Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling

Optical nonlinearities at the single-photon level are explored in a quadratically coupled optomechanical system, where the cavity frequency is coupled to the square of the mechanical displacement. The effective nonlinear interaction between photons and phonons is enhanced by a strong driving field, which allows one to implement the single-photon nonlinearities even if the single-photon coupling strength $g_0$ is much lower than the cavity decay rate $\kappa$. The photon statistical properties are discussed by calculating the second-order correlation function both analytically and numerically. The results show that the single-photon nonlinearities are robust against mechanical thermal noise in the strong-coupling and sideband-resolved regime, and photon blockade and photon-induced tunneling can be realized with experimentally accessible parameters.

Figure 1

(a) Schematic of a strongly driven optomechanical system with quadratic coupling. (b) Anharmonic level diagram for the relevant zero-photon, one-photon, and two-photon states. States are labeled $|nm\rangle$, where $n$ $\left(m\right)$ denotes the photon (phonon) number. The effective coupling $g$ splits the degeneracy between state $|nm\rangle$ and state $|n-1,m+2\rangle$. Dotted green arrows describe the one-photon transition; solid red arrows, the two-photon transition.

Figure 2

Equal-time second-order correlation function $g^{\left(2\right)}\left(0\right)$ versus $\kappa t$ at detuning $\mathrm{\Delta }=\pm \sqrt{2}g$. The solid red curve is the numerical solution based on master Eq. (13) with $g/\kappa =1$; the dashed blue curve is the exact numerical result corresponding to the full Hamiltonian $H_1$ replacing the effective Hamiltonian $H_{\text{eff}}$ in Eq. (13). Parameters are chosen as $g_0/\kappa =0.01,{\omega }_m/\kappa =100$, and $\alpha =100$ (corresponding to the driving amplitude $\mathrm{\Omega }/\kappa \simeq 6.0\times {10}^4)$. Other parameters are ${\overline{n}}_{\mathrm{th}}=0,\varepsilon /\kappa =0.1$, and $\gamma /\kappa =0.01$.

Figure 3

(a) Normalized mean intracavity photon number $〈n〉/n_0$ with $n_0=4{\varepsilon }^2/{\kappa }^2$ and (b) equal-time second-order correlation function $g^{\left(2\right)}\left(0\right)$ as a function of $\mathrm{\Delta }/g$ at zero temperature. Solid red curves show approximate analytical results based on Eqs. (11) and (12); blue circles show numerical solutions of the master equation. Parameters are taken as $g/\kappa =4,\varepsilon /\kappa =0.1$, and $\gamma /\kappa =0.01$.

Figure 4

(a) At detuning $\mathrm{\Delta }=\pm \sqrt{2}g$, the second-order correlation function $g^{\left(2\right)}\left(0\right)$ as a function of $g/\kappa$ with $\varepsilon /\kappa =0.1$. Inset: $g^{\left(2\right)}\left(0\right)$ as a function of $\varepsilon /\kappa$. (b) Second-order differential correlation function $C^{\left(2\right)}\left(0\right)$ versus $\mathrm{\Delta }/g$ for different values of $\varepsilon /\kappa$. Other parameters are $\gamma /\kappa =0.01$ and $g/\kappa =4$.

Figure 5

(a) Second-order correlation function $g^{\left(2\right)}\left(0\right)$ versus $\mathrm{\Delta }/g$ for various thermal phonon occupancy ${\overline{n}}_{\mathrm{th}}$. Parameters are selected as $\varepsilon /\kappa =0.1,\gamma /\kappa =0.001$, and $g/\kappa =10$. (b) Second-order correlation function $g^{\left(2\right)}\left(0\right)$ versus ${\overline{n}}_{\mathrm{th}}$ at $\mathrm{\Delta }=\pm \sqrt{2}g$ for different coupling strengths $g/\kappa$. Parameters are chosen as $\varepsilon /\kappa =0.1$ and $\gamma /\kappa =0.01$.