Spin Order and Phase Transitions in Chains of Polariton Condensates

We demonstrate that multiply coupled spinor polariton condensates can be optically tuned through a sequence of spin-ordered phases by changing the coupling strength between nearest neighbors. For closed four-condensate chains these phases span from ferromagnetic (FM) to antiferromagnetic (AFM), separated by an unexpected crossover phase. This crossover phase is composed of alternating FM-AFM bonds. For larger eight-condensate chains, we show the critical role of spatial inhomogeneities and demonstrate a scheme to overcome them and prepare any desired spin state. Our observations thus demonstrate a fully controllable nonequilibrium spin lattice.

Figure 1

(a) Lattice potential in the microcavity created by blueshifts at the pump beams (purple beams) forming magnetized condensates in the center of each site (yellow spots). (b) Below threshold PL showing the pump spots. Global NN barrier is tuned by modulating the intensity of the center spot ($u_r$, dashed circle). (c) Formation of condensates at the center of each of the 4 lattice sites above threshold. (d) Schematic of condensate spin chain comprised of two coupled Bose-Hubbard chains. Each condensate (indices 1–4) has two spin states ($+$ and $-$), which are coherently coupled by $\epsilon$. Each is also coherently coupled to its same spin NN by Josephson coupling $J$. (e) Magnetization of condensate chain (expt) above the spin-bifurcation threshold ($P_{\mathrm{c}}$), into an AFM state.

Figure 2

Steady states as a function of barrier height. The measured condensate spin $S_z$ for all possible stable states at three phases of (a) FM, (b) PFM, and (c) AFM, when the global NN barrier $u_r$ is increased.

Figure 3

Spin correlations as a function of barrier height. (a), (b) The measured correlation matrix of the spin chain at the two phases of (a) AFM and (b) PFM. Numbers show condensate indices. The autocorrelated diagonal elements are removed for clarity. (c), (d) Phase diagram of the spin chain showing spin correlation of the diagonal and side condensates vs $u_r$ in (c) experiment and (d) 2D numerical simulations.

Figure 4

Controlling magnetic order with feedback. (a) PL below threshold showing excitation pattern. (b) Emission above threshold showing eight-condensate chain. (c) The most probable spin states after optimization using FM, glass, or AFM search criteria in (d). (d) PCA components of spin-up or spin-down emission after the search procedure for different targets. Intensities show strengths of components. Plots show likelihood (variance percentage) for each order.