Amplitude-Modulated Multimode Motional Gate

• Amplitude-Modulated Multimode-Motional Coupling Gate is a scheme using shaped bichromatic drives to control spin-dependent displacements and achieve a targeted geometric phase.
• The design leverages both analytical and numerical pulse optimization methods to enforce motional closure and balance phase contributions from multiple modes, reducing sensitivity to detuning and timing errors.
• Experimental implementations demonstrate high fidelities (up to ~99%) and lower energy dissipation, proving the gate’s robustness and practical significance in trapped-ion systems.

Searching arXiv for the cited papers to ground the article in current metadata and confirm the relevant sources. arXiv query: (Duwe et al., 2021) An amplitude-modulation multimode-motional coupling gate is a trapped-ion entangling gate in which the time-dependent amplitude of a bichromatic drive is shaped so that spin-dependent displacements of collective motional modes return to zero at the gate endpoint while the enclosed phase-space area produces the target entangling phase. In the literature represented by microwave near-field and Raman implementations, the operative interaction is the Mølmer–Sørensen (MS) spin-dependent-force gate, and amplitude modulation is used to suppress residual spin-motion entanglement, reduce sensitivity to motional-frequency drift or gate-timing error, and, in explicitly multimode constructions, balance the entangling-phase contributions of several motional modes with a single fixed drive frequency (Duwe et al., 2021, Ruzic et al., 2022).

1. Conceptual scope and development

The modern form of amplitude-modulated motional-coupling gates emerged from attempts to improve the robustness of trapped-ion MS gates without sacrificing the advantages of geometric phase accumulation. In microwave hardware, near-field entangling gates avoid spontaneous emission as a fundamental source of decoherence, but their relatively slow operation makes them sensitive to fluctuations and noise of the motional mode frequency. A 2019 $^{9}\mathrm{Be}^+$ experiment therefore introduced amplitude-shaped microwave gate drives, using a single $\sin^2$ envelope to obtain resilience to mode-frequency changes without increasing the electrical energy cost per gate (Zarantonello et al., 2019).

A subsequent development recast the same design problem as a constrained quadratic optimization over the drive envelope $\Omega(t)$. In that formulation, the pulse is represented in a piecewise-linear basis, loop-closure and first-order detuning-robustness constraints are enforced by projection, and the optimal pulse is obtained as the largest-eigenvalue solution of a projected generalized eigenvalue problem. The result was a family of microwave-driven gates with improved resilience, faster operation, and approximately $25\%$ lower dissipated energy than square pulses at the same $\delta\tau$ (Duwe et al., 2021).

The explicitly multimode step was made in a 2022 Raman-beam implementation on a three-ion chain. There, a truncated Gaussian amplitude envelope suppresses residual displacement for all modes, while a constant bichromatic detuning is chosen so that the weighted derivatives of multiple modal phase contributions cancel, $d\theta/d\delta_c=0$. That operating point yields frequency-robust MS gates which exploit multimode participation rather than treating all but one mode as parasitic (Ruzic et al., 2022).

More recent analyses extended the amplitude-modulation framework in two orthogonal directions. One line used a Fourier-series parameterization to derive linear robustness constraints against gate-timing errors and to optimize average power, improving the leading timing-error infidelity scaling from $\mathcal{O}(\Delta t^2)$ to $\mathcal{O}(\Delta t^6)$ with one linear constraint and to $\mathcal{O}(\Delta t^{10})$ with two constraints (Ellert-Beck et al., 2024). Another line combined amplitude modulation with composite-pulse logic, constructing composite MS sequences that explicitly cancel displacement operators between physical subgates and thereby improve robustness to timing, detuning, and coupling errors simultaneously (Zlatanov et al., 30 Jan 2025).

2. Phase-space structure of the gate

The common structure is the MS interaction between collective spin and motional quadratures. In an explicitly multimode formulation, the interaction-picture Hamiltonian can be written as

$$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$

with

$\sin^2$0

Because each mode enters independently, the propagator factorizes as

$\sin^2$1

where the mode-$\sin^2$2 displacement trajectory is

$\sin^2$3

and the total two-qubit rotation angle is

$\sin^2$4

The ideal maximally entangling condition is

$\sin^2$5

This is the multimode statement of simultaneous trajectory closure and target geometric phase (Ruzic et al., 2022).

Single-mode microwave treatments use equivalent coordinates. One formulation writes

$\sin^2$6

with enclosed area

$\sin^2$7

and target conditions

$\sin^2$8

Another equivalent representation uses the propagator

$\sin^2$9

where $\Omega(t)$0 enforces decoupling and $\Omega(t)$1 gives the Bell-state phase (Duwe et al., 2021, Zarantonello et al., 2019).

These equivalent phase-space languages make the role of amplitude modulation explicit. The envelope $\Omega(t)$2 or $\Omega(t)$3 does not merely scale the interaction strength; it determines the geometry of the motional loops. Smoother or more tightly wound loops yield smaller endpoint displacement under perturbations, hence less residual spin-motion entanglement.

3. Amplitude-modulation design strategies

The earliest experimentally implemented amplitude envelopes were analytic. In the microwave near-field $\Omega(t)$4 gate, the drive was written as

$\Omega(t)$5

with soft start and soft stop enforced by $\Omega(t)$6. The experimentally useful case was $\Omega(t)$7, $\Omega(t)$8, giving a single $\Omega(t)$9 lobe. For this envelope, the allowed detunings satisfy

$25\%$0

where $25\%$1 is the order of the shaped gate. Increasing $25\%$2 increases the number of windings in phase space while reducing the trajectory radius, which is the central geometric explanation for the improved robustness (Zarantonello et al., 2019).

The numerical-optimization approach is more general. The pulse is expanded in a piecewise-linear hat-function basis,

$25\%$3

on a discretized time grid. Closure and first-order detuning-robustness constraints are enforced by projecting out the span of

$25\%$4

The energy functional is

$25\%$5

and smoothness is incorporated through

$25\%$6

After projection,

$25\%$7

and the pulse is chosen as the eigenvector with the largest eigenvalue. This implements fixed entangling area, exact loop closure, first-order insensitivity to detuning, reduced microwave energy, and soft turn-on and turn-off within one quadratic optimization framework (Duwe et al., 2021).

A complementary analytical parameterization expands the envelope in a Fourier series,

$25\%$8

In that framework, exact closure at the ideal gate time implies $25\%$9, and timing-robustness conditions reduce to linear equations on the Fourier coefficients. For the power-optimal $\delta\tau$0 family, one added linear constraint yields $\delta\tau$1 timing-error infidelity, and two constraints yield $\delta\tau$2, with the optimization again reduced to a Rayleigh-quotient eigenproblem (Ellert-Beck et al., 2024).

Composite constructions use amplitude shaping inside each physical MS pulse and then concatenate several such pulses with phase programming. In that setting, the central insight is that robustness cannot be assessed on the pure spin-spin rotation alone, because the same control errors also perturb the displacement operator $\delta\tau$3. The composite protocol therefore enforces pairwise displacement cancellation between neighboring physical gates and uses sequences such as

$\delta\tau$4

together with shaped couplings of the form

$\delta\tau$5

The preferred composite sequence improves timing, detuning, and coupling robustness relative to standard and two-tone references in the reported simulations (Zlatanov et al., 30 Jan 2025).

4. Multimode robustness and balanced modal participation

The explicitly multimode gate design separates two error channels:

$\delta\tau$6

If $\delta\tau$7, then

$\delta\tau$8

For symmetric couplings, the small-displacement estimate is

$\delta\tau$9

The point of amplitude modulation is to suppress $d\theta/d\delta_c=0$0 broadly enough that the dominant remaining sensitivity becomes coherent phase accumulation rather than residual displacement (Ruzic et al., 2022).

For the truncated Gaussian pulse,

$d\theta/d\delta_c=0$1

the paper derives

$d\theta/d\delta_c=0$2

Residual displacement is therefore exponentially suppressed in $d\theta/d\delta_c=0$3. Once the closure problem is softened for all relevant modes, the constant bichromatic detuning can be chosen to make the total entangling phase stationary:

$d\theta/d\delta_c=0$4

This is the formal meaning of “balanced contributions of multiple motional modes” (Ruzic et al., 2022).

The physical interpretation is that nearby modes can contribute opposite detuning sensitivity to the total phase. Instead of suppressing multimode participation completely, the gate uses amplitude shaping to keep all modal loops nearly closed and then selects $d\theta/d\delta_c=0$5 so that the weighted slopes of the modal phase contributions cancel. In the three-ion demonstration, the gate operated between the lowest two radial-b modes, and the measured fidelity of the balanced Gaussian dropped by less than $d\theta/d\delta_c=0$6 across $d\theta/d\delta_c=0$7, with a peak fidelity of $d\theta/d\delta_c=0$8 (Ruzic et al., 2022).

Not all amplitude-modulated gates are multimode in this strong sense. The 2021 microwave optimization was centered on a single target detuning and its derivative, not on explicit closure constraints for every spectator mode. Its robustness mechanism is therefore “first-order spectral nulling around the target sideband detuning,” even though the simulations included one spectator in-phase radial mode detuned by $d\theta/d\delta_c=0$9, thermal occupation, and heating for both modes (Duwe et al., 2021). This distinction is fundamental: detuning-robust single-mode design and true multimode balancing are related but not identical.

5. Implementations and reported performance

The principal implementations span microwave near-field $\mathcal{O}(\Delta t^2)$0 gates and Raman-driven $\mathcal{O}(\Delta t^2)$1 gates. Together they show that amplitude modulation is compatible with integrated microwave conductors, surface-electrode traps, and longer ion chains.

Protocol	Platform	Reported result
Analytic $\mathcal{O}(\Delta t^2)$2 amplitude shaping (Zarantonello et al., 2019)	Pair of $\mathcal{O}(\Delta t^2)$3 ions in a surface-electrode trap with integrated microwave conductors	In absence of injected noise, operation infidelity in the $\mathcal{O}(\Delta t^2)$4 range
Numerically optimized AM microwave gate (Duwe et al., 2021)	Pair of $\mathcal{O}(\Delta t^2)$5 ions in a surface-electrode trap with integrated microwave conductors	Infidelities below $\mathcal{O}(\Delta t^2)$6; conclusion states infidelity in the $\mathcal{O}(\Delta t^2)$7 range
Balanced Gaussian multimode gate (Ruzic et al., 2022)	Three-ion chain of $\mathcal{O}(\Delta t^2)$8 ions on Sandia’s QSCOUT surface trap	$\mathcal{O}(\Delta t^2)$9 reduction from peak fidelity over a $\mathcal{O}(\Delta t^6)$0 range of frequency offset

In the 2019 microwave experiment, the sideband coupling strength was $\mathcal{O}(\Delta t^6)$1, the relevant radial mode had $\mathcal{O}(\Delta t^6)$2 and $\mathcal{O}(\Delta t^6)$3, and finite-element simulations indicated that the microwave conductor reflects $\mathcal{O}(\Delta t^6)$4 of the amplitude. At equal pulse energy, the $\mathcal{O}(\Delta t^6)$5-shaped gate outperformed the square pulse at every injected-noise level, and without injected noise the best measured fidelities were $\mathcal{O}(\Delta t^6)$6 by weighted-Poissonian analysis, $\mathcal{O}(\Delta t^6)$7 SPAM-corrected by threshold/binning analysis, and $\mathcal{O}(\Delta t^6)$8 by maximum-likelihood analysis (Zarantonello et al., 2019).

In the 2021 optimized microwave gate, the maximum gate Rabi rate was again $\mathcal{O}(\Delta t^6)$9, and optimized envelopes were implemented for

$\mathcal{O}(\Delta t^{10})$0

with durations

$\mathcal{O}(\Delta t^{10})$1

The optimized pulses dissipated approximately $\mathcal{O}(\Delta t^{10})$2 less energy than square pulses for the same $\mathcal{O}(\Delta t^{10})$3, and the fastest reported gate was approximately $\mathcal{O}(\Delta t^{10})$4 (Duwe et al., 2021).

In the 2022 balanced multimode experiment, the Gaussian pulse was implemented digitally as a natural cubic spline with 13 amplitude knots on custom RFSoC hardware. The realized parameters were

$\mathcal{O}(\Delta t^{10})$5

The measured fidelity drop remained below $\mathcal{O}(\Delta t^{10})$6 across $\mathcal{O}(\Delta t^{10})$7, and numerical simulations on chains from $\mathcal{O}(\Delta t^{10})$8 to $\mathcal{O}(\Delta t^{10})$9 found total state infidelity below $$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$0 over $$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$1 kHz (Ruzic et al., 2022).

Analytical and numerical studies sharpened the reported trade-offs. The timing-robust Fourier construction found extra average-power overheads of $$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$2 for one linear constraint and $$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$3 for two constraints at $$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$4 (Ellert-Beck et al., 2024). The composite amplitude-modulated MS gate found that all amplitude-modulated $$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$5 sequences outperformed both the standard MS gate and the two-tone gate for timing errors, that $$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$6 was comparable to the two-tone implementation for detuning errors, and that the normalized power cost per gate in a sequence was typically below about $$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$7 times that of a single MS gate with the same modulation (Zlatanov et al., 30 Jan 2025).

6. Limitations, misconceptions, and extensions

A frequent misconception is that amplitude modulation by itself provides a full multimode solution. The literature does not support that generalization. The 2019 analytic $$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$8 gate is explicitly single-mode, with no simultaneous closure constraints $$H(t)= -\hbar \Omega(t)\sum_k S_{y,k}\left(a_k^\dagger e^{i\delta_k t}+a_k e^{-i\delta_k t}\right),$$9 (Zarantonello et al., 2019). The 2025 composite MS gate is also explicitly single-mode, with one bosonic mode operator and numerical simulations using “14 motional states” for that mode (Zlatanov et al., 30 Jan 2025). Even the 2021 numerical microwave optimization, though directly relevant to multimode robustness, is built around one target detuning and its first derivative rather than a full set of mode-resolved closure constraints (Duwe et al., 2021).

Another misconception is that amplitude modulation addresses every dominant error source. In the microwave near-field setting, amplitude modulation can induce an AC Zeeman term,

$\sin^2$00

so hardware-specific field asymmetries can convert amplitude shaping into time-dependent qubit-frequency shifts. In the reported $\sin^2$01 experiment, the trap and ion position were engineered so that $\sin^2$02, and the discussion identified chirping the sideband frequencies or using dynamical decoupling as mitigation routes (Zarantonello et al., 2019). The 2021 optimization likewise did not include qubit-frequency noise as an optimization target and explicitly noted that AC-Zeeman-shift fluctuations could still affect AM gates (Duwe et al., 2021).

The principal trade-off is between robustness and control cost. The 2021 optimization noted that higher derivatives of the closure conditions could be constrained for stronger robustness, but stopped at first order to avoid excessive energy cost (Duwe et al., 2021). The 2024 Fourier analysis made the same trade-off quantitative: more linear constraints improve timing-error scaling but raise the average-power minimum (Ellert-Beck et al., 2024). The 2022 balanced multimode study identified an additional chain-length trade-off: as the chain gets longer, mode crowding eventually raises displacement error because the nearest-mode detunings become too small for Gaussian suppression to remain effective (Ruzic et al., 2022).

Several extensions are explicitly suggested in the literature. The 2019 work proposed Blackman pulses, weighted sums of sines with different $\sin^2$03, and piecewise functions with enough steps as richer modulation families for spectral crowding (Zarantonello et al., 2019). The 2021 projection formalism naturally suggests extension from one detuning $\sin^2$04 to a set $\sin^2$05, enforcing displacement suppression for each relevant mode (Duwe et al., 2021). The 2024 Fourier formalism suggests a different route: stack the linear closure and derivative constraints for every mode while retaining a quadratic power cost and eigenvalue-based optimization (Ellert-Beck et al., 2024). Taken together, these results suggest that amplitude-modulation multimode-motional coupling gates are best regarded not as a single protocol, but as a control-design family whose central invariant is geometric phase-space engineering under simultaneous motional-closure and robustness constraints.