Direct observation of the optical Magnus effect with a trapped ion

Abstract

We directly observe and spatially map an optical analog of the Magnus effect, where intrinsic spin-orbit-like coupling of light generates a spin-dependent transverse displacement of the atom-light interaction profile for a $^{40}$Ca$^+$ ion. Probed on a quadrupole transition using a tightly focused beam, we observe displacements of the maximum in the profile of the effective interaction by several 100 nm originating from intrinsic longitudinal electric field components beyond the paraxial approximation. The tight focus of the beam induces additional transverse polarization gradients, which we characterize through a phase-sensitive measurement and spatial maps for different beam configurations. The results establish the physical basis of polarization-gradient interactions relevant to optical tweezer-based quantum control.

Figures (6)

• Figure 1: Linear polarization. Spatial profiles of the normalized quadrupole couplings $\ket{4S_{1/2},\,m_j}\rightarrow\, \ket{3D_{5/2},\,m_j+\Delta m_j}$, with $\Delta m_j=0,\pm1\pm2$, for linear polarization along $y$, and $\hat{\boldsymbol{\varepsilon}}\perp\mathbf{B}\perp \boldsymbol{k}$. At each position $(x,y)$, we apply the tweezer for a duration $T_{\text{probe}}$, calibrated to yield a $\pi$ pulse on each transition at the maximum Rabi frequency $\Omega_0$, and then measure the ground state population. The experimental maps (top row) are compared to simulated maps (bottom row) normalized to $\Omega_0$ of the $\Delta m_j=+2$ case (Eq. \ref{['eq:rabi freq']}), with scale factors given in each panel. Blue and green arrows in the center of the simulated maps highlight the transverse displacement arising from the optical Magnus effect. The $\Delta m_j=\pm1$ and $\Delta m_j=\pm2$ coupling profiles are displaced by approximately $\pm\lambda/2\pi\approx 115$ nm and $\pm\lambda/\pi\approx 230$ nm in the $y$ direction. We resolve the strongly suppressed $\Delta m_j=0$ transition by increasing the probe power $P_0$ by a factor of $\approx 100$. Differences between experiment and simulation likely arise from polarization angle sensitivity and beam aberrations. Vertical dashed lines mark the cuts for Fig. \ref{['fig:magnus_effect']}.
• Figure 2: Magnus effect displacement. Relative quadrupole coupling for $\hat{\boldsymbol{\varepsilon}}\perp\mathbf{B}\perp \boldsymbol{k}$ as a function of position $y$ around the maximum coupling indicated by blue and green vertical dashed lines in Fig. \ref{['fig:linear_pol']}. (a) For $\Delta m_j=\pm1$ we obtain $\Delta y=240(16)$ nm, in agreement with the theoretically predicted value of $\lambda/\pi=232$ nm. (b) For the $\Delta m_j=\pm2$ couplings we find $\Delta y= 463(20)$ nm, consistent with the calculated value $2\lambda/\pi=464$ nm. The measured ground state population is binned in steps of $0.2~\upmu$m and error bars indicate the standard deviation within each bin. The solid curves are Gaussian fits through which we find the separation between the minima caused by the optical Magnus effect.
• Figure 3: Circular polarization. Spatial profiles of the quadrupole couplings $\ket{4S_{1/2},\,m_j}\rightarrow\, \ket{3D_{5/2},\,m_j+\Delta m_j}$, with $\Delta m_j=0,\pm1\pm2$, for right hand circular polarization. The experimental tweezer maps (top row) are compared to simulated maps (bottom row) calculated using Eq. \ref{['eq:rabi freq']}. The scale factors in each panel give a relative scaling of the maximum Rabi frequency $\Omega_0$ for each transition with respect to the $\Delta m_j=+2$ case. Blue and green arrows highlight the optical Magnus effect displacement. Similar to the linear polarization case, the $\Delta m_j=\pm1$ and $\Delta m_j=\pm2$ couplings are displaced along the $y$ axis by approximately $\pm\lambda/2\pi\approx\pm 115$ nm and $\pm\lambda/\pi\approx\pm 230$ nm, respectively.
• Figure 4: Phase of the quadrupole coupling. (a) Signed Rabi frequency of the $\Delta m_j=+1$ transition with $\hat{\boldsymbol{\varepsilon}}\perp\mathbf{B}\perp \boldsymbol{k}$ as a function of tweezer position $x$, showing that the two lobes have opposite phase. (b) Pulse-sequence schematic on a Bloch sphere for detecting this phase: an initial $\pi/2$ pulse ($T_1$) is applied with the left lobe centered on the ion, followed by a second $\pi/2$ pulse ($T_2$) of equal duration at a tweezer position shifted along $x$. If the second pulse is carried out on the left lobe, it adds to an overall $\pi$ pulse; on the right lobe, it cancels the first pulse. (c) Measured ground state population versus tweezer position along $x$. The data are binned in $0.2~\upmu$m steps and error bars show the standard deviation within each bin. The shaded regions mark the positions of the two lobes.
• Figure 5: Experiment layout. A laser beam is tightly focused on a trapped $^{40}$Ca$^+$ ion confined in a linear Paul trap. The light of this optical tweezer at $\lambda = 729$ nm is tuned to the narrow $4S_{1/2}\leftrightarrow\,3D_{5/2}$ quadrupole transition. The polarization vector $\hat{\boldsymbol{\varepsilon}}$ illustrated here (with $\hat{\boldsymbol{\varepsilon}} \perp \mathbf{B}$) is the configuration used for the measurements in Fig. \ref{['fig:linear_pol']}.