Spin-orbit-dependent lifetimes of long-range Rydberg molecules

Abstract: Long-range Rydberg molecules (LRMs) form when a highly excited Rydberg electron scatters from ground-state atoms inside its orbit, creating oscillatory, long-range potentials. We present a combined theoretical and experimental study of caesium dimers correlated to 402P3/2 Rydberg states, with an emphasis on decay via autoionisation (associative ionisation). Our model includes a relativistic treatment of electron-atom scattering with spin-orbit coupling, the perturber's hyperfine structure, and coupling of vibrational levels to a continuum of short-range decay channels. Calculated potential-energy curves predict two families of wells: outer wells near the classical outer turning point supporting long-lived states, and inner wells at shorter range whose lifetimes are limited by tunneling and subsequent vibronic decay. Using photoassociation in an ultracold Cs gas and an analysis of pulsed-field-ionisation signals which are highly selective for the detection of molecules, we assign resonances by binding energy and measure lifetimes. The measured lifetimes of inner-well states increase systematically with increasing detuning and agree with calculated lifetimes; detection of Cs2+ product ions supports autoionisation as a dominant channel. We show that the lifetimes are strongly reduced by spin-orbit interactions in the transient Cs-collision complex, which lift the near-degeneracy in Omega observed for states in the outer well and control the inner-well binding. The identified states also provide promising pathways to create ultracold molecules in ion-pair states.

• The paper provides the first systematic lifetime measurements of inner-well long-range Rydberg molecules (LRMs), demonstrating that lifetimes increase systematically with binding energy and support the theory of spin-orbit interactions.
• The study uses a sophisticated theoretical model that accounts for Fermi-contact interactions, energy-dependent scattering lengths, and hyperfine interactions to predict potential energy curves and lifetimes, which align with experimental findings.
• In the experiment, pulsed-field ionisation and time-resolved electron detection reveal a correlation between molecular ion counts and inner-well LRM lifetimes, supporting associative ionisation as the primary decay mechanism. Mindful of the study’s robustness against off-table citation errors and lack of converging iterations

Overview

This paper presents a combined theoretical and experimental study of caesium long-range Rydberg molecules (LRMs) correlated to the $40\,^2P_{3/2}$ Rydberg state, focusing on their decay via autoionisation (associative ionisation). The work addresses a gap in the understanding of LRM decay dynamics: while outer-well states have been studied extensively, the authors provide the first systematic lifetime measurements of states bound in inner wells, where the binding is controlled by the spin–orbit interaction of the transient Cs$^-$ collision complex. The central result is that measured lifetimes of inner-well states increase systematically with binding energy and agree with calculated values, with detection of Cs$_2^+$ product ions supporting autoionisation as the dominant decay channel.

Theoretical model of vibronic structure

The electronic structure calculation builds on the Fermi-contact interaction between the Rydberg electron and ground-state perturbers, extended to non-vanishing collision energies following Omont. The interaction potential includes energy-dependent $s$-wave scattering lengths $a_S(k)$ and $p$-wave scattering volumes $a_P(k)$, evaluated at the semi-classical kinetic energy of the electron at internuclear separation $R$. The total Hamiltonian comprises the isolated Rydberg atom (with fine structure), the hyperfine interaction in the ground-state atom, and the Fermi-contact term, diagonalised independently for each value of $\Omega$, the projection of total angular momentum on the internuclear axis. Relativistic scattering parameters $a_{L,S,J}$ are obtained from a model-potential method using the renormalised Numerov method, with slight adjustments to improve agreement with experimental binding energies.

The calculated potential-energy curves for LRMs correlated to $^-$0 exhibit three characteristic regions:

• Region A (near the classical outer turning point at $^-$1): two wells — a "deep" well (~$^-$2 MHz dissociation energy) dominated by triplet scattering ($^-$3) and a "shallow" well (~$^-$4 MHz) from mixed singlet-triplet scattering ($^-$5). States here are nearly degenerate in $^-$6 due to approximate conservation of $^-$7.
• Region C (around $^-$8): butterfly-character wells arising from the $^-$9 shape resonance in elastic electron–caesium scattering.
• Region B (around $_2^+$0): inner wells where the $_2^+$1-degeneracy is completely lifted by spin–orbit coupling and $_2^+$2 ceases to be a good quantum number.

Vibrational levels are computed with the modified Milne phase-amplitude method using open boundary conditions, so that bound states acquire a finite width $_2^+$3 interpreted as tunnelling through surrounding barriers toward shorter range, followed by rapid vibronic decay. The authors acknowledge that calculated widths depend on the shape of the potential at short range, where the model is less reliable; they quantify this uncertainty by systematically varying $_2^+$4 and reporting the resulting modulation as an error band.

Key predictions include: outer-well $_2^+$5 levels with widths on the order of 1 Hz (effectively unlimited by tunnelling); inner-well $_2^+$6 levels forming a near-equally-spaced ladder with binding energies from $_2^+$7 to $_2^+$8 MHz and lifetimes increasing from 1.0 μs ($_2^+$9) to 17 μs ($s$0), governed by the exponential dependence of tunnelling rates on barrier width and height; and intermediate-well levels that are much shorter-lived because the $s$1 shape resonance narrows the barrier at $s$2, strongly enhancing tunnelling.

Experimental approach

LRMs are produced by photoassociation in an ultracold cloud of $s$3 caesium atoms at density $s$4 and temperature ~40 μK, optically pumped into $s$5. A frequency-doubled ring dye laser at ~319 nm excites the sample, with residual drifts ~100 kHz/day and electric fields compensated to better than 20 mV/cm.

Detection exploits pulsed-field ionisation (PFI) with a ramped field and time-resolved electron detection. The key experimental innovation is the "late-PFI signal": molecules surviving until the ionisation ramp produce electrons arriving up to 1.5 μs after the prompt atomic signal, due to LRM-specific field-ionisation dynamics along adiabatic pathways distinct from those of isolated atoms. An rf photodissociation measurement confirms that the late-PFI signal originates from intact LRMs immediately before field ionisation, excluding black-body-induced transitions or state-changing collisions prior to the ramp. Importantly, Rydberg atoms excited via Coulomb anti-blockade — facilitated by ions produced through associative ionisation — do not contribute to the late-PFI signal, making it an unambiguous measure of LRM population.

The late-PFI spectrum shows a dominant peak at −26.4 MHz detuning, assigned to the $s$6 level of the outermost well of the $s$7 states, plus four additional resonances at −60, −52, −44, and −36 MHz, confidently assigned to inner-well states with $s$8 and $s$9, respectively. The agreement between observed resonance positions and predicted binding energies of long-lived levels provides strong support for these assignments.

Lifetime measurements and decay mechanism

Lifetimes are extracted from the decay of the late-PFI signal versus delay between the UV pulse and the PFI ramp, using weighted nonlinear regression with bootstrap-estimated uncertainties. The analysis was verified to be robust against details of the offset-correction procedure.

State	Binding energy	Measured lifetime
Outer well, $a_S(k)$0, $a_S(k)$1	−26.4 MHz	58(3) μs
Inner well, $a_S(k)$2	−60 MHz	longest of inner-well set
Inner well, $a_S(k)$3	−36 MHz	shortest of inner-well set

The outer-well lifetime of 58(3) μs agrees well with the calculated black-body-radiation-limited lifetime of the parent $a_S(k)$4 atom (57.3 μs at 300 K), noting that spontaneous decay is suppressed by a Cooper minimum in the caesium photoionisation cross section. In contrast, inner-well lifetimes are significantly reduced and show the same systematic increase with binding energy predicted theoretically — both binding energy and lifetime decrease monotonically as $a_S(k)$5 decreases from $a_S(k)$6 to $a_S(k)$7.

To identify the dominant decay channel, the authors detect molecular ions: Cs$a_S(k)$8 arrives at the detector later than Cs$a_S(k)$9 owing to its larger mass. Cs$p$0 ions are observed for all inner-well LRMs, with counts increasing for shorter-lived states, compatible with associative ionisation accounting for a large fraction of the decay. State-changing collisions are ruled out as the dominant mechanism because their products ($p$1) would ionise earlier in the field ramp than the late-PFI window. Since the inner-well molecules have smaller geometric cross sections yet much shorter lifetimes than the outer-well state whose decay is BBR-limited, collisions with ground-state atoms cannot explain the observed reduction; instead, the data support the model attributing the reduction to barrier tunnelling followed by coupling to short-range intramolecular decay channels.

Role of spin-orbit coupling

An appendix-level analysis disentangles exchange and spin–orbit contributions to the Cs$p$2 collision complex using two interpolation parameters: $p$3 scales the singlet-triplet exchange splitting, and $p$4 scales the spin–orbit splitting among the $p$5 components. With both parameters zero, the $p$6 and $p$7 states are degenerate. Turning on exchange lifts the degeneracy in $p$8 but leaves states degenerate in $p$9. Turning on spin–orbit coupling additionally lifts the $a_P(k)$0-degeneracy; at the inner-well minimum ($a_P(k)$1), the spin–orbit-induced splitting exceeds the exchange contribution and $a_P(k)$2 becomes the only good quantum number. This establishes spin–orbit coupling in the electron–atom scattering complex as the factor controlling both inner-well binding energies and lifetimes, connecting LRM spectroscopy to the spin-dependent autoionisation dynamics known from molecular Rydberg states with structured ionic cores.

A further practical implication concerns the long-lived levels with $a_P(k)$3, which carry $a_P(k)$4 scattering-channel character and are therefore promising precursors for stimulated de-excitation into ultracold Cs$a_P(k)$5 ion-pair ("heavy-Rydberg") states, a route toward ultracold anions and strongly correlated pair plasmas.

Limitations and open questions

Several limitations qualify the results. The effective scattering parameters entering the Hamiltonian are constrained by the non-convergence of the delta-function Fermi-contact treatment with basis size, so they should be regarded as fitted quantities rather than free-electron scattering observables, even though they reproduce binding energies across a range of $a_P(k)$6. Calculated lifetimes depend on the assumed short-range shape of the potential-energy curves, where the model is least reliable and non-adiabatic effects dominate; this is handled by reporting an uncertainty band rather than eliminating the ambiguity. Quantitative branching ratios between associative ionisation and state-changing collisions could not be determined because of the poorly characterised mass- and velocity-dependent sensitivity of the MCP detector; only qualitative support for autoionisation dominance is provided. Finally, the precise mechanism producing the late-PFI signal — attributed to LRM-specific field-ionisation dynamics along adiabatic pathways — remains under investigation, though the rf photodissociation control confirms its origin in intact molecules. LRMs bound at still shorter internuclear distances, describable either as molecular Rydberg states or as LRMs, remain an open regime for future study.

Conclusion

This work demonstrates that the lifetimes of inner-well caesium long-range Rydberg molecules are governed by spin–orbit interactions in the Cs$a_P(k)$7 collision complex, which lift the $a_P(k)$8-degeneracy seen in outer-well states and control both binding energies and tunnelling-mediated decay rates. The agreement between measured and calculated lifetimes, together with the observed Cs$a_P(k)$9 product ions, validates a relativistic vibronic model including coupling to continuum decay channels and establishes associative ionisation as the dominant decay pathway for these states. The identified long-lived inner-well levels additionally offer a viable starting point for preparing ultracold molecules in highly excited ion-pair states.