Nontrivial Phase Coupling in Polariton Multiplets

We investigate the phase coupling between spatially separated polariton condensates under nonresonant optical pulsed excitation. In the simple case of two condensates, we observe phase locking either in symmetric or antisymmetric states. We demonstrate that the coupling symmetry depends both on the separation distance and outflow velocity from the condensates. We interpret the observations through stimulated relaxation of polaritons to the phase configuration with the highest occupation. We derive an analytic criterion for the phase locking of a pair-polariton condensate and extend it to polariton multiplets. In the case of three condensates, we predict theoretically and observe experimentally either in-phase locking or the appearance of phase winding with phase differences of $\pm 2\pi /3$ between neighbors. The latter state corresponds to a vortex of winding number $\pm 1$ across the three polariton condensates.

Popular Summary

Polaritons, quasiparticles that arise from strong coupling between light and matter, possess hybrid properties of photons and excitons. They combine the mobility and flexibility of light with the possibility of interactions due to their matter component. At high enough densities or low enough temperatures, polaritons can form a macroscopic coherent quantum state, a polariton condensate, or a polariton laser. Such a coherent state exhibits much of the same physics as Bose-Einstein condensation, as has been observed for cold atoms, but without requiring the ultralow temperatures necessary for atoms. Here, we investigate experimentally and theoretically the phase coupling between spatially separated polariton condensates at roughly 10 K.

Polariton condensates can be imprinted into any two-dimensional graph by spatial modulation of the pumping laser, offering the scalability matched only by optical lattices in ultracold atomic condensates. Here, we inject optically pumped polariton condensates into multisite configurations with arbitrary density profiles, which makes it possible to control the kinetics of the condensate at each site as well as the separation distance between any two neighboring sites. We explain the observations by the stimulated relaxation of polaritons to the phase configuration with the highest occupation across the lattice. We use interference patterns to study both polariton dyads and triads (i.e., two or three phase-locked condensates, respectively). In the case of a polariton triad, we surprisingly observe an exotic state of a lattice vortex in which the phase across the triad winds by $2\pi$.

We expect that our findings will pave the way for the realization of applications such as quantum polariton simulators, which are highly tunable systems that can emulate other not-so-easily-accessible quantum systems.

Figure 1

(Upper row) The time-integrated images of two polariton condensates in real space at three different spatial separation distances. The condensates are formed simultaneously at $P\simeq P_{\mathrm{th}}$, where $P_{\mathrm{th}}$ is the threshold power for condensation. The observation of an interference pattern between the two condensates indicates that they are phase correlated. Zero intensity of the interference pattern in the middle of the two sources indicates that the two condensates are antisynchronized. (Lower row) Time-integrated results of numerical simulations of the Ginzburg-Landau equations with random initial phase averaged over 75 realizations at separation distances between the condensates as in the experiment above. The condensates phase lock, on average, to zero or $\pi$ phase difference depending on their spatial separation distance.

Figure 2

The condensates flip from antisymmetric ($\pi$ phase difference) to symmetric states (zero phase difference), shown with black dots as a function of the product of the outflow wave vector $k_c$ and the separation distance $a$. Here, we change $a$, whereas $k_c$ remains virtually unchanged. The solid red curve is the Bessel function calculated versus the dimensionless parameter $k_{c}a$.

Figure 3

Energy-resolved real- and Fourier-space images of two condensates are shown for different separation distances. The fringes in between the condensates are at the same energy with the condensates.

Figure 4

(a) The time-integrated momentum space shows that condensation occurs at two distinctly different wave vectors. At $t=31\mathrm{ps}$ (marked by a blue arrow), the polariton dyad is in antiphase synchronization. At $t=54\mathrm{ps}$ (marked by an orange arrow), the polariton dyad is in in-phase synchronization. (b,c) Time-resolved interferograms at $t=31\mathrm{ps}$ and $t=54\mathrm{ps}$, respectively, showing the phase synchronization of the two condensates. The real-space scale bars in (b,c) apply to both horizontal and vertical directions. The color scale in (a–c) is shown as an inset bar in (b). The data in (a,b,c) are normalized. (d) The intensity profiles of the interferograms at the positions marked by the vertical lines in (b) (dashed blue) and (c) (solid orange) provide a direct experimental observation of the $pi$ phase difference in the two synchronization regimes. The solid lines are the fits of a Gaussian function convoluted with a $\cos (ky+\varphi )$, giving a phase difference of $(0.98\pm 0.01)\times \pi$ between the two states. The zero of the horizontal axis is defined at the maximum of the intensity profile extracted from (c).

Figure 5

(a–d) Time-integrated real-space tomography images of the triangular array condensates are shown for $a=4\mu \mathrm{m}$ (a) and $a=5.5\mu \mathrm{m}$ (b). The real-space scale bar of (a) also applies to (b). (c,d) Simulations of the Ginzburg-Landau equation with random initial phase, averaged over 75 realizations at different separations matching the experimental conditions are shown. In (c), all three condensates are in phase. In (d), we plot the average of vortices with $\pm 1$ winding numbers shown in (e) and (f). The phase diagrams of (e) and (f) are shown in (g) and (h), proving that they are indeed vortex states with winding numbers of $+1$ (h) or $-1$ (g). (i,j) The time-resolved interferograms resulting from the interference of neighboring condensates are shown at $t=47.2\mathrm{ps}$ (i) and $t=65.9\mathrm{ps}$ (j). The color bar of (a) and (b) applies to the data of (i) and (j). (k) The intensity profiles of the interferograms along the lines marked by the vertical lines in (i) (dashed red) and (j) (dashed black) provide a direct experimental observation of the nearly $2\pi /3$ phase difference in the two synchronization regimes. The line profiles of (i) and (j) taken at $x=-1.9\mu \mathrm{m}$. The solid lines are the fits by the convolution of a Gaussian function with $\cos (ky+\varphi )$ resulting in a $(0.78\pm 0.02)\times \pi$ phase difference between the two fits.

Figure 6

The condensates flip from antisymmetric ($\pi$ phase difference) to symmetric states (zero phase difference), shown with black segments as a function of the separation distance for two values of the outflow wave number (a) $k_c=4$ and (b) $k_c=1$. The solid red curves are the Bessel functions, $J_0(k_{c}d)$.