Polariton XY-simulators revisited

Abstract

Arrays of bosonic condensates of exciton-polaritons have emerged as a promising platform for simulating classical XY models, capable of rapidly reaching phase-locked states that may be mapped to arrays of two-dimensional classical spins. However, it remains unclear whether these states genuinely minimize the corresponding XY Hamiltonian and how the convergence time scales with the system size. Here, we develop an analytical model revealing that an array of $N$ condensates possesses $N$ stable phase configurations. The system selectively amplifies a specific configuration dependent on the pump power: at low power, the state with the smallest eigenvalue of an effective XY Hamiltonian is favored, while at high power, the state with the largest eigenvalue prevails. At intermediate pump powers, the system visits all eigenstates of the Hamiltonian. Crucially, the formation rate for any of these phase-locked states remains on the order of 100 ps, independent of the size of the array, demonstrating the exceptional speed and scalability of polariton-based XY simulators.

Figures (8)

• Figure 1: Schematic showing an array of coupled exciton-polariton condensates in a planar semiconductor microcavity. Non-resonant pump beams generate incoherent exciton reservoirs, which supply quasiparticles to the exciton-polariton modes. The phase locking of individual polariton condensates governed by coherent and dissipative coupling enables the system to function as an XY Hamiltonian simulator.
• Figure 2: (a) Pump profile of a one-dimensional chain of exciton-polariton condensates. (b) The growth rate of the population of the polariton mode $N_\nu$ in a one-dimensional chain of polariton condensates with $N=15$. When pump power is low, the mode with a smallest eigenvalue $N_1$ exhibits the largest growth rate (blue curve). The increase of the pump power will eventually make the mode with the largest eigenvalue $N_N$ the fastest growing (purple curve). (c) Pump profile of a two-dimensional triangular zigzag polariton lattice. (d) The growth rate of the population of the polariton mode $N_\nu$ in the triangular lattice with $N=100$.
• Figure 3: (a-d) Spatial distributions of polariton densities characterizing the multi-condensate eigenmodes in a one dimensional chain of polariton condensates for $N_1$ mode with $k=\pi/a$, $N_5$ mode with $k=5\pi/7a$, for $N_{10}$ mode with $k=2\pi/7a$, and for $N_{15}$ mode with $k=0$. (e-h) Corresponding phase patterns and schemes which show the orientations of effective spins obtained by mapping the system to an XY-Hamiltonian. The calculation is performed assuming $k_c=2\ \mu \mathrm{m}^{-1}$.
• Figure 4: (a) A schematic of pump power distribution in real space that is used to create a zigzag triangular polariton lattice, characterized by the lattice constant $a$. (b) The first Brillouin zone of the corresponding lattice of polariton condensates. At $\Gamma$ point the eigenvalue of the overlap matrix $D$ is $E_+(\mathbf{k})$, while at two-fold degenerate $K$ and $K^\prime$ points the corresponding eigenvalue is $E_-(\mathbf{k})$. (c) The structure of the sparse overlap matrix for an open-boundary $33\times 33$ zigzag triangular lattice. (d) A magnified structure of the overlap matrix.
• Figure 5: (a) The dependence of the overlap integral $\mathrm{Re}(\beta)$ on $k_c$. The sign of $\mathrm{Re}(\beta)$ determines the ground state of the polariton lattice. (b) The dependence of the extreme values of the eigenenergies $E_\pm$ on the wavevector $k_c$. The lowest eigenvalue of the system (below the yellow dashed line) is either $E_+$ ($\Gamma$-point) or $E_-$ ($K$-point), depending on the value of $k_c$, which corresponds to the ferromagnetic phase or frustrated anti-ferromagnetic phase, respectively. (c) The spatially dependent polariton density $|\varphi|^2$ of the multi-condensate eigenmode at $k_c=1.95\ \mathrm{\mu m}^{-1}$ and (d) the corresponding phase pattern. The phases of the condensates at neighboring sites differs by $2\pi/3$, indicating a frustrated anti-ferromagnetic phase. The white circles in the figure below represent the lattice sites where the optical pump beams are focused. Directions of effective spins for each individual condensate are shown on a selected hexagonal unit cell. (e) The polariton density $|\varphi|^2$ of the multi-condensate eigenmode at $k_c=2.90\ \mathrm{\mu m}^{-1}$ and (f) the corresponding phase pattern. Here phases of the condensates at different lattice sites are the same, which indicates a ferromagnetic phase. Directions of the effective spins corresponding to each individual condensate are shown by arrows on a selected hexagonal unit cell.