Pulsed quantum interaction between two distant mechanical oscillators

Feasible setup for pulsed quantum nondemolition interaction between two distant mechanical oscillators through an optical or microwave mediator is proposed. The proposal uses homodyne measurement of the mediator and feedforward control of the mechanical oscillators to reach the interaction. To verify the quantum nature of the interaction, we investigate the Gaussian entanglement generated in the mechanical modes. We evaluate it under influence of mechanical bath and propagation loss for the mediator and propose ways to optimize the interaction. Finally, both currently available optomechanical and electromechanical platforms are numerically analyzed. The analysis shows that implementation is already feasible with current technology.

Figure 1

The protocol of QND interaction between two mechanical modes. (a) Simplified scheme: $S$, squeezing operation; HD, homodyne detector. (b) Possible experimental implementation with the imperfections: $\eta$, optical losses between the cavities; $n_{\mathrm{th}}$, thermal mechanical environment.

Figure 2

Entanglement between the two mechanical modes as a function of optical presqueezing. Thick lines correspond to adiabatic solution; thin lines with markers to solution with cavity mode. Different colors and dashes are used for different ratio of the gains $K_1$ and $K_2$. Losses are absent: $\eta =1$. Highlighted is the region of squeezing magnitudes not exceeding the value of $12.7\mathrm{dB}$ reported in Ref. [62].

Figure 3

Entanglement as a function of squeezing in the presence of a mechanical bath with mean number of phonons $n_{\mathrm{th}}$ and optical losses with transmittivity $\eta$. The optomechanical gains equal $K_1=1,K_2=8$, the same as for the blue dot-dashed line in Fig. 2.

Figure 4

Maximal entanglement achievable with the coupling rate $g$ (in units of $\kappa$), for optomechanical parameters [52] (blue dot-dashed and violet dotted lines) and electromechanical [53] (brown dashed and green solid).

Figure 5

Entanglement (logarithmic negativity) as a function of squeezing of the input state computed from the full solution (thick lines) and with help of RWA (thin lines with markers). For details, see the caption of Fig. 2.