Spectroscopy of Rydberg states in erbium using electromagnetically induced transparency

We present a study of the Rydberg spectrum in ${}^{166}\mathrm{Er}$ for series connected to the $4f^{12}\left({}^{3}H_6\right)6s$, $J_c=13/2$, and $J_c=11/2$ ionic core states using an all-optical detection based on electromagnetically induced transparency in an effusive atomic beam. Identifying approximately 550 individual states, we find good agreement with a multichannel quantum defect theory (MQDT) which allows assignment of most states to $ns$ or $nd$ Rydberg series. We provide an improved accuracy for the lowest two ionization thresholds to $E_{\mathrm{IP},J_c=13/2}=49260.750\left(1\right){\mathrm{cm}}^{-1}$ and $E_{\mathrm{IP},J_c=11/2}=49701.184\left(1\right){\mathrm{cm}}^{-1}$ as well as the corresponding quantum defects for all observed series. We identify Rydberg states in five different isotopes, and states between the two lowest ionization thresholds. Our results open the way for future applications of Rydberg states for quantum simulation using erbium and exploiting its special open-shell structure.

Figure 1

Electronic levels and experimental setup. (a) Excitation scheme involving the probe transition at about $401\mathrm{nm}$ with single photon detuning ${\mathrm{\Delta }}_p$ and the coupling transition around $411\mathrm{nm}$. Also shown, the two-photon detuning ${\mathrm{\Delta }}_c$. (b) Schematic Rydberg level scheme of the two lowest ionization thresholds for different total $J$. Red (blue) lines indicate the $ns$ ($nd$) series. (c) Schematic drawing of the experimental setup with the vacuum apparatus, atomic beam, and probe, reference, and couple laser beams.

Figure 2

EIT spectroscopy and survey of Rydberg states. (a) Exemplary EIT resonance around $49147.967{\mathrm{cm}}^{-1}$. Here, the coupling beam is kept fixed while ${\mathrm{\Delta }}_p$ is scanned, showing the typical EIT signal of reduced absorption when on two-photon resonances. (b) EIT spectroscopy over a broad range of energies covering about 100 individual Rydberg states. ${\mathrm{\Delta }}_p$ is fixed to zero while the coupling laser frequency is scanned. The shaded area indicates the noise floor as a guide to the eye. (c) Energy of all observed Rydberg states extracted from the data as a function of the assigned effective principle quantum number. Ry states that are either above the first ionization threshold or comparatively broad are assigned to the second-lowest $J_c=11/2$ ionization threshold and are plotted against their corresponding principle quantum number (grey). Solid lines show the expected Rydberg energies using the simple Rydberg formula with the derived $E_{\text{ion},j}$.

Figure 3

Rydberg series: (a) Lu-Fano-style plot, showing the calculated ${\mu }_{13/2}$ of all observed states below the lowest ionization threshold as a function of the effective quantum number ${\nu }_{11/2}$ of the second-lowest ionization threshold. (b) Zoom into the EIT spectrum around $49148{\mathrm{cm}}^{-1}$, showing the states marked as red data points in (a). This bundle of resonances is identified as the $31d$ Ry state where the fine structure splitting leads to six features with $J=6,7$, or 8, see Sec. 4b for the assignment.

Figure 4

Zeeman mapping of the $27d$ fine structure multiplet. Blue thick lines show the photo diode signal of multiple averaged experimental data traces scanning the couple laser frequency. Red thin lines show the estimated spectral pattern using the fitted $g$ factors.

Figure 5

Lu-Fano plot of the $s$ series states. (a) All experimentally observed states assigned to the $J=6$ (blue circles) and $J=7$ (green diamonds) $s$ series attached to the lowest $13/2$ ionization threshold. The lines show the result of the MQDT calculations with energy dependence for the $ns$ series; the black bar indicates the ionization threshold. The dashed line indicates a broad and strong EIT feature, assigned to the $ns$ series of the $11/2$ threshold. The open orange circles indicate theoretically predicted $J=6$ states found experimentally in an additional survey. (b) Absolute values of the difference between the measured data and the MQDT model, with (grey circles) and without (red squares) energy dependence (see Sec. 5a for details on the model).

Figure 6

MQDT assignment of $nd$ Rydberg series. The lines show the results of the frame transformation approximation fitting. Small symbols show all experimental data, colored filled symbols show all states where we assigned $J$ experimentally. Large grey circles (plotted with their respective ${\mu }_{11/2}$) represent states which we assign to the $J_c=11/2$ threshold. Four of these are above the first threshold, while states for four different $n$ are identified as perturbers of the $d$ series attached to the $J_c=13/2$ threshold.

Figure 7

EIT spectrum of the possible $13g$ state at $49052.771{\mathrm{cm}}^{-1}$. A fine-structure splitting of four resonances is visible, the inset shows a second measurement run confirming the fourth resonance feature.

Figure 8

Absolute values of the transition matrix elements for the first transition (a) $|J_{\text{gs}}=6,m_{\text{gs}}\rangle \to |J_{\text{ex}}=7,m_{\text{ex}}\rangle$ as well as for the three possibilities of the second transition, (b) $|J_{\text{ex}}=7,m_{\text{ex}}\rangle \to |J=6,m_J\rangle$, (c) $|J_{\text{ex}}=7,m_{\text{ex}}\rangle \to |J=7,m_J\rangle ,$ and (d) $|J_{\text{ex}}=7,m_{\text{ex}}\rangle \to |J=8,m_J\rangle$. Circles (squares) encode ${\sigma }^{-}$ (${\sigma }^{+}$) transitions, diamonds display $\pi$ transitions.

Figure 9

Estimation of the spectral pattern given the relative two-photon transition strength for (a) $|J=6,m_J\rangle$, (b) $|J=7,m_J\rangle ,$ and (c) $|J=8,m_J\rangle$. Different colors and symbols encode the transition type (see legend). The inset shows the assumed relative distribution of populations among the ground-state manifold.