Quadratic optomechanical coupling in an active-passive-cavity system

We investigate an active-passive-cavity system where a membrane is quadratically coupled to the passive cavity. When the passive cavity is strong coherently driven to enhance the optomechanical coupling, the optical field interacts with the mechanical oscillator through the two-phonon process. By calculating the eigenfrequencies of system in Keldysh framework, we show the significant splitting of exceptional point in the presence of the optomechanical coupling. We also calculate the transmission rate to show how this system responds to an additional weak probe field. We find the gain effect will induce the two-phonon process amplification window. However, the gain effect becomes weak when the photon-tunneling strength increases. Then the transmission rate of the probe field changes from amplification to absorption. These results have potential applications in quantum metrology, quantum information processing, and quantum optical devices.

Figure 1

Schematic illustration of our optomechanical system consisting of a passive cavity (left) and an auxiliary active cavity (right). ${\widehat{a}}_1$ and ${\widehat{a}}_2$ denote the intracavity field, respectively, $J$ is the hopping rate. A membrane is located in the antinode (or node) of the passive cavity field. Thus ${\widehat{a}}_1$ is quadratically coupled to the displacement of the membrane. ${\kappa }_1({\kappa }_2^l$) denotes the loss rate of passive (active) cavity and $\gamma$ is the mechanical decay rate. The passive optical cavity is coherently driven by a strong control field with strength ${\varepsilon }_l$. Instead, we strongly pump the atoms present in the active cavity. Since the atoms will spontaneously decay to the ground state by emission photons. It provides an effective incoherent pumping of active cavity with the effective pump rate ${\kappa }_2^p$. A weak field with amplitude ${\varepsilon }_p$ is used to probe the passive cavity, and ${\varepsilon }_{\text{out}}$ is the amplitude of output field.

Figure 2

Nonequilibrium diagrammatic expansion of our model to second order in $g$. The full lines indicate classical fields and the dashed lines indicate quantum fields. (a) The (1,1) and (3,3) components of unperturbed Green's function. (b) Diagrammatic expansion of $S_{\text{I}}$. (c) Nonzero elements of the retard self-energy matrix.

Figure 3

The lines in (a) describe the real parts of the eigenfrequencies as a function of $J/{\kappa }_1$. The corresponding lines in (b) and (c) describe the imaginary parts. The insets in (b) and (c) zoom the details. In this simulation, the parameters are ${\kappa }_2=0.1{\kappa }_1$, ${\kappa }_2^l={\kappa }_1$, $\gamma =1.6\times {10}^{-4}{\kappa }_1$, $g=3.7\times {10}^{-6}{\kappa }_1$, ${\omega }_b=5.1{\kappa }_1$, ${\mathrm{\Delta }}_c=10{\kappa }_1$, and $n=1.8\times {10}^7$.

Figure 4

(a) Real and (b) imaginary parts of supermodes as a function of $J/{\kappa }_1$ in the presence and absence of optomechanical coupling. The effective optomechanical coupling strength is $g=0$ (cyan dashed lines) and $g=3.7\times {10}^{-6}{\kappa }_1$ (red solid lines). Other parameters are the same as in Fig. 3.

Figure 5

${\upsilon }_p$ and ${\tilde{\upsilon }}_p$ in the passive-passive case as a function of the normalized frequency $\delta /{\omega }_m$. In (a) and (b), the effective optomechanical coupling strength is $g=0$ (black dashed lines) and $g=7.1\times {10}^{-6}{\kappa }_1$ (red solid lines), the decay rate of passive cavity is ${\kappa }_1=4\pi \times {10}^4\mathrm{Hz}$. The effective pump rate of the auxiliary cavity is ${\kappa }_2=-{\kappa }_1$ and the photon-tunneling strength between two cavities is $J=0.4{\kappa }_1$. Other parameters are $g_0=2.16\times {10}^{-4}\mathrm{Hz}$, ${\varepsilon }_l=4.13\times {10}^{-4}{\kappa }_1$, ${\kappa }_1^e=0.8{\kappa }_1$, ${\kappa }_2^l={\kappa }_1$, ${\omega }_m=2\pi \times {10}^5\mathrm{Hz}$, $\gamma =40\mathrm{Hz}$, $T=90\mathrm{K}$, ${\mathrm{\Delta }}_c=2{\omega }_m$, and ${\omega }_b=5.06{\kappa }_1$. The insets in (a) and (b) zoom the EIT-like dip and the abnormal dispersion, respectively. Panels (c) and (d) illustrate the changes of ${\upsilon }_p$ and ${\tilde{\upsilon }}_p$ near $\delta /{\omega }_m=2$ under different photon-tunneling strength $J$. The effective optomechanical coupling strength is $g=7.1\times {10}^{-6}{\kappa }_1$. Other parameters are the same as in (a) and (b).

Figure 6

Transmission rate $\eta$ in the passive-passive case as a function of the normalized frequency $\delta /{\omega }_m$. The effective optomechanical coupling strength is $g=0$ (black dashed lines) and $g=7.1\times {10}^{-6}{\kappa }_1$ (red solid lines), the decay rate of the passive cavity is ${\kappa }_1=4\pi \times {10}^4\mathrm{Hz}$. The effective pump rate of the auxiliary cavity is ${\kappa }_2=-{\kappa }_1$. The photon-tunneling strength between two cavities is (a) $J=0.4{\kappa }_1$ and (b) $J=0.8{\kappa }_1$. The other parameters are the same as in Figs. 5 and 5. The insets zoom the two-phonon process OMIT.

Figure 7

${\upsilon }_p$ and ${\tilde{\upsilon }}_p$ in the active-passive case as a function of the normalized frequency $\delta /{\omega }_m$. In (a) and (b), the effective optomechanical coupling strength is $g=0$ (black dashed lines) and $g=7.1\times {10}^{-6}{\kappa }_1$ (red solid lines), the decay rate of passive cavity is ${\kappa }_1=4\pi \times {10}^4\mathrm{Hz}$. The effective pump rate of auxiliary cavity is ${\kappa }_2=0.05{\kappa }_1$ and the photon-tunneling strength between two cavities is $J=0.4{\kappa }_1$. The other parameters are the same as in Figs. 5 and 5. Panels (c) and (d) illustrate the changes of ${\upsilon }_p$ and ${\tilde{\upsilon }}_p$ near $\delta /{\omega }_m=2$ under different photon-tunneling strength $J$ . The effective optomechanical coupling strength is $g=7.1\times {10}^{-6}{\kappa }_1$. Other parameters are the same as in (a) and (b).

Figure 8

(a) Transmission rate $\eta$ in the active-passive case as a function of the normalized frequency $\delta /{\omega }_m$. The effective optomechanical coupling strength is $g=0$ (black dashed lines) and $g=7.1\times {10}^{-6}{\kappa }_1$ (red solid lines), the decay rate of passive cavity is ${\kappa }_1=4\pi \times {10}^4\mathrm{Hz}$. The photon-tunneling strength between two cavities is $J=0.4{\kappa }_1$ and the effective pump rate of auxiliary cavity is ${\kappa }_2=0.05{\kappa }_1$. Other parameters are the same as in Fig. 7. The inset zooms the two-phonon process induced amplification effect. (b) The changes of transmission rate $\eta$ under different strengths of ${\kappa }_2$. The effective optomechanical coupling strength is $g=7.1\times {10}^{-6}{\kappa }_1$. Other parameters are the same as in (a).

Figure 9

(a) Transmission rate $\eta$ in the active-passive case as a function of the normalized frequency $\delta /{\omega }_m$. The effective optomechanical coupling strength is $g=0$ (black dashed lines) and $g=7.1\times {10}^{-6}{\kappa }_1$ (red solid lines), the decay rate of passive cavity is ${\kappa }_1=4\pi \times {10}^4\mathrm{Hz}$. The photon-tunneling strength between two cavities is $J=0.8{\kappa }_1$, the effective pump rate of the auxiliary cavity is ${\kappa }_2=0.05{\kappa }_1$. Other parameters are the same as in Fig. 7. The inset in (a) zooms the two-phonon process induced absorption effect. Panel (b) illustrates the changes of $\eta$ near $\delta /{\omega }_m=2$ under different strengths of ${\kappa }_2$. The effective optomechanical coupling strength is $g=7.1\times {10}^{-6}{\kappa }_1$. Other parameters are the same as in (a).