Capacitive coupling of two transmission line resonators mediated by the phonon number of a nanoelectromechanical oscillator

Detection of quantum features in mechanical systems at the nanoscale constitutes a challenging task, given the weak interaction with other elements and the available technology. Here we describe the interaction between two monomodal transmission-line resonators (TLRs) mediated by vibrations of a nanoelectromechanical oscillator. This scheme is then employed for quantum nondemolition detection of the number of phonons in the nanoelectromechanical oscillator through a direct current measurement in the output of one of the TLRs. For that to be possible an undepleted field inside one of the TLRs works as an amplifier for the interaction between the mechanical resonator and the remaining TLR. We also show how the nonclassical nature of this system can be used for generation of tripartite entanglement and conditioned mechanical coherent superposition states, which may be further explored for detection processes.

Figure 1

Capacitive coupling of two TLRs mediated by an oscillatory NEMS. A double clamped mechanical resonator (green) is electrically coupled to two transmission line resonators (grey).

Figure 2

Schematic circuit corresponding to Hamiltonian (1); a single-mode approximation was considered for both the TLRs and the NEMS.

Figure 3

The photonic current $\langle I_1\rangle \left(t\right)$ in TLR 1, as given by Eq. (15) for three values of the NEMS Fock states with average phonon number $n_b=1,2,3$. For $t\gg 2/(k_1+\Gamma )$, all three curves reach a stationary threshold given by the phonon number $n_b$ in TLR 1 (remembering that $\theta <0$).

Figure 4

Linear entropy of the partition $N|12$ quantifying the entanglement between the NEMS and the two TLRs as a function of time $t$ (here normalized with $\Gamma$) and $\left|\alpha \right|$. We choose $\alpha =\beta =\gamma \in \mathbb{R}$. Note the entanglement's recurrence due to the $2\pi$-periodic behavior of the functions in Eqs. (19).

Figure 5

Linear entropy of the partitions $N|12$ (solid) and $1|N2$ (dashed) for $\alpha =\beta =\gamma =2$ as a function of time $t$ (here normalized with $\Gamma$). Since $\beta =\gamma$, $E_{1|N2}=E_{2|N1}$. Inset: The same linear entropies for distinct values of the initial state for $\alpha =2$ and $\beta =\gamma =2$ (blue); $\beta =3,\gamma =4$ (green); $\beta =\gamma =1+2i$ (yellow); and $\beta =3+4i,\gamma =1+2i$ (red).