Abstract
We show that a lattice of phase oscillators with random natural frequencies, described by a generalization of the nearest-neighbor Kuramoto model with an additional cosine coupling term, undergoes a phase transition from a desynchronized to a synchronized state. This model may be derived from the complex Ginzburg-Landau equations describing a disordered lattice of driven-dissipative Bose-Einstein condensates of exciton polaritons. We derive phase diagrams that classify the desynchronized and synchronized states that exist in both one and two dimensions. This is achieved by outlining the connection of the oscillator model to the quantum description of localization of a particle in a random potential through a mapping to a modified Kardar-Parisi-Zhang equation. Our results indicate that long-range order in polariton condensates, and other systems of coupled oscillators, is not destroyed by randomness in their natural frequencies.
Highlights
Synchronization of coupled oscillators is a phenomenon that appears regularly throughout nature
Such condensation has been realized for exciton polaritons [8], which are bosonic quasiparticles formed from the strong coupling of excitons and photons in semiconductor microcavities
Nearest-neighbor Kuramoto model that includes an additional coupling term that is even in the relative phases. Such models have previously been considered [22,23] in the case where the natural frequencies are identical and frequency synchronization is expected, but the phases may become disordered due to the space- and time-dependent noise associated with gain and loss [24,25,26,27]
Summary
Synchronization of coupled oscillators is a phenomenon that appears regularly throughout nature. If the condensates are well separated, they oscillate independently, but their mutual coupling can lead them to synchronize to a common frequency and phase [15] Such effects have been observed for polaritons trapped in double-well potentials [10,15], in the intrinsic random potential of the samples [13,16], and in the wells of a weakly disordered two-dimensional (2D) lattice [14]. Nearest-neighbor Kuramoto model that includes an additional coupling term that is even in the relative phases Such models have previously been considered [22,23] in the case where the natural frequencies are identical and frequency synchronization is expected, but the phases may become disordered due to the space- and time-dependent noise associated with gain and loss [24,25,26,27]. Our conclusions apply more generally to coupled oscillator systems, implying there are other settings [1] in which this synchronization could occur
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