Robust entanglement with a thermal mechanical oscillator

We consider a protocol to entangle an electromagnetic pulse with a mechanical oscillator at high temperature. We show this protocol to be capable of entangling currently existing experimental systems at relatively high (above the available cryostat) temperatures of the mechanical part. We also predict a possibility of conditional squeezing of the mechanical mode below the shot noise level at the cryostat temperatures.

Figure 1

Protocol to create and verify entanglement. First the blue-detuned entangling pulse with length $\tau$ enters the cavity, then after the delay time ${\tau }_{\mathrm{del}}$ the red-detuned pulse lasting ${\tau }_{\mathrm{tr}}$ performs the read-out of the state of the mechanical subsystem. Mechanical system is initially in a thermal state with occupation $n_0$ and is coupled at the rate $\gamma$ to the thermal bath with occupation $n_{\mathrm{th}}$.

Figure 2

Different types of entanglement (logarithmic negativity) versus the temperature (occupation number) of the mechanical subsystem. (a) Entanglement between the quadratures of the entangling and the verification pulses. (b) Entanglement between the entangling pulse and the mechanical mode. Darker (blue) traces, electromechanical experimental setup reported by Palomaki et al. [13]; lighter (green) traces, optomechanical crystal by Chan et al. [18]. Dashed lines, lossless case; solid lines, realistic damping in both modes, $\eta =0.2$. (Inset) Nonzero elements of covariance matrix simulating the experiment from Ref. [13] (initial mechanical occupation $n_0=0$, mechanical bath occupation $n_{\mathrm{th}}\approx 40$). Black lines in (b) represent logarithmic negativity $E_{N,a}$ and EPR variance ${\mathrm{\Delta }}_{\text{EPR},a}$ of the optomechanical system calculated after the adiabatic elimination of the cavity mode. For the parameters utilized for estimations see Table 1.

Figure 3

Dependence of the entanglement between the two pulses on the time span between them for the parameters of experiment by Palomaki et al. [13]. Dashed and solid lines correspond to the lossless case and the case of losses $\eta =0.2$, respectively.

Figure 4

Minimal coupling (in relative units) allowing conditional squeezing ${\zeta }_s$ and entanglement ${\zeta }_e$ for parameters of experiment by Palomaki et al. [13] (${\zeta }^{\left(P\right)}$) and by Chan et al. [18] (${\zeta }^{\left(C\right)}$). (Solid line) Minimal variance $\sigma$ of the conditional mechanical state versus mechanical occupation (parameters from [13]). The conditional squeezing is possible if $\sigma <1/2$.