Production of trilobite Rydberg molecule dimers with kilo-Debye permanent electric dipole moments

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Science  03 Apr 2015:
Vol. 348, Issue 6230, pp. 99-102
DOI: 10.1126/science.1260722

Making a molecular fossil lookalike

Atoms are generally compact objects. However, if one of the electrons orbiting the nucleus is given an extra boost of energy so that it's barely still attached, you get a much larger Rydberg atom. Booth et al. created an even more exotic species: a giant molecule consisting of a regular cesium atom bound within a Rydberg atom of the same element. The molecules are named “trilobites” because their electronic density visually resembles fossils of these extinct marine creatures.

Science, this issue p. 99


Permanent electric dipole moments are important for understanding symmetry breaking in molecular physics, control of chemical reactions, and realization of strongly correlated many-body quantum systems. However, large molecular permanent electric dipole moments are challenging to realize experimentally. We report the observation of ultralong-range Rydberg molecules with bond lengths of ~100 nanometers and kilo-Debye permanent electric dipole moments that form when an ultracold ground-state cesium (Cs) atom becomes bound within the electronic cloud of an extended Cs electronic orbit. The electronic character of this hybrid class of “trilobite” molecules is dominated by degenerate Rydberg manifolds, making them difficult to produce by conventional photoassociation. We used detailed coupled-channel calculations to reproduce their properties quantitatively. Our findings may lead to progress in ultracold chemistry and strongly correlated many-body physics.

Electric dipole moments of molecules are fundamentally important for the control of chemical reactions (1, 2), precision spectroscopy, realization of certain strongly correlated many-body gases (3), quantum information processing (4), and tests of fundamental symmetries (2). Permanent electric dipole moments (PEDMs) in molecules are a manifestation of symmetry breaking. They form in quantum systems by charge separation and mixing of opposite-parity eigenstates. Homonuclear molecules are therefore not expected to possess PEDMs (5).

Ultralong-range rubidium (Rb) Rydberg dimers correlating to nS + 5S molecular asymptotes, where n is the Rydberg atom’s principal quantum number, possess sizable PEDMs (~1 Debye) and a linear Stark map (6). Because of the smaller noninteger component of the Cs S-state quantum defect μSS(Cs) = 4.05, whereas μS(Rb) = 3.13], Cs Rydberg molecules correlating to nS + 6S asymptotes are predicted to have a PEDM as large as ~15 Debye (6). The quantum defect is the amount by which the principal quantum number of an atomic state is shifted— that is, n* = n – μl. In contrast to states with angular momentum quantum numbers l ≤ 2 (713), which are most easily addressed by laser excitation, the original theory of ultralong-range Rydberg molecules (14) predicted that huge kilo-Debye PEDMs would form in the excitation of a completely l-degenerate Rydberg manifold. An ultralong-range Rydberg molecule that is a hybrid of the low and high l-type molecules would possess a giant PEDM but still be accessible via conventional two-photon laser excitation. Because of their energy level structure and ground-state electron scattering properties, Cs atoms are an ideal system for observing such a hybrid species.

We found that for Cs atoms, the peculiarly small noninteger fraction of the S-state quantum defect strongly admixes the (n – 4)l > 2 degenerate electronic manifold with spherically symmetric nondegenerate nS states to form ultralong-range Rydberg molecules with kilo-Debye PEDMs, which are spectroscopically accessible. The Rydberg electron probability distributions for the observed Cs(nS + 6S) 3Σ+ Rydberg molecules are predominantly of the “trilobite” type (Fig. 1A). Here, the label in parentheses denotes the separated atom limit and the superscripts refer to the total electronic spin (triplet) and the homonuclear molecule reflection symmetry (+) through a plane containing the internuclear axis. The fractional mixing of states with high angular momentum can be as large as 90%. This contrasts with the 0.01% admixture of hydrogenic state character that occurs in Rb(nS) Rydberg molecules as studied in (6). The large admixture of a nearly degenerate electronic manifold localizes the electron density on the Cs(6S) perturber (Fig. 1A). We measured the PEDM by monitoring how the molecular Rydberg lines broaden when subjected to an external electric field of strength F ≈ 30 mV/cm (Fig. 1C). Quantitative calculations of potential energy curves (PECs) with complicated nonadiabatic avoided crossings, vibrational energy levels, and PEDMs corroborated the observations (Fig. 1B, Fig. 2A, and Fig. 3A).

Fig. 1 Trilobite states in Cs molecules.

(A) The electron probability distribution |Ψ(Re;r)|2 (Eq. 1) for the molecular states red-detuned from Cs(n = 37S, 39S, and 40S) Rydberg lines, each marked in (B) with an arrow, in cylindrical coordinates. The Rydberg ion is located near the center of each plot, and the Rydberg electron is localized near the location of the ground-state perturber. The distributions are shown at the equilibrium separations at Re = 102.4, 115.9, and 123.0nm for the corresponding BO PECs, given in (B). (B) The BO PECs (black, red, and blue curves) correlate asymptotically to the nS + 6S, (n – 4)F + 6S, and (n – 4)G + 6S states, respectively. The dashed and solid PECs have MJ = 0 and MJ = ±1 projection symmetry, respectively. The vibrational wave functions in the outermost wells are indicated in black. The vibrationally averaged PEDMs are listed on the right for each value of v. (C) Linewidths as a function of Fa for the arrow-marked states in (B). The error bars for the PEDM are determined as the background field and two-photon laser linewidth are varied within the measurement error. The error bars on the linewidth data are the statistical error of the linewidth fit.

Fig. 2 Energy levels.

(A) Comparison between the calculated vibrational levels in the outermost PEC well superposed with the associated wave functions (left) and observed spectra (right) for states correlating to the 37S + 6S limit. Even-parity (v = 0, 2, …) vibrational levels have stronger signals because the de Broglie wavelength of the ground-state wave function, λdB ≈ 35 nm, is much longer than the width of the outermost potential well. (B) Level diagram for the two-photon excitation scheme.

Fig. 3 Molecular spectra.

(A) Comparison between calculated spectra (red) and experimental spectra (blue) for the states correlating to the 37S + 6S, 39S + 6S, and 40S + 6S dissociation limits. The centroids of the vibrational levels are shown as sticks underneath the calculated spectra. Bound states corresponding to the MJ = 0 and MJ = ±1 projections are indicated with red and blue sticks, respectively. (B) Ion counts for the v = 4 level in the PEC correlating to the 37S + 6S limit, and for the v = 0 level in PECs correlating to the 39S + 6S and 40S + 6S limits, as a function of the delay between excitation and ionization. The uncertainties in the delay time are due to the width of the 5-μs laser pulses used. The vertical error bars are the statistical error in the ion counts.

Excitation into Rydberg states in a quantum gas has the potential for probing many-body effects with high precision and creating exotic states of matter. A recent observation of Rydberg electron orbital excitation to sizes comparable to or exceeding the extent of a Bose-Einstein condensate heralds possibilities for charged impurity research with extremely low mass ratios (15). Because of Rydberg blockade (16), only one Rydberg atom is excited in the condensate and single-impurity studies can be conducted. The formation of ultralong-range Rydberg molecules can be likened to localization in solids (17): Ultralong-range Rydberg molecules are formed through multiple scattering of electrons from perturbers, leading to the localization of the electronic wave packet. The Cs states produced in this work are precursors to states where the electron is strongly localized at the position of several ground-state atoms. Such states will have exotic properties; they can involve dipolar and spin degrees of freedom, as well as interactions between Rydberg atoms if more than one Rydberg atom is present.

The interaction between a Rydberg electron and a ground-state perturber located within the Rydberg atom can be described by a Fermi contact interaction with energy-dependent scattering lengths (18, 19). For Cs, there is a large p-wave spin-orbit splitting, manifested in the 3PJ electron-cesium scattering phase shifts that must be considered. To calculate the molecular states for the Cs dimer, we diagonalized the electronic Hamiltonian that results from the electron-atom interaction for a range of internuclear distance R, using a large basis set of Rydberg electron orbital wave functions (20). The calculation yielded a set of Born-Oppenheimer (BO) PECs, U(R), and their corresponding Rydberg electron wave functions, Ψ(R;r). Examples of these PECs and electron density distributions calculated from the Ψ(R;r) are shown in Fig. 1, A and B. The resulting set of coupled Schrödinger equations were solved directly to extract the vibrational states and the full spectrum of Rydberg molecular states.

The depth of the BO PECs with respect to the nS + 6S molecular asymptotes shows extreme sensitivity to the value of the zero-energy s-wave scattering length. This is because the depth of the PECs with respect to the hydrogenic manifold is approximately proportional to the scattering length. Because the hydrogenic manifold lies several GHz above the Cs Rydberg S state, a 1% variation in the electron ground-state atom s-wave scattering length results in a change of ~100 MHz in the PEC depth. We used this sensitivity to adjust the value of the s-wave scattering length so that the lowest vibrational level in the outer well of the PEC correlating to 40S + 6S was in agreement with the experimentally observed resonance peak. The s-wave scattering length obtained was –21.3 ± 0.1 a0, where a0 is the Bohr radius. This value is 2% smaller than the theoretically calculated value of –21.7a0 used in (9, 21). The error bars are set conservatively to correspond to an energy level shift approximately equal to the spectral width of the observed vibrational states.

The p-wave electron-perturber scattering creates a set of narrow avoided crossings in the BO PECs, evident in Fig. 1B. These features correspond to metastable p-wave Cs states (22). The PEC crossings can have a large impact on the overall behavior of the molecular PECs and the resulting vibrational states, particularly for those states lying nearby in energy. Far from the crossings, the PECs are dominated by s-wave scattering. We focus on states in the outermost PEC wells (Fig. 1B), where the p-wave scattering produces only a small energy shift and the spin-orbit splitting between the MJ = ±1 and MJ = 0 bound-state energies is smaller than the experimental spectral resolution of 3 MHz.

The experiment was performed in a far-off resonance trap (FORT). The crossed FORT was loaded to a peak density of 5 × 1013 cm–3 at a temperature of 40 μK. A two-photon excitation was used to photoassociate the molecules (Fig. 2B). The molecules were ionized using the FORT beams. The ions were detected with a microchannel plate detector. The molecular spectra were acquired by counting Cs+ and Cs2+ ions arriving at the detector as a function of excitation laser frequency. No Cs2+ ions were detected.

Molecular spectra were acquired red-detuned from the n = 37, 39, and 40S1/2 atomic states. Three spectral absorption lines were selected for Stark shift measurements: the excited vibrational level at ~ –277 MHz, v = 4, in the PEC correlating to the 37S + 6S limit, and the ground vibrational levels, v = 0, in the PECs correlating to the 39S + 6S and 40S + 6S limits. These states are indicated in Fig. 1B with arrows, and their ion yield spectra are shown in Fig. 2A and Fig. 3A. For the Stark shift measurements, external electric field (Fa) scans were performed on the spectral lines to determine their positions and widths. Because the electric field plates can only apply electric fields normal to the plates in our apparatus, a constant horizontal background electric field of Fh ≈ 15 mV cm–1 was present for all the measurements.

At electric fields of Fa ≈ 15 to 30mV cm–1, the Stark shift appeared as a broadening of the spectral line that increases linearly as a function of Fa (Fig. 1C) (20). The broadening increased nonlinearly at very small Fa ≈ 10 mV cm–1 because of the presence of Fh ≈ 15 mV cm–1. From the broadenings shown in Fig. 1C, we determined dipole moments for the measured molecular states. For the –277 MHz vibrational peak near the 37S + 6S asymptote, the measured electric dipole moment was D = 2330 ± 400 Debye, whereas we obtained D = 2310 ± 250 Debye and D = 1910 ± 150 Debye for the v = 0 level in the outer wells shown in Fig. 1B near the 39S + 6S and 40S + 6S asymptotes, respectively. The observed PEDMs are 100 to 1000 times those measured in previous experiments (6, 9), primarily because of the greater degenerate admixture present in the states observed here. Electron density distributions for each molecular state, |Ψ(Re;r)|2, whose PEDM was measured are shown in Fig. 1A. In contrast to the Rb results (6), it was not necessary to subtract the Cs nS-state contributions from Ψ(R;r) to observe the “trilobite.”

The highly localized electron density shown in Fig. 1A combined with the large internuclear separation (~120nm) yields an extremely large PEDM, Embedded Image, where Embedded Image is the R-dependent dipole moment, and Embedded Image is the vth vibrational wave function. The R-dependent dipole moment d(R) is almost entirely a consequence of the hybridization by the high-l degenerate Rydberg manifold. Ψ(R;r) can be written asEmbedded Image (1)where the ψ(n–4)T(R;r) trilobite state is a linear combination of l > 2 states that maximally localizes the Rydberg electron near the ground-state perturber. Here, cS(R) and cT(R) are the probability amplitudes for finding the electron in an nS state or a trilobite state, respectively. The R-dependent dipole moment is d(R) ≈ |cT(R)|2dT(R), where dT(R) ∝ (n – 4)2R is the dipole moment of the bare trilobite molecule. Far from the avoided crossings, cS(R) is approximately proportional to [E(n–4)l>2EnS]/U(R), where Enl is the atomic Rydberg energy. Therefore, the PEDM becomes monotonically smaller with increasing v as the higher-lying vibrational wave functions progressively maximize their probability amplitude at the outer turning points. The calculated PEDMs (Fig. 1B) are within 13% of the corresponding experimental values (Fig. 1C). The main source of error in the theoretical PEDMs is the uncertainty in the position of the vibrational state energy levels. For example, a shift in the binding energy of 40 MHz in v = 0 for Cs(37S + 6S) 3Σ results in an 80 Debye change in the PEDM. Taking this uncertainty into account, the experimental and theoretical values agree to within one standard deviation.

Figure 2A shows a comparison between the experimental spectrum and the PEC correlating to the 37S + 6S dissociation limit, with Fh = 15 mV cm–1. By applying the same functional form employed for fitting the Stark shifts, we calculated the theoretical spectra shown in Fig. 3A. There is sizable modulation in the peak strength between even and odd v. The modulation of the vibrational state amplitude is most clearly observed for the molecular states correlating to the 37S + 6S asymptote, as well as near the minima of each of the different PEC wells. The modulation is a consequence of odd-parity cancellation in the Franck-Condon factors, akin to the Cooper minima in atomic ionization spectra (23). The de Broglie wavelength of the Cs atoms is λdB(T = 40 μK) ≈ 35 nm, whereas the width of the potential wells is w ≈ 5 nm. Over this span, the ground-state wave function is effectively flat.

The observed lifetimes of each state are shown in Fig. 3B. The lifetimes are shorter than the Cs(nS) states. For example, the lifetime of the 40S state limited by radiative and blackbody decay at 300 K is 37 μs (24). The shorter lifetimes are indicative of mixing states, other than the nS states, such as (n – 4)F Rydberg state, which has a small, yet nonzero, quantum defect. The corresponding lifetime of Cs(36F) is τ36F ≈ 18 μs (25). The fact that the decay of the molecular state is different from the purely radiative decay of the Cs(nS) Rydberg state, and closer to the nearby (n – 4)F state, is another indication of the degenerate mixing involved in the formation of the hybrid trilobite molecules.

The class of ultralong-range molecule observed here is an advantageous cross between a traditional “trilobite” molecule and a low-l ultralong-range Rydberg molecule (712). The discovery of the trilobite ultralong-range Rydberg molecule could open opportunities in ultracold chemistry and strongly correlated many-body physics, as these exotic states require engineered mesoscopic localization and kilo-Debye permanent electric dipole moments.

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References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: Supported by NSF grant PHY-1205392 (D.B., J.Y., and J.P.S.) and by an NSF grant through ITAMP at the Harvard-Smithsonian Center for Astrophysics (H.R.S.). S.T.R. thanks B. M. Peden and B. L. Johnson for insightful discussions. All data are available upon request.
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