Magic Rydberg-Rydberg transitions in electric fields

Rydberg states of atoms and molecules are very sensitive to electric fields. This property makes them ideal electric-field sensors but is detrimental to applications of Rydberg states in quantum optics, quantum-information processing, and quantum simulation because of inhomogeneous Stark broadening and the resulting loss of quantum coherence. We demonstrate, with the example of Rydberg states of ${}^{39}\mathrm{K}$, the existence of Rydberg-Rydberg transitions with extremely small differential dc Stark shifts, which we call dc-field-magic Rydberg-Rydberg transitions. These transitions hardly exhibit any Stark broadening, even when the electric-field strength varies by as much as $10\mathrm{V}{\mathrm{cm}}^{-1}$ over the experimental volume. We present a systematic study of dc-field-magic Rydberg-Rydberg transitions combining experiment and calculations and classify them in three main types, which should also be encountered in the other alkali-metal atoms, in alkaline-earth-metal atoms, and even in molecules. The observed insensitivity to dc electric fields does not reduce the interactions between Rydberg atoms, even if they are dominantly electric dipole-dipole in nature. Rydberg states coupled by dc-field-magic Rydberg-Rydberg transitions, therefore, have great potential as qubits.

Figure 1

Energy-level diagram of Rydberg-Stark states of ${}^{39}\mathrm{K}$ as a function of the electric field. The Rydberg states involved in types I–III magic transitions are depicted in red in panels (a)–(c), respectively. The origin of the energy scale corresponds to the position of the $35d_{5/2}$ Rydberg state. See the text for details.

Figure 2

Fraction of transferred Rydberg atoms for the $(38,12,j^{-},1/2)\to (35,13,j^{-},1/2)$ transition as a function of the millimeter-wave frequency in an electric field of (a) $F_1=5.60\mathrm{V}{\mathrm{cm}}^{-1}$, (b) $F_2=7.51\mathrm{V}{\mathrm{cm}}^{-1}$, (c) $F_3=8.47\mathrm{V}{\mathrm{cm}}^{-1}$, and (d) $F_4=9.42\mathrm{V}{\mathrm{cm}}^{-1}$. (e) Comparison of experimental (dots) and calculated (full line) transition frequencies. The dashed line and gray area represent the average frequency and the maximal deviation over the field range ($3.4–9.4\mathrm{V}{\mathrm{cm}}^{-1}$). (f) UV-laser spectra of the $(38,12,j^{-},1/2)\leftarrow 4s_{1/2}$ transition of ${}^{39}\mathrm{K}$ recorded at the fields $F_1–F_4$.

Figure 3

Comparison of calculated Stark shift (full line) of the $(42,0,j^{+},3/2)\to (40,0,j^{+},1/2)$ transition of ${}^{39}\mathrm{K}$ to the experimentally observed Stark shifts (full circles) as a function of the applied electric-field $F$.