Anisotropic polarizability of erbium atoms

We report on the determination of the dynamical polarizability of ultracold erbium atoms in the ground and in one excited state at three different wavelengths, which are particularly relevant for optical trapping. Our study combines experimental measurements of the light shift and theoretical calculations. In particular, our experimental approach allows us to isolate the different contributions to the polarizability, namely, the isotropic scalar and anisotropic tensor part. For the latter contribution, we observe a clear dependence of the atomic polarizability on the angle between the laser-field-polarization axis and the quantization axis, set by the external magnetic field. Such an angle dependence is particularly pronounced in the excited-state polarizability. We compare our experimental findings with the theoretical values, based on semiempirical electronic structure calculations, and we observe a very good overall agreement. Our results pave the way to exploit the anisotropy of the tensor polarizability for spin-selective preparation and manipulation.

Figure 1

Calculated (solid line) and measured (filled circles) atomic polarizability ${\alpha }_{\mathrm{tot}}$ of Er in the ground state for ${\theta }_p={\theta }_k=90{}^{\circ }$ as a function of the light-field wave number and wavelength in atomic units. A divergence of the polarizability indicates an optical dipole transition. The finite amplitude of the peaks of the narrow transitions are an artifact caused by the finite number of calculated data points. The red and blue shadows indicate that there is a broad red-detuned region for long wavelengths without many resonances and also a mostly blue-detuned region in the ultraviolet range. The inset illustrates the configuration of angles ${\theta }_k$ and ${\theta }_p$ for the shown data. B denotes the orientation of the magnetic field.

Figure 2

Ground-state polarizability of Er in the proximity of a narrow optical transition at 631.04 nm with a linewidth of $2\pi \times 28$ kHz. (a) Polarizability coefficients ${\alpha }_s$ (solid line), ${\alpha }_v$ (dotted line), and ${\alpha }_t$ (dashed line) vs the laser-field wavelength. The vertical dotted line indicates the zero crossing of ${\alpha }_s$. (b) Total polarizability ${\alpha }_{\mathrm{tot}}$ as a function of $m_J$, identifying the different Zeeman sublevels of the ground-state manifold for ${\theta }_p=90{}^{\circ }$ (circles), ${\theta }_{p0}=54.7{}^{\circ }$ (squares), and ${\theta }_p=0{}^{\circ }$ (stars) calculated with Eq. (3) for ${\theta }_k=90{}^{\circ }$ at 630.7 nm, corresponding to the wavelength of the zero crossing of ${\alpha }_s$.

Figure 3

Anisotropic polarizability of Er atoms in the ground state. The plot shows the relative change of the light shift at 532.26 nm (squares) and 1064.5 nm (circles) for ${\theta }_k=90{}^{\circ }$ as a function of ${\theta }_p$. The variation of the total light shift unambiguously reveals the tensor polarizability, which vanishes for an angle of ${\theta }_p\approx 54.7{}^{\circ }$. The lines are fits to the data with Eq. (3). The error bars indicate the statistical uncertainties from the trapping-frequency measurements. The dotted lines represent the theory prediction.

Figure 4

583 nm excited-state polarizability. (a) Illustration of the energy of atoms in an optical dipole trap with Gaussian shape. The upper (lower) panel indicates the case with the excited-state polarizability negative (positive). We measure the shift of the bare atomic resonance in the optical dipole trap (see text) for different values of ${\theta }_p$ (dark-to-light red and light-to-dark blue). This shift is given by the sum of the light shifts in the ground and in the excited state ($|J,m_J\rangle =|6,-6\rangle \to |J^{\prime },m_{J^{\prime }}\rangle =|7,-7\rangle$). To extract the excited-state light shift, we subtract the ground-state shift. (b) 583 nm excited-state polarizability for 1064.5 nm (red squares) and for 1570.0 nm (gray circles). The solid lines indicate fits to the data.