Spontaneous Polariton Currents in Periodic Lateral Chains

We predict spontaneous generation of superfluid polariton currents in planar microcavities with lateral periodic modulation of both the potential and decay rate. A spontaneous breaking of spatial inversion symmetry of a polariton condensate emerges at a critical pumping, and the current direction is stochastically chosen. We analyze the stability of the current with respect to the fluctuations of the condensate. A peculiar spatial current domain structure emerges, where the current direction is switched at the domain walls, and the characteristic domain size and lifetime scale with the pumping power.

Figure 1

(a) Sketch of the proposal: polariton current of spontaneously chosen direction flows in a condensate fed by a symmetric homogenous pumping. (b) Spatial distribution of real and imaginary parts of the periodic potential $U(x)$ and distribution of the first and second band wave functions at the Brillouin zone edge. (c) Complex polariton band structure in the Kronig-Penney model. The decay rate is shown with color. The fastest (slowest) decay rate characterizes $A$ ($S$) mode.

Figure 2

(a),(b) Condensate population dependence on the pumping power. (c),(d) Imbalance of populations $s_z$ defining the value of the net polariton current. Uniformly stable (unstable) stationary solutions are plotted with solid (dotted) lines. Blue and red lines correspond to symmetry conserving condensates at $S$ and $A$ polariton modes. The two sets of parameters correspond to the presence (type I) and absence (type II) of a tristability region between the $S$ state solution and the pair of symmetry breaking solutions. In the former case, two hysteresis loops emerge. The lower two plots are zoomed to the pitchfork and saddle-node bifurcation points $w_c$ and $w_f$. Inset: phase transition type depending on $\alpha /\eta$ and $\gamma /ϵ$.

Figure 3

(a) Stability diagram. Type I and II transitions are shown with green and blue colors, respectively. Regimes of current stability with respect to short wavelength fluctuations are shown with hatching: the blue double hatched area corresponds to the unstable regime; the red single hatched area covers the parameter space where the current is stable in a range of pumping powers; the area free of hatching corresponds to the stable current regime. (b)–(d) The Lyapunov spectra of condensate current solutions. Solid and dashed lines show the real and imaginary parts, respectively. Parameters correspond to the black dots in the panel (a). The pumping power is slightly above the critical point.

Figure 4

(a), (b) Numerical simulation in the two-mode approximation. (a) Absolute value of the spatially averaged current vs pumping power. The system finite size defines the instability onset point. Parameters correspond to the type II transition with the finite system size set by $-c\le x\gamma \le c$. (b) Spatial and temporal dependence of the polariton current in the unstable regime (for $w=2.1\gamma$): domain structure emerges in an initially homogeneous condensate. (c),(d) Full numerical simulation of GPE with a periodic complex potential. (c) Condensate dispersion demonstrating spatial asymmetry. The intensity has been broadened in energy and momentum for visualization. Circles denote the bare dispersion, quantized by the finite system size. Parameters: $w=\gamma$, $-a\le x\le a$. (d) Spatial and temporal dependence of local phase gradient in the unstable regime (given by a larger system size $-80a\le x\le 80a$), with chaotic evolution of domain structure.