Polaromechanical NMS: Cavity-Magnon-Phonon Dynamics

• Polaromechanical NMS is the phenomenon where coherent coupling between a dressed polaritonic mode and a mechanical mode produces a resolvable spectral doublet.
• It distinguishes genuine pole splitting from mere spectral dips induced by coherent perfect absorption, clarifying effective linewidth modulation in hybrid systems.
• Experiments in cavity-magnon-phonon platforms reveal that strong coupling—when the hybrid coupling exceeds dissipation—enables tripartite mode hybridization and multimode applications.

Searching arXiv for recent and relevant papers on polaromechanical normal-mode splitting, CPA, and related hybrid-mode analogues. Polaromechanical normal-mode splitting (NMS) denotes the resolved spectral splitting that occurs when a mechanical mode hybridizes coherently with a polaritonic mode, so that the observed resonances correspond to dressed hybrid eigenmodes rather than uncoupled bare excitations. In the cavity-magnon platform, the relevant polariton is a cavity-magnon polariton formed by strong magnon-photon coupling, and “polaromechanical” emphasizes that the subsequent mechanics couples to this already hybrid light-matter quasiparticle rather than to a bare cavity mode alone (Shen et al., 2023). Recent literature has also made the pole-versus-zero distinction central to the subject: some works interpret narrow-band splitting near coherent perfect absorption (CPA) as genuine polaromechanical NMS, whereas others argue that CPA alters scattering zeros without producing pole splitting in the linear weak-probe regime (Ebrahimi et al., 25 Oct 2025).

1. Definition and scope

Polaromechanical NMS is defined operationally by a resolved doublet or avoided crossing associated with coherent coupling between a polariton branch and a mechanical mode. In the cavity-magnon realization, microwave cavity photons and magnons first hybridize into upper and lower cavity-magnon polaritons, and one of these polaritons is then coupled to a phonon mode through magnetostriction (Shen et al., 2023). The corresponding splitting is therefore not a bare photon-phonon or magnon-phonon phenomenon, but a polariton-mechanics hybridization.

The terminology separates several notions that are often conflated. CPA means zero reflection due to destructive interference, i.e. a zero of the scattering amplitude; cavity polariton splitting refers to ordinary magnon-photon hybridization governed by the poles $\tilde{\omega}_\pm$; and polaromechanical NMS would mean true splitting of a polariton due to coupling to the mechanical mode, again diagnosed by hybrid-mode poles or by a resolvable doublet in linear response (Ebrahimi et al., 25 Oct 2025). This pole-based usage aligns with broader hybrid-mode literature, where NMS is the spectral consequence of coherent mode mixing that outpaces dissipation, rather than merely a deep notch created by a scattering zero (Asjad, 2012).

A broader literature shows that the same logic extends beyond cavity magnonics. In a Bose–Einstein-condensate cavity system, fluctuations of the intracavity field hybridize with a collective Bogoliubov mode, and the resulting NMS appears in both the matter-wave displacement spectrum and the cavity output spectrum when coupling is sufficiently strong (Asjad, 2012). In multimode optomechanics, hybridization of two optical fluctuation modes with one mechanical mode can produce three resolved peaks in the mechanical displacement spectrum, illustrating that NMS in composite systems need not reduce to a simple two-peak doublet (Aggarwal et al., 2013).

2. Canonical cavity-magnon-phonon formulation

The clearest explicit realization of polaromechanical NMS is the cavity-magnon-phonon system of Shen et al. (Shen et al., 2023). The experimental platform consists of a 3D oxygen-free copper cavity operated in the $\mathrm{TE}_{102}$ mode, with resonance frequency $\omega_a/2\pi = 7.213~\mathrm{GHz}$, containing a $0.25$-mm-diameter YIG sphere placed at the magnetic-field antinode (Shen et al., 2023). A static bias field $B_0$ tunes the magnon frequency according to

$$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$

and the YIG sphere supports a mechanical mode at

$\omega_b/2\pi = 10.9565~\mathrm{MHz},$

with bare magnetostrictive coupling

$$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$

The cavity and magnon strongly couple with

$g_{ma}/2\pi = 6.63~\mathrm{MHz},$

forming upper and lower polaritons. The paper defines the polariton operators as

$$p_+ = a\cos\theta + m\sin\theta,\qquad p_- = -a\sin\theta + m\cos\theta,$$

with mixing angle

$\mathrm{TE}_{102}$0

The complex polariton eigenfrequencies are

$\mathrm{TE}_{102}$1

where

$\mathrm{TE}_{102}$2

In the polariton basis, the linearized polaromechanical Hamiltonian is written as

$\mathrm{TE}_{102}$3

with

$\mathrm{TE}_{102}$4

where $\mathrm{TE}_{102}$5 is the steady-state magnon amplitude (Shen et al., 2023). This form makes the essential point explicit: the phonon couples to the magnon sector, so its coupling to each polariton is weighted by that polariton’s magnon content.

Under red-sideband driving of the upper polariton, the interaction reduces to a beam-splitter-type exchange between the driven polariton and the phonon. The paper uses the criterion

$\mathrm{TE}_{102}$6

for entry into the triple strong-coupling regime, and it identifies the observed splitting as approximately $\mathrm{TE}_{102}$7 for the upper branch (Shen et al., 2023). In this usage, polaromechanical NMS is the spectroscopic signature of triply hybridized photon-magnon-phonon normal modes.

3. Spectral signatures and strong-coupling criteria

The primary signatures of polaromechanical NMS are a resolved doublet and an avoided crossing. In the cavity-magnon-phonon experiment, a red-detuned microwave drive pumps the upper polariton so that the anti-Stokes sideband is resonant with the polariton: $\mathrm{TE}_{102}$8 At low power, the experiment shows magnomechanically induced transparency rather than split modes. At $\mathrm{TE}_{102}$9 dBm, the reported values are

$\omega_a/2\pi = 7.213~\mathrm{GHz}$0

so $\omega_a/2\pi = 7.213~\mathrm{GHz}$1, and the system remains in weak coupling (Shen et al., 2023).

At higher drive power, the system enters the strong-coupling regime. At $\omega_a/2\pi = 7.213~\mathrm{GHz}$2 dBm, the extracted parameters are

$\omega_a/2\pi = 7.213~\mathrm{GHz}$3

so that

$\omega_a/2\pi = 7.213~\mathrm{GHz}$4

and the spectrum displays normal-mode splitting (Shen et al., 2023). At $\omega_a/2\pi = 7.213~\mathrm{GHz}$5 dBm the coupling reaches

$\omega_a/2\pi = 7.213~\mathrm{GHz}$6

and the split peaks become more evident.

The complementary hallmark is avoided crossing. By fixing the drive at

$\omega_a/2\pi = 7.213~\mathrm{GHz}$7

the anti-Stokes sideband is fixed at

$\omega_a/2\pi = 7.213~\mathrm{GHz}$8

while the upper polariton is tuned through resonance by the magnon self-Kerr effect. The observed repulsion of the two branches is interpreted as direct evidence of hybridization (Shen et al., 2023). The same logic appears in other bosonic systems: in the BEC–cavity analogue, NMS appears when the coupled response denominator develops separated poles,

$\omega_a/2\pi = 7.213~\mathrm{GHz}$9

and the splitting vanishes for weak coupling (Asjad, 2012).

A recent comment defending the cavity-magnon-phonon experiment uses a narrow-band strong-coupling criterion centered on the CPA frequency. It reports an observed splitting of

$0.25$0

and compares it with

$0.25$1

while quoting

$0.25$2

to argue that $0.25$3 and that the splitting is genuine in a narrow frequency range around $0.25$4 (Shen et al., 18 Mar 2026).

4. Coherent perfect absorption and linewidth engineering

CPA occupies a central place in the recent polaromechanical NMS literature because it has been used as a route to drastic polariton linewidth reduction. In the cavity-magnon-phonon experiment, the cavity decay rate is decomposed as

$0.25$5

with $0.25$6. Under CPA, the cavity outputs vanish,

$0.25$7

and the paper writes an effective cavity gain rate

$0.25$8

For the resonant case $0.25$9, CPA requires

$B_0$0

so that in the ideal limit

$B_0$1

for the polaritons (Shen et al., 2023).

Experimentally, the cavity is tuned to

$B_0$2

with

$B_0$3

At the CPA frequency the smallest symmetric polariton linewidths are reported as

$B_0$4

and by further detuning the magnon frequency the upper-polariton linewidth is reduced to

$B_0$5

The paper further uses the Wigner time delay

$B_0$6

and reports

$B_0$7

corresponding to

$B_0$8

in agreement with the theory-derived $B_0$9 (Shen et al., 2023).

The controversy arises because a later analysis revisits CPA using standard input-output theory and concludes that CPA creates zeros of the measured reflection amplitude but does not move the poles of the linear response (Ebrahimi et al., 25 Oct 2025). For a one-port bare cavity with total decay

$$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$0

the reflection coefficient is

$$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$1

and CPA occurs at

$$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$2

But the pole remains

$$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$3

so the full width at half maximum remains set by $$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$4, not by the numerator (Ebrahimi et al., 25 Oct 2025).

The same paper extends the argument to a cavity-magnon hybrid. For $$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$5, it finds split-frequency CPA zeros at

$$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$6

provided

$$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$7

Yet the hybrid-mode poles remain

$$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$8

and the observable linewidths are fixed by

$$\omega_m=\gamma B_0,\qquad \gamma/2\pi = 28~\mathrm{GHz/T},$$9

On this basis the paper argues that CPA alone is not a route to linewidth suppression or to polaromechanical mode splitting in the linear weak-probe regime (Ebrahimi et al., 25 Oct 2025).

5. Pole-based versus zero-based interpretations

The recent debate over polaromechanical NMS is fundamentally a debate over what counts as physically relevant spectral structure. One side argues that genuine splitting requires distinct poles or eigenfrequencies; the other argues that near CPA the relevant linewidth is local and zero-based rather than pole-based.

The critique states that in cavity magnomechanics, logarithmic plots over narrow frequency spans can show what looks like a doublet, but linear-scale spectra over a sufficiently wide span show only a single polariton resonance with a narrow magnomechanically induced transparency feature superimposed. In its formulation, the observed structure is “a single mode with a transparency window, not a genuine splitting,” and true NMS would require actual pole separation and a resolvable doublet in linear spectra (Ebrahimi et al., 25 Oct 2025). The same paper explicitly attributes the apparent splitting to “visual sharpening” caused by logarithmic representation of a near-zero minimum. Because $\omega_b/2\pi = 10.9565~\mathrm{MHz},$0 or $\omega_b/2\pi = 10.9565~\mathrm{MHz},$1 magnifies changes near a deep CPA dip, small perturbations can appear as two shoulders around a zero.

The reply adopts a different criterion. It argues that in CPA-induced NMS the relevant physics is confined to a narrow frequency range around

$\omega_b/2\pi = 10.9565~\mathrm{MHz},$2

and that the proper strong-coupling condition is

$\omega_b/2\pi = 10.9565~\mathrm{MHz},$3

where $\omega_b/2\pi = 10.9565~\mathrm{MHz},$4 is an effective decay rate under CPA rather than the total linewidth extracted from the pole of the full output spectrum (Shen et al., 18 Mar 2026). It states that the total or intrinsic decay rate defined via the pole of the spectrum cannot characterize the vanishing effective decay rate at the CPA frequency, and it distinguishes

$\omega_b/2\pi = 10.9565~\mathrm{MHz},$5

from the imaginary part of a pole of the total output spectrum $\omega_b/2\pi = 10.9565~\mathrm{MHz},$6 and

$\omega_b/2\pi = 10.9565~\mathrm{MHz},$7

from the imaginary part of a zero of $\omega_b/2\pi = 10.9565~\mathrm{MHz},$8. It further reports the statement that the FWHM of $\omega_b/2\pi = 10.9565~\mathrm{MHz},$9 gives $$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$0, whereas the FWHM of

$$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$1

gives $$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$2 (Shen et al., 18 Mar 2026).

The disagreement is therefore not simply about a plotted figure; it is about the diagnostic standard. A pole-based definition treats CPA as amplitude suppression without modal restructuring. A zero-based definition treats the monochromaticity of CPA as the relevant ingredient that enables local strong coupling and narrow-band NMS. The former perspective is explicit in the statement that CPA changes amplitudes, not spectral poles (Ebrahimi et al., 25 Oct 2025). The latter is explicit in the claim that the splitting is visible in both logarithmic and linear scales once one isolates the proper narrow spectral window (Shen et al., 18 Mar 2026).

6. Analogues and generalizations beyond cavity magnomechanics

The conceptual structure of polaromechanical NMS has close analogues in other hybrid bosonic systems. In a cavity optomechanical analogue with a Bose–Einstein condensate, the Bogoliubov density mode acts as an effective mechanical oscillator with frequency

$$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$3

and the linearized coupling

$$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$4

produces normal-mode frequencies

$$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$5

The system shows NMS in both the condensate displacement spectrum and the cavity output spectrum, with splitting absent at weak coupling and visible for larger $$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$6 (Asjad, 2012). This provides a direct analogue of mechanics coupling to a dressed electromagnetic-matter mode.

Tripartite and multimode generalizations further broaden the notion. In the atom-assisted optomechanical system with intensity-dependent coupling, the mirror displacement spectrum evolves from ordinary two-mode splitting to three resolved peaks when the Lamb–Dicke parameter is large enough, corresponding to simultaneous hybridization of mechanical, optical, and atomic fluctuation modes (Barzanjeh et al., 2011). In a two-optical-mode plus one-mechanical-mode system, the mechanical displacement spectrum can also show three peaks, because one mechanical mode hybridizes with two optical fluctuation modes (Aggarwal et al., 2013). A double-cavity system with two mechanically coupled movable mirrors and two optical modes produces four visible branches in the displacement spectrum, illustrating that normal-mode splitting in driven multimode systems is a full susceptibility problem rather than a simple two-level anticrossing (Kumar et al., 2011).

A different but conceptually relevant case is hyperfine-mediated spin-photon hybridization in MnCO$$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$7. There the observed low-frequency resonance is a hybridized nuclear-electron magnon, and the cavity coupling is inherited from the electron-spin component rather than from a bare nuclear spin (Abdurakhimov et al., 2014). This mediated-coupling structure is useful for polaromechanics because it shows that the mode exhibiting NMS may itself be composite and may inherit electromagnetic visibility from an internally hybridized subsystem.

These analogues suggest a common principle: the observable mode splitting depends on hybrid eigenmodes of the coupled linear response, not merely on bare microscopic couplings. A plausible implication is that polaromechanical NMS should be treated as one member of a wider class of hybrid-mode phenomena in which mechanics couples to dressed bosonic excitations of varying composition.

7. Experimental metrics, misconceptions, and open issues

The most prominent experimental metric in the cavity-magnon-phonon realization is the polaromechanical cooperativity

$$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$8

reported as

$$g_{mb}/2\pi = 1.40~\mathrm{mHz},\qquad \kappa_b/2\pi = 155~\mathrm{Hz}.$$9

at the maximum coupling (Shen et al., 2023). The paper further defines the quantum cooperativity

$g_{ma}/2\pi = 6.63~\mathrm{MHz},$0

and states that $g_{ma}/2\pi = 6.63~\mathrm{MHz},$1 would be achievable at cryogenic temperatures rather than in the present room-temperature experiment (Shen et al., 2023). This distinguishes demonstrated classical strong coupling from projected quantum-coherent operation.

A recurrent misconception is to equate a deep dip or transparency feature with genuine NMS. The pole-based critique explicitly warns that a zero in $g_{ma}/2\pi = 6.63~\mathrm{MHz},$2 can suppress reflection without narrowing linewidth and that logarithmic plotting can exaggerate a narrow transparency notch into an apparent doublet (Ebrahimi et al., 25 Oct 2025). The opposing comment responds that in CPA-induced phenomena the relevant splitting is intrinsically narrowband and can be hidden in broad linear-scale plots by a large dynamic range, so local windowing is essential (Shen et al., 18 Mar 2026). The dispute therefore concerns not only interpretation but also data presentation.

Another issue is the meaning of “effective” parameters. In the cavity-magnon-phonon experiment, CPA is described as producing an effective cavity gain rate $g_{ma}/2\pi = 6.63~\mathrm{MHz},$3 and vanishing polariton decay in the ideal limit (Shen et al., 2023). The later critique argues that substituting the CPA condition back into the equation of motion to infer a globally modified decay is incorrect, because $g_{ma}/2\pi = 6.63~\mathrm{MHz},$4 holds only at a single frequency and does not redefine the dynamical pole structure over frequency (Ebrahimi et al., 25 Oct 2025). The reply instead maintains that the local effective decay near the CPA zero is the relevant quantity for the observed narrow-band NMS (Shen et al., 18 Mar 2026). This remains the central unresolved interpretive issue.

Finally, the literature suggests that plotting choices and basis choices matter. In systems with more than two coupled modes, splitting can appear as a triplet or as several branches rather than a single symmetric doublet [(Aggarwal et al., 2013); (Kumar et al., 2011)]. In such settings, linear-scale spectra, logarithmic spectra, avoided-crossing maps, and pole analyses need not emphasize the same aspects of the same underlying response. This suggests that future work will likely focus on more explicit three-mode pole calculations in cavity magnomechanics, clearer experimental discrimination between transparency windows and pole splitting, and a more standardized criterion for CPA-assisted hybrid-mode spectroscopy.