Cross-correlation between photons and phonons in quadratically coupled optomechanical systems

We study photon, phonon statistics, and the cross-correlation between photons and phonons in a quadratically coupled optomechanical system. Photon blockade, phonon blockade, and strong anticorrelation between photons and phonons can be observed in the same parameter regime with the effective nonlinear coupling between the optical and the mechanical modes, enhanced by a strong optical driving field. Interestingly, an optimal value of the effective nonlinear coupling strength for the photon blockade is not within the strong nonlinear coupling regime. This abnormal phenomenon results from the destructive interference between different paths for two-photon excitation in the optical mode with a moderate effective nonlinear coupling strength. Furthermore, we show that phonon (photon) pairs or correlated photons and phonons can be generated in the strong nonlinear coupling regime with a proper detuning between the weak mechanical driving field and mechanical mode. Our results open up a way to generate anticorrelated and correlated photons and phonons, which may have important applications in quantum information processing.

Figure 1

The schematic energy spectrum of the linearized quadratically coupled optomechanical system [see the Hamiltonian in Eq. (8)] given in the noncoupling basis (left) and in the diagonal basis (right).

Figure 2

${log}_{10}{g}_{ij}^{\left(2\right)}\left(0\right)(ij=bb,aa,ab)$ are plotted (a) as functions of the effective coupling strength $J/{\gamma }_c$; (b) as functions of the detuning $\delta /{\gamma }_c\equiv ({\omega }_c-{\omega }_L-2{\omega }_d)/{\gamma }_c$; (c) as functions of the detuning ${\mathrm{\Delta }}_m/{\gamma }_c$. (d) Mean phonon number $n_b$ and photon number $100\times n_a$ are plotted as functions of the detuning ${\mathrm{\Delta }}_m/{\gamma }_c.{log}_{10}{g}_{ij}^{\left(2\right)}\left(\tau \right)$ is plotted as a function of the normalized time delay $\tau /(2\pi /{\gamma }_c)$ in (e) and (f). $\mathrm{\Delta }={\mathrm{\Delta }}_m=0$ in (a); ${\omega }_0={\omega }_d$ and $J=0.406{\gamma }_c$ in (b); $J=0.406{\gamma }_c$ and $\mathrm{\Delta }=2{\mathrm{\Delta }}_m$ in (c) and (d); $\mathrm{\Delta }={\mathrm{\Delta }}_m=0$ and $J=0.406{\gamma }_c$ in (e) and (f). The other parameters are $\varepsilon =0.05{\gamma }_c,{\gamma }_m={\gamma }_c/10$, and $n_{\mathrm{th}}={10}^{-4}$.

Figure 3

${log}_{10}{g}_{ij}^{\left(2\right)}\left(0\right)$ [(a) $ij=bb$, (b) $ij=aa$, and (c) $ij=ab]$ is plotted as a function of the driving strength $\varepsilon /{\gamma }_c$ for different mean thermal phonon number $n_{\mathrm{th}}$'s [solid curve for $n_{\mathrm{th}}={10}^{-3}$; dashed curve for $n_{\mathrm{th}}={10}^{-2}$; dotted curve for $n_{\mathrm{th}}={10}^{-1}]$. (d) Mean phonon number $n_b$ and photon number $n_a$ are plotted as functions of the driving strength $\varepsilon /{\gamma }_c$ for mean thermal phonon number $n_{\mathrm{th}}={10}^{-2}$. The other parameters are $\mathrm{\Delta }={\mathrm{\Delta }}_m=0,J=0.406{\gamma }_c$, and ${\gamma }_m={\gamma }_c/10$.

Figure 4

(a) Contour plot of ${log}_{10}{g}_{aa}^{\left(2\right)}\left(0\right)$ as a function of the damping rate ${\gamma }_m/{\gamma }_c$ and the effective coupling strength $J/{\gamma }_c$ for $\varepsilon =0.05{\gamma }_c$; (a) contour plot of ${log}_{10}{g}_{aa}^{\left(2\right)}\left(0\right)$ vs the damping rate ${\gamma }_c/\left(10{\gamma }_m\right)$ and the effective coupling strength $J/\left(10{\gamma }_m\right)$ for $\varepsilon =0.5{\gamma }_m$. The white dashed line refers to Eq. (22). The other parameters are $\mathrm{\Delta }={\mathrm{\Delta }}_m=0$ and $n_{\mathrm{th}}={10}^{-4}$.

Figure 5

(a) ${log}_{10}{g}_{ij}^{\left(2\right)}\left(0\right)(ij=bb,aa,ab)$ is plotted as a function of the detuning ${\mathrm{\Delta }}_m/{\gamma }_c$; (b) and (c) are the local enlarged drawings of (a). The blue dashed-dotted lines in (b) refer to ${\mathrm{\Delta }}_m/{\gamma }_c=5/\sqrt{2},5\sqrt{6}/3$, and 5. ${log}_{10}{g}_{ij}^{\left(2\right)}\left(\tau \right)$ [(d) $ij=bb$, (e) $ij=aa$, and (f) $ij=ab]$ is plotted as a function of the normalized time delay $\tau /(2\pi /{\gamma }_c)$ in (d)–(f) for ${\mathrm{\Delta }}_m=3.9{\gamma }_c$. The other parameters are $\varepsilon =0.05{\gamma }_c,J=5{\gamma }_c,{\omega }_L={\omega }_c-2{\omega }_0,\mathrm{\Delta }=2{\mathrm{\Delta }}_m,{\gamma }_m={\gamma }_c/10$, and $n_{\mathrm{th}}={10}^{-4}$.

Figure 6

${log}_{10}{g}_{ij}^{\left(2\right)}\left(0\right)$ [(a) $ij=bb$, (b) $ij=aa$, and (c) $ij=ab]$ is plotted as a function of the driving strength $\varepsilon /{\gamma }_c$ for different mean thermal phonon numbers $n_{\mathrm{th}}$ [solid curve for $n_{\mathrm{th}}={10}^{-4}$; dashed curve for $n_{\mathrm{th}}={10}^{-3}$; dotted curve for $n_{\mathrm{th}}={10}^{-2}]$. (d) Mean phonon number $n_b$ (solid curve) and photon number $n_a$ (dashed curve) are plotted as functions of the driving strength $\varepsilon /{\gamma }_c$ for mean thermal phonon number $n_{\mathrm{th}}={10}^{-4}$. The other parameters are ${\mathrm{\Delta }}_m=3.9{\gamma }_c,\mathrm{\Delta }=2{\mathrm{\Delta }}_m,J=5{\gamma }_c$, and ${\gamma }_m={\gamma }_c/10$.