Optomechanically induced transparency associated with steady-state entanglement

We theoretically investigate a two-cavity optomechanical system in which a cavity (cavity $a$) couples to a mechanical resonator via radiation pressure and to another cavity (cavity $c$) via a common waveguide. In the excitation of a strong pump filed to cavity $a$, the steady-state entanglement between cavity $a$ and $c$, as a quantum channel, can be generated, which provides an indirect optical pathway to excite cavity $c$ by means of the pump filed. Quantum interference between the direct and indirect optical pathways gives rise to an optomechanically induced transparency appearing in the probe transmission of cavity $c$. Unlike in a typical optomechanically induced transparency effect, the electromagnetical control of the transmission is implemented by resorting to the quantum channel. Furthermore, the coupling strength of the two cavities is an important factor of the quantum channel, which can influence the width of the transparency window and the bistable behavior of the mean photon number in cavity $a$. We also illustrate that the electromagnetical control via quantum channel can be exploited to implement the optical switch and the slow light.

Figure 1

(a) Setup of two coupled cavities via a common waveguide in the presence of a strong pump field and a weak probe field. (b) Level diagram of the two-cavity optomechanical system. $|n_{\text{tot}}\rangle$ is an entangled state satisfying the total photon number of the two cavities $n_{\text{tot}}=n_a+n_c$. (c) The probe transmission spectrum showing optomechanically induced transparency.

Figure 2

(a) The probe transmission spectrum with the pump strengths ${\Omega }_{\text{pu}}=0,50,80$ ${\omega }_m$. (b) The photon number of cavity $a$ $n_a$ as a function of the pump strength. Left inset: Quantum interference between two routs from $|n_{\text{tot}},n_m\rangle$ to $|(n_{\text{tot}}+1),n_m\rangle$ . Right inset: The photon number of cavity $c$ $n_c$ as a function of the pump strength. Other parameters are $J=2$ GHz, ${\omega }_{\text{pu}}={\omega }_a$.

Figure 3

The probe transmission spectrum showing the widening transparency window with the increasing coupling strength $J$ ($J$ = 2, 2.5, 3 GHz). Other parameters are ${\Omega }_{\text{pu}}$ = 80 ${\omega }_m,{\omega }_{\text{pu}}={\omega }_a$.

Figure 4

The photon number $n_c$ as a function of the coupling strength $J$. Other parameters are ${\Omega }_{\text{pu}}$ = 80 ${\omega }_m,{\omega }_{\text{pu}}={\omega }_a$.

Figure 5

Optical bistability of the photon number $n_a$ with the pump strength ${\Omega }_{\text{pu}}$ = 80 ${\omega }_m$. Inset: The photon number $n_a$ as a function of the coupling strength $J$ at resonance ${\omega }_{\text{pu}}={\omega }_a$.

Figure 6

The probe transmission spectrum for ${\Omega }_{\text{pu}}$ = 0 (pump off) and ${\Omega }_{\text{pu}}$ = 80 ${\omega }_m$ (pump on). Other parameters are the coupling strength $J$ = 2 GHz, $\varepsilon \approx 6\times {10}^{-3}$ ${\omega }_m,{\omega }_{\text{pu}}={\omega }_a$.

Figure 7

The transmission group delay ${\tau }_g$ as a function of the pump strength ${\Omega }_{\text{pu}}$ for different coupling strengths $J=$ 2, 2.5, 3 GHz. Other parameters are the probe frequency ${\omega }_{\text{pr}}={\omega }_c+\varepsilon ,\varepsilon \approx 6\times {10}^{-3}$ ${\omega }_m,{\omega }_{\text{pu}}={\omega }_a$.