the axial dynamics: for the cavity system, we will have a much smaller tranverse frequency relative to the axial frequency, i.e. $\omega_t \ll \omega_a$ and there is little mixing between these transverse and axial degrees of freedom.

We consider the dynamics of the following Hamiltonian: $$\begin{aligned}
\frac{\hat H}{\hbar}&=& -\Delta_1{\hat a}_1^\dagger {\hat a}_1  -\Delta_2 {\hat
a_2}^\dagger {\hat a}_2
      +  \frac{{\hat P}^2}{2m\hbar}   -A  {\hat a}_1^\dagger  {\hat a}_1 \cos^2
(k_1 {\hat x}-\phi_1)
         \nonumber \\
 &- & A{\hat a}_2^\dagger {\hat a}_2\cos^2 (k_2{\hat x}-\phi_2)
  + E_1({\hat a_1}^\dagger+ {\hat a}_1) + E_2({\hat a}_2^\dagger + {\hat a}_2).
\label{Hamiltonian}\end{aligned}$$

Two optical field modes of a high finesse cavity ${\hat a}_{1,2}$ are coupled to a nanosphere with centre-of-mass position $x$. The parameter $A$ (dependent on the nanosphere polarizability), determines the depth of the optical standing-wave potentials. We investigate the case where both modes competitively cool and trap the nanosphere, in contrast to previous schemes [@Isart; @Zoller] where one optical field is exclusively responsible for trapping, while the other is exclusively responsible for cooling. $\hat H$ is given in the rotating frame of the laser which drives the modes with amplitudes $E_1$ and $E_2=R E_1$ respectively, where $R$ represents the ratio of driving amplitudes for the two modes. We restrict 