ourselves to the regime $0 \leq R \leq 1$, since we consider the most general case where both optical modes contribute to the trapping as well as the cooling. Thus we can define mode 1 simply as the mode which is more strongly driven and mode 2 as the mode which is (except where $R=1$) more weakly driven. The detunings $\Delta_{j}=\omega^j_L-\omega^{j}_c$ for $j=1,2$ are between the input lasers and the corresponding cavity mode of interest, and $\phi_{1,2}$ represents the phases of the optical potentials.

The two fields could represent two modes generated by the same laser field, or they could be generated by two independent lasers. Nonetheless, since the particle motion is confined to within one wavelegth, one can make the approximation $k_1\approx k_2 \equiv k$. Previous studies generally consider $\phi_1=0 , \phi_2=\pi/4$ to be convenient, since then the anti-node of one field coincides with a purely linear potential of the other optical field, but we may also consider general values of $\phi_1-\phi_2$. One can write corresponding equations of motion:

$$\begin{aligned}
{\ddot {\hat x}} & = & -\frac{\hbar k A}{m}\sum_j  {\hat a}_j^\dagger {\hat a}_j
\sin 2(k {\hat x}-\phi_j)
 -\Gamma_M  {\dot  {\hat x}} \nonumber\\
{\dot {\hat a}_j}& = & i \Delta_j 