$\mathcal{PT}$-assisted control of Goos-Hänchen shift in cavity magnomechanics

TL;DR

This work addresses how to control the Goos-Hänchen shift in a non-Hermitian cavity magnomechanical system by exploiting PT symmetry and higher-order exceptional points. The authors develop a three-mode model with magnon–photon and magnon–phonon couplings, introduce traveling-field-induced gain, and derive the non-Hermitian effective Hamiltonian $H_{\text{eff}}$, optical susceptibility $\chi$, and GHS via stationary-phase analysis. They show that the GHS is significantly enhanced in the PT-unbroken phase, suppressed at the third-order exceptional point EP3, and tunable through the intracavity length and the magnomechanical coupling $G_b$; without $G_b$, an EP2 emerges and a phase transition occurs, while large $G_b$ suppresses the transition due to strong absorption. The PT-symmetric configuration thus yields larger, more controllable lateral shifts than Hermitian counterparts, with implications for microwave switching and precision sensing in non-Hermitian magnomechanics.

Abstract

We propose a scheme to manipulate the Goos-Hänchen shift (GHS) of a reflected probe field in a non-Hermitian cavity magnomechanical system. The platform consists of a yttrium-iron-garnet sphere coupled to a microwave cavity, where a strong microwave drive pumps the magnon mode and a weak field probes the cavity. The traveling field's interaction with the magnon induces gain, yielding non-Hermitian dynamics. When the traveling field is oriented at $π/2$ relative to the cavity's $x$-axis, the system realizes $\mathcal{PT}$ symmetry; eigenvalue analysis reveals a third-order exceptional point ($\mathrm{EP}_3$) at a tunable effective magnon-photon coupling. Under balanced gain-loss and finite effective magnomechanical coupling, we demonstrate coherent control of the GHS by steering the system across the $\mathcal{PT}$-symmetric transition and through $\mathrm{EP}_3$ via the effective magnon-photon coupling, enabling pronounced enhancement or suppression of the lateral shift. Furthermore, we show that without effective magnomechanical coupling, the system exhibits a second-order exceptional point ($\mathrm{EP}_2$) with a distinct GHS phase transition. This phase transition vanishes when the effective magnomechanical coupling exceeds a parametric threshold, where strong absorption at resonance suppresses the GHS. We also identify the intracavity length as an additional control parameter for precise shift tuning. Notably, the $\mathcal{PT}$-symmetric configuration yields substantially larger GHS than its Hermitian counterpart. These results advance non-Hermitian magnomechanics and open a route to GHS-based microwave components for quantum switching and precision sensing.

Figures (7)

• Figure 1: Schematic illustration of a non-Hermitian CMM system featuring an embedded YIG sphere. The cavity mode ($\hat{a}$ with frequency $\omega_a$ and dissipation rate $\kappa_a$) is subjected to a bias magnetic field $B_z$ along the $z$-axis, which excites the magnon mode ($\hat{m}$ with frequency $\omega_m$ and gain $\kappa_m$). Magnon-photon coupling arises from magnetic dipole interactions. Magnetostrictive interactions induce the phonon mode ($\hat{b}$ with frequency $\omega_b$ and dissipation rate $\gamma_b$), facilitating magnon-phonon coupling that is further enhanced by an $x$-axis microwave drive field (frequency $\omega_0$). The perpendicular magnetic fields—cavity ($B_y$), drive ($B_x$), and bias ($B_z$)—are depicted. Non-Hermiticity emerges from a traveling field characterized by incident angle $\theta$ and coupling strength $\Gamma$. A TE-polarized probe field $E_p$ is incident on mirror $M_1$ at angle $\theta_i$. The lateral displacement experienced by the reflected probe field during total internal reflection constitutes the GHS, denoted by $S_{r}$.
• Figure 2: (a) Absolute value of the reflection coefficient $|r(k_{z}, \omega_{p})|$ and (b) the normalized GHS $S_{r}/\lambda$ versus incident angle $\theta_{i}$ for three effective magnon-photon coupling regimes: (i) $G_{a} = 0.039\,\omega_{b}$ (green, broken $\mathcal{PT}$ phase), (ii) $0.139\,\omega_{b}$ (red, third-order exceptional point $\mathrm{EP}_3$), and (iii) $0.239\,\omega_{b}$ (blue, unbroken $\mathcal{PT}$ phase) at the resonance condition ($x=0$). Fixed parameters: $\omega_{a}/2\pi = 13.2~\mathrm{GHz}$, $\kappa_{a}/2\pi= 2.1~\mathrm{MHz}$, $\gamma_{b}/2\pi= 150~\mathrm{Hz}$, $\kappa_{m}=\kappa_{a} + \gamma_{b}$, $\omega_{b}/2\pi =15.101~\mathrm{MHz}$, $G_{b}/2\pi=0.001~\mathrm{MHz}$, $\epsilon_{0}=1$, $\epsilon_{1} =\epsilon_{3} = 2.2$, $d_{1} = 4~ \mathrm{mm}$, and $d_{2} = 45~\mathrm{mm}$.
• Figure 3: Normalized GHS $S_r/\lambda$ as a function of the incident angle $\theta_i$, parametrized by the effective magnon-photon coupling strength $G_a$. The curves are shown for the three dynamical phases: broken $\mathcal{PT}$ phase, $\mathrm{EP}_3$, and the unbroken $\mathcal{PT}$ phase, respectively, at resonance ($x=0$). The effective coupling strengths are: $G_a/\omega_b = 0.039$, $0.049$, $0.058$ (green solid/dashed/dotted for the broken $\mathcal{PT}$ phase); $0.139$ (red solid at the $\mathrm{EP}_3$); and $0.22$, $0.229$, $0.239$ (blue solid/dashed/dotted for the unbroken $\mathcal{PT}$ phase). Fixed parameters are: $\omega_{a}/2\pi = 13.2~\mathrm{GHz}$, $\omega_{b}/2\pi =15.101~\mathrm{MHz}$, $\kappa_{a}/2\pi= 2.1~\mathrm{MHz}$, $\gamma_{b}/2\pi= 150~\mathrm{Hz}$, $\kappa_{m}=\kappa_{a} + \gamma_{b}$, $G_{b}/2\pi=0.001~\mathrm{MHz}$, $\epsilon_{0}=1$, $\epsilon_{1} =\epsilon_{3} = 2.2$, $d_{1} = 4~\mathrm{mm}$, and $d_{2} = 45~\mathrm{mm}$.
• Figure 4: Contour plots of normalized GHS $S_{r}/\lambda$ versus incident angle $\theta_{i}$ and normalized effective detuning $x/\omega_{b}$ for: (a) broken $\mathcal{PT}$ phase ($G_{a} = 0.039\,\omega_{b}$), (b) $\mathrm{EP}_3$ ($G_{a} = 0.139\,\omega_{b}$), and (c) unbroken $\mathcal{PT}$ phase ($G_{a} = 0.239\,\omega_{b}$). Fixed parameters: $\omega_{a}/2\pi = 13.2~\mathrm{GHz}$, $\omega_{b}/2\pi = 15.101~\mathrm{MHz}$, $\kappa_{a}/2\pi = 2.1~\mathrm{MHz}$, $\gamma_{b}/2\pi = 150~\mathrm{Hz}$, $\kappa_{m} = \kappa_{a} + \gamma_{b}$, $G_{b}/2\pi = 0.001~\mathrm{MHz}$, $\epsilon_{0}=1$, $\epsilon_{1} =\epsilon_{3} = 2.2$, $d_{1} = 4~\mathrm{mm}$, and $d_{2} = 45~\mathrm{mm}$.
• Figure 5: Normalized GHS $S_{r}/\lambda$ versus incident angle $\theta_{i}$ for two cases: (a) $G_{b} =0$ and (b) $G_{b}/2\pi = 0.05~\mathrm{MHz}$ at resonance ($x=0$). Fixed parameters: $G_{a}= 0.039\,\omega_{b}$ (broken $\mathcal{PT}$ phase, green), $G_{a}=0.139\,\omega_{b}$ ($\mathrm{EP}_2$, red), $G_{a}=0.239\,\omega_{b}$ (unbroken $\mathcal{PT}$ phase, blue), $\omega_{a}/2\pi = 13.2~\mathrm{GHz}$, $\kappa_{a}/2\pi= 2.1~\mathrm{MHz}$, $\gamma_{b}/2\pi= 150~\mathrm{Hz}$, $\kappa_{m}=\kappa_{a} + \gamma_{b}$, $\omega_{b}/2\pi =15.101~\mathrm{MHz}$, $\epsilon_{0}=1$, $\epsilon_{1} =\epsilon_{3} = 2.2$, $d_{1} = 4~\mathrm{mm}$, and $d_{2} = 45~\mathrm{mm}$.