Figure 2: Initialization of the mechanical state.
From: Enhanced nonlinear interactions in quantum optomechanics via mechanical amplification
Fast turn on of mechanical parametric driving using the TD scheme (see main text). (a) Time dependence of the parametric driving strength λ(t)=λ0(t)+iλ1(t), corresponding to a Gaussian profile for the instantaneous amplification factor r(t) [tanh 2r(t)=λ0(t)/Δ]. The final value of λ(t) corresponds to e2r=20 dB. The pulse is chosen to ramp up the parametric drive in a time much shorter than the inverse Bogoliubov-mode energy Eβ. (b) Evolution of the mechanical state, as characterized by the population of the Bogoliubov mode 
. The solid red line is for the TD approach, showing preparation of a pure squeezed state (characterized by no β-mode excitations) in a time 
. In contrast, a sudden (step-function) turn on of the parametric drive results in β mode being far from its ground state (red dashed line). The TD protocol plays a crucial role in our scheme, as it allows a rapid turn on of the mechanically mediated photon–photon interaction, without any spurious effects resulting from a large initial β-mode population. Neither a purely adiabatic protocol nor a sudden diabatic approach would be sufficient. Here the mechanical dissipation is γ=10−4Eβ and g=0, but the results are unchanged for g0 and a sufficiently small γ.
