Cavity magnonics and bound states in the continuum with Bragg and anti-Bragg mirrors

AI summary

Overview

Research area: Cavity magnonics, waveguide quantum electrodynamics (waveguide QED), and non-Hermitian cavity physics, with a focus on bound states in the continuum (BICs).

Technical level: Intermediate. The experimental concept and the physical picture are accessible with undergraduate quantum optics background; the theoretical treatment (effective non-Hermitian Hamiltonian, master equation, scattering theory, supermode decomposition) is advanced and is largely relegated to the Supplementary Material.

Scope: An experimental and theoretical study showing that two periodic arrays of yttrium iron garnet (YIG) spheres coupled to a waveguide can act as Bragg or anti-Bragg mirrors forming a cavity whose dark supermodes are BICs, probed position-dependently by a movable third YIG sphere.

What This Paper Is About

A periodic array of resonant emitters in a 1D waveguide can reflect light like a mirror, and two such mirrors form a cavity. Theory has predicted that the localized "dark" cavity modes in these systems are bound states in the continuum, but their spatial field profiles have never been measured, so the picture has remained phenomenological. The authors build this cavity out of ferrimagnetic YIG spheres (magnons instead of atoms) in a dual-waveguide setup, and use a movable probe sphere to map the cavity field directly, for both Bragg and anti-Bragg mirror spacings.

Key Contributions

1. First experimental realization of magnonic BIC cavities with two different mirror types. Two periodic YIG-sphere arrays and a movable probe YIG sphere are integrated in a dual-open-waveguide architecture, forming both Bragg cavities (spacing d₁ = nλ₀/2) and anti-Bragg cavities (spacing d₁ = λ₀/4 + nλ₀/2), whose mirrors have different lattice constants.

2. Quantitative characterization of Bragg-cavity polaritons and their collective scaling. A single dark supermode couples coherently to the probe magnon, producing two polaritons whose splitting scales with both the probe-sphere size and the number of mirror spheres, following the predicted √N law, with the scaling measured for two probe-sphere diameters (0.5 mm and 1.0 mm).

3. Demonstration that the anti-Bragg cavity behaves qualitatively differently. It hosts degenerate bright and dark supermodes inside a photonic bandgap; the probe magnon couples coherently to the dark supermode and dissipatively to the bright one, which introduces level attraction and modifies the observed gap, and the coherent coupling becomes independent of cavity size.

4. Position-dependent detection of the cavity field, i.e., direct mapping of BIC spatial profiles. By translating the probe sphere, the authors map the standing-wave pattern of the Bragg BIC and the cavity-confined field of the single anti-Bragg BIC, showing where the field is maximal and where it vanishes.

Main Findings

• The Bragg mirror works as expected. With mirror spheres spaced by d₁ = λ₀/2, the coherent exchange coupling J_{j,l} between mirror magnons vanishes and the dissipative coupling |γ_{j,l}| is maximized; transmission |S₂₁| shows a broadened Lorentzian line shape from constructive interference.

• Transverse-waveguide detection fails, so a second waveguide is added. Detecting through the transverse waveguide (|S₂₁|) does not reveal the expected polariton splitting because the magnonic mirrors strongly reflect the incident light. Adding a longitudinal waveguide directly on top of the probe sphere (measuring |S₄₃|) eliminates the spectral influence of the bright supermode and reveals two well-resolved absorption dips.

• N = 2 Bragg cavity numbers. With a 0.5 mm probe sphere and two mirror spheres, fitting gives 2J_D/2π = 6.8 MHz, total probe-magnon decay Γ_PM/2π = 5.5 MHz, total cavity decay Γ_C/2π = 2.7 MHz, and cooperativity C = (2J_D)²/(Γ_C Γ_PM) = 3.1.

• N = 8 Bragg cavity reaches strong coupling. With a 1.0 mm probe sphere and eight mirror spheres (four per mirror), 2J_D/2π = 20.4 MHz, Γ_PM/2π = 10.1 MHz, Γ_C/2π = 4.7 MHz, and C = 8.8, which satisfies J_D > {Γ_PM, Γ_C}.

• √N scaling is confirmed. J_D follows √(N κ_M κ_PT)/2 for symmetric cavities; the fitted J_D/(√N π) is 4.8 MHz for the 0.5 mm probe sphere and 7.2 MHz for the 1.0 mm probe sphere, attributed to the larger probe–waveguide coupling κ_PT of the bigger sphere.

• Asymmetric Bragg cavities deviate slightly. With |N_L − N_R| = 1 and odd N, the coupling vector is no longer parallel to the modified dark supermode, the probe magnon also couples to the bright supermode, and J_D = √(N κ_M κ_PT (1 − 1/N²))/2, falling below √N scaling at N = 3; the deviation diminishes as N grows.

• The anti-Bragg cavity has a different supermode structure. For N = 4 it supports one bright, one dark, and N − 2 subradiant supermodes inside a photonic bandgap; the single dark supermode is the BIC, with a magnonic part (1/√2)(m†{M,N/2} + m†{M,N/2+1})|G⟩ and an optical field localized inside the cavity region.

• Anti-Bragg coupling is size-independent. At d₂ = 3λ₀/4 the coherent coupling is J_D = √(2 κ_M κ_PT)/2, independent of cavity size, while the dissipative coupling to the bright supermode produces level attraction.

• Anti-Bragg gap values. An avoided-crossing gap Δω/2π = 12.2 MHz is measured for N_L = N_R = 2; comparing with the N_L = N_R = 1 case (coherent coupling only) shows the gap is reduced and the dips deepen. For N_L = N_R = 4 the gap is reduced to 11.2 MHz, and numerics indicate it approaches a constant as N → ∞.

• Side features come from subradiant supermodes and loss. Faint avoided crossings on both sides of the main anti-crossing, plus non-Lorentzian dips away from resonance, arise from coupling to subradiant supermodes; they are much more pronounced in lossless calculations, so the faintness is attributed primarily to large intrinsic losses.

• Bragg BIC field is mapped. For N_L = N_R = 2, d₀ = 3λ₀/2, d₁ = 3λ₀/2, the rescaled field |φ̃(x)| of the BIC associated with |D₁⟩ has three antinodes in the cavity region and weaker peaks in the mirror region; at x_P = 0 mm a splitting 2J_D/2π = 14.6 MHz is measured, which disappears at x_P = 15 mm where the BIC field is zero.

• Linewidth narrowing appears where the BIC field vanishes. At x_P = 15 mm the probe couples only dissipatively to the bright supermode with J_B = i√(N κ_M κ_PT)/2, forming two hybridized modes, one of which has low dissipation via destructive interference, giving a narrower line than the bare probe response.

• The mirror-region coupling is not a simple field ratio. The maximum coupling in the mirror region (at x_P = −90 mm) is smaller than at the cavity center, but the ratio differs from the corresponding ratio of maximum |φ̃(x)| because four supermodes (|D₁⟩, |D₂⟩, |D₃⟩, |B⟩) interact with the probe there.

• The anti-Bragg BIC is confined to the cavity. For N_L = N_R = 2 with d₁ = 9λ₀/4, the field lies within −45 mm < x < 45 mm and vanishes at x = ±15 mm; moving the probe from x_P = 0 to ±15 mm drives the gap to zero, consistent with theory.

Methodology in Plain English

The cavity mirrors are not mirrors in the usual sense: they are rows of small ferrimagnetic YIG spheres, each of which supports a uniform spin-precession mode (the Kittel mode) that behaves like a single artificial atom at microwave frequencies. Each sphere sits on a shared coplanar waveguide, so photons traveling between spheres mediate long-range couplings — a real exchange part and a dissipative part — that depend sinusoidally on the sphere-to-sphere distance. Choosing the spacing to be a half-integer multiple of the wavelength (Bragg) maximizes the dissipative coupling and suppresses exchange; choosing a quarter-wave offset (anti-Bragg) does the opposite.

Two such rows form a cavity, and the collective eigenstates of the rows split into one bright (superradiant, leaky) supermode and several dark (subradiant) supermodes. The dark ones are the BICs. To see them, the authors place a third, movable YIG sphere — the probe — in the cavity, on a second, longitudinal waveguide that runs above it. Driving and reading out through that longitudinal waveguide avoids the problem that the mirror rows reflect the probe light. Measuring the transmission |S₄₃| as a function of drive frequency and of the magnetic field tuning the probe magnon frequency reveals absorption dips whose splitting gives the probe–cavity coupling; by sliding the probe along the transverse waveguide, that coupling is measured point by point, which is effectively a scan of the cavity field intensity. The spheres' resonance frequencies are tuned individually with local external magnetic fields.

Why This Matters

• It turns a phenomenological picture into a measured one. BICs in waveguide QED had been inferred from spectra but never spatially resolved; the scanning probe supplies the missing field profile.
• It establishes magnons as a practical testbed for atomic cavity QED. Spin ensembles can be positioned, tuned, and made much larger than single atoms, relaxing the experimental constraints of true atomic or superconducting artificial-atom arrays.
• It distinguishes these cavities from ordinary optical cavities. Superradiant supermodes coexisting with BICs produce non-Hermitian effects (dissipative coupling, level attraction, linewidth narrowing) that conventional Fabry–Pérot or photonic-crystal cavities do not exhibit.
• It gives a clean, parameter-controlled comparison. Bragg versus anti-Bragg mirrors differ only in lattice constant yet yield qualitatively different cavity physics, which makes the role of mirror reflection directly testable.

Applications:

• Hybrid quantum networks and interconnects: YIG magnons have been operated at millikelvin temperatures, so array-based cavities could interface with superconducting qubits.
• Quantum sensing and field mapping: the movable probe sphere acts as a local field sensor for microwave cavities.
• Non-Hermitian photonics and magnonics design: the Bragg/anti-Bragg contrast provides design rules for engineered loss, level attraction, and dark-mode storage.
• Cryogenic microwave signal routing and filtering: mirror arrays act as narrowband reflectors and bandgap elements in waveguide circuits.

Industry relevance: the building blocks are commodity microwave components — coplanar waveguides, YIG spheres, vector network analyzers — and the platform is compatible with superconducting circuit fabrication, which matters for quantum-computing hardware, microwave filtering for cryogenic electronics, and magnonic device engineering.

Future Directions

• Porting the scheme to other emitters. The authors propose applying the same architecture to superconducting circuits, quantum dots, and natural atoms, where the mirror arrays would be genuinely quantum two-level systems.
• Accessing non-Hermitian physics more cleanly. The faint subradiant-supermode features are currently masked by large intrinsic losses (α_M/2π = 6.7 MHz, α_P/2π = 13.1 MHz in the numerics); lower-loss samples would expose the predicted pronounced side anti-crossings.
• Using the scanning probe to find other exotic states. The authors specifically name emitter-photon bound states as a target, arguing the movable probe generalizes to any light-matter state with spatial structure.
• Interfacing with superconducting qubits at millikelvin temperatures. The stated route is a scalable quantum network via the dual-waveguide setup, which remains to be demonstrated experimentally.

Target Audience

Researchers and graduate students in cavity magnonics, waveguide QED, and non-Hermitian photonics; experimentalists building hybrid quantum platforms with spin ensembles or superconducting circuits; and theorists interested in BICs, subradiance, and collective light-matter coupling who want an experimentally grounded reference case. Readers without background in waveguide QED will need to work through the Supplementary Material for the supermode and scattering formalism.

Authors’ abstract

A periodic resonant emitter array coupled to a one-dimensional waveguide can act as a mirror. A typical example is a Bragg mirror, where emitters are spaced by multiples of half wavelengths and collectively enhance light reflection. Two such mirrors form an effective cavity that hosts bound states in the continuum (BICs). However, generating and detecting the spatial profiles of such BICs has proven challenging. Here, we experimentally demonstrate BICs in cavity magnonics using two periodic ferrimagnetic-sphere arrays and a probe sphere in a dual-open-waveguide architecture. We realize both Bragg and anti-Bragg cavities, whose mirrors have different lattice constants. In the Bragg cavity, there are degenerate supermodes formed by mirror spheres. We show that a single dark supermode coherently couples to the probe magnon in the cavity region, forming two polaritons whose splitting scales with both the probe-sphere size and the number of mirror spheres. By contrast, the anti-Bragg cavity has bright and dark supermodes in a bandgap, substantially changing the magnon-cavity interaction. Moreover, by moving the probe sphere, we perform position-dependent detection of the cavity field, highlighting the role of BICs in these cavities. Our flexible experimental setup with a scanning probe opens possibilities to detect other exotic states created by light-matter interaction, to interface with superconducting circuits in hybrid quantum networks, and to study non-Hermitian physics with Bragg and anti-Bragg cavities.