Fig. 3: Logarithmic negativity as a function of efficiency.

a The squeezing \({S}_{{\mathrm {opt}}}(\eta )\) is optimized for each value of the efficiency \(\eta\). The inset illustrates the dependence of the optimized value of squeezing on the efficiency \(\eta\) (the shape of this curve does not differ a lot for any case, but for the adiabatic parameters the increase is much sharper). b Limited role of the optimization. Solid curves assume optimized squeezing \({S}_{{\mathrm {opt}}}(\eta )\) and the dotted ones illustrate the case with fixed values \({S}_{{\mathrm {opt}}}(\eta =0.999)\), while the dotdashed curve shows the case with \({S}_{{\mathrm {opt}}}(\eta =0.9)\) for \({n}_{0}=0\). As the efficiency \(\eta\) approaches the lossless case, \({S}_{{\mathrm {opt}}}(\eta \to 1)\to \infty\) and the dotted lines become vertical. Dashed lines (in the inset) are limiting expressions (\(S\to \infty\)) of the expansions of the logarithmic negativity to first order around \(S\) near \(\eta =1\) for \({n}_{0}=0,\ 1\) that can be analytically obtained only for the adiabatic approximation. Near \(\eta =1\) these lines lie very close to the logarithmic negativity calculated with optimized squeezing for the adiabatic parameters. The thickness of the curves shows different initial occupations (\({n}_{0}=0,\ 1,\ 10\)). The colors of the curves mark the case: purple curves correspond to the adiabatic case with parameters \({\Gamma }_{{\mathrm {m}}}=1{0}^{-5}{\kappa }_{{\mathrm {m}}},\ \tau {\kappa }_{{\mathrm {m}}}=100,\ G=g=0.085{\kappa }_{{\mathrm {m}}}\), black and blue, for the non-adiabatic one with \({\Gamma }_{{\mathrm {m}}}=0.01{\kappa }_{{\mathrm {m}}},\ G=g=0.4{\kappa }_{{\mathrm {m}}}\), moreover, \(\tau {\kappa }_{{\mathrm{m}}}=50\) for the black and \(\tau {\kappa }_{{\mathrm {m}}}=20\) for the blue. Parameters used: \({\gamma }_{{\mathrm {m}}}=1{0}^{-5}{\kappa }_{{\mathrm {m}}}\), \({\kappa }_{{\mathrm {a}}}=2{\kappa }_{{\mathrm {m}}}\).