Phase locking of ring-shaped exciton-polariton condensates to coherent optical drive

The effect of an additional quasiresonant drive on the dynamics of the ring-shaped incoherently pumped polariton condensates carrying angular momentum (vorticity) is studied theoretically. Numerical simulations of the 2D and 1D Gross-Pitaevskii equations show that the difference of the topological charges (vorticities) $\mathrm{\ensuremath{\Delta}}n$ of the condensate and the quasiresonant coherent drive plays a crucial role in the synchronization dynamics. It is shown that in an axially symmetric system, synchronization can only occur if $|\mathrm{\ensuremath{\Delta}}n|=0$, whereas in the other cases the phase of the condensate cannot be locked to the phase of the coherent drive. To explain this effect observed in the numerical simulations a perturbation theory is developed. The theory shows that the phase slip between the condensate and the coherent drive can be understood in terms of the motion of $2\ensuremath{\pi}$ kinks. It is shown that the breaking of the axial symmetry can stop the motion of the kinks, allowing the phase locking of the condensate to the coherent drive.