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) $\mathrm{\Delta }n$ 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 $\left|\mathrm{\Delta }n\right|=0$, 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 $2\pi$ 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.

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• Bose-Einstein condensates
• Exciton polariton
• Light-matter interaction
• Polaritons
• Semiconductor microcavities
• Synchronization
• Polariton condensate
• Sine-Gordon equation
• Perturbative methods
• Schroedinger equation