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Physical Review A

Photon statistics of a cavity-QED laser: A comment on the laser–phase-transition analogy

Perry R. Rice, H. J. Carmichael

Published Nov 1, 1994• 291 citations
Publisher

Abstract

We develop a birth-death model for a cavity QED laser and analyze the dependence of the lasing threshold on the fraction, \ensuremath{\beta}, of spontaneous emission directed into the laser mode. We define threshold in terms of the Fano factor for the intracavity phonon number. We emphasize the role of \ensuremath{\beta} as a parameter characterizing the ``system size'' and show that the concept of laser threshold is well-defined in the ``thermodynamic'' limit, ${\mathrm{\ensuremath{\beta}}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}\ensuremath{\infty}. An ideal cavity-QED laser operates far outside this limit and therefore, in contrast to a conventional laser, is not a threshold device. We present numerical results showing how this difference affects the noise properties of the device.

DOI
10.1103/physreva.50.4318

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We develop a birth-death model for a cavity QED laser and analyze the dependence of the lasing threshold on the fraction, \ensuremath{\beta}, of spontaneous emission directed into the laser mode. We define threshold in terms of the Fano factor for the intracavity phonon number. We emphasize the role of \ensuremath{\beta} as a parameter characterizing the system size'' and show that the concept of laser threshold is well-defined in the thermodynamic'' limit, \ensuremathβ\ensuremath−1{\mathrm{\ensuremath{\beta}}}^{\mathrm{\ensuremath{-}}1}\ensuremathβ\ensuremath−1\ensuremath{\rightarrow}\ensuremath{\infty}. An ideal cavity-QED laser operates far outside this limit and therefore, in contrast to a conventional laser, is not a threshold device. We present numerical results showing how this difference affects the noise properties of the device.
We consider a laser model consisting of a single four-level or three-level atom, an optical cavity, and an incoherent pump. Results for photon statistics for varying pump levels are obtained using a quantum trajectory algorithm. In particular, we calculate the mean photon number, Fano factor (which is the variance over the mean). We examine that the behavior of the single-atom device as \ensuremath{\beta}, the fraction of spontaneous emission into the lasing mode, is varied. Typical values considered for \ensuremath{\beta} are $0.01<\ensuremath{\beta}<1.0.$ We find that for large enough \ensuremath{\beta}, lasing action, with properties similar to those predicted by semiclassical theories that factorize atom-field correlations and use a small-noise approximation, can occur. Squeezing can occur as \ensuremath{\beta} is increased. There is no evidence of a sharp phase transition from weakly excited thermal light to coherent light at a particular pump power. This is consistent with work on many-atom lasers with \ensuremath{\beta} values in the range considered here. As \ensuremath{\beta} is increased, the output goes from quasithermal light to coherent and finally to squeezed light, progressing into a fully quantum-mechanical regime. We also consider the effects of cavity damping and spontaneous emission rates on these results.
Nanocavity lasers are commonly characterized by the spontaneous coupling coefficient β that represents the fraction of photons emitted into the lasing mode. While β is conventionally discussed in semiconductor lasers where the photon lifetime is much shorter than the carrier lifetime (class-B lasers), little is known about β in atomic lasers where the photon lifetime is much longer than the other lifetimes and only the photon degree of freedom exists (class-A lasers). We investigate the impact of the spontaneous coupling coefficient β on lasing properties in the class-A limit by extending the well-known Scully–Lamb master equation. We demonstrate that in the class-A limit all the photon statistics are uniquely characterized by β and that the laser phase transition-like analogy becomes transparent. In fact, β perfectly represents the “system size” in phase transition. Finally, we investigate the laser-phase transition analogy from the standpoint of a quantum dissipative system. Calculating a Liouvillian gap, we clarify the relation between β and the continuous phase symmetry breaking.
Do Nanolasers Benefit From a High β\betaβ-Factor?
The realization of high-β\betaβ lasers is one of the prime applications of cavity-QED promising ultra-low thresholds, integrability and reduced power consumption in the field of \textit{green photonics}. In such nanolasers spontaneous emission can play a central role even above the threshold. By going beyond rate-equation approaches, we revisit the definition of a laser threshold in terms of the input-output characteristics and the degree of coherence of the emission. We demonstrate that there are new regimes of cavity-QED lasing, realized e.g. in high-QQQ nanolasers with extended gain material, for which the two can differ significantly such that coherence is reached at much higher pump powers than required to observe the thresholdlike intensity jump. Against the common perception, such devices do not benefit from high-β\betaβ factors in terms of power reduction, as a significant amount of stimulated emission is required to quieten the spontaneous emission noise.
In a recent Letter [Phys. Rev. Lett. 126, 063902 (2021)], M. A. Carroll et al. derived a model to analytically determine regimes of thermal, collective anti-bunching, and laser emission for emitters in a cavity. According to their model, nanolasers exhibit a distinct threshold at which the coherent laser field emerges from a bifurcation at a finite pump rate. The amplitude of this coherent field increases with a further increase in the pump rate. Such a behavior contrasts the usual view of the transition to lasing in single-mode high-β\betaβ nanolasers, according to which strong spontaneous emission into the lasing mode results in a smooth transformation from the thermal state to the coherent state as the pump rate increases. Here, we demonstrate that the authors have ignored important terms in the equations of their model, which caused the bifurcation and the ideally monochromatic field to emerge at a finite pump rate.
In the literature two different simplified expressions exist for the spontaneous emission factor \ensuremath{\beta}, being the fraction of spontaneous emission radiated into a specific optical mode. These expressions are valid in two mutually exclusive regimes. By treating the coupling between one discrete cavity mode and the outside optical continuum quasi one dimensionally in the spirit of Fano [Phys. Rev. 124, 1866 (1961)], we extend the validity range of both expressions and show that they lead to essentially the same result. The relevance and limitation of this result are discussed. \textcopyright{} 1996 The American Physical Society.
The laser--phase-transition analogy is considered for a system of identical emitters interacting with a cavity mode. As in the case of equilibrium phase transitions, the thermodynamic limit is shown to be an essential condition for the appearance of an abrupt transition, with a well-defined threshold. The proof is discussed first in the simpler formalism of the rate equation and then at the higher level of the master equation for probabilities. The latter is derived from the von Neumann--Lindblad evolution of the density operator. In the thermodynamic limit and for two-level emitters the master equation becomes a two-dimensional dynamic flow with two possible fixed points, related to the normal and the lasing regimes, respectively. The master equation analysis confirms the threshold value deduced from the rate equation, but also provides further evidence of lasing by showing that all photonic autocorrelation functions converge to unity.