Multipartite quantum entanglement in $\mathcal{PT}$-symmetric molecular optomechanics: Nonreciprocal enhancement and thermal resilience to \SI{500}{\kelvin}

Abstract

We present a theoretical framework for a $\mathcal{PT}$-symmetric double-cavity molecular optomechanical system demonstrating nonreciprocal enhancement of multipartite quantum entanglement at elevated temperatures. All bipartite entanglement channels ($E_{ac}$, $E_{aB_1}$, $E_{cB_2}$, $E_{B_1B_2}$) simultaneously maximize at optimal nonreciprocal asymmetry $J_1/J_2 \approx 5$, with entanglement persisting to $T \sim \SIrange{400}{500}{\kelvin}$ (material-limited ceiling) two orders of magnitude beyond conventional optomechanical systems. This thermal resilience and balanced enhancement across all channels arise from synergistic combination of ultra-high-frequency molecular vibrations ($ω_m/2π= \SI{30}{\tera\hertz}$), collective $\sqrt{N}$ coupling enhancement with $N=\num{e6}$ molecules, and directional nonreciprocal coupling shielding entanglement-generating interactions from backaction noise. Unlike optical parametric amplifier schemes where vibration-vibration enhancement suppresses optical-vibration correlations, our $\mathcal{PT}$-symmetric architecture circumvents this fundamental trade-off, validated through rigorous stability analysis via Routh-Hurwitz criterion.

Figures (9)

• Figure 1: Schematic of the $\mathcal{PT}$-symmetric double-cavity molecular optomechanical system. An ensemble of $N$ molecules with collective vibrational modes ($B_1$, $B_2$, frequency $\omega_m$) resides in active cavity $c$ (optical gain rate $\kappa_c$), which is nonreciprocally coupled to passive cavity $a$ (optical loss rate $\kappa_a$) with directional rates $J_1 \gg J_2$. Both cavities are driven by external lasers (amplitudes $\mathcal{E}_a$, $\mathcal{E}_c$). This architecture combines $\mathcal{PT}$-symmetric gain-loss engineering, ultra-high-frequency molecular vibrations, and nonreciprocal coupling to achieve robust multipartite entanglement at elevated temperatures.
• Figure 2: Eigenvalue spectra revealing $\mathcal{PT}$-symmetry phase transitions and exceptional points (EPs). (a-b) Real and imaginary parts of eigenvalues $\lambda_\pm$ versus nonreciprocal coupling $J_1/\omega_m$ for unbalanced gain-loss ($\kappa_a/\omega_m=0.02$, $\kappa_c/\omega_m=0.001$). The EP occurs at $J_1/\omega_m \approx 0.027$, marking the transition between unbroken-$\mathcal{PT}$ (real eigenvalues, stable oscillations) and broken-$\mathcal{PT}$ (complex conjugate eigenvalues, exponential growth/decay) phases. (c-d) Same analysis for balanced gain-loss ($\kappa_a/\omega_m=\kappa_c/\omega_m=0.02$), showing EP shifted to higher coupling $J_1/\omega_m \approx 0.1$ due to enhanced effective coupling threshold. Parameters: $\omega_m/2\pi=30THz$, $J_2/\omega_m=0.004$, others in \ref{['tab:parameters']}.
• Figure 3: Dynamical stability analysis via Routh-Hurwitz criterion, identifying stable (purple) and unstable (green) operational regimes. (a) Stability boundaries in $(N, \mathcal{E}/\omega_m)$ parameter space: larger molecular ensembles reduce stability threshold due to enhanced collective coupling, requiring lower driving strengths to avoid instability. (b) Stability boundaries in $(J_1/\omega_m, \mathcal{E}/\omega_m)$ parameter space: stronger nonreciprocal coupling expands stable region by shielding system from detrimental backaction. All entanglement results presented in subsequent figures correspond to parameter values within the purple stable regions, ensuring physical validity. Parameters: $N=e6$, $J_1/\omega_m=0.02$, others in \ref{['tab:parameters']}.
• Figure 4: Optical bistability driven by collective optomechanical nonlinearity. (a) Mean intracavity photon number $\mathcal{I}_c$ versus bare detuning $\Delta_c$ for varying molecular ensemble size $N$, showing tilted resonance curves. Larger $N$ enhances nonlinearity via stronger collective coupling $G \propto \sqrt{N}$, shifting and tilting the resonance peaks. (b) $\mathcal{I}_c$ versus driving amplitude $\mathcal{E}$ for different $\Delta_c$ values, exhibiting characteristic S-shaped bistable curves with hysteretic switching behavior between two stable branches (upper and lower) and one unstable intermediate branch. Stability of all branches confirmed by Routh-Hurwitz criterion. Parameters: $N=e6$, $\omega_m/2\pi=30THz$, $g_m/2\pi=50MHz$, others in \ref{['tab:parameters']}.
• Figure 5: Entanglement versus cavity detunings: resonance-enhanced quantum correlations. Bipartite entanglement for (a) inter-cavity $E_{ac}$, (b) passive cavity-molecule $E_{aB_1}$, (c) active cavity-molecule $E_{cB_2}$, and (d) vibration-vibration $E_{B_1B_2}$ channels as functions of normalized detunings $\Delta_a/\omega_m$ and $\Delta_c/\omega_m$. All channels exhibit maximum entanglement at resonance ($\Delta_a = \Delta_c = \omega_m$), where strong effective optomechanical coupling enhanced by collective molecular interaction optimizes quantum state transfer. Channel hierarchy: $E_{cB_2} > E_{ac} > E_{B_1B_2} > E_{aB_1}$ reflects direct vs. indirect coupling strengths. Parameters: $T=300K$, $N=e6$, others in \ref{['tab:parameters']}.