Synchronization and spacetime vortices in one-dimensional driven-dissipative condensates and coupled oscillator models

Driven-dissipative condensates, such as those formed from polaritons, expose how the coherence of Bose-Einstein condensates evolves far from equilibrium. We consider the phase and frequency ordering in the steady-states of a one-dimensional lattice of condensates, described by a coupled oscillator model with nonodd couplings, including both time-dependent noise and a static random potential. We present numerical results for the phase and frequency distributions, and discuss them in terms of the Kardar-Parisi-Zhang equation and the physics of spacetime vortices. We find that the nucleation of spacetime vortices causes the breakdown of the single-frequency steady state and produces a variation in the frequency with position. Such variation would provide an experimental signature of spacetime vortices. More generally, our results expose the nature of synchronization in oscillator chains with nonodd couplings, random frequencies, and noise.

Figure 1

Phases in a chain of 400 oscillators at the time $tJ=6000$ for various noise strengths. $\alpha =1, \sigma /J=0.2$, and noise strengths $D/J=0$ (red), 0.032 (blue), 0.064 (yellow), 0.128 (purple). Noise increases from top to bottom in the center of the figure. The zero of phase is chosen to be that of the first oscillator.

Figure 2

Standard deviation of the time-averaged frequencies, $\mathrm{\Delta }\omega$, for chains of 400 oscillators with static disorder $\sigma$, noise strength $D$, and $\alpha =1$. The frequencies are computed over a time interval $T=1500/J$. Each point is an average of the frequency width over 32 realizations of the disorder and noise.

Figure 3

Phases in a chain of $N=128$ oscillators, without noise (top panel) and with noise (center panel). Position is along the horizontal axis, with time along the vertical axes, increasing from top to bottom. $\alpha =1, \sigma /J=0.4$, and $D=0$ (top) and $D/J=0.07$ (center). The colored points in the center panel mark spacetime vorticity $+1$ (red) or $-1$ (yellow). The bottom panel is an enlargement showing the vorticity in the region of the center panel marked with the red box, with vorticity $+1$ and $-1$ in black and white.

Figure 4

Dependence of the frequency width $\mathrm{\Delta }\omega$ on disorder strength $\sigma$ for a chain of $N=400$ oscillators with $\alpha =1$ (colored points). The same data is shown on a logarithmic (top) and linear (bottom) scale. Colors indicate the noise strengths $D/J$ of 0.128 (green), 0.096 (purple), 0.064 (yellow), 0.032 (blue), and 0 (red). The curves are fits to Eq. (9) (top panel) and Eq. (12) (bottom panel), as described in the text.