Motivated by experimental studies [1], we propose a theoretical study of the synchronization in coupled Bose-Einstein condensed polaritons considering a dynamic interaction of two polaritonic modes with a dynamic phonon mode of 20 GHz. The considered hopping between the polaritons involves a linear or quadratic coupling to the phonon mode.  Our model shows that nonlinear polariton-polariton and polariton-reservoir interactions are important to extend the synchronization regions characterized by integer and fractional locking to the phonon frequency taking the form of a ladder known in dynamical systems as the “Devil’s Ladder” [2]. Going beyond the situation in which the phonon population is assumed, we look for cases in which the phonon can oscillate due to the interplay with the polaritons. This involves self consistently solving the dynamics of the coupled polaritons and the phonon. Also an energy balance is introduced to identify the zones of self sustained phonon population. Stochastic calculations also demonstrate that phonon cooling effects can also be realized.

[1] D. L. Chafatinos, A. S. Kuznetsov, A. A. Reynoso, G. Usaj, P. Sesin, I. Papuccio, A. E. Bruchhausen, P.V. Santos, A. Fainstein, Nature Communications. 14, 3485. (2023). 

[2] I. A. Ramos-Pérez, I. Carraro-Haddad, F. Fainstein, D. L. Chafatinos, G. Usaj, G. B. Mindlin, A.Fainstein, and A. A. Reynoso, Phys. Rev. B. 109, 165305 (2024).