Electric field control in ultralong-range triatomic polar Rydberg molecules

We theoretically explore the external electric field control of a species of ultralong-range molecules that emerge from the interaction of a ground-state polar molecule with a Rydberg atom. The external field mixes the Rydberg electronic states and, therefore, strongly alters the electric field seen by the polar diatomic molecule due to the Rydberg electron. As a consequence, the adiabatic potential energy curves responsible for the molecular binding can be tuned in such a way that an intersection with neighboring curves occurs. The latter leads to admixture of $s$-wave character in the Rydberg wave function and will substantially facilitate the experimental preparation and realization of this particular class of Rydberg molecule species.

Figure 1

A qualitative sketch (not to scale) of the Rydberg atom—polar perturber system is shown. We adopt the minimal energy configuration where the total dipole $\vec{d}$ is oriented by an external electric field ${\vec{F}}_{\mathrm{ext}}$. The dipole ${\vec{d}}_0$ of the polar perturber, on the other hand, possesses two different parity states, namely, being parallel or antiparallel to the external field, respectively. Here we depict the expected ground state configuration when the dipole resides between the Rydberg electron and the ionic core.

Figure 2

(a) Illustration of the basis sets used for calculating the potential energy surfaces stemming from the $n=34$ Rydberg manifold. The zero-order basis includes the $\{n=34,l>2\}$ manifold as well as the $37s$ quantum defect split state, energetically close to the $n=34$ manifold; the Stark effect of the $37s$ state has been included. The larger basis sets include the $n-\Delta n,\cdots ,n+\Delta n$ manifolds as well as the $\left\{\right(n+\Delta n+1,s,p,d),(n+\Delta n+2,s,p),(n+\Delta n,s\left)\right\}$ states and are labeled by $\Delta n$. For all the basis sets, only the $m_l=0$ is considered. (b,c) Convergence of the potential energy surfaces with respect to the basis set for zero external electric field and for $F_{\mathrm{ext}}=8.0$V/cm, respectively. In both panels, the solid black line corresponds to the largest basis set considered, $\Delta n=6$; the solid gray lines are for decreasing $\Delta n=5,4,3,2$. The red dashed line illustrates the zero-order results, for which already a good agreement with the $\Delta n=6$ result can be found.

Figure 3

Overview of the potential energy surfaces resulting from the interaction of a $n=34$ Rydberg atom with a polar perturber of strength $d_0=0.566$ Debye and a parity state splitting of $\Delta =2.228$ GHz. Solid lines: zero external field; dashed lines: $F_{\mathrm{ext}}=8.0$V/cm. The energy surfaces are calculated using the zero-order basis; the dotted lines indicate the results from a calculation using the $\Delta n=6$ basis set for the lowest three energy surfaces. The two lowest asymptotes are split by $\Delta$ due to parity.

Figure 4

The BO potentials for the Rydberg-polar molecule system are shown, calculated for Rb $(n=34)$ using the zero-order basis and the molecular parameters $d_0=0.566$ Debye, $\Delta =2.228$ GHz. The external electric field strengths considered are (a) $F_{\mathrm{ext}}=6.0$ V/cm, (b) $F_{\mathrm{ext}}=7.0$ V/cm, and (c) $F_{\mathrm{ext}}=8.0$ V/cm. In the outermost potential wells the modulus squared of the first few vibrational wave functions are indicated, calculated assuming the mass of the KRb molecule for the polar perturber. The insets show density plots of the electronic wave function that gives rise to the BO potential where the vibrational states are localized; the wave functions are depicted in cylindrical coordinates and for a fixed polar perturber position of $R=2250a_0$, which corresponds to the minimum of the outermost well.

Figure 5

(a) Stark effect $E_R$ of the outermost potential well (solid line), which follows the Stark effect of the unperturbed $n=34,l>2$ Rb Rydberg manifold (dashed lines); for the latter, the small $f$-wave quantum defect is included. Panel (b) shows the Stark effect of the outermost two potential wells (solid: $E_R$/right well, dashed: $E_L$/left well), as well as of the first two vibrational states supported by these wells, offset by the linear Stark effect Eq. (13) of the $n=34,l>2$ manifold. The parameters ($n=34$, $d_0=0.566$ Debye, and $\Delta =2.228$ GHz) are taken to mimic a KRb molecule, where the parity splitting corresponds to the splitting of the first and second rotational states; accordingly, the KRb reduced mass has been used for the determination of the vibrational states.

Figure 6

$s$-wave character $\beta$, cf. Eq. (14), of the first (solid lines) and second (dashed lines) vibrational states of the right (black) and left (red/gray) outer well; the first and second vibrational states show a very similar behavior and hence partially overlap. The zero-order basis set has been used for the determination of $\beta$. The gray shaded area corresponds to the regime of field strengths where the wells cross through the $37s$ states and hence the vibrational states could not be determined in a single-channel picture.