Cross-Kerr nonlinearity in optomechanical systems

We consider the response of a nanomechanical resonator interacting with an electromagnetic cavity via a radiation-pressure coupling and a cross-Kerr coupling. Using a mean-field approach we solve the dynamics of the system and show the different corrections coming from the radiation pressure and the cross-Kerr effect on the usually considered linearized dynamics.

Figure 1

Schematic picture of the system. A cavity and a mechanical resonator coupled via a radiation-type coupling $g$ and a cross-Kerr coupling $g_{ck}$. The number of photons in the cavity $n_c$ is coupled to the oscillations of the mechanical resonator $\widehat{x}$ and the number of phonons in the mechanical resonator $n_m$.

Figure 2

Cooling (heating) process. The cavity is driven with a frequency ${\omega }_d={\omega }_c-{\omega }_m$ $\left({\omega }_d={\omega }_c+{\omega }_m\right)$. The drive does not allow a transition from $|n_m,n_c\rangle$ to $|n_m,n_c+1\rangle$ but allows the transition from $|n_m,n_c\rangle$ to $|n_m-1,n_c+1\rangle$ $(|n_m,n_c\rangle$ to $|n_m+1,n_c+1\rangle )$. The cavity relaxes then to the state $|n_m-1,n_c\rangle$ $\left(|n_m+1,n_c\rangle \right)$, resulting in cooling (heating) of the mechanical resonator.

Figure 3

Schematic picture of the red sideband with and without cross-Kerr coupling for $\Delta <0$. For $g_{ck}>0$ the sideband peak is shifted to lower values, while for $g_{ck}<0$ the sideband peak is shifted to higher values.

Figure 4

Steady-state phonon number in the resonator and ${\Gamma }_{\mathrm{opt}}$ (inset) as a function of the ratio ${\omega }_m/\kappa$ at the red sideband for the optimal case $\tilde{\Delta }=-{\tilde{\omega }}_m$ with $\gamma ={10}^{-3}\kappa ,g={10}^{-2}\kappa$. The number of photons pumped into the cavity is fixed to ${\left|\alpha \right|}^2=100$, and the bath temperature corresponds to $n^{th}=100$.

Figure 5

Steady-state phonon number in the resonator and ${\Gamma }_{\mathrm{opt}}$ (inset) as a function of the ratio ${\omega }_m/\kappa$ at the blue sideband for the optimal case $\tilde{\Delta }={\tilde{\omega }}_m$ with $\gamma ={10}^{-2}\kappa ,g={10}^{-2}\kappa$. The number of photons pumped into the cavity is fixed to ${\left|\alpha \right|}^2=100$, and the bath temperature corresponds to $n^{th}=10$. The dashed lines indicate the onset of the parametric instability for ${\Gamma }_{\mathrm{opt}}=-\gamma$.

Figure 6

(a) Steady-state phonon number in the resonator and (b) the optical damping ${\Gamma }_{\mathrm{opt}}$ as a function of the number of photons pumped into the cavity in the case where $\Delta =-{\omega }_m$. The values for the parameters are $\gamma ={10}^{-4}{\omega }_m,\kappa ={10}^{-1}{\omega }_m,g={10}^{-3}{\omega }_m$, and the bath temperature corresponds to $n^{th}=10$.

Figure 7

(a) Steady-state phonon number in the resonator and (b) the optical damping ${\Gamma }_{\mathrm{opt}}$ as a function of the number of photons pumped into the cavity in the case where $\Delta ={\omega }_m$. The values for the parameters are $\gamma ={10}^{-3}{\omega }_m,\kappa ={10}^{-1}{\omega }_m,g={10}^{-3}{\omega }_m$, and the bath temperature corresponds to $n^{th}=10$.

Figure 8

Frequency shift as a function of the number of photons pumped into the cavity when $\Delta =-{\omega }_m$ with $\gamma ={10}^{-3}{\omega }_m,\kappa ={10}^{-1}{\omega }_m,g={10}^{-2}{\omega }_m$ and the bath temperature corresponds to $n^{th}=100$.

Figure 9

Frequency shift as a function of the number of photons pumped into the cavity when $\Delta ={\omega }_m$ with $\gamma ={10}^{-3}{\omega }_m,\kappa ={10}^{-1}{\omega }_m,g={10}^{-2}{\omega }_m$ and the bath temperature corresponds to $n^{th}=100$.