Predissociation dynamics of charged long-range Rydberg molecules

Abstract: We investigate predissociation in homonuclear ($^{{87}$Rb$^{*$^{87}RbRb^+)}}</sup></sup>andheteronuclear()</sup></sup></sup></sup> and heteronuclear (^{87}RbRb^*<sup>7<sup>{7}Li<sup>+<sup>+) long-range Rydberg atom-ion molecules. Owing to their micron-scale bond lengths, these dissociate on time scales far removed from those of more tightly bound diatomic molecules. We employ the eigenchannel $R$-matrix method to compute predissociation rates for a broad range of principal quantum numbers $n$. The rates depend strongly on the mass, but more remarkably display a rapid and periodic variation as a function of $n$ as well as within a single vibrational ladder. A semiclassical Landau-Zener-Stückelberg analysis reveals that Stückelberg interference governs the decay process and produces the observed variation in the molecular lifetime. Although the heavy mass of the homonuclear Rb molecule constrains its predissociation rates to a sub-kHz level, the lighter molecule $<sup>{87}$Rb$<sup>*$^{7}LiLi^+$ dissociates on time scales competitive with radiative and collisional decay. This can enable in situ study of non-adiabatic decay via ion microscopy.

• The paper combines eigenchannel R-matrix calculations with Landau-Zener-Stückelberg analysis to show that interference between adiabatic and diabatic pathways controls predissociation widths and lifetimes.
• Rb*Li+ lifetimes span about 0.15–166 microseconds across the studied states, making predissociation experimentally observable, while Rb*Rb+ lifetimes remain above 1 millisecond and are dominated by other decay processes.
• The results identify strong sensitivity to near-singular derivative couplings, predict a mass- and quantum-number-dependent crossover near n ≈ 80 for Rb*Rb+, and highlight unresolved discrepancies requiring improved numerical and time-dependent treatments.

This paper presents a combined quantum-mechanical and semiclassical study of non-adiabatic predissociation in long-range Rydberg atom–ion molecules (RAIMs), comparing the homonuclear $^{87}\mathrm{Rb}^*{}^{87}\mathrm{Rb}^+$ and heteronuclear $^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$ systems. Using an eigenchannel $R$-matrix treatment of the coupled-channel Schrödinger equation together with a Landau-Zener-Stückelberg (LZS) semiclassical analysis, the authors compute vibronic resonance positions and widths over a broad range of principal quantum numbers ($n = 32$–$65$) and vibrational levels ($\nu = 0$–$9$), and show that Stückelberg interference between adiabatic and diabatic decay pathways governs the molecular lifetimes.

Physical system and electronic structure

The RAIM consists of a Rydberg atom interacting with a distant, structureless alkali ion via a multipole expansion truncated at $L=6$. Because the internuclear separation exceeds the LeRoy radius, the Born-Oppenheimer potential energy curves (PECs) are determined entirely by the Rydberg atom's electronic structure; changing the ion species changes only the reduced mass $\mu$, providing an independent handle on the non-adiabatic coupling strength. The molecule of interest forms in a well of the PEC asymptotically connected to the $nP_{1/2}$ state, which bends away from repulsive high-$^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$0 polar states (asymptotically $^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$1) at an avoided crossing located at $^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$2. Derivative couplings $^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$3 are sharply localized at these avoided crossings.

A central numerical difficulty is the presence of "almost dark" states: adiabatic curves that couple to the rest of the manifold only through extremely narrow avoided crossings. The resulting near-singular $^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$4 matrix element requires very dense radial grids, and the computed lifetimes are extraordinarily sensitive to it — a one-percent error in its peak value shifts the Rb$^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$5Rb$^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$6 predissociation lifetimes by three orders of magnitude. The authors circumvent this by constructing an effective two-channel model in which the narrow avoided crossing is treated strictly diabatically and the coupling to the upper curve is fitted by a Lorentzian profile. They note that such almost dark states arise generically in ultralong-range Rydberg molecules (trilobite, butterfly, and nonpolar molecules) and that a more rigorous framework than interpolation would be desirable.

Quantum and semiclassical methodology

The eigenchannel $^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$7-matrix method partitions configuration space at a matching radius $^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$8 beyond which all couplings vanish and the upper channel is closed. Open-channel solutions are matched to regular and irregular Bessel-function solutions appropriate to the $^{87}\mathrm{Rb}^*{}^{7}\mathrm{Li}^+$9 dipole tail, backpropagated with Numerov's algorithm. Resonance parameters are extracted from the largest eigenvalue of the Wigner-Smith time-delay matrix, which traces a Lorentzian about each resonance seeded by the Born-Huang bound-state energies.

The complementary LZS analysis expresses the semiclassical $R$0-matrix through five fundamental phases: the left and right WKB phases on each adiabatic curve, and the Stokes phase $R$1 from the non-adiabatic transition, with the Landau-Zener parameter $R$2 built from the gap $R$3 and classical velocity $R$4 at the avoided crossing. Linearizing the phases about each resonance yields closed-form Breit-Wigner expressions for the width and shift:

$R$5

where $R$6 is half the phase difference between the two dominant collision pathways (fully adiabatic versus diabatic passage). The width therefore vanishes whenever $R$7: destructive Stückelberg interference decouples the bound state from the dissociation continuum. The same phase controls the Fano asymmetry parameter, $R$8, so that suppressed-width states exhibit highly asymmetric profiles with strong $R$9-reversal, while broad states appear as window resonances with $n = 32$0. Notably, this $n = 32$1-reversal occurs here in a strictly two-channel setting, driven by pathway interference rather than by an interloper resonance as in conventional three-channel atomic autoionization.

Lifetimes of the Rb$n = 32$2Li$n = 32$3 molecule

For the light heteronuclear molecule, the Landau-Zener probability is on the order of $n = 32$4, and the semiclassical time delay reproduces the quantum calculation closely, with the largest discrepancies for the narrowest resonances. The predicted lifetimes vary dramatically: for $n = 32$5, they range from roughly $n = 32$6s ($n = 32$7) to $n = 32$8s ($n = 32$9), and across the full $65$0 grid they span nearly three orders of magnitude, from $65$1s at $65$2 to $65$3s at $65$4. Adjacent vibrational levels can differ in lifetime by more than an order of magnitude. The short end of this distribution lies below both radiative and typical collisional timescales, meaning predissociation becomes experimentally observable; the long-lived states, conversely, decay predominantly through collisions.

Three systematic trends emerge. First, lifetimes increase weakly overall with $65$5, superimposed with a semi-periodic modulation traced to the slow variation of $65$6 with $65$7. Second, lifetimes decrease slightly with $65$8, consistent with the increased velocity at the avoided crossing enhancing the hopping probability. Third, the lifetime map exhibits diagonal streaks of enhanced longevity, corresponding to contours of approximately constant path-difference phase along which the destructive-interference condition is satisfied. The microsecond-scale dynamics, combined with ion microscopy, should permit in situ observation of the non-adiabatic decay process.

The homonuclear Rb$65$9Rb$\nu = 0$0 molecule and discrepancy with prior work

For the heavier homonuclear molecule, the calculated predissociation lifetimes never fall below $\nu = 0$1 s — orders of magnitude longer than radiative or collisional decay — confirming the earlier conclusion that predissociation is irrelevant to Rb$\nu = 0$2Rb$\nu = 0$3 stability. However, the present lifetimes exceed those of the prior time-dependent calculation [Duspayev & Raithel, Phys. Rev. A 105, 012810 (2022)] by one to two orders of magnitude, while the resonance positions agree excellently. The authors could not identify a definitive cause but offer two candidate explanations: (i) extreme sensitivity of the coupled-channel solution to sub-percent errors in the near-singular $\nu = 0$4 coupling — artificially reducing it by one percent shrinks lifetimes to tens of microseconds; and (ii) possible power-law rather than exponential population decay due to wave-packet interference, which would bias extrapolation from a time-dependent complex-absorbing-potential propagation that assumes exponential decay. This unresolved quantitative discrepancy is stated plainly and remains an open point.

The LZS analysis also resolves an apparent even–odd parity dependence of lifetimes on $\nu = 0$5 at certain $\nu = 0$6 values: the effect is not vibronic parity but a coincidental 3:2 ratio between the accumulation rates of the path-difference and adiabatic phases, which reverses phase at other $\nu = 0$7. Semiclassical agreement is noticeably worse than for Rb$\nu = 0$8Li$\nu = 0$9, attributed to the strongly adiabatic crossings ($9$0), whose exponential sensitivity amplifies semiclassical errors; agreement improves when the crossing is artificially widened.

Mass and principal-quantum-number scaling

Replacing Rb$9$1 by Li$9$2 reduces $9$3 by nearly a factor of three, raising the Landau-Zener hopping probability by three to four orders of magnitude — the essential reason the lighter molecule predissociates competitively. A fitted scaling law,

$9$4

with $9$5 and $9$6, captures the coarse-grained behavior. Its logarithm contains competing $9$7-dependent terms, yielding a crossover scale

$9$8

below which the width grows with $9$9 and above which it decreases. Since $L=6$0 ranges from roughly 25 for Rb$L=6$1Li$L=6$2 to roughly 80 for Rb$L=6$3Rb$L=6$4, the observed decrease of lifetime with $L=6$5 in the rubidium system is a finite-$L=6$6 effect expected to reverse near $L=6$7. The mass-scaling estimate neglects destructive interference and thus overestimates decay rates, serving only as a qualitative guide.

Limitations and open questions

Several caveats qualify these results. The treatment of almost dark states relies on a numerically validated but ad hoc interpolation/diabatization procedure lacking a rigorous general framework. The homonuclear lifetime discrepancy with the prior time-dependent calculation is unexplained, though bounded by sensitivity tests. The semiclassical method degrades quantitatively in the deeply adiabatic regime relevant to Rb$L=6$8Rb$L=6$9. All calculations assume zero rotational angular momentum and fields parallel to the molecular axis; arbitrarily oriented external fields, trap-induced quadrupole modifications of the PECs in hybrid atom–ion platforms, and extension to Rydberg–Rydberg–ion trimers remain untreated questions raised by the work.

Conclusion

By combining multichannel $\mu$0-matrix scattering with an analytically tractable LZS formulation, this work establishes Stückelberg interference as the organizing principle behind the strongly non-monotonic predissociation rates of charged long-range Rydberg molecules. It predicts that the lighter Rb$\mu$1Li$\mu$2 molecule decays on microsecond timescales competitive with radiative and collisional channels — enabling direct experimental access to non-adiabatic decay dynamics via ion microscopy — while the heavier Rb$\mu$3Rb$\mu$4 molecule remains stable against predissociation throughout the studied parameter range, with its apparent $\mu$5-dependent trend reversing only beyond $\mu$6.