Analysis of collision shift assessments in ion-based clocks

TL;DR

The paper addresses collision-induced frequency shifts in ion-based clocks by unifying classical Langevin and quantum treatments within a simple interaction model, showing that the shift is bounded by the Langevin collision rate diminished by a recoil-induced suppression factor. It introduces the Ramsey Suppression Factor (RSF) to quantify laser–ion decoupling due to recoil and derives bounds that apply to any single-ion clock, with consistent results across hard-sphere and Lennard-Jones potentials. The study demonstrates that glancing collisions contribute negligibly to the clock shift, while the dominant effect arises from the loss of laser coupling during recoil, enabling a practical, analytic bound without heavy molecular-potential calculations. The bound is robust against reasonable short-range potential details and provides a clear path to experimentally measure collision rates via RSF, with important implications for achieving accuracies near the $10^{-19}$ level and for evaluating pressures in vacuum systems. Overall, the work offers a tractable framework for estimating and bounding collision shifts in ion clocks, bridging classical intuition and quantum scattering theory, and guiding future refinements through molecular-potential physics.

Abstract

We consider back-ground gas collision shifts in ion-based clocks. We give both a classical and quantum description of a collision between an ion and a polarizable particle with a simple hard-sphere repulsion. Both descriptions give consistent results, which shows that a collision shift bound is determined by the classical Langevin collision rate reduced by a readily calculated factor describing the decoupling of the clock laser from the ion due to the recoil motion. We also show that the result holds when using a more general Lennard-Jones potential to describe the interaction between the ion and its collision partner. This leads to a simple bound for the collision shift applicable to any single ion clock without resorting to large-scale Monte-Carlo simulations or determination of molecular potential energy curves describing the collision. It also provides a relatively straightforward means to measure the relevant collision rate.

Figures (11)

• Figure 1: Plots of the factors $\mathcal{R}$ (solid) and $\mathcal{A}$ (dashed) given in the text. We have taken a background gas of molecular hydrogen, a collision rate of $\Gamma=1/1000\,\mathrm{s}^{-1}$, an interrogation time $T=5\,\mathrm{s}$ and $\omega_x=\omega_y=2\pi\times 500\,\mathrm{kHz}$.
• Figure 2: Collision shift bound for different interrogation times using a hard-sphere collision model. Other parameters are as for Fig. \ref{['fig:Ramsey']}.
• Figure 3: Typical collision for an attractive potential showing the impact parameter $b$, distance of closest approach $r_\mathrm{min}$, and angles $\phi_m$ and $\theta$ given in the text.
• Figure 4: Orbits for scattering angles $\theta=\pi/2,\pi,3\pi/2,$ and $2\pi$ for different scenarios with a fixed energy. In all cases axes are scaled by the critical parameter $b_c$: (left) $b>b_c$. The dashed curve has $b=1.2b_c$; (right) $b<b_c$ where it is assumed there is a sharp repulsion that reflects the particle at small $r$; (right) $b<b_c$ where it is assume the trajectory passes through the origin.
• Figure 5: Scattering angle as a function of impact parameter where the impact parameter is scaled by $b_c$. For $b<b_c$ the scattering angle assumes a sharp repulsion at small $r$.