Room temperature exciton–polariton Bose–Einstein condensation in organic single-crystal microribbon cavities

Exciton–polariton Bose–Einstein condensation (EP BEC) is of crucial importance for the development of coherent light sources and optical logic elements, as it creates a new state of matter with coherent nature and nonlinear behaviors. The demand for room temperature EP BEC has driven the development of organic polaritons because of the large binding energies of Frenkel excitons in organic materials. However, the reliance on external high-finesse microcavities for organic EP BEC results in poor compactness and integrability of devices, which restricts their practical applications in on-chip integration. Here, we demonstrate room temperature EP BEC in organic single-crystal microribbon natural cavities. The regularly shaped microribbons serve as waveguide Fabry–Pérot microcavities, in which efficient strong coupling between Frenkel excitons and photons leads to the generation of EPs at room temperature. The large exciton–photon coupling strength due to high exciton densities facilitates the achievement of EP BEC. Taking advantages of interactions in EP condensates and dimension confinement effects, we demonstrate the realization of controllable output of coherent light from the microribbons. We hope that the results will provide a useful enlightenment for using organic single crystals to construct miniaturized polaritonic devices.

The mixture was then diluted with ethanol, followed by the addition of HCl (2 M, 200 mL). The precipitate thus formed was collected through vacuum filtration and further purified by silica gel chromatography using dichloromethane as the eluent. The desired product (PDI-O) was obtained in 75% yield (2.70 g). The components of transition dipole moment along the x, y and z axes are μ x = 9.3172696 D, μ y = 0.0000587 D and μ z = 0.0000214 D, respectively. The calculated result reveals that PDI-O molecule has a transition dipole moment of 9.317 D, whose direction is along the N-N axis. The large transition dipole moment is favor for strong coupling because, according to the strong coupling theory, the coupling strength is proportional to the transition dipole moment 3 .     The PDI molecules in the crystal adopt a J-type stacking mode, as evidenced by the slipping angle between adjacent molecules θ =35.89°, which is less than the so-called "magic angle" θ M = 54.7°. The J-type stacking mode would reduce non-radiative decay caused by π-π interactions, which is beneficial for providing sufficient number of excitons to form polaritons.

Strong exciton-photon coupling in PDI-O microribbon cavities
The polariton emission is located near 580 nm, which indicates that the energetic separation between polariton ground state (ky = 0) and PDI-O molecular ground state (S00) is 2.14 eV. This value equals to the energetic separation between exciton reservoir (S10) and the first vibronic sublevel of the molecular ground state (S01),  Meanwhile, the emission spectrum is a mirror image of the absorption spectrum showing the main emission peak at 2.32 eV (535 nm) and a vibronic replica at 2.14 eV (580 nm), which correspond to the S10→S00 and S10→S01 transitions respectively.
The vibronic energy was thus estimated to be 2.32 eV -2.14 eV = 180 meV. As shown in Supplementary Figure 12, the energetic separation between the polariton ground state (ky = 0) and the PDI-O molecular ground state (S00) is 2.14 eV (580 nm, from the polariton emission spectrum), and therefore the polariton ground state is energetically lower than the exciton reservoir by 180 meV, indicating that the polariton ground state is directly populated with emission of a vibron. The PL wavelength is in the range from 620 to 680 nm, which is ~100 nm longer than the absorption maximum of PDI-O (525 nm). Exciton-photon strong coupling did not occur due to the large energy separation between them, and therefore the measured dispersion corresponds to the uncoupled waveguide cavity modes. By using the equations discussed in Supplementary Note 1, we can give a best fit to the measured results with fitting background index n bg = 2.2.
Where E c , E ex and E p are the uncoupled cavity mode energies, exciton energy, and polariton energies, respectively. Ω is Rabi splitting energies indicating the coupling strength. The mixing coefficients α c 2 and α ex 2 describe the relative photonic and excitonic weightings of the polaritons. In our PDI-O microribbon, the exciton energy extracted from the absorption spectrum is 2.36 eV.
The thickness of the microribbons is in the range of one to a few hundred nanometers, which is large enough for the microribbon to function as a slab waveguide and support guided modes. For the guided modes which travel inside the microribbon in x-direction by total internal reflection in the zigzag fashion, the sum of all phase shifts after each round trip of the wave must be equal to a multiple of 2π, which can be expressed by the characteristic equation: 9 : Where k z is the wave vector inside the microribbon along the z-axis, and φ t and φ b are phase shifts on total reflection from the microribbon's top and bottom facets, respectively. m and d are the mode order of the guided modes and thickness of the microribbon.
The z-component of the mode wave vector k z can be expressed by: Where k0 = 2π/λ and nbg are the wave vector in free space and the background refractive index of the microribbon, respectively. nbgk0 and β are the wave vector of the guided modes and the propagation constant along the x-axis, respectively. For a microribbon with a width of W, β satisfies the F-P resonance condition β= Nπ W (4) Where N is integer number for modes.
For TE modes, we can extract from the Fresnel formulas the following expressions for the phase shifts φt and φb: Where nair =1 and nsub = 1.46 are the refractive index of the air and glass substrate, respectively.
For the waveguide cavity modes in the x-direction, considering the long length of the microribbon, the y-component of the wave vector ky is free and can be expressed by 10 k y =k 0 tan(arcsin sinθ n bg ) (7) Therefore, the energy of the uncoupled F-P type waveguide cavity mode is S25 E c = hc 2π k 0 2 +k y 2 (8) Using these equations, we can give a best fit to the AR μ-PL mapping data for all cavity modes simultaneously with fitting parameters n bg = 2.20 and Ω = 530 meV.

S26
Supplementary Note 2 | Calculation of refractive index from fitting parameters in the coupled oscillator model.

Calculation of refractive index from experimental data
In F-P microcavities, the refractive index is calculated by 11 n λ = λ 2 2W∆λ (9) Where λ is the wavelength of light, Δλ is the space between the F-P mode peaks, and W = 5 μm is the width of PDI-O microribbon.
Calculation of refractive index from fitting parameters in the coupled oscillator model The dielectric constant under strong coupling effect can be calculated by the following equations 10,11 Ω= 2E ex (ω L -ω T ) (10) Where ω L and ω T are the transverse and longitudinal resonance energies, respectively.     In steady-state polariton condensates, the macroscopic wave function ψ(r) of polaritons can be described by the following mean-field Gross-Pitaevskii equation 13 : h 2 2m ∇ 2 +g res n res (r)+g|ψ r | 2 ψ r =υψ r (12) Where m is the effective mass of polaritons, and υ is chemical potential of polaritons.
g and g res describe the strength of polariton -polariton and polariton-exciton reservoir interactions, respectively. n res (r) is exciton reservoir density controlled by the pump and reservoir decay rate. The spatial distribution of exciton reservoir n res (r) is given by that of pumping due to the small diffusion coefficient of excitons. The inhomogeneous spatial distribution of exciton reservoir can induce spatial modifications of polariton wave function. Specifically, as there are no excitons outside the excitation area, the repulsive interactions between polaritons and excitons in the excitation area create a force, which expels polaritons from the excitation area. As a result, the polariton condensate undergoes a lateral acceleration and acquires a wave vector along the PDI-O microribbon 13 .
Due to the small width and thickness of the microribbon, polaritons are confined in two dimensions, and therefore we can use a one-dimensional (1D) model to describe the polariton condensate with interaction potential. We have fitted the PL intensity