Enhancing quadratic optomechanical coupling via a nonlinear medium and lasers

We propose a scheme to significantly increase quadratic optomechanical couplings of optomechanical systems with the help of a nonlinear medium and two driving lasers. The nonlinear medium is driven by one laser, and the optical cavity mode is driven by a strong laser. We derive an effective Hamiltonian using squeezing transformation and rotating wave approximation. The effective quadratic optomechanical coupling strength can be larger than the decay rate of the cavity mode by adjusting the two optical driving fields. The thermal noise of squeezed cavity mode can be suppressed totally with the help of a squeezed vacuum field. Then a driving field is applied to the mechanical mode. We investigate the equal-time second-order correlations and find there are photon, phonon, and photon-phonon blockades even when the original single-photon quadratic coupling is much smaller than the decay rate of the optical mode. In addition, the sub-Poissonian window of the two-time second-order correlations can be controlled by the mechanical driving field. Finally, we show the squeezing and entanglement of the model could be tuned by the driving fields of the nonlinear medium and mechanical mode.

Figure 1

The quantities $g_{\mathrm{eff}}/\kappa$, ${\omega }_{m,\mathrm{eff}}/\kappa$, ${\tilde{\mathrm{\Delta }}}_c/{\tilde{\omega }}_m$, and ${\tilde{\mathrm{\Delta }}}_c/g_p$ are plotted as functions of $g_0/\kappa$ for $\eta =0.9$ (red solid lines), $\eta =0.99$ (green dashed lines), and $\eta =0.999$ (blue dash dot lines). The parameters are ${\omega }_m=100\kappa$ and ${\tilde{A}}_s=1000$. Throughout this work, we consider the resonant case with ${\mathrm{\Delta }}_a=2{\mathrm{\Delta }}_b$. If ${\omega }_m$, $g_0$, $\eta$, and ${\tilde{A}}_s$ are given, then ${\mathrm{\Delta }}_c$ is determined by Eq. (20). Using the relation ${\tilde{\mathrm{\Delta }}}_c={\mathrm{\Delta }}_c\sqrt{1-{\eta }^2}$, we can calculate ${\tilde{\mathrm{\Delta }}}_c$. Note that we assume $\eta <1$.

Figure 2

Equal-time second-order correlations as functions of $g_0/\kappa$ are plotted for analytical results calculated using Eqs.  (26, 27, 28) (dashed lines) and numerical results by solving Eq. (30) (solid lines). The parameters are ${\omega }_m=100\kappa ,\gamma =0.1\kappa ,{\mathrm{\Delta }}_a={\mathrm{\Delta }}_b=0,E=0.05\kappa ,\eta =0,{\tilde{A}}_s=500\text{,}\text{and}{N}_{th}={10}^{-4}$.

Figure 3

Equal-time second-order correlations $g_{ij}^{\left(2\right)}\left(0\right)$ versus $g_0/\kappa$ for $\eta =0$ (upper panel), $\eta =0.9$ (middle panel), and $\eta =0.99$ (lower panel). $g_{aa}^{\left(2\right)}\left(0\right)$, $g_{bb}^{\left(2\right)}\left(0\right)$, and $g_{ab}^{\left(2\right)}\left(0\right)$ are represented by red solid, green dashed, and blue dash-dot lines, respectively. The parameters are ${\omega }_m=100\kappa ,\gamma =0.1\kappa ,{\mathrm{\Delta }}_a={\mathrm{\Delta }}_b=0,E=0.05\kappa ,{\tilde{A}}_s=1000\text{,}\text{and}{N}_{th}={10}^{-4}$.

Figure 4

Equal-time second-order correlations $g_{ij}^{\left(2\right)}\left(0\right)$ versus ${\mathrm{\Delta }}_b/\kappa$ for $\eta =0$ (upper panel), $\eta =0.9$ (middle panel), and $\eta =0.99$ (lower panel). $g_{aa}^{\left(2\right)}\left(0\right)$, $g_{bb}^{\left(2\right)}\left(0\right)$, and $g_{ab}^{\left(2\right)}\left(0\right)$ are represented by red solid, green dashed, and blue dash-dot lines, respectively. The parameters are ${\omega }_m=100\kappa ,g_0={10}^{-4}\kappa ,\gamma =0.1\kappa ,{\mathrm{\Delta }}_a=2{\mathrm{\Delta }}_b,E=0.05\kappa ,{\tilde{A}}_s=1000\text{,}\text{and}{N}_{th}={10}^{-4}$.

Figure 5

Equal-time second-order correlations $g_{ij}^{\left(2\right)}\left(0\right)$ versus $N_{th}$ for $\eta =0$ (upper panel) and $\eta =0.9$ (lower panel). $g_{aa}^{\left(2\right)}\left(0\right)$, $g_{bb}^{\left(2\right)}\left(0\right)$, and $g_{ab}^{\left(2\right)}\left(0\right)$ are represented by red solid, green dashed, and blue dash-dot lines, respectively. The parameters are ${\omega }_m=100\kappa ,g_0={10}^{-4}\kappa ,\gamma =0.1\kappa ,{\mathrm{\Delta }}_a={\mathrm{\Delta }}_b=0,E=0.05\kappa \text{,}\text{and}{\tilde{A}}_s=1000$.

Figure 6

Mean phonon number $n_{b,ss}$ versus ${\mathrm{\Delta }}_b/\kappa$ for $\eta =0$ (red solid line), $\eta =0.9$ (green dashed line), and $\eta =0.99$ (blued dash-dot line). The parameters are ${\omega }_m=100\kappa ,g_0={10}^{-4}\kappa ,\gamma =0.1\kappa ,{\mathrm{\Delta }}_a=2{\mathrm{\Delta }}_b,E=0.01\kappa ,{\tilde{A}}_s=1000\text{,}\text{and}{N}_{th}=0.1$.

Figure 7

Time-delayed second-order correlations $g_{ij}^{\left(2\right)}\left(\tau \right)$ versus $\kappa \tau$ for $E=0.03\kappa$ (red solid lines), $E=0.06\kappa$ (green dashed lines), and $E=0.1\kappa$ (blue dash-dot lines). The parameters are ${\omega }_m=100\kappa ,g_0={10}^{-4}\kappa ,\gamma =0.1\kappa ,{\mathrm{\Delta }}_a=2{\mathrm{\Delta }}_b,\eta =0.9,{\tilde{A}}_s=1000\text{,}\text{and}{N}_{th}={10}^{-4}$.

Figure 8

${\left(\mathrm{\Delta }{X}_a\right)}^2\approx {\left(\mathrm{\Delta }{Y}_a\right)}^2\approx 0.25$ (red solid lines), ${\left(\mathrm{\Delta }{X}_b\right)}^2$ (green solid lines), ${\left(\mathrm{\Delta }{Y}_b\right)}^2$ (green dashed lines), ${\left(\mathrm{\Delta }{X}_{ab}\right)}^2$ (blue lines), and ${\left(\mathrm{\Delta }{Y}_{ab}\right)}^2$ (blue dashed lines) versus detuning ${\mathrm{\Delta }}_b$ for $\eta =0$ (upper panel), $\eta =0.99$ (middle panel), and $\eta =0.999$ (lower panel). The parameters are ${\omega }_m=100\kappa ,g_0={10}^{-4}\kappa ,\gamma =0.1\kappa ,{\mathrm{\Delta }}_a=2{\mathrm{\Delta }}_b,E=0.3\kappa ,{\tilde{A}}_s=1000\text{,}\text{and}{N}_{th}={10}^{-4}$.

Figure 9

Entanglement criteria ${\mathcal{D}}_{ab}$ (red solid lines), ${\mathcal{E}}_{1,ab}$ (green dashed lines), and ${\mathcal{E}}_{2,ab}$ (blue dash-dot lines) versus detuning ${\mathrm{\Delta }}_b$ for $\eta =0$ (upper panel), $\eta =0.997$ (middle panel), and $\eta =0.999$ (lower panel). The parameters are ${\omega }_m=100\kappa ,g_0=3\times {10}^{-4}\kappa ,\gamma =0.1\kappa ,{\mathrm{\Delta }}_a=2{\mathrm{\Delta }}_b,E=2\kappa ,{\tilde{A}}_s=1000\text{,}\text{and}{N}_{th}={10}^{-4}$.