Optically trapped polariton condensates as semiclassical time crystals

We analyze nonequilibrium phase transitions in microcavity polariton condensates trapped in optically induced annular potentials. We develop an analytic model for annular optical traps, which gives an intuitive interpretation for recent experimental observations on the polariton spatial mode switching with variation of the trap size. In the vicinity of polariton lasing threshold we then develop a nonlinear mean-field model accounting for interactions and gain saturation, and identify several bifurcation scenarios leading to formation of high angular momentum quantum vortices. For experimentally relevant parameters we predict the emergence of spatially and temporally ordered polariton condensates (time crystals), which can be witnessed by frequency combs in the polariton lasing spectrum or by direct time-resolved optical emission measurements. In contrast to previous realizations, our polaritonic time crystal is spontaneously formed from an incoherent excitonic bath and does not inherit its frequency from any periodic driving field.

Figure 1

Results of the linear non-Hermitian rectangular trap model. (a) Reservoir pumping rate at the lasing threshold for states with $\mathrm{m}=0,1,...,8$ as a function of the trap size $\rho =R\sqrt{2m_p\mathrm{\Gamma }}$. Condensate switching points are marked with circles and dashed lines. (b) Condensate angular quantum number $m$ at the polariton lasing threshold as a function of the only two linear model parameters.

Figure 2

(a) Phase diagram of polariton condensate bifurcations. Areas of parameters corresponding to Andronov-Hopf (pitchfork) bifurcation type of the trivial stable solution branch $S^{+}\left(\mathcal{P}\right)$ are shown in green (yellow) color. The dashed line separates the repulsion (above) and attraction (below) regimes. The hatched blue area corresponds to the regime where limit cycles are the only orbitally stable solutions. The unhatched blue area corresponds to bistability between the trivial and the symmetry breaking solution. (b) Stationary condensate population dependence on pumping power. Stable (unstable) fixed points are plotted with solid (dashed) lines. Trivial $S^{+}\left(\mathcal{P}\right)$ and $S^{-}\left(\mathcal{P}\right)$ and the symmetry-breaking solutions are plotted with red (upper straight), blue (lower straight), and black (curved) lines. The gray area highlights the instability range, and the frequency of limit cycles is plotted with the green dash-dotted line.

Figure 3

Limit cycles of the condensate evolution. (a) Pseudospin trajectories for pumping power spanning the instability range. (b) Spectral and temporal (inset) dependence of the pseudospin projections in the anharmonic limit cycle regime, corresponding to the blue line in panel (a).

Figure 4

(a)–(c) Spatial density profiles of an elliptically pumped polariton condensate and (d)–(f) corresponding phase profiles. In (a) and (d) the pump power is 5% above threshold and a stable petal pattern forms due to the geometric ellipticity of the pump. At 10% above threshold (b),(e) the petal pattern bifurcates (loses stability) and evolves into one of two counter-rotating vortex states (c),(f) with equal probability. The slight modulation in the density and finite distance between phase singularities in the vortex center (c),(f) stems from the elliptic profile of the pump.

Figure 5

Numerical solution of GPE in the time crystal regime. (a) Polariton density time shot. (b) Evolution of instantaneous polariton density and current (shown with arrows) throughout the period $T$.