Entanglement of distant optomechanical systems

We theoretically investigate the possibility to generate nonclassical states of optical and mechanical modes of optical cavities, distant from each other. A setup comprised of two identical cavities, each with one fixed and one movable mirror and coupled by an optical fiber, is studied in detail. We show that with such a setup there is potential to generate entanglement between the distant cavities, involving both optical and mechanical modes. The scheme is robust with respect to dissipation, and nonlocal correlations are found to exist in the steady state at finite temperatures.

Figure 1

Sketch of the physical setup to entangle distant optomechanical modes. Two optomechanical cavities pumped by classical laser fields are coupled to each other by an optical fiber. As a result of indirect coupling mediated by the two cavity modes, the two movable mirrors become entangled. Furthermore, two initially uncorrelated auxiliary cavity modes interact independently with the two entangled movable mirrors, which induces nonlocal correlations between the two modes. Using standard homodyne measurement techniques nonlocal correlations between the two auxiliary cavity modes can be read out giving an indirect signature of quantum correlations between the two mirrors.

Figure 2

Time evolution of the degree of entanglement, as measured by the logarithmic negativity, as a function of initial temperature of the movable mirrors, measured in terms of ${\overline{n}}_{\mathrm{thermal}}$. The dimensionless parameters are chosen such that $\Omega =1$, $g={10}^{-2}$, $\lambda ={10}^{-1}$, ${\alpha }_A=4$, and ${\alpha }_B=1$.

Figure 3

Temporal evolution of the degree of entanglement between two indirectly coupled movable mirrors as measured by the logarithmic negativity. Compared with Fig. 2, here losses in all modes have been considered and the degree of entanglement is consequently somewhat smaller, but importantly, it survives for a reasonably long time. Each mirror is initially assumed to be in its ground state and the dimensionless parameters used are chosen such that $\Omega =1$, $g={10}^{-2}$, $\lambda ={10}^{-1}$, ${\alpha }_A=4$, ${\alpha }_B=1$, $\kappa ={10}^{-3}$, $\Gamma ={10}^{-4}$, and $\overline{n}=0$.

Figure 4

Logarithmic negativity as a measure of entanglement between (a) two distant cavity mirrors, (b) a mirror and adjacent cavity mode, and (c) a mirror and distant cavity mode, plotted as a function of detuning $\Delta$ and average thermal occupancy of the two mirrors ${\overline{n}}_1={\overline{n}}_2=\overline{n}$. We have chosen the different physical parameters such that $\Omega =1$, $g_a^s=g_b^s=2.5$, $\lambda =20$, $\kappa =0.08$, ${\gamma }_m=0.01$, and ${\Delta }_a={\Delta }_b=\Delta$.