Coupling phase interference effects in a multimode cavity magnonics system

AI summary

Overview

Research area: Cavity magnonics — the interaction between microwave cavity photons and magnons (spin-wave excitations) in ferrimagnetic materials such as Yttrium Iron Garnet (YIG), with a focus on multimode cavity systems and nonreciprocal microwave transmission.

Technical level: Advanced. The paper combines a second-quantized input–output Hamiltonian model, finite-element (COMSOL) simulation, and room-temperature vector-network-analyzer measurements.

Scope in one sentence: The paper shows that the phases associated with photon–magnon (internal) and cavity–probe (external) couplings, rather than coupling magnitudes alone, determine the distinct transmission signatures and the nonreciprocity observed in a four-post re-entrant cavity coupled to two YIG spheres.

What This Paper Is About

Most cavity magnonics models keep only a few cavity modes near the magnon frequency and ignore the coupling phase. In a multimode cavity with multiple YIG spheres, this reduced description fails to explain features such as an apparently uncoupled resonance that depends on where the spheres sit, and strong nonreciprocal transmission at antiresonance frequencies. The authors build an input–output model that explicitly retains both internal coupling phases (set by the orientation of the microwave magnetic field inside each sphere) and external coupling phases (set by the parity of the cavity field sensed by each loop probe), and use it to explain these effects in a quadruple-post re-entrant cavity with two YIG spheres.

Key Contributions

1. A phase-complete input–output model for multimode cavity magnonics. The authors write a 6 × 6 internal Hamiltonian (4 cavity modes plus 2 magnon modes) coupled to 2 probe baths, with every photon–magnon coupling carrying an internal phase factor and every cavity–probe coupling carrying an external phase factor, yielding a 2 × 2 scattering matrix.
2. Identification of a positionally dependent "uncoupled" mode. They show that a mode that appears uncoupled to the magnons in the diagonal arrangement (DA) is not intrinsically decoupled; it emerges from interference of internal coupling phases, quantified by an accumulated physical phase θ_p = θ21 − θ11 − (θ20 − θ10).
3. Explanation of nonreciprocity at antiresonance frequencies. They report experimentally observed large nonreciprocity at antiresonances and attribute it to the odd-parity external coupling phase of far-detuned cavity modes interfering with the internal coupling phases.
4. A reduced 4 × 4 analytical model. By retaining only the two degenerate cavity modes and two magnon modes, they derive a closed-form S21 that reproduces the three-peak (DA) and four-peak (PA) responses and yields analytical resonance, antiresonance, and effective-coupling expressions for θ_p = 0, θ_p = π, and arbitrary θ_p.

Main Findings

• Two sphere arrangements give qualitatively different spectra. In the diagonal arrangement (DA), three resonances appear at μ0H = 0.170 T, one of them an uncoupled mode at ω/2π = 5.03 GHz, with effective coupling g_DA^exp/2π = 160 MHz. In the parallel arrangement (PA), four resonances appear, with g_PA^−,exp/2π = 106 MHz and g_PA^+,exp/2π = 127 MHz.
• Interference of internal coupling phases produces the uncoupled mode. For the DA the extracted physical phase is θ_p ≈ 0 rad; the resonance condition then gives three solutions — a pure magnon resonance (dark to the probe), an uncoupled cavity resonance at Δc = 0, and a polariton branch with ΔcΔm = 4g², i.e. g_eff² = 4g².
• A near-π phase suppresses the uncoupled mode and weakens coupling. For θ_p ≈ 3.09 rad (the PA case) the response shows two asymmetric polariton pairs with no uncoupled mode, and in the ideal θ_p = π limit the resonance condition becomes ΔcΔm = 2g², giving g_eff² = 2g² — a smaller effective coupling than the θ_p = 0 case. The antiresonance in that limit sits at the FMR, ω_ar = ω_m.
• An arbitrary phase lifts the polariton degeneracy. The authors state that a value other than θ_p = 0 or π lifts the degeneracy of the cavity-magnon polaritons and explains the two dips between the resonance pairs in the PA.
• Far-detuned modes cannot be neglected. The nonreciprocity at antiresonances is traced to the odd-parity external coupling phase of modes far from the polaritons, which interferes with the internal coupling phases; this is why reduced few-mode descriptions miss the effect.
• Experiment and model agree on the main spectral features but not perfectly on coupling strength. Simulated effective couplings are g_DA^sim/2π = 128 MHz, g_PA^+,sim/2π = 121 MHz and g_PA^−,sim/2π = 92 MHz, corresponding to offsets of 21.45%, 4.83% and 11.28% from experiment. The authors attribute this to possible YIG position mismatch (affecting the filling factor η), post-to-lid gap mismatch, and higher magnetostatic modes not included in the model.
• Cavity characterization. The empty four-post cavity supports a degenerate pair D0↑0↓ and D↓0↑0 at 4.96 GHz with total line width γ_tot/2π = 21 MHz, plus a lower D↓↓↓↓ mode at 2.00 GHz and a higher D↑↓↑↓ mode at 6.31 GHz; modes beyond 6.31 GHz start at 17.5 GHz and are treated as negligible.

Methodology in Plain English

The team machined a brass microwave cavity from a design generated in COMSOL Multiphysics, with four cylindrical posts arranged symmetrically around the center and the resonant frequency set mainly by the small gap d between the post tops and the lid. Two magnetic loop probes, made from 1.19 mm coaxial cable cores, are inserted at opposite ports and connected to a Rohde & Schwarz ZNB40 vector network analyzer, so the full scattering matrix is measured in amplitude and phase versus frequency and applied magnetic field at room temperature.

Two YIG spheres are held in a 3D-printed resin sample holder that locks onto the four posts, allowing repeatable placement at the midpoints between adjacent posts; the two arrangements studied are the diagonal arrangement and the parallel arrangement. Finite-element eigenfrequency studies give the microwave magnetic field maps, from which the filling factor η, the coupling strength, the internal coupling phase θ_uv and the external phase φ_up are extracted; finite-domain studies that include the probe drive are used instead for parameters such as the coupling strength and internal phase, because an active excitation port obscures decay-rate and external-phase extraction. The measured parameters are fed into the input–output scattering matrix, and a symmetry-based unitary transformation reduces the 6 × 6 problem to a 4 × 4 matrix from which an explicit S21 formula is obtained.

Measurements span 4.00 < ω/2π < 7.25 GHz with a 5 MHz frequency step and 0.114 < μ0H < 0.248 T with a 0.25 mT field step, using a 1.0 kHz intermediate-frequency bandwidth and −10 dBm source power. The post-to-lid gap was calibrated to 130.5 μm (nominal geometry: post radius r = 0.5 mm, d = 130 μm) by matching simulated eigenfrequencies to the measured spectrum, and perfect-electric-conductor boundary conditions were applied to reduce simulation time.

Why This Matters

Impact on research. The work argues that coupling-phase engineering should be treated as a first-class design principle in cavity magnonics, alongside coupling strength and dissipation. It provides a concrete mechanism — interference between internal and external coupling phases, including contributions from far-detuned modes — for two effects that reduced few-mode models do not capture: position-selective coupling to degenerate whispering-gallery-like modes and nonreciprocal transmission at antiresonances. It also offers a compact 4 × 4 analytical model that reproduces the full 6 × 6 result.

Real-world applications (as motivated in the paper):

• Nonreciprocal microwave components such as isolators, circulators and signal routers for quantum computing architectures and microwave technology, including source protection.
• Field-tunable, ferrimagnet-based microwave devices exploiting the antisymmetric Polder susceptibility tensor.
• Quantum sensors and dark matter detection, which the paper lists among the broad technological targets of cavity magnonics.
• Quantum memories and hybrid bosonic quantum hardware built from cavity-magnon polaritons.

Industry relevance. The results give device designers a route to engineer nonreciprocity and mode selectivity through geometry and sphere placement — via the phases — rather than through added components, which is relevant to microwave isolation, signal routing and the integration of ferrimagnetic components into quantum hardware.

Future Directions

• Phase-engineered nonreciprocal devices. The conclusion frames the results as a step toward deliberately designed, phase-engineered nonreciprocal magnonics devices; turning the demonstrated antiresonance nonreciprocity into a usable isolator or router is the obvious next step.
• Extending the mode set. The model currently truncates the cavity at four modes and two magnon modes and ignores higher magnetostatic modes, which the authors identify as a likely source of residual mismatch in antiresonance frequency; a fuller treatment would test the model further.
• Exploiting more sphere positions and configurations. The sample holder accommodates up to four spheres but only two are used here in two arrangements, so the positional design space for tuning θ_p remains largely unexplored.
• Improving simulation–experiment agreement. The offsets in effective coupling strength (21.45%, 4.83% and 11.28%) point to the need for better control or modelling of sphere positioning, filling factor and fabrication tolerances such as the post-to-lid gap.

Target Audience

This paper is aimed at researchers and graduate students in cavity magnonics, magnon–photon hybrid quantum systems and microwave engineering who model or fabricate multimode cavity devices. It is most useful to readers already comfortable with input–output formalism and scattering parameters, since the central argument depends on the phase structure of the coupling matrices; experimentalists building re-entrant cavity setups will also find the characterization and parameter-extraction details directly relevant.

Authors’ abstract

Coupling phases play a decisive yet often overlooked role in cavity magnonics, particularly in complex multimode systems. Here, we investigate phase-mediated interference effects in a cavity magnonics system comprising a four-post re-entrant microwave cavity coupled to Yttrium Iron Garnet (YIG) spheres. Using an input-output model that explicitly accounts for both internal and external coupling phases, we achieve agreement with experimental microwave transmission measurements. Our results unravel the emergence of a positionally-dependent uncoupled mode due to interference of cavity photon-magnon (internal) coupling phases. Further, we experimentally observed large nonreciprocity at the antiresonance frequencies and show that this feature arises due to the far-detuned modes' odd parity cavity photon-probe (external) coupling phase interfering with the internal coupling phases. Bridging theory, simulation and experiment, these results establish coupling-phase engineering as a key principle for accurately modeling and designing multimode cavity magnonics devices.