Multi-path vector entanglement engineering via dark mode control in optomechanics

TL;DR

This work tackles the challenge of generating robust, multi-path entanglement in an optomechanical network subject to thermal noise. It introduces polarization-controlled dark-mode engineering in a cavity with two mechanically coupled resonators, using the polarization angle $\phi$ and the phonon-hopping rate $J_m$ (and modulation phase $\theta$) to switch between dark-mode unbreaking and breaking (ODMU/ODMB). The authors formulate a full Hamiltonian with TE/TM polarization components, linearize it, and compute steady-state covariance matrices to quantify bipartite entanglement via logarithmic negativity $E_N$ and tripartite entanglement via residual contangle $R^{min}$. They demonstrate that dark-mode breaking enables abundant multi-path bipartite and tripartite entanglement, with enhanced resilience to thermal noise (up to about two orders of magnitude) in the DMB regime, and show a special case at $\phi=\pi/4$ that yields twin entangled states. The results point to a flexible, noise-tolerant resource-generation platform for quantum information processing and communication, with potential extensions to microwave and hybrid optomechanical systems.

Abstract

We propose a scheme to generate multi-paths entanglement in an optomechanical system by exploiting polarized electromagnetic fields and dark mode control. Our system consists of two mechanically coupled mechanical resonators, which are driven by a common electromagnetic field. An inclusion of a polarizer induces linear polarizations of the electromgnetic field corresponding to the vertical (transverse electric ($\rm{TE}$) and horizontal (transverse magnetic [($\rm{TM}$]) modes, which drive the mechanical resonators. Without the mechanical coupling $J_m=0$, the polarization angle ($φ$) controls dark mode in the system. The breaking of this dark mode leads to multi-paths engineering of bipartite optomechanical entanglements. By switching on the phonon hopping rate ($J_m\neq0$), both the polarization angle and the modulation phase of the mechanical coupling allow a further control of the dark mode. The simultaneous Dark Mode Breaking (\rm{DMB}) conditions under these two parameters leads to multi-paths bipartite and tripartite entanglements. For a fine tuning of the polarization angle ($φ=π/4$) this scheme enables a generation of twin entangled states, where the bipartite/tripartite generated entangled states are degenerated and might be of great interest for quantum information processing, quantum communication and diverse quantum computational tasks. The generated entanglements are more resilient against thermal fluctuations in the \rm{DMB} regime, i.e., up to two order of magnitude robust than in the Unbreaking regime. Our work sheets light on new possibilities to generate noise-tolerant quantum resources that are useful for plethora of modern quantum technologies.

Figures (11)

• Figure 1: Sketch of our proposal. Two mechanically coupled mechanical resonators are placed inside an optical resonator, which is driven by polarized electromagnetic fields.
• Figure 2: Stability diagram of the system depending on the effective coupling $G_m$ and the mechanical coupling $J_m$. The blue/dark region is stable, while the red/light area represents the unstable zone. The used parameters are $\tilde{\Delta}=-\omega_m$, $\kappa=0.2\omega_m$, $\omega_j=\omega_m$, $\gamma_j=10^{-5}\omega_m$, $\phi=\frac{\pi}{4}$, and $\theta=\frac{\pi}{2}$.
• Figure 3: (a) Coupling strength $G_-$ in polar-coordinate versus the polarized angle $\phi$ for $G_m=0.2\omega_m$, $\tilde{\Delta}_{\updownarrow}=-\omega_m$, and $\tilde{\Delta}_{\leftrightarrow}=-(1+10^{-3})\omega_m$. The other parameters are $J_m=0$, $\kappa=0.2\omega_m$, $\omega_j=\omega_m$, and $\gamma_j=10^{-5}\omega_m$.
• Figure 4: (a) Polar entanglement representation and its (b) $2D$-representation versus $\phi$ for $J_m=0$. The quantity $\rm{E_N^{\alpha_v-M1/M2}}$ ($\rm{E_N^{\alpha_h-M1/M2}}$) is the bipartite entanglement between the $\rm{TE}$ ($\rm{TM}$) mode and each mechanical resonator. In these figures, $G_m=0.2\omega_m$, $n_{th}^j=100$, and the other parameters are the same as in \ref{['fig:fig3']}.
• Figure 5: (a) Contour plot of the entanglement between $\rm{TE}$ mode and each mechanical resonator ($E_N^{\rm{\alpha_v-M1/M2}}$) versus $J_m$ and $\phi$ for $\theta=\frac{\pi}{2}$. (b) Contour plot of the entanglement between $\rm{TM}$ mode and each mechanical resonators ($E_N^{\rm{\alpha_h-M1/M2}}$) versus $J_m$ and $\phi$ for $\theta=\frac{\pi}{2}$. (c) Polar entanglement representation versus $\theta$ for $J_m=0.2\omega_m$ and $\phi=\frac{\pi}{4}$. (d) $2D$-representation of (c) where $E_N^{\rm{\alpha_{h,v}-M1}}$ ($E_N^{\rm{\alpha_{h,v}-M2}}$) reaches it optimal value for $\theta=(n+1/2)\pi$, $n$ being an even integer ($\theta=(n+1/2)\pi$, $n$ being an odd integer). For all these figures, $G_m=0.2\omega_m$, $n_{th}^j=100$, and the other parameters are the same as in \ref{['fig:fig3']}.