MMIG-Bench: Benchmark for Magnomechanical Gratings

• MMIG-Bench is a first-principles protocol that defines the system Hamiltonian and quantum Langevin dynamics for benchmarking multi-order diffraction in cavity–magnon–phonon setups.
• It leverages spatially modulated control fields to engineer and measure diffraction patterns, providing quantitative recipes to optimize parameters like cooperativities and drive amplitudes.
• The protocol offers practical guidelines on coupling strengths and interaction lengths, enabling reproducible benchmarks for applications in quantum memory, transduction, and multiplexed information processing.

MMIG-Bench (Magnomechanically Induced Grating Benchmark) provides a rigorous, first-principles protocol for benchmarking magnomechanical grating phenomena in hybrid cavity–magnon–phonon systems. Exploiting the interplay of collective spin (magnon), mechanical (phonon), and electromagnetic (microwave photon) modes, MMIG-Bench delivers quantitative recipes for engineering, measuring, and interpreting multi-order diffraction induced by spatially modulated control fields. This unified framework formalizes the system Hamiltonian, open-system quantum Langevin dynamics, optical propagation and spectral response, and yields concrete operating regimes for maximal diffraction efficiency, reproducible across cavity-based platforms (Liu et al., 2024).

1. Cavity Magnomechanical System: Physical Model

The cavity magnomechanical system considered in MMIG-Bench consists of three coupled modes: (1) a microwave cavity photon mode (annihilation operator $a$), (2) a localized magnon mode in a ferrimagnet (operator $m$), and (3) a vibrational phonon mode (position operator $x$, momentum $p$), typically in a YIG sphere. The operator dynamics are governed by the Hamiltonian in a pump-rotating frame: $H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$ where:

• $$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$
• $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$
• $H_{\mathrm{dr}}$ describes the drives: standing-wave pump ($E_\ell$), probe ($E_p e^{-i\delta t}$), and magnon control ($m$0).

Key parameters include cavity detuning ($m$1), magnon–cavity ($m$2) and magnon–phonon ($m$3) couplings, and the spatially modulated control $m$4, which sets the fundamental grating period.

The open-system dynamics are described by linearized quantum Langevin equations including dissipation ($m$5) and corresponding input noise channels. Solutions yield the steady-state response of the probe field and the position-dependent transmission spectrum $m$6, which encodes the grating via spatial modulation in the absorption ($m$7) and dispersion ($m$8) profiles.

2. Grating Formation, Diffraction, and Figures of Merit

A central component of MMIG-Bench is the introduction of a transverse standing wave control field in the cavity, which imprints a periodic potential on the magnon–photon–phonon system. This produces a spatial modulation of the probe transmission described by the transfer function

$m$9

with $x$0 the interaction length. The resultant far-field diffraction pattern in the Fraunhofer regime follows

$x$1

where

$x$2

$x$3 is the number of spatial periods, $x$4 the probe wavelength.

Dimensionless figures of merit:

• Magnon–photon cooperativity: $x$5
• Magnon–phonon cooperativity: $x$6
• Quality factors: $x$7
• Normalized drive: $x$8, normalized length $x$9

High diffraction efficiency requires operation in regimes $p$0, $p$1, $p$2, and $p$3.

3. Parameter Dependencies and Benchmarking Protocols

MMIG-Bench systematically details how grating diffraction efficiencies depend on principal system parameters, with concrete numerical benchmarks for each case.

Variations:

• Magnon–photon coupling $p$4 (with fixed $p$5): increasing $p$6 from 1 MHz to 4 MHz raises first-order efficiency ($p$7) from ~5% to ~45%; higher-order channels ($p$8) also increase but with diminishing returns.
• Magnon–phonon coupling $p$9: At fixed $H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$0, increasing $H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$1 boosts higher-order efficiencies (e.g., $H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$2 from ~12% to 18% as $H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$3 goes 1→3 MHz).
• Control amplitude $H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$4: $H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$5 to 6.0 MHz: $H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$6 reaches 48-50%; second- and third-order ($H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$7, $H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$8) scale to 22-25% and 12%.
• Interaction length $H = H_0 + H_{\mathrm{int}} + H_{\mathrm{dr}}$9: first-order efficiency plateaus around $$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$0 mm at ~50%; further increase yields moderate decline.

Sensitivity analysis shows robust efficiency to moderate ($$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$1) fluctuations in coupling strengths and drive fields.

Representative benchmarks:

Parameter	First-order Efficiency ($$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$2)	Second-order ($$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$3)	Third-order ($$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$4)
$$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$5 MHz, $$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$6 MHz, $$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$7 MHz, $$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$8 mm	~50%	~25%	~12%

4. Practical Guidelines and Regimes for Quantum Information Protocols

MMIG-Bench specifies quantitative guidelines for operations required by quantum memory and information retrieval applications:

• For quantum memory/readout with high selectivity, aim for $$H_0 = \Delta_a a^\dagger a + \Delta_m m^\dagger m + \omega_b (x^2 + p^2)/2$$9 (first-order), with $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$0 and $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$1 enabling multiplexed storage.
• Figures of merit for quantum memory include the storage lifetime (driven by mechanical Q-factor $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$2), grating order selectivity, and noise-to-signal in desired diffraction orders.
• Information retrieval is optimized at high cooperativities $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$3, and interaction lengths $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$4 mm.
• Frequency comb and multi-channel operation are accessible by tuning grating period $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$5 and drive amplitude $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$6.

Recommended regime for high-fidelity operation is $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$7 MHz, $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$8 MHz, $$H_{\mathrm{int}} = g_{am}(a^\dagger m + m^\dagger a) + g_{mb} m^\dagger m x$$9 MHz, $H_{\mathrm{dr}}$0 mm.

5. Significance, Applications, and Benchmark Scope

MMIG-Bench offers a replicable, Hamiltonian-level protocol for benchmarking cavity magnomechanics in quantum transduction and hybrid information processing:

• Enables comparative studies of diffraction efficiency and bandwidth in different material platforms or device scales.
• Calibrates performance for quantum memory, where order selectivity (>10:1 signal-to-noise) and lifetimes ($H_{\mathrm{dr}}$1 μs for $H_{\mathrm{dr}}$2) are crucial.
• Defines achievable parameter spaces for integrating MMIG into quantum networks, frequency-multiplexed processing, or coherent optical-microwave conversion.
• Provides a systematic reference for parameter sweeps, device engineering, and reproducibility across groups.

6. Limitations and Extensions

All numerical protocols assume sideband-resolved, weak-probe conditions and neglect strong-mode hybridization or higher pump-induced nonlinearities. The protocol is based on photonic and magnonic quality factors achievable in contemporary YIG microcavities; further scaling may require device-specific recalibration. MMIG-Bench prescribes benchmarks only for spatial grating orders and does not encompass time-bin or frequency-comb protocols unless additional sideband control is imposed.

Potential future extensions include explicit treatment of thermal noise, strong-drive beyond quadratic response, and integration with optomechanical/circuit-based platforms, but the primary MMIG-Bench protocol as presented is self-contained and comprehensive for spatial diffraction in canonical cavity–magnon–phonon architectures (Liu et al., 2024).