Electrically dressed ultra-long-range polar Rydberg molecules

We investigate the impact of an electric field on the structure of ultra-long-range polar diatomic rubidium Rydberg molecules. Both the $s$-wave and $p$-wave interactions of the Rydberg electron and the neutral ground-state atom are taken into account. In the presence of the electric field the angular degree of freedom between the electric field and the internuclear axis acquires vibrational character and we encounter two-dimensional oscillatory adiabatic potential energy surfaces with an antiparallel equilibrium configuration. The electric field allows shifting of the corresponding potential wells in such a manner that the importance of the $p$-wave interaction can be controlled and the individual wells are energetically lowered at different rates. As a consequence the equilibrium configuration and corresponding energetically lowest well move to larger internuclear distances for increasing field strength. For strong fields the admixture of nonpolar molecular Rydberg states leads to the possibility of exciting the large angular momentum polar states via two-photon processes from the ground state of the atom. The resulting properties of the electric dipole moment and the vibrational spectra are analyzed with varying field strength.

Figure 1

Energy-dependent triplet phase shifts ${\delta }_1$ and ${\delta }_0$ for $e^{-}{-}^{87}$Rb($5s$) scattering. For $E_{\mathrm{kin}}=24.7$ meV the phase shift ${\delta }_1=\pi /2$, i.e., the (cubed) energy-dependent $p$-wave scattering length $A_p^3\left(k\right)=-\tan \left[{\delta }_1\left(k\right)\right]/k^3$ possesses a resonance at this energy. This can be clearly seen in the inset.

Figure 2

Two-dimensional potential energy surfaces for the electrically dressed polar trilobite states for $E=150\frac{V}{m}$ (a) and $300\frac{V}{m}$ (b). We observe a potential minimum at $\theta =\pi$. An increase of the electric field goes along with a stronger confinement of the angular motion and an increase of the diatomic equilibrium distance $R_{\mathrm{eq}}$. Thus, the electric field stabilizes the $s$-wave-dominated molecular states.

Figure 3

(a) Intersections through the two-dimensional PES for $\theta =\pi$ for the 9th to 15th excitation for $E=300\frac{V}{m}$. For the two lowest PESs arising from the high-$\ell$ degenerate manifold, a strongly oscillatory behavior is visible. The region of avoided crossing is clearly visible at $R\sim 1400–1500a_0$. In addition, we present the potential curves provided by the Borodin-Kazansky model given by Eq. (8). The inset (i) shows the field-free trilobite and first $p$-wave PES. The inset (ii) shows the $38s$ split PES, which is oscillatory in the megahertz regime and hardly affected by the electric field. (b) Same as in panel (a) but with varying $E=150,300,$ and $450\frac{V}{m}$. The diatomic equilibrium distance $R_{\mathrm{eq}}$ is moving away from the region of the avoided crossing at $R=1500a_0–1700a_0$.

Figure 4

The electric dipole moment as a function of the electric field $E$ (blue [gray] points). For comparison we show a semiclassical prediction (green [gray] points). The inset shows the behavior of the equilibrium distance $R_{\mathrm{eq}}$ with varying electric field strength.

Figure 5

Spectrum of coefficients of the electronic eigenvector $\psi (\mathbf{r};{\mathbf{R}}_{\mathrm{eq}},\mathbf{E})$ at $E=300\frac{V}{m}$ (a) and $600\frac{V}{m}$ (b). For increasing field strength the eigenstates gain a finite admixture of the quantum defect split states. For $600\frac{V}{m}$ we clearly see a major contribution provided by the $38s$ state.

Figure 6

Shown are the 11 lowest vibrational energies as functions of the field strength $E$. The dips around $E=100,200$, and $385\frac{V}{m}$ are caused by the change of potential wells determining the diatomic equilibrium distance $R_{\mathrm{eq}}$. In the inset we show the offset corrected potential curves for $E=300\frac{V}{m}$ and $E=380\frac{V}{m}$ ($\theta =\pi$). For $E=300\frac{V}{m}$ bound states in the middle well with energies larger than 200 MHz can tunnel into the neighbored potential wells. This causes a reduction of their level spacings. For $E=380\frac{V}{m}$ we nearly get a double potential well and states with energies less than 200 MHz possess a higher tunneling probability. Correspondingly, this leads to a denser spectrum.

Figure 7

Scaled probability densities $|F_{\nu m}{(\rho ,z)|}^2$ for rovibrational wave functions. Both wave funtions belong to the trilobite PES for $E=300\frac{V}{m}$ with an azimuthal quantum number $m=0$. In panel (a) we observe a deformed Gaussian-like density profile for the ground state ($\nu =0$) centered at $Z=-1938a_0$ and $\rho =72a_0$. In $Z$ ($\rho$) direction the density distribution has an extension of approximately $50a_0$ ($100a_0$). In panel (b) we show the density profile for the second excited state ($\nu =2$). This density profile provided three peaks at $(Z,\rho )=(-1939a_0,32a_0),(-1934a_0,122a_0)$, and $(-1924a_0,234a_0)$.