Remote macroscopic entanglement on a photonic crystal architecture

The outstanding progress in nanostructure fabrication and cooling technologies allows what was unthinkable a few decades ago: bringing single-mode mechanical vibrations to the quantum regime. The coupling between photon and phonon excitations is a natural source of nonclassical states of light and mechanical vibrations, and its study within the field of cavity optomechanics is developing lightning fast. Photonic crystal cavities are highly integrable architectures that have demonstrated the strongest optomechanical coupling to date and should therefore play a central role for such hybrid quantum-state engineering. In this context, we propose a realistic heralding protocol for the on-chip preparation of remotely entangled mechanical states, relying on the state-of-the-art optomechanical parameters of a silicon-based nanobeam structure. Pulsed sideband excitation of a Stokes process, combined with single-photon detection, allows the writing of a delocalized mechanical Bell state in the system, signatures of which can then be read out in the optical field. A measure of entanglement in this protocol is provided by the visibility of a characteristic quantum interference pattern in the emitted light.

Figure 1

Three-dimensional model of the system under study, showing the nanobeam photonic crystal (upper) and the clogged waveguide (lower), together with a sketch of the excitation and detection conditions. Cavity 1 is driven under pulsed excitation and its output is collected by a single-photon detector. The yellow arrow highlights the waveguide-induced optical coupling of strength $J_c$ between the two cavities. The deformation of the cavities illustrate the slight transverse vibrations induced by the optical field that are computed in Figs. 2 and 2.

Figure 2

(a) Antisymmetric and (b) symmetric normal modes of the electromagnetic field in the coupled cavity system. The color map expresses the $\mathrm{Re}\left(E_y\right)$ component of the electric field. (c) Simulated optical spectrum after individual or simultaneous excitation of the optical modes. The inset shows the time dependence of the electric field, as recorded at the center of one cavity when both modes are excited. (d) Antisymmetric and (e) symmetric confined mechanical modes. The color map expresses the amplitude of the displacement field.

Figure 3

(a) Schematic representation of the model system. (b) Write and readout pulses sequences for the interference reconstruction. (c) The computed power spectrum in log scale of the symmetric and antisymmetric cavity modes, revealing the Stokes and anti-Stokes sidebands.

Figure 4

Write procedure: Log scale (a) classical and (b) fluctuation occupations versus time. Blue and dashed-red lines represent cavity modes 1 and 2, yellow and dashed purple lines represent mechanical modes 1 and 2. The vertical black line marks the time $t_q$, starting from which quantum fluctuations become dominant. (c) Parametric plot showing the computed heralding rate ${\nu }_{\mathcal{H}}\left(t\right)$ versus the concurrence $\mathcal{C}\left(t\right)$. (d) Absolute value of the heralded mechanical density matrix ${\widehat{\rho }}_{\mathcal{H}}^{\left(1\right)}$ at the ${\mathcal{C}}_{\max }$ time [vertical red dashed line in panels (a) and (b)] conditioned on the detection of a single photon out of cavity 1.

Figure 5

Readout procedure: Logarithmic scale (a) classical and (b) fluctuation contributions to the state occupations versus time, as computed for a given realization of the readout procedure. The inset of panel (b) shows the mechanical occupation in linear scale. (c) The field intensity in cavity 1, averaged over several runs, as a function of the readout $t_R$ and shifted evolution time $t-\mathrm{\Delta }{t}_R$ where $\mathrm{\Delta }{t}_R=t_R^{\left(j\right)}-t_R^{\left(1\right)}$ for run number $j$. The inset shows a normalized cut along the vertical red dashed line, from which the visibility $\mathcal{V}$ can be inferred. The horizontal white dashed line corresponds to the evolutions in panels (a) and (b) where the intensity is at a maximum. (d) Simulated temperature dependence of the negativity and the fringe visibility.

Figure 6

(a) Negativity $\mathcal{N}$ versus ${\overline{n}}_{\mathrm{th}}$ for several values of $N$. The black dashed line shows the concurrence $\mathcal{C}$ in the case $N=2$. (b) Corresponding visibility computed from Eq. (B3). The arrows point the direction of increasing values of $N$.