Exciton–polariton condensates, due to their nonlinear and coherent characteristics, have been employed to construct spin Hamiltonian lattices for potentially studying spin glass, critical dephasing, and even solving optimization problems. Here, we report the room-temperature polariton condensation and polaritonic soft-spin XY Hamiltonian lattices in an organic–inorganic halide perovskite microcavity. This is achieved through the direct integration of high-quality single-crystal samples within the cavity. The ferromagnetic and antiferromagnetic couplings in both one- and two-dimensional condensate lattices have been observed clearly. Our work shows a nonlinear organic–inorganic hybrid perovskite platform for future investigations as polariton simulators.

Exciton–polaritons are quasiparticles that arise from the strong coupling between excitons in semiconductors and photons in microcavities. Due to their small effective mass inherited from photons and strong nonlinearity inherited from excitons, exciton–polaritons have been utilized to achieve Bose–Einstein condensation (BEC) at very high temperatures than cold atoms [

Semiconductor lead halide perovskites, as a new room-temperature polariton platform, have been researched. Because of the large Wannier–Mott exciton binding energy [_{3} and FAPbBr_{3}, used in the research for the next generation of solar cells [

In this work, we realized room-temperature polariton condensation with organic–inorganic halide perovskite MAPbBr_{3}. Large and uniform single-crystal perovskites were grown directly in prebonded microcavities by solution method, as shown in our previous work [

The cavity structure is shown schematically in _{3} was grown by a solution method under the confinement of the microcavity (∼330 nm). With this method, high-quality, large-sized perovskite single crystals can be grown in the cavity, as shown in _{3} plate grown in the 200-nm thick empty bonded quartz substrates. Previous studies show that excitons in MAPbBr_{3} perovskite have an exciton binding energy of about 25–42 meV at room temperature [

Characterization of exciton–polaritons in organic–inorganic halide perovskite microcavity. (a) Schematic of the perovskite microcavity, consisting of a solution-grown MAPbBr_{3} single crystal sandwiched between two 12.5 pairs DBR mirrors with cavity quality factor Q ∼700–800. (b) Optical image of a MAPbBr_{3} single crystal in the cavity with a transmission illumination. Scale bar: 50 μm. (c) The normalized absorption and photoluminescence (PL) spectra of a ∼200-nm thick MAPbBr_{3} single-crystal at room temperature. The absorption spectrum in the top panel was fitted by the Elliott model [

For the perovskite sample in the cavity, a sharp polariton dispersion can be observed via the angle-resolved PL spectrum shown in

Here, the cavity mode is represented as _{0} at _{‖} = 0 and effective mass _{
ex
} is the exciton energy that can be extracted from the absorption spectrum in

The strong nonlinear properties of MAPbBr_{3} enable us to achieve polariton condensation. A 450-nm, 250-fs pulsed laser was used to excite the sample nonresonantly with a laser spot size of ∼28 μm. _{th}, _{th}, and 1.5 _{th} from left to right, where _{th} is the threshold of ∼42.2 μJ cm^{−2}. A pinhole at the real-space imaging plane was used to only extract the PL from the condensation area. Above the threshold, the polariton ground state at _{||} = 0 exhibits a strong nonlinear emission enhancement and linewidth narrowing, clearly evidences of the condensation. This is also illustrated in

Exciton–polariton condensation at room temperature. (a) Angle-resolved power-dependent PL dispersions under nonresonant excitation at 0.1 _{th}, _{th}, and 1.5 _{th} from left to right, where _{th} (∼42.2 μJ cm^{−2}) is the pumping power of condensation threshold. With the power increasing, the polaritons condensate at the ground state of the polariton mode. The dashed red lines are the fitted curve of the lower polariton mode with low pumping power. (b) Log–log plot of the integrated PL intensities of the ground polariton mode at _{||} = 0 and the full width at half maximum versus pulse energy. The nonlinear increase of the intensity and the narrowing of the linewidth clearly demonstrate the polariton condensation. (c) PL peak blueshifts of the mode at _{||} = 0. Due to the strong exciton reservoir interactions, a stronger blueshift was observed below the threshold.

By utilizing the long-range coherence of the condensates, we can create multiple condensates and study their interactions. Due to the repulsion interaction between polaritons, condensation can occur at a high energy state with a nonzero in-plane momentum when the pumping laser spot is small [

Phase configuration between two polariton condensates. (a) The time-integrated real-space images of the two polariton condensates with different separation distances of 1.78, 2.5, and 3.55 μm from left to right, respectively. With the separation distances increase, the interference fringes increase from zero to three, demonstrating the transitions from antiphase to the in-phase and back to the antiphase synchronization between the two condensates. The red arrows marked at the condensates represent the relative spin directions. Scale bars: 2 μm. (b) The corresponding angle-resolved PL dispersions of the condensates in (a). The detuning of this sample is smaller than that shown in

To validate the capability of our sample to construct a large-scale polariton condensate lattice, we assembled a 2 × 2 square lattice to simulate the two-dimensional soft-spin XY Hamiltonian. As shown in

Construction of the two-dimensional polariton spin lattices. (a–c) The time-integrated real-space images of the 2 × 2 polariton condensate square lattices. With increasing lattice distance, the condensate lattices undergo a transition from ferromagnetic to antiferromagnetic and then to the next ferromagnetic coupling states in (a)–(c), respectively. The red arrows marked at the condensates represent the relative spin directions. Scale bars in (a)–(c): 2 μm.

We model the lattice of polariton condensates as the system of coupled oscillators [_{
i
} (_{
i
}) is its number density (phase), _{
c
} represents the linear losses, _{
ij
} is the coupling strength between _{
ij
} between the pumping sites [_{
c
} and the coupling strength varies as a Bessel function of the first kind

The system of _{
loss
} for ^{inj} acts as the bifurcation (annealing) parameter, taking the system through the Andronov–Hopf bifurcation [_{
i
} ∈ {0, _{
i
} = 0 for all

Therefore, _{
loss
} can become non-convex) when ^{inj} increases, the loss landscape becomes nonconvex and new minima appear along the direction defined by the eigenvector associated with the largest eigenvalue of the coupling matrix _{
i
} = ^{inj} − _{
c
} to the leading order while the phases minimize the classical XY Hamiltonian _{
XY
} becomes a weak perturbation of the uncoupled system of oscillators. In an analogy with the soft-spin Ising Hamiltonian [_{
loss
} defined in _{
XY
} as “hard-spin” XY Hamiltonians.

Finally, more information about the dynamics can be gathered by separating real and imaginary parts of

The fixed point of ^{inj} yields_{
i
}; however, the coupling strengths between the Kuramoto oscillators are modified by their amplitudes. A constant relative phase between them will be achieved only if amplitudes are not vastly different.

To summarize, our analysis indicates that as ^{inj} increases, the system undergoes several transitions and for each fixed value of ^{inj} solves a different problem. Close to the condensation threshold when _{
i
} = 0 state into the state along the negative curvature direction dictated by the principle eigenvector of the coupling matrix ^{inj} when amplitudes are sufficiently homogeneous. This fixed point corresponds to the minimum of the loss function, which is a soft-spin version of the classical XY Hamiltonian. In the high injection limit, ^{inj} − _{
c
} with the relative phases reaching a fixed point corresponding to the minimum of the hard-spin XY Hamiltonian. However, the minimization comes as a perturbation. In our experiments, therefore, we do not reach the high intensity limit and observe the minimization of the soft-spin XY Hamiltonian represented by the loss function given by

To illustrate the evolution and the steady state of our system, we solved the system of _{12} = _{23} = _{34} = _{14} = 1 ferromagnetic (F) and _{12} = _{23} = _{34} = _{14} = −1 antiferromagnetic (AF) interactions along the sides of the square formed by four condensates. Without loss of generality, we took _{
c
} = 5. _{1}. _{1} condensate is depleted with respect to the densities of the other condensates, while the phases deviate from {0, _{1} and _{2} and become {0, _{1}, _{2}, _{1}}, as seen in _{1} and ± _{2} are equally likely to be realized, so on the time-averaged experiments, like ours, the state {0,

The time evolution of the phases and the squares of the complex amplitudes of four condensates. (a–c) The phases of the condensates represented by _{12} = _{23} = _{34} = _{14} = 1 (a) and _{12} = _{23} = _{34} = _{14} = −1 (b, c) interactions along the sides of the square formed by four condensates. Here _{
c1} = 1.2_{
ci
}, _{1} = ±0.110_{2} = ±0.154

In summary, we realized room temperature polariton condensation and an analog simulation of a two-dimensional soft-spin XY spin Hamiltonian by polariton condensate lattices with high-quality organic–inorganic halide perovskite single crystals for the first time. Due to the higher solubility of the precursors, the resulting single-crystal hybrid perovskites tend to be much larger in lateral sizes compared to CsPbBr_{3}. On the other hand, due to the relatively higher condensation threshold and vulnerability to nanofabrication compared to inorganic perovskites, this real-time tuning of the pumping laser method is well-suited for establishing condensate lattices in organic perovskite materials. As a new nonlinear room-temperature system, our cavity can be used to realize large-scale soft-spin XY Hamiltonians [

12.5 pairs of SiO_{2}/Ta_{2}O_{5} DBR mirror were deposited on six-inch well-clean quartz wafers by electron beam evaporation with an advanced plasma source. A gold pad array of 250 μm side length was deposited on the DBR surface by electron beam evaporation. Then, two of the same DBR wafers were aligned by the gold pads and bonded together in a wafer bonder. The bonded wafers were diced into 2-cm chips. More details can be found in our previous work [

The perovskites were grown by an inverse-temperature crystallization method [_{2} in

A home-built optical setup in a transmission configuration was used for all the optical measurements. A 450-nm, 250-fs optical parametric amplifier pulsed laser with a 2 kHz repetition rate was used to pump the sample nonresonantly. A phase-only reflective liquid-crystal spatial light modulator was used to generate pumping laser patterns by holograms computed by a Gerchberg–Saxton algorithm. The laser pattern was transferred to an objective (Nikon 40× Plan Fluor ELWD, N.A. = 0.6) with a Fourier imaging configuration. The PL was collected in a transmission configuration with another objective (Nikon 40/60× Plan Fluor ELWD, N.A. = 0.6/0.7). The angle-resolved dispersions were measured by a Fourier imaging system with two achromatic tube lenses and an Andor spectrometer equipped with a two-dimensional CCD. The real-space images were obtained by a two-dimensional Princeton instrument EMCCD.

We thank A. Gao from SVOTEK Inc. for assisting with the high-quality DBR mirror coating.