Probing the isolated vector magnetic field of structured laser beams by atoms

Abstract

Electric and magnetic fields are inherently coupled in an electromagnetic wave. However, structured light beams enable their spatial separation. In particular, azimuthally polarized laser beams exhibit a localized magnetic field on-axis without the electric counterpart. Recent study by Martin-Domene et al. [App. Phys. Lett. 124, 211101 (2024)] has shown that combining these beams enables the generation of locally isolated magnetic fields with a controllable direction and phase. In the present paper we propose a method to probe and characterize such magnetic fields by studying their interaction with a single trapped atom. In order to theoretically investigate magnetic sublevel populations and their dependence on the relative orientation and phase -- i.e. the polarization state -- of the isolated magnetic field, we use a time-dependent density-matrix method based on the Liouville-von Neumann equation. As illustrative cases, we consider the $2s^2 2p^2 \, {}^3P_0 \, - \, 2s^2 2p^2 \, {}^3P_1$, the $1s^2 2s^2 \, {}^1S_0 \, - \, 1s^2 2s 2p \, {}^3P_2$, and the $2 s^2 2p \, {}^2 P_{1/2} \, - \, 2 s^2 2p \, {}^2 P_{3/2}$ transitions in ${}^{40}$Ca$^{14+}$, ${}^{10}$Be, and ${}^{38}$Ar$^{13+}$, respectively. Our results indicate that monitoring atomic populations serves as an effective tool for probing isolated vector magnetic fields, which opens avenues for studying laser-induced processes in atomic systems where electric field suppression is critical.

Figures (6)

• Figure 1: Geometry for the excitation of a single atom by a superposition of two azimuthally polarized VB, which propagate in the laboratory frame $x$ and $y$ direction, respectively. The intersection point of the beam axes is chosen as the origin of the coordinate system. The position of a target atom with respect to the origin is characterized by the impact parameter $\boldsymbol{b}$. The quantization $(z)$ axis is perpendicular to the propagation directions of both beams. The bottom inset shows the squared magnetic (top row) and electric (bottom row) fields given by Eqs. \ref{['eq:CombinedEField']} and \ref{['eq:CombinedBField']} for weights $c_1=c_2=1/\sqrt{2}$, opening angle $\theta_k = 0.54^\circ$, and photon energy $\hbar \omega = 2.73$ eV corresponding to a wavelength of $455$ nm. The arrows indicate the polarization of the magnetic field at the origin. The electromagnetic field is shown only in the vicinity of the intersection point, since the target atom is localized there.
• Figure 2: (a) The $2s^2 2p^2 \, {}^3P_0 \, - \, 2s^2 2p^2 \, {}^3P_1$ magnetic dipole transition in ${}^{40}$Ca$^{14+}$. The arrows indicate the interaction with "magnetic light" (solid) and plane waves (dashed). Shown are $M_g=0 \rightarrow M_e=+1$ (black) and $M_g=0 \rightarrow M_e=-1$ (green) transitions. The atom is located at the origin of the coordinate system. (b) Normalized Rabi frequencies for these transitions as a function of the relative phase $\phi$ which determines the light polarization (top). Here, the weights are $c_1 = c_2 = 1/\sqrt{2}$, the photon energy is $\hbar \omega = 2.18$ eV, and the polar opening angle is $\theta_k = 0.54^\circ$. The values for "magnetic light" are increased by a factor of 50.
• Figure 3: (a) The $1s^2 2s^2 \, {}^1S_0 \, - \, 1s^2 2s 2p \, {}^3P_2$ magnetic quadrupole transition in ${}^{10}$Be. "Magnetic light" induces $M_g=0 \rightarrow M_e=\pm 2$ (red solid line) and $M_g=0 \rightarrow M_e=0$ (blue solid line) transitions. Plane waves induce $M_g=0 \rightarrow M_e=+1$ (black dashed line) and $M_g=0 \rightarrow M_e=-1$ (green dashed line) transitions. (b) Phase dependence of normalized Rabi frequencies. The photon energy is $\hbar \omega = 2.73$ eV. All other parameters are the same as in Fig. \ref{['fig:CaClikeRabiFrequencies']}.
• Figure 4: (a) The $2 s^2 2p \; {}^2 P_{1/2} \, - \, 2 s^2 2p \; {}^2 P_{3/2}$ magnetic dipole transition in ${}^{38}$Ar$^{13+}$. The arrows represent the interactions with "magnetic light" (\ref{['eq:VP_Admixture']}), and the wavy lines represent spontaneous decay. (b) Time dependence of the orientation $\mathcal{A}_{10}$ of the $2 s^2 2p \; {}^2 P_{1/2}$ ground state for different relative phases: $\phi = 0^\circ$ (red), $\phi = 45^\circ$ (blue), $\phi = 90^\circ$ (green), $\phi = 225^\circ$ (black), and $\phi = 270^\circ$ (magenta). (c) Steady-state orientation $\mathcal{A}_{10}$ as a function of the relative phase $\phi$. The photon energy is $\hbar \omega = 2.81$ eV. All other parameters are the same as in Fig. \ref{['fig:CaClikeRabiFrequencies']}.
• Figure 5: Same as Fig. \ref{['fig:SystemTimeAlignmentOrientation']} (c), but for the coefficients $c_\mathrm{1}=c_\mathrm{2}$ (black solid line), $c_\mathrm{1}=2c_\mathrm{2}$ (red dashed line), $c_\mathrm{1}=3c_\mathrm{2}$ (blue dash-dotted line), and $c_\mathrm{1}=4c_\mathrm{2}$ (green dotted line).