| Summary: | Deeply subwavelength lasers (or nanolasers) are highly demanded for compact
on-chip bioimaging and sensing at the nanoscale. One of the main obstacles for
the development of single-particle nanolasers with all three dimensions shorter
than the emitting wavelength in the visible range is the high lasing thresholds
and the resulting overheating. Here we exploit exciton-polariton condensation
and mirror-image Mie modes in a cuboid CsPbBr$_3$ nanoparticle to achieve
coherent emission at the visible wavelength of around 0.53~$\mu $m from its
ultra-small ($\approx$0.007$\mu$m$^3$ or $\approx\lambda^3$/20) semiconductor
nanocavity. The polaritonic nature of the emission from the nanocavity
localized in all three dimensions is proven by direct comparison with
corresponding one-dimensional and two-dimensional waveguiding systems with
similar material parameters. Such a deeply subwavelength nanolaser is enabled
not only by the high values for exciton binding energy ($\approx$35 meV),
refractive index ($>$2.5 at low temperature), and luminescence quantum yield of
CsPbBr$_3$, but also by the optimization of polaritons condensation on the Mie
resonances. Moreover, the key parameters for optimal lasing conditions are
intermode free spectral range and phonons spectrum in CsPbBr$_3$, which govern
polaritons condensation path. Such chemically synthesized colloidal CsPbBr$_3$
nanolasers can be easily deposited on arbitrary surfaces, which makes them a
versatile tool for integration with various on-chip systems.
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