Photoexcitation of atoms by Laguerre-Gaussian beams

In a recent experiment, Schmiegelow et al. [Nat. Commun. 7, 12998 (2016)] investigated the magnetic sublevel population of ${\mathrm{Ca}}^{+}$ ions in a Laguerre-Gaussian light beam if the target atoms were just centered along the beam axis. They demonstrated in this experiment that the sublevel population of the excited atoms is uniquely defined by the projection of the orbital angular momentum of the incident light. However, little attention has been paid so far to the question of how the magnetic sublevels are populated when atoms are displaced from the beam axis by some impact parameter $\mathit{b}$. Here, we analyze this sublevel population for different atomic impact parameters in first-order perturbation theory and by making use of the density-matrix formalism. Detailed calculations are performed especially for the $4s{}^{2}S_{1/2}\to 3d{}^{2}D_{5/2}$ transition in ${\mathrm{Ca}}^{+}$ ions and for the vector potential of a Laguerre-Gaussian beam in Coulomb gauge. It is shown that the magnetic sublevel population of the excited ${}^{2}D_{5/2}$ level varies significantly with the impact parameter and is sensitive to the polarization, the radial index, as well as the orbital angular momentum of the incident light beam.

Figure 1

Comparison of the transverse momentum distribution $|v_{pm}\left(k_{\perp }\right)|$ for Laguerre-Gaussian beams with different radial indices $p$ and OAM $m$. Results in arbitrary units are shown for beams with waist $w_0=2.7\mu \mathrm{m}$ and photon energy $\hslash \omega =1.699$ eV.

Figure 2

Geometry for the excitation of a single atom by a Laguerre-Gaussian beam. The quantization axis ($z$ axis) is chosen along the propagation direction of the incident beam, and the origin of the coordinate system lies on the beam axis. The atom is supposed to lie at the beam focus ($z=0$), and its nucleus is shifted from the beam axis along the $x$ axis by the impact parameter $\mathit{b}$.

Figure 3

Top: Intensity profile $|u\left(r_{\perp }\right){|}^2$ of a Laguerre-Gaussian beam for different radial indices $p=0$ (left) and $p=1$ (right), while the helicity $\lambda =+1$ and OAM $m=1$ are kept constant. Results are shown in arbitrary units for the beam waist $w_0=2.7\mu \mathrm{m}$ and the photon energy $\hslash \omega =1.699$ eV at the beam focus ($z=0$). Bottom: Relative partial cross sections (32) for the $4s{}^{2}S_{1/2}{M}_i=-1/2\to 3d{}^{2}D_{5/2}$ excitation of a ${\mathrm{Ca}}^{+}$ ion by beams from above as a function of the atomic impact parameter $b$. The relative cross sections are compared for sublevels with $M_f=\pm 1/2$, $+3/2$, while those with $M_f=\pm 5/2,-3/2$ are very small and are not shown here.

Figure 4

Same as Fig. 3, but for the Laguerre-Gaussian beams with different OAM $m=0$ and $m=1$, while the helicity $\lambda =+1$ and the radial index $p=0$ are kept fixed.

Figure 5

Same as Fig. 3, but for the Laguerre-Gaussian beams with different helicities $\lambda =\pm 1$, when the OAM $m=1$ and the radial index $p=2$ are kept constant. In the case of helicity $\lambda =+1$ ($\lambda =-1$), the relative cross sections are compared for sublevels with $M_f=\pm 1/2,+3/2$ ($M_f=-5/2,-3/2,-1/2$), while those with $M_f=\pm 5/2,-3/2$ ($M_f=+1/2,+3/2,+5/2$) are very small and are not shown here.

Figure 6

Density matrix $\langle {\alpha }_f{J}_f{M}_f\left|{\widehat{\rho }}_f\right|{\alpha }_f{J}_f{M}_f^{\prime }\rangle$ of the excited $3d{}^{2}D_{5/2}$ state of a ${\mathrm{Ca}}^{+}$ ion following the $4s{}^{2}S_{1/2}{M}_i=-1/2\to 3d{}^{2}D_{5/2}$ excitation by a Laguerre-Gaussian beam. Here, the absolute values of density matrix elements (31) are displayed for three different impact parameters $b$. Calculations were performed for a beam with helicity $\lambda =+1$, radial index $p=0$, OAM $m=1$, beam waist $w_0=2.7\mu \mathrm{m}$, and photon energy $\hslash \omega =1.699$ eV.