Highly polar states of Rydberg atoms in strong magnetic and weak electric fields

We study the spectra of diamagnetic rubidium Rydberg atoms in strong magnetic and weak electric parallel fields, in the $n$-mixing regime. Our emphasis is on isolated pairs of near-degenerate, opposite-parity, diamagnetic states that become mixed by the weak electric field. Such level pairs allow for the generation of nondegenerate states with large, tunable permanent electric dipole moments and large optical excitation cross sections from the atomic ground state. We investigate how the dipole moments and the zero-electric-field energy defects of these level pairs can be tuned using small variations of the electric and magnetic fields. Using calculations, we explore the abundance of such level pairs over wide spectral regions for several magnetic quantum numbers. Applications of polar, diamagnetic Rydberg states in Rydberg-atom interaction experiments are briefly discussed.

Figure 1

Electron in a one-dimensional, symmetric double-well potential perturbed by an electric field $F$. For $F=0$ the double-well system has discrete ${\Pi }_z$ symmetry, which is lost when $F\ne 0$. As explained in the text, this simple model system mimics the essential properties of electric-field-coupled vibrator states of Rydberg atoms in strong magnetic and weak parallel electric fields.

Figure 2

(a) Rydberg levels observed in the vicinity of scaled energy $\varepsilon \approx -0.51$, probed by scanning the upper-transition laser frequency, for $B_{\mathrm{expt}}=2.577\mathrm{T}$, and at the indicated electric field values. Level energies are shown with respect to the field-free photoionization threshold. Some close-lying level pairs of the same $m_J$ exhibit large and opposite electric dipole moments (slopes of the dotted lines). (b) Calculated line spectra, as detailed in Sec. 6, for $B_{\mathrm{th}}=2.581\mathrm{T}$. In the calculations states of both parities are shown for all optically accessible $m_J$ levels. Hence, numerous calculated levels are experimentally undetectable.

Figure 3

High-resolution contour plot showing the pair of Rydberg states labeled $X$ in Fig. 2 at $-55.1{\mathrm{cm}}^{-1}$, with $B_{\mathrm{expt}}=2.576\mathrm{T}$ and excitation pulse lengths of 10 $\mu$s. Excitation frequency versus applied electric field (V/m) is plotted, with color representing the observed excitation rate (white 0, red 14 counts per shot). The electric field is varied in steps of 0.3 V/m, and a few selected scans are overlaid as black lines. The weak state at a fixed frequency offset of about 40 MHz has $m_J=5/2$, and is excited due to imperfect polarization. The $m_J=5/2$ state has a negligible Stark shift and does not couple to the $m_J=3/2$ pair of states.

Figure 4

(a) Scans of the state pair labeled $X$ in Fig. 2 for the indicated magnetic field values and a weak parallel electric field $F\approx 1$ V/m. We show the line shifts relative to that of the $5P_{3/2}|m_I=5/2,m_J=3/2\rangle$ level. The two states of opposite parity have different magnetic dipole moments, of $-$9.2${\mu }_{\mathrm{B}}$ ($-$11.7${\mu }_B$) for the odd- (even-) parity peak. (b) Levels of the $X$ pair of states, which both have $m_J=3/2$, relative to the $m_J=5/2$ level, which is used as a convenient reference line. The symbol size qualitatively represents the line strength. The $X$ pair of states exhibits an anticrossing (instead of an exact crossing) because of the coupling induced by the weak electric field $F$.

Figure 5

(a) Histograms of all magnetic dipole moments in the interval from $-60$ to $-50{\mathrm{cm}}^{-1}$, for $B_{\mathrm{th}}=2.581$ T. For each value of $m_J$, the total number of Rydberg states in this interval is indicated below the legend. (b) Histograms of all electric dipole moment magnitudes, for the same energy interval and magnetic field value, with an applied electric field of $F=3\mathrm{V}/\mathrm{m}$. For clarity, the bars with dipole moments $<$$100e{a}_0$ are cut off at height 40, with the number of occurrences indicated on the bar.

Figure 6

Scatter plot of line pairs with electric dipole moments $>$$100e{a}_0$ at $B_{\mathrm{th}}=2.581$ T and $F=3\mathrm{V}/\mathrm{m}$, for the energy range between $-60$ and $-50{\mathrm{cm}}^{-1}$. The pairs are plotted versus dipole moment and excitation amplitude. The dashed line indicates our experimental excitation-amplitude limit above which lines can be easily observed.

Figure 7

Calculated normalized electron wavefunction probabilities $P={\left|\psi \left(\mathbf{r}\right)\right|}^{2}r\sin \theta$ for the $X$ line pair at $-55.1$ cm${}^{-1}$ with $B_{\mathrm{th}}=2.589$ T and $F=0$ V/m, on a grayscale gradation. Midlevel gray represents zero probability, while both white and black correspond to maximum probability. Dark (light) lobes in the probability distributions correspond to a negative (positive) wavefunction sign, to show whether ${\Pi }_{\mathrm{z}}$ is even or odd.

Figure 8

Calculated normalized electron wavefunction probabilities $P={\left|\psi \left(\mathbf{r}\right)\right|}^{2}r\sin \theta$ for the $X$ line pair at $\sim -55.1$ cm${}^{-1}$ with $B_{\mathrm{th}}=2.589$ T, with a weak parallel electric field of $F=3$ V/m. The $F$-induced coupling leads to the displayed states with large permanent electric dipole moments.

Figure 9

Classical electron trajectories at an energy of $\sim$$-55.1{\mathrm{cm}}^{-1}$, $B_{\mathrm{th}}=2.580$ T. The simulated trajectory duration is 137 ns and the electron position is plotted every 8.2 ps. At $F=0$, the sum of the point densities of two equivalent trajectories (a) mimics the wave-function probabilities in Fig. 7. At small but nonzero $F$, the point densities of individual, slightly nondegenerate trajectories (b) and (c) mimic the wave-function probabilities in Fig. 8.