Synchronization in disordered oscillator lattices: Nonequilibrium phase transition for driven-dissipative bosons

We show that lattices of phase oscillators with random natural frequencies undergo a transition from a desynchronized to a synchronized state for dimensions $d<4$. The oscillators are described by a generalization of the nearest-neighbor Kuramoto model with an additional cosine coupling term. This model may be derived from the complex Ginzburg-Landau equations for a lattice of driven-dissipative Bose-Einstein condensates of exciton polaritons. We derive phase diagrams that classify the desynchronized and synchronized states, focusing on the behavior in 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 synchronization in coupled polariton condensates and other examples of low-dimensional lattices of coupled oscillators is not destroyed by randomness in their natural frequencies.

Figure 1

Probability of synchronization of chains of condensates of varying lengths, determined by solving Eq. (3) with $\alpha =1$. Each probability is estimated from 100 disorder realizations, each simulated up to a time $t\sigma =1.8\times {10}^4$. The curves are fits to the Gumbel distribution.

Figure 2

Phase (left column) and frequency (right column) profiles in a chain of 800 coupled oscillators, with $J/\sigma =3.33$ and $\alpha =1$. The oscillators initially have a uniform phase. The profiles are shown after times $t\sigma =100$ (top row), 600 (middle row), and $24000$ (bottom row).

Figure 3

Phase (left column) and frequency (right column) profiles in a two-dimensional lattice of $512\times 512$ coupled oscillators, with $J/\sigma =3.33$ and $\alpha =1$. The oscillators initially have a uniform phase. The profiles are shown after times $t\sigma =50$ (top row), 300 (middle row), and 1700 (bottom row).

Figure 4

Phase boundary for synchronization for (a) a one-dimensional (1D) chain of 800 oscillators, and (b) a two-dimensional (2D) square lattice of $128\times 128$ oscillators. $J_c$ is defined as the value of $J$ corresponding to a probability of synchronization of 0.5. The points are numerical values, computed as discussed in the text. The solid lines show fits to the predicted slopes of (a) 0.5 or (b) 1. The dashed line in (a) is the phase boundary for synchronization for a chain of 800 Kuramoto oscillators.