Coherent Control of the Goos-Hänchen Shift in Polariton Optomechanics

TL;DR

The paper addresses controlling the Goos-Hänchen shift (GHS) in a hybrid polariton optomechanical system by engineering a tripartite interaction among an optical cavity mode, a molecular vibrational mode, and $N$ excitonic transitions. Using a collective bright-mode reduction and a linearized Heisenberg–Langevin framework under red-sideband driving, it derives a closed-form expression for the probe response and a corresponding optical susceptibility that encapsulates the effective exciton–vibration coupling $G_v$. The main finding is that $G_v$ acts as a dynamic switch: when absent, the system exhibits pronounced GHS at resonance due to exciton-mediated OMIT, while activating $G_v$ suppresses the GHS and breaks detuning symmetry, with additional tunability provided by the effective cavity detuning and intracavity length; increasing the collective exciton–optical coupling $G_a$ further enhances the GHS. This work provides a theoretical framework for probing and exploiting beam-displacement phenomena in polariton optomechanics, offering pathways for novel optical devices in sensing and quantum information processing.

Abstract

We propose a theoretical scheme for controlling the Goos-Hänchen shift (GHS) of a reflected probe field in a polariton optomechanical system. The system comprises an optical mode, a molecular vibrational mode, and $N$ excitonic modes, where excitons couple to molecular vibrations via conditional displacement interactions and to photons through electric dipole interactions. We show that the effective exciton-vibration coupling provides a powerful mechanism for coherent GHS control: in its absence, the system exhibits a pronounced GHS at resonance, while activating it strongly suppresses the shift. The effective cavity detuning and the cavity length serve as additional tunable parameters for GHS manipulation. Furthermore, increasing the collective exciton-optical coupling enhances the GHS. Our results establish a framework for probing the GHS in polariton optomechanical systems and offer new avenues for designing optical devices that exploit beam-displacement phenomena.

Figures (7)

• Figure 1: Schematic illustration of the polariton optomechanical system. (a) A single-mode optical cavity (frequency $\omega_{a}$, decay rate $\kappa_{a}$) interacts with $N$ excitonic modes (frequency $\omega_{b}$, decay rate $\kappa_{b}$) through electric-dipole coupling. The excitonic modes further couple to the vibrational mode of the molecule (frequency $\omega_v$, decay rate $\gamma$) via conditional displacement interactions. A pump field (frequency $\omega_{0}$, amplitude $\eta$) drives the excitonic modes. A transverse electric (TE)-polarized probe field $E_{p}$ incident on mirror $M_{1}$ at an angle $\theta_{i}$ acquires a lateral displacement upon reflection, referred to as the GHS, and represented by $S_{r}$. (b) Simplified energy level diagram illustrating the conditional displacement interaction between the electronic transition and the vibrational state.
• Figure 2: (a) Absorption spectra $\mathrm{Re}[E_T]$ and (b) dispersion spectra $\mathrm{Im}[E_T]$ of the probe field versus the normalized effective detuning $x/\omega_v$. The solid blue curves represent the absence of effective exciton-vibration coupling ($G_v=0$), while the dashed blue curves correspond to $G_v=2\pi\times0.3~\mathrm{THz}$ at a fixed collective exciton–optical coupling strength $G_{a}/2\pi = 1.0~\mathrm{THz}$. The insets in Figs. \ref{['fig2']}(a) and (b) show the transparency-window width for $G_{a}/2\pi=0.6~\mathrm{THz}$ (green) and $G_{a}/2\pi=0.8~\mathrm{THz}$ (red). Fixed parameters: $\omega_v/2\pi = 30~\mathrm{THz}$, $\kappa_{a}/2\pi = \kappa_{b}/2\pi = 0.2~\mathrm{THz}$, and $\gamma/2\pi = 1.0~\mathrm{GHz}$.
• Figure 3: (a) Absolute value of the reflection coefficient $|r(k_{z}, \omega_{p})|$ and (b) the normalized GHS $S_{r}/\lambda$ versus incident angle of the probe field $\theta_{i}$, for different effective exciton–vibration coupling strengths: $G_{v} = 0$ (green), $2\pi\times0.06~\mathrm{THz}$ (blue), and $2\pi\times0.3~\mathrm{THz}$ (red), respectively, at the resonance condition ($x=0$). Fixed parameters: $\omega_a/2\pi = 300~\mathrm{THz}$, $\omega_v/2\pi = 30~\mathrm{THz}$, $G_{a}/2\pi = 1.0~\mathrm{THz}$, $\kappa_{a}/2\pi = \kappa_{b}/2\pi = 0.2~\mathrm{THz}$, $\gamma/2\pi = 1.0~\mathrm{GHz}$, $\epsilon_0=1$, $\epsilon_1=\epsilon_3=2.22$, $d_1=0.2~\mu\mathrm{m}$, and $d_2=5~\mu \mathrm{m}$.
• Figure 4: Normalized GHS $S_{r}/\lambda$ as functions of incident angle $\theta_{i}$ and effective exciton–vibration coupling strength $G_{v}$ (units: $2\pi\times1.0~\mathrm{THz}$) at resonance ($x=0$). Fixed parameters: $\omega_a/2\pi = 300~\mathrm{THz}$, $\omega_v/2\pi = 30~\mathrm{THz}$, $G_{a}/2\pi = 1.0~\mathrm{THz}$, $\kappa_{a}/2\pi = \kappa_{b}/2\pi = 0.2~\mathrm{THz}$, $\gamma/2\pi = 1.0~\mathrm{GHz}$, $\epsilon_0=1$, $\epsilon_1=\epsilon_3=2.22$, $d_1=0.2~ \mu\mathrm{m}$, and $d_2=5~\mu\mathrm{m}$.
• Figure 5: Normalized GHS $S_r / \lambda$ as functions of effective detuning $x / \omega_v$ and probe field incident angle $\theta_i$ for effective exciton-vibration coupling (a) $G_v = 0$, (b) $G_v = 2\pi \times 0.06~\mathrm{THz}$, and (c) $G_v = 2\pi \times 0.3~\mathrm{THz}$. Fixed parameters: $\omega_a / 2\pi = 300~\mathrm{THz}$, $\omega_v / 2\pi = 30~\mathrm{THz}$, $G_a / 2\pi = 1.0~\mathrm{THz}$, $\kappa_a / 2\pi = \kappa_b / 2\pi = 0.2~\mathrm{THz}$, $\gamma / 2\pi = 1.0~\mathrm{GHz}$, $\epsilon_0 = 1$, $\epsilon_1 = \epsilon_3 = 2.22$, $d_1 = 0.2~\mu\mathrm{m}$, and $d_2 = 5~\mu\mathrm{m}$.