Nonlinear Coupling between Motional Modes in Trapped Ion Quantum Processors

TL;DR

This work tackles NoMoCou, a coherent error channel arising from third-order Coulomb interactions that couples trapped-ion motional modes as systems scale. It develops a unified framework using the Tressian tensor to identify near-resonant triads, then quantizes the dynamics with an $\\epsilon_0$ expansion and a two-level reduction to quantify NoMoCou across linear chains and 2D crystals in rf and Penning traps. The study shows NoMoCou can significantly degrade Molmer–Sørensen gate fidelity near nonlinear resonances and when spectator modes are thermally occupied, but yields concrete mitigations: detune away from resonances, engineer the axial spectrum with mild anharmonicity, cool soft spectator modes, and bound bus displacement via multi-loop or shaped gates. Across architectures, three-mode couplings grow rapidly with system size, while spectral shaping and faster, displacement-bounded gates markedly reduce NoMoCou, guiding scalable, high-fidelity trapped-ion quantum processors. Overall, NoMoCou represents a controllable source of coherent error and a lever for exploring nonequilibrium dynamics in engineered quantum matter.

Abstract

Trapped-ion crystals are a leading platform for quantum information science, but achieving the high-fidelity entangling gates required for fault-tolerant quantum computing becomes harder as system size increases. As systems scale, spectral crowding makes low-order nonlinear resonances between collective motional modes increasingly common and can limit gate performance, especially in monolithic or global-mode architectures. We develop a general model to identify and simulate nonlinear motional-mode coupling (NoMoCou) arising from third-order Coulomb terms and quantify its impact on the Molmer-Sorensen gate across linear chains and 2D crystals in rf and Penning traps. We delineate the regimes where NoMoCou dominates the error budget and provide design rules: detune operating points from low-order resonances, tune trap anisotropy to reshape spectra, and shape gate waveforms.

Figures (14)

• Figure 1: Two-dimensional ion crystals exhibit larger motional-energy fluctuations than linear chains, with Penning-trap 2D arrays showing enhanced axial (drumhead) variability while their highest-frequency cyclotron modes remain the most stable. Top row: normal-mode spectra. Bottom row: per-mode mean energy and standard deviation over a $T_\mathrm{sim}=10~\mathrm{ms}$ classical MD evolution (all $3N$ modes initialized at $100~\mu\mathrm{K}$ with random phases; $N=53$). (a,d) Linear chain in an rf trap: small fluctuations about initial energies. (b,e) 2D rf array: increased fluctuations, especially among planar modes. (c,f) 2D Penning array: larger axial fluctuations than (b,e) while cyclotron modes remain comparatively stable. These trends indicate stronger nonlinear energy exchange (NoMoCou) in 2D geometries---most prominently in Penning traps---motivating a quantitative assessment of their impact on gate performance later in the paper.
• Figure 2: Schematic of the two-ion crystal showing the nonlinear coupling between the tilt mode (red, $\omega_\text{T}$) and the breathing (bus) mode (blue, $\omega_\text{B}$). The interaction involves products of the orthogonal modes' ladder operators and enables energy exchange when the resonance condition $\omega_\text{B} \approx 2\omega_\text{T}$ is satisfied.
• Figure 3: Dynamics of the two-ion crystal showing nonlinear mode coupling (NoMoCou) between the tilt (blue) and breathing (green) modes when the resonance condition $\Delta_{\mathrm{TTB}}=\omega_\mathrm{B}-2\omega_\mathrm{T} = 0$ is satisfied. (a) Molecular dynamics simulation with the full Coulomb potential. All modes except the breathing mode are initialized to amplitudes corresponding to a temperature of $100~\mu\mathrm{K}$, while the breathing mode is initialized with zero amplitude. All other modes are plotted in black. (b) Classical reduced model evolving the four canonical variables of the tilt and breathing modes. (c) Quantum reduced model with (solid lines) and without (dashed lines) the rotating-wave approximation (RWA). (Note: the dashed RWA lines are indistinguishable from the solid lines on this scale.) All simulations are initialized with energy in the tilt mode and the breathing mode in its ground state. The close agreement between (a) and (b) validates the classical reduced model, while (c) shows the quantum oscillations and the accuracy of the RWA.
• Figure 4: Fidelity of a Mølmer--Sørensen gate nonlinear motional mode coupling (NoMoCou) in a two-ion crystal. The breathing mode serves as the bus, and the trap is tuned so that the tilt mode is nearly resonant with the bus via the three-phonon (three-wave) interaction ($\omega_\mathrm{B}\approx 2\omega_\mathrm{T}$). We plot $\Delta\omega_y\equiv\omega_y-\omega_{y,\mathrm{res}}$, where $\omega_{y,\mathrm{res}}=(\sqrt{7}/2)\,\omega_z$ [Eq. (\ref{['eq:two_ion_resonance']})]; this tunes the nonlinear detuning $\Delta_{\mathrm{TTB}}\equiv\omega_\mathrm{B}-2\omega_\mathrm{T}$ through resonance ($\Delta_{\mathrm{TTB}}=0$). The gate detuning is set by $\delta_\mathrm{gate}=2\pi/T_{\mathrm{gate}}$ for a single-loop gate ($k=1$). (a) Bell-state fidelity vs. $T_\mathrm{gate}$ and $\Delta\omega_y\equiv\omega_y-\omega_{y,\mathrm{res}}$; contours at $\mathcal{F}=\{0.90,0.99,0.999\}$. (b) Spin-subsystem entropy $S_2(\rho_\mathrm{spin})$ for the same scan; contours at $\{0.01,0.1,1\}$. Off resonance ($|\Delta_{\mathrm{TTB}}|\gg g$ with $g=\epsilon_0|C^{\mathrm{RWA}}_{\mathrm{TTB}}|$), the dominant effect is a dispersive shift $\propto g^2/|\Delta_{\mathrm{TTB}}|$ that changes the bus-mode frequency and produces gate over- or under-rotations. (c) Final spectator (tilt) energy $\langle \mathcal{H}_T\rangle/\hbar\omega_z$; higher values coincide with the low-fidelity band near $\Delta_{TTB}\approx 0$; contour at $\langle \mathcal{H}_\text{T}\rangle/\hbar\omega_z=.1$ (white solid). Here, the NoMoCou is resonant, and there is direct competition between the Mølmer--Sørensen drive at longer $T_\text{gate}$ and NoMoCou ($g\sim\eta\Omega_r$).
• Figure 5: Fidelity of a two-ion Mølmer--Sørensen gate with nonlinear mode coupling when the spectator (tilt) mode is thermally occupied. The bus (breathing) mode is initialized in the ground state and the spins in $\left| 00 \right\rangle$. (a) Spectator initialized in a thermal state with mean occupation $\bar{n}_{\mathrm{spec}}=1$; the scan over nonlinear detuning and gate time $T_\mathrm{gate}$ matches Fig. \ref{['fig:n2_motional_detuning_vs_gate_time']}. (b) Same scan with $\bar{n}_{\mathrm{spec}}=2$. Relative to the ground-state case, the parameter region with degraded fidelity is substantially larger even though the bus begins in $\left| 0 \right\rangle$. Contours are shown at 0.90 (red dashed), 0.99 (blue dotted), and 0.999 (black solid).