Motional-Mode Separation Technique

• Motional-mode separation is a set of methods that represent complex dynamics in a modal basis, enabling isolation and decoupling of interactions in trapped-ion and molecular systems.
• The technique employs methods like Hessian diagonalization, sideband-selective addressing, and full state tomography to precisely identify and control individual modes.
• Pulse optimization and tailored canonical transformations mitigate cross-mode coupling and nonlinear effects, enhancing fidelity and robustness in quantum operations.

Searching arXiv for relevant papers on motional-mode separation and related trapped-ion methods. I’ll look for the exact arXiv records and closely related work to ground the article in the cited literature. Motional-mode separation denotes a family of procedures that isolate, characterize, or suppress couplings between dynamical modes after a system has been resolved into normal coordinates or coherent components. In trapped-ion and molecular settings, the central operations are Hessian diagonalization, sideband-selective addressing, spin-to-motion mapping, pulse shaping that nulls off-resonant couplings, and canonical transformations that identify the single collective coordinate coupled to an interaction Hamiltonian (Kalis et al., 2016, Jia et al., 2022, Liang et al., 2023, Lu et al., 19 May 2026). In adjacent signal-processing and structural-dynamics literatures, closely related separation problems are treated with dynamic mode decomposition, adaptive local-frequency extraction, second-order separation, and time-invariant beamforming (Chávez-Dorado et al., 2024, Li et al., 2020, Panda et al., 2023, Koldovský et al., 2020). This suggests that the term is best understood not as one fixed algorithm, but as a recurring methodological pattern: represent the dynamics in a modal basis, identify the subset that carries the desired physics, and either reconstruct or decouple that subset.

1. Normal-mode foundations

In trapped-ion systems, motional-mode separation begins with the normal-mode decomposition of the coupled ion crystal. The full Hamiltonian contains kinetic energy, the radio-frequency and static trapping potentials, and Coulomb interactions. Expanding about the equilibrium positions to second order yields a quadratic Hamiltonian whose Hessian matrix $H$ satisfies

$$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$

so that the small-oscillation dynamics become a sum of independent harmonic oscillators in normal-mode coordinates $Q_k$ (Kalis et al., 2016). For a chain of $N$ ions this gives $3N$ collective modes, with eigenvectors $e_k$ that define the mode axes.

Two-ion and two-particle problems admit a more explicit factorization. Sutherland et al. define center-of-mass and stretch coordinates for two same-species ions,

$$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$

with corresponding uncoupled Hamiltonians

$$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$

in the displaced frame (Sutherland et al., 2021). In the molecular gate analysis of two polar molecules, the analogous transformation to

$$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$

produces an uncoupled center-of-mass mode $(+)$ and a relative mode $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$0, with the crucial result that the dipole-dipole interaction depends only on the relative coordinate (Lu et al., 19 May 2026).

These constructions show that “separation” can mean either diagonalization of a quadratic form or identification of the physically relevant collective coordinate after a canonical transformation. A plausible implication is that the modal basis is not merely descriptive: it determines which degrees of freedom must be simulated, cooled, driven, or tomographically reconstructed.

2. Spectral addressing and mode identification

Once the normal modes are known, trapped-ion experiments separate them spectroscopically through sideband selectivity. In the Lamb–Dicke regime $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$1, two-photon Raman transitions couple the internal qubit to motional modes through Jaynes–Cummings-type interactions. For mode $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$2, a red sideband is driven at detuning $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$3 and couples $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$4, while a blue sideband is driven at $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$5 and couples $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$6. The selection rule is $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$7 and $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$8, and spectral separation requires $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$9 (Jia et al., 2022).

The same objective appears in the two characterization methods of “Motional-Mode Analysis of Trapped Ions.” In the weak-binding limit, electric-field “tickling” drives the ion with a small oscillating voltage and reveals secular resonances as fluorescence dips when $Q_k$0. In the strong-binding limit, resolved sideband spectroscopy measures sideband Rabi flops and extracts $Q_k$1 from the transition strengths (Kalis et al., 2016). The paper reports demonstration with single $Q_k$2 ions confined $Q_k$3 above a surface-electrode trap array, with representative secular frequencies $Q_k$4.

These protocols address a common misconception: a motional mode is not directly measured as an independent observable in the same way that an optical spectrum line is read off a detector. In the trapped-ion implementations summarized here, the motional populations are inferred through fluorescence or spin-state measurements after a mode-selective interaction has mapped motional information onto an internal-state signal (Kalis et al., 2016, Jia et al., 2022).

3. Multi-mode state reconstruction

Jia et al. extend spectral selectivity into full tomography of arbitrary multi-mode motional states. After preparing the $Q_k$5-mode target state $Q_k$6, one initializes $Q_k$7 probe ions in $Q_k$8 and simultaneously drives the blue sideband of mode $Q_k$9 on probe ion $N$0 with

$N$1

The measured joint spin-down probability is

$N$2

where $N$3 (Jia et al., 2022). By scanning $N$4, and in practice truncating $N$5, one fits the known basis functions $N$6 and recovers the joint Fock-state populations.

Off-diagonal density-matrix elements are obtained by displaced-Fock tomography. Before the blue-sideband scan, each mode is coherently displaced by $N$7. For a grid of phases $N$8, $N$9, with $3N$0, one measures the displaced populations

$3N$1

takes the $3N$2-dimensional discrete Fourier transform, and inverts the resulting linear relations to obtain all matrix elements $3N$3 up to the chosen cutoff (Jia et al., 2022).

The protocol is experimentally verified with different entangled states of multiple radial modes in a 5-ion chain. Its practical validity rests on the Lamb–Dicke limit, mode-frequency separations $3N$4, finite cutoffs $3N$5, negligible motional decoherence during the short blue-sideband scans, and statistical error propagation through least-squares fits and linear inversion. Because the reconstructed $3N$6 may not be strictly positive, the procedure replaces it by the closest positive-semi-definite density matrix within the experimental error bars (Jia et al., 2022).

4. Pulse-optimized suppression of cross-mode coupling

A distinct use of motional-mode separation appears in mode characterization itself. Liang et al. emphasize that a finite-amplitude sideband pulse intended for one mode $3N$7 also excites neighboring modes, producing cross-mode coupling (CMC). In the interaction picture, the leading first-order Magnus term is

$3N$8

and the unwanted coupling to non-target modes is $3N$9 for $e_k$0 (Liang et al., 2023). The conventional single-mode model therefore fails whenever $e_k$1.

The proposed remedy expands the pulse in a discrete Fourier basis,

$e_k$2

and chooses the coefficients so as to null all $e_k$3, maximize $e_k$4 under fixed average power, and optionally stabilize against frequency offsets (Liang et al., 2023). In matrix form,

$e_k$5

and the first-order cancellation condition is enforced by taking $e_k$6 in the null space of the reduced matrix $e_k$7 with the $e_k$8 row removed. Frequency-drift robustness is added by also nulling low-order derivatives $e_k$9, with the paper stating that even-order nullings are most effective because the qubit-population error depends on $$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$0, an even function of detuning.

The three-ion-chain benchmarks quantify the benefit. For target mode $$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$1 on ion $$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$2, square pulses give $$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$3–$$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$4 for typical parameters $$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$5, $$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$6. Moment-0 shaped pulses suppress $$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$7 to $$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$8, giving $$\hat x_c=\frac{\hat x_1+\hat x_2}{2},\qquad \hat x_s=\frac{\hat x_1-\hat x_2}{2},$$9–$$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$0 improvement at $$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$1. With moment-2 stabilization and $$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$2, $$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$3, the error reaches $$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$4–$$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$5, an order-of-magnitude better than square pulses (Liang et al., 2023).

5. Nonlinear coupling, spectral crowding, and design rules

Motional-mode separation is also limited by nonlinear resonances that survive after quadratic diagonalization. The NoMoCou model expands the Coulomb potential to third order and, after quantization and a rotating-wave approximation, obtains

$$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$6

(Johnson et al., 8 Oct 2025). A near-resonant triad is classified as dynamically significant when

$$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$7

For practical suppression, the paper recommends choosing a detuning $$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$8 to obtain $$\hat H_{t,c}(t)=\frac{\hat p_c^2}{2M}+\frac12 M\omega_c^2(t)\hat x_c^2,\qquad \hat H_{t,s}(t)=\frac{\hat p_s^2}{2M}+\frac12 M\omega_s^2(t)\hat x_s^2$$9.

The reported simulations make the spectral-crowding problem explicit. In a two-ion $$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$0 example, the relevant RWA two-mode coupling strength is $$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$1–$$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$2, with fidelity $$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$3 for detunings $$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$4 or $$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$5 when the spectator is in the ground state. For thermally occupied spectators $$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$6–$$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$7, fidelity dips below $$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$8 over a $$x_\pm=\frac{x_i-x_i^{(0)}\pm(x_{ii}-x_{ii}^{(0)})}{\sqrt2}$$9 window around resonance (Johnson et al., 8 Oct 2025).

Scaling studies show that in linear chains up to $(+)$0, radial-axial couplings turn on as $(+)$1 approaches the zigzag transition; empirically, no significant triads are found for $(+)$2, whereas at $(+)$3 the median $(+)$4, overlapping typical Mølmer–Sørensen gate times. In 2D crystals with $(+)$5, fidelities remain $(+)$6 for $(+)$7 and axial ground state, but Doppler-cooled radial spectators $(+)$8–$(+)$9 reduce fidelity to $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$00–$$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$01 unless the gate time is shortened or the bus displacement is bounded (Johnson et al., 8 Oct 2025). The resulting design rules are to detune operating points from low-order resonances, tune trap anisotropy to reshape spectra, and shape gate waveforms.

6. Canonical separation in transport, splitting, and molecular gates

In ion transport and splitting, mode separation is embedded in the control protocol rather than only in the analysis stage. Sutherland et al. show that Gaussian evolution under time-dependent quadratic Hamiltonians can be written with motional squeeze and displacement operators,

$$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$02

combined with a classical trajectory for the center-of-mass and stretch equilibria (Sutherland et al., 2021). Their separation protocol for two same-species ions consists of diagonalizing into center-of-mass and stretch modes, pre-squeezing both modes, ramping the confining well down, allowing Coulomb-driven separation, and ramping up two independent catching wells. By construction, the final displacement vanishes and each mode returns to the instantaneous ground state, so that

$$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$03

The paper gives a realistic example with $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$04 ions, $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$05, separation by $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$06 in $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$07, and parametric-modulation strengths $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$08, $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$09 (Sutherland et al., 2021).

The molecular controlled-phase gate study applies the same logic to motion-induced uncertainty in dipole-dipole interactions. After transforming to $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$10, the position-dependent interaction expands as

$$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$11

with only the relative mode $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$12 appearing; the center-of-mass mode $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$13 is completely decoupled (Lu et al., 19 May 2026). This reduces the quantum simulation to a single oscillator while retaining all orders relevant to fidelity. Using QuTiP with up to $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$14 excitations in $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$15, the study reports $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$16 for $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$17 and thermal population $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$18, fidelity above $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$19 under a $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$20 static uncertainty in $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$21 when $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$22, and $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$23 for a one-phonon input at $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$24 (Lu et al., 19 May 2026).

A recurring limitation is explicit in both works: the harmonic approximation must remain valid over the wave-packet extent, and the small Lamb–Dicke parameter or low-order expansion must be quantitatively justified (Sutherland et al., 2021, Lu et al., 19 May 2026).

7. Related mode-separation frameworks beyond trapped ions

Outside trapped-ion physics, the same structural problem appears when coherent modes overlap in frequency or evolve in time. In oceanographic time series, simple bandpass filters cannot effectively separate wave motion from turbulence because the frequencies overlap. The DMD-based method constructs snapshot matrices, performs an SVD with rank truncation, computes the low-rank propagator $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$25, and classifies modes by their continuous-time frequencies $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$26 and coherence. The reconstructed wave component is

$$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$27

with sensitivity dominated by the rank truncation $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$28 (Chávez-Dorado et al., 2024). The paper states that, in synthetic and laboratory tests, the DMD-based separation outperforms EEMD and synchrosqueezed wavelet transform, typically producing $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$29–$$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$30 less error in turbulence recovery.

Li–Chui–Jiang–Ji’s adaptive signal separation operation (ASSO) addresses multi-component AM-FM signals for which the usual “divide-and-conquer” decomposition can fail. ASSO approximates each component locally by a linear chirp, detects ridges in an adaptive STFT, and reconstructs components with the corrected formula

$$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$31

thereby removing the $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$32 bias of direct ridge sampling (Li et al., 2020). The reported mono-component linear-frequency-modulation example improves the max reconstruction error from $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$33 to $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$34 when the linear-chirp correction is used with the true $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$35.

Structural-dynamics and blind-source-separation work use related modal logic. The real-time complex-mode identification algorithm of “Mastering Complex Modes” forms analytic signals $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$36, updates the covariance recursively, uses first-order eigen-perturbation for eigenspace tracking, whitens the data, and then jointly diagonalizes lagged covariances to isolate complex modes (Panda et al., 2023). Reported performance includes Modal Assurance Criterion values exceeding $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$37, modal-frequency errors within $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$38, and convergence within $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$39–$$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$40 samples. Koldovský et al.’s Dynamic ICA/IVA study treats time-variant mixtures that remain separable by time-invariant beamformers under the CSV model, extending FastICA-style extraction to moving sources and showing improved recovery for moving-speaker separation when $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$41 blocks are used instead of $$H\,e_k=\lambda_k\,e_k,\qquad \omega_k=\sqrt{\lambda_k/m},$$42 (Koldovský et al., 2020).

These related literatures do not describe the same physical hardware as trapped-ion mode control. They do, however, instantiate the same technical motif: identify a representation in which coherent dynamics become low-rank, narrowband, or jointly diagonalizable, then reconstruct the desired component while quantifying leakage, residual coupling, or model mismatch.