Multimode cavity magnonics in mumax+: from coherent to dissipative coupling in ferromagnets and antiferromagnets

This paper introduces a two-tier extension for the GPU-accelerated micromagnetic framework mumax+ that enables efficient, spatially resolved simulation of multimode cavity magnonics in both ferromagnets and antiferromagnets, successfully validating the tool through eight benchmarks covering phenomena ranging from coherent coupling and mode-selective addressing to dissipative interactions and level attraction.

The Gist

Imagine you have two very different musical instruments: a microwave oven (which creates invisible waves of energy) and a tiny, spinning magnet (specifically, a crystal called YIG).

Usually, these two don't talk to each other. But in the world of quantum physics, if you get them close enough, they can start a duet. They can "dance" together, swapping energy back and forth so perfectly that they become a single hybrid creature. Physicists call this creature a magnon-polariton.

This paper is about building a digital playground (a computer simulation) to watch this dance happen, but with a twist: they wanted to make the simulation fast, flexible, and able to handle complex scenarios like "anti-magnets" (antiferromagnets).

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Slow Messenger" vs. The "Super-Runner"

To simulate this dance, you need to calculate two things at the exact same time:

1. How the magnet spins (the LLG equation).
2. How the microwave waves bounce around the cavity (the Cavity ODE).

The Old Way: Imagine a runner (the magnet simulation) and a messenger (the microwave calculation). Every time the runner takes a step, they have to stop, run all the way to the office (the computer's CPU), hand a note to the messenger, wait for the messenger to calculate the next move, and then run back. This "stop-and-go" is incredibly slow and wastes time.

The New Way (This Paper): The authors built a two-tier system:

• Tier 1 (The Super-Runner): They wrote special code (CUDA kernels) that lets the runner and the messenger live in the same house (the computer's GPU). They never have to leave the house to talk. They calculate the next step together instantly. This is for the heavy-duty, professional simulations.
• Tier 2 (The Quick Sketch): They also built a Python tool that acts like a "quick sketch." It doesn't run as fast as the Super-Runner because it still has to send notes back and forth, but it's super easy to use. You don't need to be a coding wizard to use it; you just want to test an idea quickly.

2. The Dance Moves: From "Repulsion" to "Attraction"

The paper tests this new playground with eight different "dance routines" (benchmarks) to prove it works.

• The Avoided Crossing (The "Level Repulsion"):
  Imagine two dancers approaching each other. Usually, if they have the same energy, they bump into each other and bounce apart. In physics, this creates a "gap" where they can't exist. The simulation perfectly recreated this gap, proving the math is right.
• The Rabi Oscillation (The "Energy Swap"):
  Imagine one dancer starts with a ball (energy). They toss it to the other, who tosses it back. The paper showed that the ball goes back and forth perfectly, like a pendulum. If the simulation was wrong, the ball would get lost or the rhythm would be off. It wasn't; the rhythm was perfect.
• The "Cooperativity" (The "Crowd Control"):
  Sometimes the dancers are too shy (weak coupling) to notice each other, and sometimes they are too loud (strong coupling). The simulation showed exactly when the "shy" phase turns into the "loud" phase, matching real-world physics.

3. The Special Tricks

The authors didn't just stop at the basics. They showed off some advanced moves:

• The "Ghost" Mode (Dark Polaritons):
  Imagine three dancers. Two are talking to a third one. But if they time it just right, the two can talk to each other without the third one even knowing they are there. The simulation found this "ghost" connection, which is crucial for future quantum computers.
• The "Anti-Magnet" (Antiferromagnets):
  Most magnets have all their tiny spins pointing the same way. But "anti-magnets" have spins pointing in opposite directions, canceling each other out. The simulation successfully modeled these tricky materials, showing how they dance with microwaves in a unique way (using something called the "Néel vector," which is like the "net mood" of the anti-magnet).
• The "Abnormal" Dance (Dissipative Coupling):
  Usually, dancers push apart when they get close. But the authors showed that if you change the rules (add "friction" or dissipation), the dancers might actually pull toward each other and merge. This "level attraction" is weird and counter-intuitive, and the simulation captured it perfectly.

4. Why This Matters

Think of this paper as releasing a new, upgraded video game engine for scientists.

• Before: Scientists had to build their own game engines from scratch or use slow, clunky tools to study how magnets and microwaves interact.
• Now: They have a free, open-source toolkit (mumax+) that is already built.
  • If you want to do a quick experiment, you use the Python tool (Tier 2).
  • If you want to simulate a massive, complex crystal with millions of tiny details, you use the GPU-native tool (Tier 1).

The Bottom Line

The authors built a bridge between the world of microwaves and the world of spinning magnets. They proved their bridge is strong, fast, and capable of handling weird, complex scenarios like "anti-magnets" and "energy-sucking" friction. This allows scientists to design better quantum computers and sensors without having to build expensive physical labs for every single test—they can just run the simulation first.

Technical Summary

Here is a detailed technical summary of the paper "Multimode cavity magnonics in mumax+: from coherent to dissipative coupling in ferromagnets and antiferromagnets."

1. Problem Statement

Cavity magnonics explores the coherent coupling between microwave cavity photons and magnon excitations in magnetic materials. This interaction enables advanced quantum technologies such as quantum transduction, magnon-mediated entanglement, and single-magnon detection.

• Simulation Challenge: Accurately simulating photon-magnon coupling requires self-consistently integrating cavity photon dynamics (Ordinary Differential Equations, ODEs) with micromagnetic magnetization dynamics (Landau-Lifshitz-Gilbert, LLG, equation).
• Existing Limitations:
  • Full Electromagnetic Solvers (FDTD): While physically complete, coupling a full FDTD solver to the LLG equation is computationally prohibitive (orders of magnitude slower).
  • Lumped Cavity Models: Previous extensions (e.g., for mumax3) often required modifying core solvers or lacked support for complex scenarios like multimode coupling, spatially resolved mode profiles, and antiferromagnetic systems.
  • Performance vs. Flexibility: There is a trade-off between high-performance GPU-native implementations (hard to prototype) and flexible CPU-based Python scripts (slow due to data transfer overhead).

2. Methodology

The authors present a two-tier extension for mumax+, a GPU-accelerated, open-source micromagnetic framework. This architecture allows for both rapid prototyping and high-performance production simulations.

Tier 1: CUDA-Native Solver (High Performance)

• Integration: The cavity ODEs are integrated directly inside the GPU-based RK45 adaptive time-stepper as a DynamicEquation.
• Mechanism:
  • Multi-mode Support: A custom CUDA kernel (multimode_cavity.cu) solves $N$ coupled cavity ODEs simultaneously with the LLG equation.
  • Spatial Resolution: A second kernel (cavity_mode_profile.cu) handles spatially resolved mode profiles $u_n(\mathbf{r})$. It uses CUDA reduction kernels to compute weighted magnetization averages $\langle u_n m_\alpha \rangle$ and applies spatially varying RF feedback fields $H_{rf}(\mathbf{r}, t) = \sum h_{0,n} u_n(\mathbf{r}) \text{Re}[a_n(t)]$.
• Benefit: Eliminates per-step GPU $\leftrightarrow$ CPU data transfers, enabling simulations of large grids and non-uniform spin-wave modes ($k \neq 0$).

Tier 2: Python Co-Simulation (Rapid Prototyping)

• Implementation: A lightweight Python class (CavityMagnon) wraps the cavity ODE and a co-simulation loop.
• Mechanism:
  • Uses the operator-splitting method: The CPU solves the cavity ODE (RK4) using the spatially averaged magnetization read from the GPU.
  • The resulting RF field is applied back to the GPU magnet object.
• Benefit: Requires no recompilation of the C++/CUDA codebase, allowing researchers to rapidly test new physics (e.g., nonlinear terms, parametric drives) by editing only the Python ODE.

Theoretical Framework

• Coupling Convention: Uses the co-rotating component $m_- = m_x - i m_y$ to ensure physically correct Rabi dynamics.
• Self-Consistency: Enforces the condition $h_0 = 2g/\gamma$ to ensure the back-action field strength matches the forward coupling strength, preserving energy conservation and correct oscillation periods.
• Dissipative Coupling: Extends the model to include dissipative coupling ($g_d$), allowing the simulation of "level attraction" and abnormal anticrossing phenomena.

3. Key Contributions

1. Dual-Tier Architecture: A novel implementation strategy that offers a high-performance CUDA-native path for production and a zero-compilation Python path for prototyping, both sharing identical physics.
2. Multimode & Spatial Resolution: Support for $N$ coupled cavity modes and spatially resolved mode profiles, enabling the study of selection rules and dark polariton states.
3. Antiferromagnetic Support: Extension to two-sublattice antiferromagnets, enabling the simulation of Néel vector spectroscopy and field-split magnon branches.
4. Dissipative Coupling Implementation: Successful modeling of the transition from coherent (level repulsion) to dissipative (level attraction) coupling regimes.
5. Open Source: The complete source code (CUDA kernels, Python classes, and benchmark scripts) is made publicly available.

4. Results (Eight Benchmark Simulations)

The implementation was validated against analytical theory through eight distinct benchmarks:

1. Anticrossing Spectra: Demonstrated magnon-polariton splitting with a Root Mean Square Error (RMSE) of 17 MHz (17% of $2g$), attributed to FFT frequency resolution limits rather than physics errors.
2. Vacuum Rabi Oscillations: Verified coherent energy exchange between photons and magnons. The Rabi period error was $<6\%$ across coupling strengths, confirming the self-consistency condition $h_0 = 2g/\gamma$.
3. Cooperativity Phase Diagram: Mapped the transition from weak ($C < 1$) to strong coupling ($C > 1$). At $C=6.7$, a splitting of 21.3 MHz was measured against a theoretical 20.0 MHz.
4. Mode-Profile Selection Rules: Confirmed that uniform cavity modes couple to uniform Kittel magnons, while higher-order modes (with nodes) have zero overlap with uniform magnons in symmetric geometries.
5. Multimode Polariton Hybridization: Simulated two cavity modes coupling to one magnon, observing the formation of two bright polariton branches and one dark mode (decoupled from the magnon) that mediates indirect photon-photon energy transfer.
6. Mode-Selective Coupling: Demonstrated engineering of spatial overlap to selectively address specific spin-wave modes (e.g., coupling a TE102 mode to a $k=1$ standing wave while decoupling it from the $k=0$ Kittel mode).
7. Antiferromagnetic Magnonics: Validated the simulation of antiferromagnets, showing the Néel vector spectrum tracks bare magnon frequencies without hybridization gaps, while the cavity transmission shows the expected hybridized splitting.
8. Abnormal Anticrossing: Successfully simulated the transition from normal level repulsion (coherent coupling) to level attraction (pure dissipative coupling), where the frequency gap closes but linewidths split.

Numerical Convergence:

• Time-step studies ($\Delta t = 0.25 - 4$ ps) confirmed errors remain below 3% for all step sizes.
• Performance analysis showed that for small grids, the Python tier is limited by GPU $\leftrightarrow$ CPU transfer overhead (~4.4 ms/step), while the CUDA tier scales efficiently for larger, spatially resolved simulations.

5. Significance

This work bridges the gap between theoretical cavity magnonics and practical micromagnetic simulation.

• Accessibility: By providing a Python co-simulation tier, it lowers the barrier to entry for researchers to explore complex cavity-magnon physics without needing deep expertise in CUDA programming or core solver modification.
• Scalability: The CUDA-native tier enables the simulation of realistic experimental setups involving large samples, complex cavity geometries, and non-uniform spin waves, which were previously computationally intractable with lumped models.
• Versatility: The framework's ability to handle both ferromagnetic and antiferromagnetic systems, as well as coherent and dissipative coupling, makes it a comprehensive tool for designing next-generation quantum transducers and magnonic devices.
• Validation: The rigorous benchmarking against analytical theory establishes the framework as a reliable standard for future research in hybrid quantum systems.