Symmetry-controlled thermal activation in pyramidal Coulomb clusters: Testing Kramers-Langer theory

TL;DR

The paper investigates thermally activated inversions in a square-pyramidal cluster of five trapped Ca^+ ions to test the multidimensional Kramers-Langer theory. The authors combine tunable trap anisotropy, real-time tracking of inversion via the parity-odd octupole moment, and isotope substitution to break permutation symmetry, enabling a symmetry-controlled test of reaction pathways. They demonstrate a Berry-type pseudo-rotation that lowers the barrier for identical ions and show that substituting the apex with a heavier isotope suppresses inversions by forcing a turnstile rotation with a higher barrier; MD simulations and a parameter-free K-L calculation show quantitative agreement across conditions, yielding a cluster temperature of about $1.8 \pm 0.1$ mK. The work establishes laser-cooled Coulomb clusters as a platform to study symmetry-controlled collective dynamics and to validate high-dimensional rate theories, with potential implications for reaction kinetics and quantum control.

Abstract

Laser-cooled ions confined in electromagnetic traps provide a unique, tunable mesoscopic system where the interplay of the trapping potential, nonlinear Coulomb interactions, and laser-ion scattering generates rich, collective dynamics. In this work, we engineer thermally activated switching between two oppositely oriented, square-pyramidal configurations of five laser-cooled ions in a Paul trap. For identical ions ($^{40}\mathrm{Ca}^{+}$), the inversions proceed via a \textit{Berry pseudo-rotation} mechanism with a low activation barrier, enabled by the permutation symmetry, in contrast to the \textit{umbrella inversion} observed in ammonia. The experimentally measured inversion rates, spanning two orders of magnitude, are accurately captured by the multidimensional Kramers-Langer theory, enabling thermometry of the Doppler-cooled ion cluster at $1.8 \pm 0.1$ mK. By substituting the apex ion with a heavier isotope ($^{44}\mathrm{Ca}^{+}$), we break the permutation symmetry and observe a suppression of thermally activated inversions. Numerical analysis reveals that this symmetry breaking closes the low-barrier channel, forcing the system to invert through a high-barrier \textit{turnstile rotation}. Thus, we demonstrate a structural analogue of molecular kinetic isotope effects, establishing trapped ions as a versatile platform to explore symmetry-controlled collective dynamics.

Figures (8)

• Figure 1: Inversion mechanisms for a pyramidal ion cluster. (a) Fluorescence images of the two symmetry-broken pyramidal configurations. (b, c) Experimental time traces of the parity-odd octupole moment $\psi_{3,0}$, showing inversions for an identical-ion cluster at two different settings for the trap aspect ratio $\alpha$. (d) Suppression of thermal activation after the apex ion is replaced by a heavier isotope. The rare transitions are triggered by background-gas collisions. (e, f) Inversion pathway for the identical cluster is a low-barrier pseudo-rotation. (g, h) For the isotope-substituted cluster, the rate-limiting path is a turnstile rotation with a much higher activation barrier. (i, j) The canonical umbrella inversion involves enormous energy barriers.
• Figure 2: (a) Trajectory obtained from an MD simulation at experimentally relevant temperature of $2$ mK, showing the z coordinates of five $^{40}$Ca$^+$ ions. At the inversion seen at $t \approx 10$ ms, the apex ion (green solid line) moves to the square base, while that base ion (purple dotted line) moves to the inverted apex, completing the pseudo rotation. The system remains in this new state for a duration $\tau \approx 25$ ms before inverting back to the initial state, where a different ion (red dashed line) becomes the apex. At $t \approx 48$ ms, we observe rapid barrier recrossings. (b, c) A closer look at cooperative reconfiguration of all five ions during the inversion at $t \approx 35$ ms, capturing the interchange in the roles of apex ions (purple dotted line and red dashed line) and the nearly uniform rotation of the base ions about the $z$ axis. The modulations are due to thermal excitation of the collective modes.
• Figure 3: Experimentally measured pyramidal inversion rates fitted to the K-L theory to determine the cluster temperature $T$ and the inversion rate $R_{\rm bg}$ due to background collisions. (Inset) Agreement between the inversion rates obtained from MD simulations at four different temperatures to the corresponding K-L theory predictions without any fit parameters.
• Figure SF1: Estimation of the experimental inversion rates at different $\alpha$ values. The dwell times obtained from the time series data are binned into histograms. We obtain the inversion rate and its uncertainty from the exponential fits.
• Figure SF2: Equilibrium configurations of $^{44}\mathrm{Ca}^+(^{40}\mathrm{Ca}^+)_4$ cluster. The global (local) minima are represented by filled (empty) markers, respectively.