Atom-mirror entanglement via cavity dissipation

We present a cavity dissipation scheme to prepare a hybrid system of an atomic ensemble and a moving mirror in squeezed and entangled states. This scheme is based on four-wave mixing in the atoms and radiation pressure on the mirror, both of which combine to lead to Bogoliubov interactions of the atoms and the mirror with the cavity fields. Under adiabatic conditions, the cavity fields cause the hybrid Bogoliubov modes to evolve into the vacuum states, which correspond to the two-mode squeezed and entangled states. The dependence of the hybrid entanglement on the system parameters is also presented in and beyond the Bogoliubov interactions.

Figure 1

Setup for our scheme. The atoms $a$ are placed at the cross site of two cavities and the vibrating mirror $b$ is used as the respective reflecting ends of them. The two cavities $c_{1,2}$ are pumped by the external fields ${\varepsilon }_{1,2}$, respectively. The insets on the right and left top represent the quantum transitions for the cavity-mirror interactions of the down-conversion form $b{c}_1+b^{†}{c}_1^{†}$ and of the beam-splitter-like form $b{c}_2^{†}+b^{†}{c}_2$, respectively, and the inset at the bottom shows the four-wave-mixing interactions of the two types $a{c}_1^{†}+a^{†}{c}_1$ and $a{c}_2+a^{†}{c}_2^{†}$ (see Sec. 2).

Figure 2

Diagrammatic sketch for the Bogoliubov interactions of the atoms and the mirror with the two cavity fields. The solid lines represent the down-conversion processes for the squeezing and the dashed lines denote the beam-splitter-like interactions for the state transfer. The linked processes in a quadrilateral contour determine the Bogoliubov collective interactions of the atoms and the mirror with the cavity fields when $r_1=r_2$ and $\mathrm{\Phi }=0$. Generally, squeezing and entanglement are obtained for $r_1\ne r_2$ and/or $\mathrm{\Phi }=0$. Even in the absence of either down-conversion process $(r_1=0$ or $r_2=0)$, the squeezing and entanglement are still possible. The simultaneous existence of the two down-conversion processes is not necessarily required for the entanglement, but leads to a remarkable enhancement of entanglement.

Figure 3

Two-mode variance $\langle {\left(\delta X_{-}\right)}^2\rangle$ versus $r_2$ for $q=(0,0.2,0.5,1,3,5)$ and (a) $\mathrm{\Phi }=0$, (b) $\mathrm{\Phi }=\pm \frac{\pi }{12}$, (c) $\mathrm{\Phi }=\pm \frac{\pi }{8}$, and (d) $\mathrm{\Phi }=\pm \frac{\pi }{4}$. We have used the logarithmic axis for $r_2$ in order to show clearly the dependence on $r_2$ when $r_2$ is very small.

Figure 4

Diagrammatic sketch for the interaction of three-level atoms with two driving fields ${\mathrm{\Omega }}_{1,2}$ and two cavity fields $c_{1,2}$ on Raman two-photon resonances but far off one-photon resonances.

Figure 5

Two-mode variance $\langle {\left(\delta X_{-}\right)}^2\rangle$ when the Raman interactions are involved. The parameters are the same as in Fig. 3 except for $C_{1,2}/{\gamma }_a=5$ and the ground-state spin decoherence rate ${\gamma }_s/{\gamma }_a={10}^{-4}$.