Photonic spin Hall effect in $\mathcal{PT}$-symmetric non-Hermitian cavity magnomechanics

TL;DR

This work analyzes PSHE in a PT-symmetric non-Hermitian cavity magnomechanical system that couples magnon–photon and magnon–phonon modes in a YIG-sphere–cavity setup. By deriving the effective non-Hermitian Hamiltonian $H_{ ext{eff}}$, applying pseudo-Hermiticity conditions, and solving a cubic eigenvalue problem, the authors show a robust third-order exceptional point $ ext{EP}_3$ at $G_a/ ext{omega}_b \,ig|_{ heta=rac{ ext{pi}}{2}} \, ext{≈ }0.139$ under balanced gain–loss. Optical susceptibility and PSHE are computed via a transfer-matrix framework, revealing phase-dependent transverse shifts: enhanced PSHE in the $ ext{PT}$-symmetric phase, suppressed PSHE at $ ext{EP}_3$, and tunability with magnon–phonon coupling $G_b$ and intracavity length. These results link PSHE to the non-Hermitian spectrum and suggest routes to spin-selective photonic devices and precision microwave sensing in engineered non-Hermitian platforms.

Abstract

Non-Hermitian cavity magnomechanics (CMM), which incorporates the magnon-photon and magnon-phonon interactions simultaneously, enables rich physical phenomena, including exceptional-point-enhanced sensing, and offers pathways toward topological transitions and nonreciprocal quantum transformation. These interactions exert a pivotal influence on the optical response of a weak probe field and pave the way for novel applications in quantum technologies. In this work, we consider a yttrium-iron-garnet (YIG) sphere coupled to a microwave cavity. The magnon mode of the YIG sphere is directly excited through microwave field coupling, whereas the cavity mode is probed via a weak-field interrogation scheme. The direct interaction of a traveling field with the magnon mode induces gain in the system, thereby establishing non-Hermitian dynamics. The parity-time (PT)-symmetric behavior of a hybrid non-Hermitian CMM is designed and investigated. Eigenvalue spectrum analysis demonstrates that a third-order exceptional point (EP_3) emerges under tunable effective magnon-photon coupling when the traveling field is oriented at an angle of π/2 relative to the cavity's x-axis. The photonic spin Hall effect (PSHE) in a reflected probe field is subsequently examined in such a system. Under balanced gain and loss conditions and in the presence of effective magnon-phonon coupling, tunable effective magnon-photon coupling enables coherent control of the PSHE across the broken PT-symmetric phase, at the EP_3, and in the PT-symmetric phase. Investigation reveals that the PSHE can be significantly enhanced or suppressed via effective magnon-photon coupling. The influence of intracavity length on the PSHE is further explored, providing an additional parameter for fine-tuning the transverse shift. These findings establish a direct correspondence between the PSHE and the underlying non-Hermitian eigenvalue spectrum.

Figures (9)

• Figure 1: Schematic illustration of a non-Hermitian cavity magnomechanical system with an embedded YIG sphere. A microwave cavity (photon mode $\hat{a}$, resonance frequency $\omega_a$, dissipation $\kappa_a$) hosts the sphere under $z$-axis bias field $B_z$ (exciting magnon mode $\hat{m}$, resonance frequency $\omega_m$, gain $\kappa_m$). Magnon-photon coupling occurs via magnetic dipole interaction. Magnetostriction excites phonon mode $\hat{b}$ ($\omega_b$, dissipation $\gamma_b$), enabling magnon-phonon coupling enhanced by $x$-axis microwave drive field ($\omega_0$). Orthogonal cavity ($B_y$), bias ($B_z$), and drive ($B_x$) magnetic fields are shown. Non-Hermiticity arises from a traveling field (incident angle $\theta$, coupling $\Gamma$). A TM-polarized probe field $E_p$ incident on mirror $M_1$ at $\theta_i$ undergoes spin-dependent splitting; reflected transverse shifts $\delta^{\pm}_p$ for left/right circular components are measured.
• Figure 2: Eigenvalues of $H_{\text{eff}}$ [Eq. (\ref{['H_eff']})] versus normalized effective magnon-photon coupling $G_{a}/\omega_{b}$. (a) Real part: $(\operatorname{Re}[\lambda]-\omega_{b})/\omega_{b}$; (b) imaginary part: $\operatorname{Im}[\lambda]/\omega_{b}$. Parameters: $\kappa_{a}/2\pi= 2.1~\mathrm{MHz}$, $\gamma_{b}/2\pi= 150~\mathrm{Hz}$, $\kappa_{m}=\kappa_{a} + \gamma_{b}$, $\omega_{b}/2\pi =15.101~\mathrm{MHz}$, $\Delta_{s}/2\pi=\Delta_{a}/2\pi=15.10~\mathrm{MHz}$, and $G_{b}/2\pi=0.001~\mathrm{MHz}$.
• Figure 3: Eigenvalues of $H_{\text{eff}}$ [Eq. (\ref{['H_eff']})] as a function of $G_{a}/\omega_{b}$. (a,b) Real parts: $(\operatorname{Re}[\lambda]-\omega_{b})/\omega_{b}$; (c,d) Imaginary parts: $\operatorname{Im}[\lambda]/\omega_{b}$. Columns correspond to effective magnon–phonon coupling strengths $G_{b}=0$ and $G_{b}/2\pi=0.05~\mathrm{MHz}$, respectively. The inset in Fig. \ref{['fig2b']}(d) provides a magnified view of $\operatorname{Im}[\lambda]/\omega_{b}$ as a function of $G_{a}/\omega_{b}$. Fixed parameters: $\kappa_{a}/2\pi = 2.1~\mathrm{MHz}$, $\gamma_{b}/2\pi = 150~\mathrm{Hz}$, $\kappa_{m} = \kappa_{a} + \gamma_{b}$, $\omega_{b}/2\pi = 15.101~\mathrm{MHz}$, $\Delta_{s}/2\pi =\Delta_{a}/2\pi = 15.10~\mathrm{MHz}$, and at $G_{b}=0$, $\Delta_{m}/2\pi = \Delta_{a}/2\pi = 15.10~\mathrm{MHz}$.
• Figure 4: (a-c) Absorption spectra (real part of the output field $E_T$) and (d-f) dispersion spectra (imaginary part of $E_T$) versus normalized effective detuning $x/\omega_{b}$. Columns correspond to $G_{a} = 0.039\,\omega_{b}$ (broken $\mathcal{PT}$-symmetry, red), $0.139\,\omega_{b}$ (third-order exceptional point $\mathrm{EP_3}$, green), and $0.239\,\omega_{b}$ ($\mathcal{PT}$-symmetry, blue). Fixed parameters: $\kappa_{a}/2\pi= 2.1~\mathrm{MHz}$, $\gamma_{b}/2\pi= 150~\mathrm{Hz}$, $\kappa_{m}=\kappa_{a} + \gamma_{b}$, $\omega_{b}/2\pi=15.101~\mathrm{MHz}$, and $G_{b}/2\pi=0.001~\mathrm{MHz}$.
• Figure 5: (a) Absolute values of the reflection coefficients $|r^{s}|$ (dashed) and $|r^{p}|$ (solid), and (b) their ratio $|r^{s}|/|r^{p}|$ versus incident angle $\theta_{i}$, at $G_{a} = 0.039\,\omega_{b}$, $0.139\,\omega_{b}$, and $0.239\,\omega_{b}$, corresponding to the broken $\mathcal{PT}$-symmetric (red), $\mathrm{EP_3}$ (green), and $\mathcal{PT}$-symmetric (blue) phases at the resonance condition $x=0$, respectively. Other parameters are $\kappa_{a}/2\pi= 2.1~\mathrm{MHz}$, $\gamma_{b}/2\pi= 150~\mathrm{Hz}$, $\kappa_{m}=\kappa_{a} + \gamma_{b}$, $\omega_{b}/2\pi =15.101~\mathrm{MHz}$, $G_{b}/2\pi=0.001~\mathrm{MHz}$, $\omega_{a}/2\pi = 1.32~\mathrm{GHz}$, $w_{0}=50\,\lambda$, $d_{1} = 4~\mathrm{mm}$, $d_{2} = 45~\mathrm{mm}$, $\epsilon_{0}=1$, and $\epsilon_{1} =\epsilon_{3} = 2.2$.