Amplification and generation bounds of gravity-induced entanglement in pulsed optomechanical systems

We investigate gravity-induced entanglement between the output optical fields of two red-detuned pulsed optomechanical systems with their masses coupled by mutual gravitational interaction. For each individual system, the optomechanical interaction realizes a beam-splitter state swap between an incident optical pulse and its mechanical mode. Using two rectangular pulses for each system -- the first to imprint a nonclassical state on the mechanical modes and the second to read the gravitationally generated entanglement back onto the outgoing light -- we show that the amount of entanglement can be amplified by preparing the input in a squeezed or Fock state. However, the threshold for entanglement generation is set by the competition between the gravitational coupling and thermal decoherence, $g_G>2\gamma_m N_{\rm th}$, and cannot be lowered by any choice of input state. We prove this bound for two-mode Gaussian inputs and show that it continues to hold for Fock-state inputs. We further analyze how imperfect detection modifies the threshold and identify the entanglement-annihilating and entanglement-breaking regimes, which are set by the thermal decoherence accumulated over the interaction time, independent of the gravitational coupling.