Observation of a Topological Berry Phase with a Single Phonon in an Ion Microtrap Array

TL;DR

This work demonstrates precise, site-resolved quantum control of three trapped ions in a triangular 2D microtrap array. By adiabatically tuning local curvatures, a single phonon is coherently shared among the ions and driven around a conical intersection in the motional spectrum, yielding a topological Berry phase of $\pi$ observable via single-phonon interference. The Berry phase is shown to vanish when the path fails to enclose the intersection or becomes non-adiabatic, illustrating the interplay between topology and dynamics in a controllable bosonic system. The platform promises scalable exploration of quantum many-body effects, tunable lattice geometries, and spin-boson models in larger ion arrays.

Abstract

Controlled quantum mechanical motion of trapped atomic ions can be used to simulate and explore collective quantum phenomena and to process quantum information. Groups of cold atomic ions in an externally applied trapping potential self-organize into "Coulomb crystals" due to their mutual electrostatic repulsion. The motion of the ions in these crystals is strongly coupled, and the eigenmodes of motion all involve multiple ions. While this enables studies of many-body physics, it limits the flexibility and tunability of the system as a quantum platform. Here, we demonstrate an array of trapped ions in individual trapping sites whose motional modes can be controllably coupled and decoupled by tuning the local applied confining potential for each ion. We show that a single motional quantum, or phonon, can be coherently shared among two or three ions confined at the vertices of an equilateral triangle 30 $μ$m on a side. We can adiabatically tune the ion participation in the motional modes around a closed contour in configuration space, observing that the single-phonon wavefunction acquires a topological Berry phase if the contour encircles a conical intersection of motional eigenvalue surfaces. We observe this phase by single-phonon interference and study its breakdown as the motional mode tuning becomes non-adiabiatic. Our results show that precise, individual quantum control of ion motion in a two-dimensional array can provide unique access to quantum multi-body effects.

Figures (8)

• Figure 1: (a) Rendering of the triangular 2D ion microtrap array. Electrodes are shown in gold, gaps in black. Black spheres labeled A, B, and C mark the positions of the potential wells where $^9$Be$^+$ ions are trapped. Colored arrows labeled $\hat{\mathbf{e}}_{\rm RA}$ (purple), $\hat{\mathbf{e}}_{\rm VA}$, (white), $\hat{\mathbf{e}}_{\rm TA}$ (gray) indicate the radial, vertical, and tangential principal axis directions in site A, respectively. The principal axes in the other sites are shown but not labeled. The electrodes closest to site A and site B (purple) are used to fine-tune the site curvatures. (b) The radial motion eigenvalue surfaces vs. tuning parameters $s_A$ and $s_B$. The lower two manifolds (blue and red) form a conical intersection at $s_A = s_B = 0$. The semitransparent gray plane indicates a cut through the surfaces along $s_B=0$ where the eigenmode frequencies (related to the curvature eigenvalues by $\delta f_j = \tfrac{1}{2 \pi}\delta k_j/(2 m\, \omega_R)$) were experimentally measured, with data shown in (c). The observed normalized fluorescence rate is shown as a function of the curvature tuning voltage $\delta V_A$ and the difference $\delta f$ of the excitation frequency from the eigenfrequency conical intersection $\omega_R/(2 \pi)$. Larger motional excitation leads to higher normalized fluorescence rates. The colored lines are the best fit to theory with eigenfrequencies colored in analogy to Fig. \ref{['fig:mode-surfaces']} (for details see \ref{['sec:methods']}).
• Figure 2: Single phonon interferences. (a) Probability of finding ion C in $\left| {\downarrow} \right>$ as a function of the evolution time between $\pi$-pulses that inject and remove a single phonon. With ions in sites A and C and tuned on resonance, a single phonon injected into C is coherently exchanged hundreds of times. (b)--(e) All three sites occupied with the ions in B and C on resonance ($\delta V_B \approx 0$), (b) $\delta V_A =$ -1.1, (c) -0.51, (d) 0.00, and (e) 0.92 mV, respectively. Solid lines are fits to the data. Error bars represent $1\sigma$ of $P_{\downarrow}$. (f) Frequency differences extracted from single phonon interference in (b)--(e), labeled by matching colors; and from additional experiments with different $\delta V_A$. The frequency differences are plotted relative to the eigenfrequency that changes linearly with $\delta V_A$ and are in good agreement with the theoretical eigenmode frequencies for the parameters found from fitting the data in Fig. \ref{['fig:spectrum']} (solid, dashed and dashed-dotted lines, for details see \ref{['sec:methods']}). Error bars represent 95% confidence intervals obtained from 5000 bootstrapping trials.
• Figure 3: Observation of a topological Berry phase via single-phonon interference. (a) The path on surface $\delta k_2$ enclosing the conical intersection as a function of $\{s_A,s_B\}$ and (b) not enclosing the conical intersection. The paths approximate contours of constant curvature (also shown in grey). Surface $\delta k_2$ is mirror-symmetric about the plane indicated by the white dashed line. (c) Observation of $P_\downarrow$ as a function of evolution time after tuning the eigenmodes along an adiabatic, closed path in parameter space that encloses the conical intersection (gold disks) vs. a closed path that does not enclose the intersection (pink squares). The curves show a phase difference of $\Delta \phi = \pi \times(0.99 \pm0.01)$ as expected for the Berry phase acquired on the enclosing path. Solid lines are fits and error bars are $1\sigma$ of the mean. The eigenmodes were tuned with a transit duration of $T=$ 780 µ s. (d) The observed phase difference vs. the duration of traversing the enclosing and non-enclosing paths. The solid line is the phase difference predicted by numerical integration of the Schrödinger equation of the 2D-array for the enclosing and non-enclosing paths in parameter space. Error bars are 68 % confidence intervals obtained from 5000 bootstrapping trials.
• Figure S1: Single phonon interferences not shown in Figs. \ref{['fig:exchange-b']}-\ref{['fig:exchange-e']} for the remaining frequency data points as a function of $\delta \text{V}_\text{A}$ shown in Fig. \ref{['fig:exchange-f']}. As the voltage is changed slightly on the electrode under site A, the observed interference pattern changes. By fitting these data to a sum of sinusoids with different amplitudes (pink lines), the motional frequency differences of the system can be extracted and are shown in Fig. \ref{['fig:exchange-f']}. Error bars represent $1\sigma$ of $P_{\downarrow}$.
• Figure S2: Observation of a topological Berry phase via single-phonon interference. (a) The path on surface $\delta k_2$ as a function of $\{s_A,s_B\}$ enclosing the conical intersection but with a larger enclosed area than that of Fig. \ref{['fig:berry-enclosed-path']} and no longer traversing points of equal curvature. (b) The corresponding path not enclosing the conical intersection. Contours of constant curvature are shown in gray. Surface $\delta k_2$ is mirror-symmetric about the plane indicated by the white dashed line. The eigenmodes were tuned with a transit duration of $T=$ 1.8 ms. (c) Observation of $P_\downarrow$ as a function of evolution time after tuning the eigenmodes along the closed path described above that encloses the conical intersection (gold disks) vs. the closed path that does not enclose the intersection (pink squares). The phase difference is $\Delta \phi = \pi \times(0.96 \pm0.02)$. Solid lines are fits and error bars are $1\sigma$ of the mean.