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Applied Physics Letters

Threshold behavior of super-radiance under continuous pumping

C. T. Lee

Published Feb 15, 1983• 0 citations
Publisher

Abstract

Recognizing that the super-radiance emission rate is proportional to the product of the number of cooperative atoms in the upper level and that in the lower level, we propose simple deterministic and stochastic equations to describe the evolution of the super-radiant atomic system pumped continuously at a constant rate. The steady-state solutions of these equations display threshold behavior very similar to the laser. This provides yet another example of nonequilibrium phase transition of the second order.

DOI
10.1063/1.93935

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Recognizing that the super-radiance emission rate is proportional to the product of the number of cooperative atoms in the upper level and that in the lower level, we propose simple deterministic and stochastic equations to describe the evolution of the super-radiant atomic system pumped continuously at a constant rate. The steady-state solutions of these equations display threshold behavior very similar to the laser. This provides yet another example of nonequilibrium phase transition of the second order.
Super-radiance is a striking phenomenon resulting from the collective interaction of emitters with light, and it is fundamentally related to the appearance of long-range correlations between the dipole moments of the emitters. As such, it represents a distinct phase of dissipative many-body systems compared to the case of independent emitters, similarly to how ferromagnetism stands out in magnetic materials in contrast to paramagnetism. In this work we address the conditions under which the transition to super-radiance can occur in ensembles of emitters subject to dephasing and individual decay -- a situation which is particularly relevant to the case of emitters in the solid state. The simplifying assumption of an ensemble of permutationally invariant emitters allows for the efficient solution of both the dissipative dynamics after a pulsed excitation, as well as of the steady state under incoherent pumping. This exact solution allows us to benchmark a truncated cumulant expansion approach, which can give predictions for arbitrarily big system sizes. We show that super-radiance is fundamentally robust to sizable dephasing and non-radiative decay rates, both under a pulsed excitation, as well as under continuous pumping. This robustness is the result of the collective acceleration effect of super-radiant emission with respect to the individual coupling to a local environment. We establish the universal critical scaling laws at the transition between normal radiance and super-radiance; and we show that, in the super-radiant phase, significant finite-size crossovers can be observed before reaching the asymptotic scaling regime. Our results pave the way for future experiments to provide a quantitative characterization of the scaling properties of super-radiance, seen as a distinct non-equilibrium many-body phase in ensembles of incoherently pumped emitters.
We consider the problem of two-photon cooperative emission in systems of two-level atoms. Two physically distinct regimes are analyzed. First, we investigate the case of a small number of atoms. We study the evolution of two-photon super- and sub-radiant states and associated two-photon spectra. Second, we investigate the problem of a constant density of atoms confined to a spherical volume. We analyze separately the cases in which the radius is small or large in comparison to the resonant wavelength.
General properties of super-radiance--cooperative spontaneous emission of multi-atomic systems are explained on the basis of some simple models. First we consider spontaneous emission of two atoms separated from each other by a distance less than the wavelength. Then the kinetics of super-radiance of N-atomic system in a small volume are described. The main attention we put into super-radiance photon statistics. We show the Poissonian statistics for the main part of super-radiance pulse and demonstrate the special conditions when it can be Sub-Poissonian one.
A model of three-level laser, in which the pumping excites the atoms from the lower lasing level, is considered. The statistical properties of the field are calculated using full quantum treatment. It is shown that apart from threshold there is additional critical value of the pump rate, at which the laser field behaves in a phase-transition-like manner: the photon number drops down, the Fano factor has a strong peak, the relative photon number variance grows enormously, and the second-order coherence function jumps to 2, which is typical for thermal light.
We arrive at transition probabilities for a laserlike system by neglecting wave properties of matter and radiation, and treating the problem, in an analogy to chemical systems, entirely in a number representation. Thus, for example, stimulated emission is idealized as a collision between a photon and an excited atom and the transition rate is proportional to the amount of each species present in the cavity. In all, transition probabilities for stimulated emission, absorption of radiation, cavity loss, atomic deexcitation (other than stimulated emission), and pumping of atoms are included in the model. The resulting stochastic master equation analogous to that employed for the discussion of fluctuations in chemical systems is then used to obtain the deterministic behavior of atoms and field. This agrees with that of the quantum mechanical laser model of Casagrande and Lugiato; thus the rate parameters of the stochastic model may be related to the parameters of their model. Returning to the master equation, we project out the atomic variable and use adiabatic elimination to arrive at a closed, single-variable equation for P(n), the probability of having n photons in the cavity. The stationary solution of this equation is obtained and graphs of (n) and log[(n2)/(n)2] versus the pumping parameter show close agreement with the microscopic results of the Casa-grande-Lugiato model. A similar discussion of the fluctuations can also be given for a laser with saturable absorption. It shows a first-order phase transition. A comparison with the microscopic results of Lugiato et al. is outlined. Again a remarkable agreement of the stochastic approach via the master equation and the microscopic theory is obtained.
Perturbation theory is used to show that that super-radiance as envisaged by Dicke, in which the radiation rate goes as N 2 , the square of the number of atoms, can be interpreted in the same terms as classical diffraction theory. The rate goes as N 2 from samples small in three dimensions compared with a wavelength but goes essentially as N from larger samples. There is a natural distinction between coherent and incoherent contributions to the total radiation rate from larger samples prepared (supposedly by an incident pulse) in Dicke states phased with a vector k.