Enhancing Optomechanical Entanglement and Mechanical Squeezing by the Synergistic Effect of Quadratic Optomechanical Coupling and Coherent Feedback

Quantum entanglement and squeezing associated with the motions of massive mechanical oscillators play an essential role in both fundamental science and emerging quantum technologies, yet realizing such macroscopic nonclassical states remains a formidable challenge. In this paper, we investigate how to achieve strong optomechanical entanglement and mechanical squeezing in a membrane-embedded cavity optomechanical system incorporating a coherent feedback loop, where the membrane interacts with the cavity mode through both linear and quadratic optomechanical couplings. This hybrid optomechanical architecture offers a flexible tunability of intrinsic system parameters, thereby enabling controlled stiffening or softening of the mechanical mode through adjusting quadratic optomechanical coupling, as well as effective modulation of the cavity decay rate via feedback control. More importantly, the synergistic interplay effect allows for a strategic reconfiguration of the system’s stability regime, which in turn permits the presence of significantly enhanced effective optomechanical coupling strengths before entering the unstable regime. Exploiting these unique features, we demonstrate that optomechanical entanglement can be substantially enhanced with positive coupling sign and suitable feedback parameters, while strong mechanical squeezing beyond the $3$dB limit is simultaneously achieved over a broad parameter range with negative coupling sign, reaching squeezing degree above $10$dB under optimized conditions. Our proposal, establishing an all-optical method for generating highly entangled or squeezed states in cavity optomechanical systems, opens up a new route to explore macroscopic quantum effects and to advance quantum information processing.

💡 Research Summary

The authors propose a hybrid cavity‑optomechanical platform that combines a membrane‑in‑the‑middle Fabry‑Perot cavity with a coherent‑feedback loop. The membrane couples to the intracavity field through both linear (LOC) and quadratic (QOC) radiation‑pressure interactions, the relative strengths of which are set by the membrane’s equilibrium position. By embedding a coherent feedback circuit—three high‑reflectivity mirrors and a tunable beam‑splitter—the optical output is redirected back into the cavity, allowing the effective cavity decay rate κ̃ to be continuously tuned via the feedback reflectivity r_B and phase shift θ.

Starting from the full Hamiltonian
H = ℏω_c a†a + ℏω_m/2(p²+q²) + ℏg₁ a†a q + ℏg₂ a†a q² + iℏε_d(e^{-iω_dt}a† – e^{iω_dt}a),
the authors derive quantum Langevin equations that include mechanical damping γ_m, cavity losses κ₁, κ₂, and thermal Brownian noise. Under strong driving they linearize the dynamics around steady‑state amplitudes (α_s, q_s, p_s) and obtain a set of linearized equations for the fluctuations. The drift matrix’s stability is analyzed using the Routh‑Hurwitz criterion, yielding a stability parameter C₁ that depends on detuning Δ̄, the ratio g₂/g₁, and the feedback parameters (r_B, θ).

Key findings are illustrated in Fig. 2. (a) The effective mechanical frequency Ω_m = ω_m + 2g₂|α_s|² can be stiffened (g₂<0) or softened (g₂>0), providing a direct knob for mechanical resonance. (b) The decay‑ratio η = κ̃/(κ₁+κ₂) can be increased or decreased by adjusting r_B and θ, effectively amplifying the optomechanical coupling G_eff = g₁α_s·(κ₁+κ₂)/κ̃. (c‑f) Stability maps show that the combined action of QOC and feedback dramatically reshapes the stable region, allowing much larger G_eff before the system becomes unstable.

When the quadratic coupling is positive, the authors find that the logarithmic negativity E_N (a measure of optical‑mechanical entanglement) can be enhanced by a factor of ≈5 compared with a pure LOC configuration, provided the feedback parameters are chosen near r_B≈0.9 and θ≈3π/2. Conversely, for negative quadratic coupling the effective mechanical frequency is raised, which suppresses thermal noise and enables mechanical quadrature squeezing well beyond the 3 dB limit. Numerical simulations with realistic parameters (κ₁=κ₂=2π·1.5 MHz, ω_m=2π·10 MHz, g₁≈−1.35 kHz, g₂/g₁≈6×10⁻⁵, laser power 5 mW, wavelength 810 nm, temperature ≈300 mK) predict squeezing levels exceeding 10 dB (ΔX²/ΔX²_vac < 0.1).

The paper emphasizes that neither QOC nor coherent feedback alone can achieve this performance; it is their synergistic interplay that shifts the instability threshold and permits simultaneously strong entanglement and deep squeezing. The proposed scheme is fully compatible with current experimental technology: membrane‑in‑the‑middle devices have already demonstrated both linear and quadratic couplings, and coherent‑feedback loops have been realized in optical and microwave platforms.

In conclusion, the work provides a clear theoretical blueprint for generating highly nonclassical states of massive mechanical resonators using all‑optical means. By exploiting tunable quadratic coupling and feedback‑controlled cavity loss, one can access parameter regimes previously deemed unstable, opening new avenues for quantum‑enhanced sensing, secure communication, and hybrid quantum networks where mechanical modes serve as quantum memories or transducers. Future directions include experimental validation, analysis of feedback‑induced time delays, and extension to multimode or multimembrane configurations.