Enabling entanglement distillation via optomechanics

Quantum networking based on optical Gaussian states, although promising in terms of scalability, is hindered by the fact that their entanglement cannot be distilled via Gaussian operations. We show that optomechanics, and particularly the possibility to measure the mechanical degree of freedom in an integrable system, can address this problem. Here, one of the optical modes of a two-mode squeezed vacuum is injected into a single-sided Fabry-Pérot cavity and nonlinearly coupled to a mechanical oscillator. Afterwards, the position of the oscillator is measured using pulsed optomechanics and homodyne detection. We show that this measurement can supply non-Gaussian entangled states frequently enough to enable scalable entanglement distillation. Moreover, it can conditionally increase the initial entanglement under an optimal radiation-pressure interaction strength, which corresponds to an effective unsharp measurement of the photon number inside the cavity. We show how the resulting entanglement enhancement can be verified by using a standard teleportation procedure.

Figure 1

Concentration scheme for two-mode squeezed vacuum (TMSV) states. In (a) we show the general idea, in which one mode ${\widehat{a}}_1$ interacts with a damped ($\gamma$) mechanical harmonic oscillator with strength $g$. Subsequently, we proceed to measure the position ($x$) of the mechanical oscillator, thus increasing the initial TMSV entanglement. In general, we model the position measurement considering an ideal detector preceded by a beam splitter (BS) of transmissivity $\nu$, being ${\delta }_q=(1-\nu )/\left(4\nu \right)$. In (b) we have substituted the above scheme with a typical optomechanical setup, modeling the injection of the mode ${\widehat{a}}_1$ into the cavity via a BS of reflectivity $r$ and where the position of the mirror is to be measured by pulsed optomechanics [60].

Figure 2

Upper panel (a): We plot in the left $y$ axis the ratio of the negativity $N_f/N_0$ as a function of the oscillator's position $q$ (solid line), where $N_f$ ($N_0$) stands for the distilled (initial) negativity. In the right $y$ axis we show the PDF as a function of $q$ (dashed line). In the middle panel (b) we illustrate the concentration success probability $\left[\mathrm{Pr}{(g,\lambda )}_s\right]$ corresponding to the shaded region in the upper panel. Finally, in the bottom panel (c) we show the ratio of negativity as function of $\lambda$ and $g$ for a specific oscillator's position $q=1.5$.

Figure 3

Standard quantum teleportation procedure [85]. We later show in Fig. 4 that the teleportation of an arbitrary coherent state $\left|\beta \right.〉$ is enhanced by using our concentrated state.

Figure 4

Ratio between the fidelity $\langle \mathcal{F}\rangle$ of the teleportation protocol that uses as a resource the concentrated state $\widehat{\rho }{\left(q\right)}_{12}$ [in Eq. (8)] and the fidelity $\langle {\mathcal{F}}_{\mathrm{TMSV}}\rangle$ obtained when an ideal TMSV is used. The ratio is plotted as a function of the outcome $q$ of the oscillator's position measurement. Other parameters are $\gamma =0.01,\nu =0.95,r=0.1,t=\pi , \lambda =0.3$, and $g=0.2$.

Figure 5

Right panel shows the fidelity between two adjacent concentrated states after measuring the oscillator's position where we have fixed $\mathrm{\Delta }q=0.5$. Left panel illustrates the corresponding PDF for the position measurement. Other parameters are $\gamma =0.01,\nu =0.95,r=0.1,t=\pi , \lambda =0.3$, and $g=0.2$.