Fig. 2: Logarithmic negativity for adiabatic and full solutions as a function of squeezing \(S\).

a Adiabatic regime of parameters: dashed curves (adiabatic solution) are close to the solid ones (full solution). Here, \(\Gamma_{\mathrm{m}}=10^{-5}{\kappa}_{\mathrm{m}},\ {\gamma}_{\mathrm{m}}=10^{-5}{\kappa}_{\mathrm{m}},\ g=G=0.085 {\kappa}_{\mathrm{m}},\ \tau {\kappa}_{\mathrm {m}}=100\). Interaction gains \({\bf{K}}_{1,2}\) are given by Eq. (9) and calculated by \(\tau\), \(g\), and \(G\) assuming \(\kappa_{\mathrm{a}}=2\kappa_{\mathrm{m}}\) (here, \({\bf{K}}_{1}={0.85},{\bf{K}}_{2}={1.20}\)), \(\mathfrak{G}_{\mathrm{a},\mathrm{m}}\) are defined by Eq. (6). For the lossless case the logarithmic negativity saturates, while for the case of any loss it has a maximum. For the adiabatic case the entangling power (with low population and high efficiency) always has a positive value even with no squeezing of the light pulse. b Full solution calculated using non-adiabatic parameters \(\Gamma_{\mathrm{m}}=0.01 {\kappa}_{\mathrm{m}},\ \tau {\kappa}_{\mathrm{m}}=50,\ {\gamma}_{\mathrm {m}}=10^{-5}{\kappa}_{\mathrm {m}},\ {\kappa}_{\mathrm {a}}=2 {\kappa}_{\mathrm{m}}\), \(n_{0}=0\), \(G=g\). Here, thick curves correspond to the lossless case (\(\eta =1\)), thin ones correspond to the case of \(10{\%}\) loss (\(\eta =0.9\)). In contrast to a squeezing of the pulse can be insufficient to reach positive entangling power even for \(n_{0}=0\) with \(\eta =1\). In both these cases logarithmic negativity as a function of squeezing with \(\eta \ <\ 1\) has a maximum that depends on the efficiency and initial population.