Quantum communication between remote mechanical resonators

Mechanical resonators represent one of the most promising candidates to mediate the interaction between different quantum technologies, bridging the gap between efficient quantum computation and long-distance quantum communication. Here, we introduce an interferometric scheme where the interaction of a mechanical resonator with input-output quantum pulses is controlled by an independent classical drive. We design protocols for state teleportation and direct quantum state transfer, between distant mechanical resonators. The proposed device, feasible with state-of-the-art technology, can serve as a building block for the implementation of long-distance quantum networks of mechanical resonators.

Figure 1

Sketch of the proposed interferometric scheme. An optomechanical device, composed of two identical optical cavities sharing a vibrating mirror, is embedded in a common-path Sagnac interferometer. In our protocol, a classical drive sent through the control port ${\widehat{\beta }}_B\left(\omega \right)$ modulates the optomechanical interactions with input-output quantum signals ${\widehat{\alpha }}_B\left(\omega \right)$.

Figure 2

Teleportation protocol. Where not specified, here $T_m=25$ mK, $g=32.5$ MHz, ${\omega }_c/2\pi =194$ THz, ${\omega }_m/2\pi =5.3$ GHz, $\gamma /2\pi =6.5$ kHz, $r=0.3$, and ${\kappa }_L/2\pi =455$ MHz. (a) EPR variance as a function of the squeezing parameter $r=G\tau$. Black full line: no mechanical dissipation $\gamma$ nor photon losses ${\kappa }_L$; red dotted line: no photon losses; green dashed line: ${\kappa }_L/\kappa =0.3$ and $\kappa /2\pi =1.52$ GHz. The state is entangled if ${\mathrm{\Delta }}_{\text{EPR}}<2$. (b) EPR variance as a function of the mechanical bath temperature. (c) Teleportation fidelity $\mathcal{F}$ of a coherent state, as a function of its size $X_{\text{in}}$ (here, $P_{\text{in}}=0$) for a displacement efficiency $\eta =0.99$. The vertical gray line shows the value of $X_{\text{in}}=10$ used in (d). (d) Teleportation fidelity as a function of the mechanical bath temperature, for $\eta =0.99$. Color code: from lighter to darker shades of blue: ${\kappa }_L/\kappa =\{0.7,0.45,0.3,0.18,0.1\}$ and $r=\{0.2,0.3,0.4,0.58,0.7\}$. As the ${\kappa }_L/\kappa$ ratio is decreased, the total dissipation rate $\kappa +{\kappa }_L$ is increased.

Figure 3

Schemes of the teleportation and state transfer protocols. Circled numbers represent different temporal steps. Top panel: Teleportation protocol. (1) Bob generates an optomechanical EPR state. (2) Alice implements a beam-splitter interaction and the required measurements. (3) The results of the measurements are sent over a classical communication channel to Bob. Bottom panel: State transfer process between remote mechanical oscillators. The classical drive must be time modulated in order to optimize the time envelope of the quantum signal.

Figure 4

State transfer protocol. Fraction of the initial state $\widehat{m}\left(0\right)$ that have been transferred to the second mechanical oscillator ${\widehat{m}}_2\left(\tau \right)$ ($W_{\mathrm{TM}}^2$), and that have been destroyed on the first mechanical oscillator ${\widehat{m}}_1\left(\tau \right)$ ($W_D^2$), as functions of (a) the pulses duration $\tau$ and (b) the ratio ${\kappa }_L/\kappa$. The parameters are ${\tau }_{\text{max}}/2\pi =0.04/\gamma , g=32.5$ MHz, ${\omega }_c/2\pi =194$ THz, ${\omega }_m/2\pi =5.3$ GHz, $\gamma /2\pi =6.5$ kHz. (a) ${\kappa }_L/\kappa =0.3, \kappa /2\pi =1.52$ GHz. (b) $\tau ={\tau }_{\text{max}}$, and as ${\kappa }_L/\kappa$ goes from 0 to 1, $\kappa /2\pi$ goes from 4.55 GHz to 650 MHz. Inset: optimal temporal shapes of the control pulses for the sender (full curve) and the receiver (dotted curve). Note that in (b), even when ${\kappa }_L/\kappa$ goes to 0, we still have $W_{\mathrm{TM}}^2<1$ due to the mechanical dissipation.

Figure 5

Teleportation fidelity from Eq. (B41) for a coherent state using the parameters in lines four [darker blue dashed line in (a) and (b)] and five [lighter blue dashed line in (a) and (b)] in Table 1. The red dots correspond to the temperature from each experiment. From lighter to darker shades of blue: $r=0.25$; $r=0.1$. (c) Experiment from line four in Table 1, with the temperature as in line five, i.e., $T_m=25$ mK. (d) Experiment from line five in Table 1. ${\kappa }_0$ corresponds to the original value from Table 1. Note that in (c) and (d) the maximum values for $\kappa /{\kappa }_0$ that are considered are limited by the resolved sideband regime where $\kappa <{\omega }_m$.

Figure 6

Teleportation fidelity from Eq. (B41) for a coherent state using the parameters in lines one to three [respectively, from darker to lighter shades of blue in (a) and (b)] in Table 1. The red dots correspond to the temperature from each experiment. Note that for the experiment from line three in Table 1, we used the same ratio ${\kappa }_L/\kappa =0.2$ as in line two. From lighter to darker shades of blue: $r=0.35, r=0.5, r=0.5$.

Figure 7

Direct state transfer efficiency $W_{\mathrm{TM}}^2$ defined in Eq. (C13), as a function of the pulses duration $\tau$, using lines one to five (respectively, from darker to lighter shades of green) in Table 1. From darker to lighter shades of green: ${\tau }_{\text{max}}/2\pi =0.417/\gamma , {\tau }_{\text{max}}/2\pi =0.467/\gamma , {\tau }_{\text{max}}/2\pi =0.0533/\gamma , {\tau }_{\text{max}}/2\pi =0.0857/\gamma$, and ${\tau }_{\text{max}}/2\pi =0.0295/\gamma$. The two darkest dashed lines using parameters from lines one and two in Table 1 are superimposed, due to the fact that for these references, the ratio $\gamma /\kappa$ in similar.