An ensemble of emitters can behave significantly different from its individual constituents when interacting coherently via a common light field. After excitation, collective coupling gives rise to an intriguing many-body quantum phenomenon, resulting in short, intense bursts of light: so-called superfluorescence 1 . Because it requires a fine balance of interaction between the emitters and their decoupling from the environment, together with close identity of the individual emitters, superfluorescence has thus far been observed only in a limited number of systems, such as atomic and molecular gases 2 and semiconductor crystals 37 , and could not be harnessed for applications. For colloidal nanocrystals, however, which are of increasing relevance in a number of opto-electronic applications 8 , the generation of superfluorescent light was precluded by inhomogeneous emission broadening, low oscillator strength, and fast exciton dephasing. Using caesium lead halide (CsPbX3, X = Cl, Br) perovskite nanocrystals 912 that are self-organized into highly ordered threedimensional superlattices 13,14 allows us to observe key signatures of superfluorescence: red-shifted emission with more than ten-fold accelerated radiative decay, extension of the first-order coherence time by more than a factor of four,

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The possibility of super'radiant spin-phonon relaxation is theoretically investigated in the framework of the Jacobsen-Stevens model for spins coupled to the lattice. The equations derived are similar to those of the optical case, so that under appropriate conditions superradiance should occur as in the optical case.As suggested by Dicke and recently observed by Skribanowitz e~al. , '3 a large assembly of molecules, initially all in the excited state, can emit all of its energy in a burst of highly directional optical radiation many orders of magnitude shorter than the single-molecule decay time T,. This effect, called superradiance, occurs because the molecules do not relax independently, but instead radiate in a collective mode of decay. The characteristic radiation rate in this process is enhanced from T, ' by a factor~X L/8m (= 10 in the experiments ), where no is the excitation density of molecules, X the optical wavelength, and L the length of the sample (L» &). The superradiant burst is preceded by a delay typically an order of magnitude longer than the duration of the burst and, under appropriate conditions, is accompanied by ringing. The basic condition for optical superradiance to occur is that all relaxation times must be long compared to the characteristic superradiant radiation time.The purpose of this paper is to explore the possibility that under appropriate conditions, superradiant behavior can also occur in the spin-phonon interaction process in paramagnetic crystals. The system would be prepared with all the spins inverted and the lattice unexcited. Then, under appropri3te conditions, 0T fluctuations of the lattice would trigger the collective de-excitation of the spins, leaving all the spins in the ground state and all the energy in lattice vibrations. As shall be shown, the characteristic rate T~' of this process would be enhanced from the spin-lattice relaxation rate T~~b y the factor no A. L/4m (which can be» 1), where now no is the spin density and X the acoustic wavelength.The condition for spin-phonon superradiance to occur is that all relaxation times for the spin and lattice processes must be long compared to the characteristic superradiant time T~. Under this condition, such a system should exhibit all the features of optical superradiance.Reported results of an experiment by Brya and Wagner on paramagnetic relaxation show a behavior which on the surface appears to be similar to that described in the previous paragraph.The time dependence of the spin-level population difference was observed after excitation of the paramagnetic transition to a negative spin temperature.It was found that after a delay much shorter than the normal spin-lattice relaxation time Tj, the spin system quickly released its energy to the resonant lattice modes ("phonon avalanche" ). In contrast to the anticipated behavior, however, no ringing was observed, and the rapid decay stopped as soon as the population of the two spin levels equalized.Since the emission rate in a superradiant process reaches its maximum wh...