Multipartite entanglement swapping and mechanical cluster states

We present a protocol for generating multipartite quantum correlations across a quantum network with a continuous-variable architecture. An arbitrary number of users possess two-mode entangled states, keeping one mode while sending the other to a central relay. Here a suitable multipartite detection is implemented, by multiple homodyne detections at the outputs of the interferometer, to conditionally generate a cluster state on the retained modes. This cluster state can be suitably manipulated by the parties and used for tasks of quantum communication in a fully optical scenario. More interestingly, the protocol can be used to create a purely-mechanical cluster state starting from a supply of optomechanical systems. We show that detecting the optical parts of optomechanical cavities may efficiently swap entanglement into their mechanical modes, creating cluster states up to five modes under suitable cryogenic conditions.

Figure 1

Multipartite entanglement swapping. We start from $N$ independent copies of the state ${\rho }_{AB}$. The $A$ systems are sent to a relay for a multipartite CV Bell detection. The latter is an interferometer with a suitable cascade of beam splitters, followed by homodyne detections ($N-1$ in the $X$ quadratures, and a final one in the $P$ quadrature). The outcomes $\gamma$ are broadcast to the users, so that their multipartite state collapses into a conditional cluster state ${\rho }_{B_1\cdots B_N|\gamma }$. The transmissivities $T_k$ of the beam splitters are chosen so that the cluster state is invariant under permutation of the users.

Figure 2

Study of the output mechanical entanglement. (a) For $N=2$ we plot the output log negativity $E_{\mathcal{N}}^{\left(2\right)}$ as a function of the log negativity $E_{\mathcal{N}}^{\text{in}}$ of the input Gaussian state which is generated by random sampling. Upper and lower bounds (solid lines) are achieved by the classes of states discussed in the main text. (b) We show the distribution of $E_{\mathcal{N}}^{\left(2\right)}$ as a function of the asymmetry parameter $d$ by randomly sampling the input state. The solid line shows the maximum achievable value. (c) We plot the output log negativity $E_{\mathcal{N}}^{\left(2\right)}$ of two mechanical modes as a function of the input log negativity $E_{\mathcal{N}}^{\text{in}}$ between the optical and the mechanical modes of two identical optomechanical systems with parameters ${\gamma }_m/2\pi =100$ Hz, ${\omega }_m/2\pi =10$ MHz, $\kappa /2\pi =31.4$ MHz, and $T=0.4$ mK. Each mechanical mode has mass $m=5$ ng. The green dashed line (1), the dashed purple (2), and the solid magenta (3), correspond to effective optomechanical coupling rates of $2\pi \times 4$ MHz, $2\pi \times 8$ MHz, and $2\pi \times 8.5$ MHz, respectively. These values match closely those typical of a photonic-crystal-based optomechanical architecture [63]. The curvilinear abscissa of each line is the detuning $\mathrm{\Delta }\in [0,1.5{\omega }_m]$. (d) We show the output log negativity $E_{\mathcal{N}}^{\left(N\right)}$ between any two modes in a cluster of $N$ mechanical modes, for $N=2–5$. Parameters as in panel (c) with an effective optomechanical coupling strength of $2\pi \times 8$ MHz. Inset: we show the time $t^{*}$ (s) at which the entanglement $E_{\mathcal{N}}^{\left(N\right)}$ disappears when assuming dissipation into independent thermal baths at temperature $T$ for each involved mechanical mode.