows that high multipole orders of the vortex beams allow excitations in the intensity minimum in the center of these beams without causing large disturbances, and thus may enable a significant reduction of the light shift in atomic transitions.
We examine the applicability and limitations of several established measures of paraxiality in the context of electric quadrupole interactions between light and trapped atoms. Focusing on Laguerre–Gaussian beams, we compare analytical predictions—particularly transition amplitudes and selection rules—derived under the paraxial assumption with full numerical solutions obtained via diffraction integrals that account for the complete vectorial field structure. This comparison allows us to assess the validity and relevance of these measures across a range of beam parameters. We also investigate the role of the power-series expansion in the paraxial parameter in bridging paraxial and nonparaxial regimes.