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arXiv:2106.15242

Comment on "Thermal, Quantum Antibunching and Lasing Thresholds from Single Emitters to Macroscopic Devices"

Andrey A. Vyshnevyy, Dmitry Yu. Fedyanin

Published Jun 29, 2021• 6 citations
arXivHTMLPDF

Abstract

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-$\beta$ 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.

DOI
10.1103/physrevlett.128.029401

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Carroll et al. reply : [Reply to the comment on "Thermal, quantum antibunching and lasing thresholds from single emitters to macroscopic devices"]

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Reply to the Comment on "Thermal, quantum antibunching and lasing thresholds from single emitters to macroscopic devices". We deconstruct and address a comment to Carroll et al. [Phys Rev Lett 126, 063902 (2021)]

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Reply to the Comment on "Thermal, quantum antibunching and lasing thresholds from single emitters to macroscopic devices"

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We deconstruct and address a comment to Carroll et al. [Phys Rev Lett 126, 063902 (2021)] (PRL) that has been posted on arXiv appearing as two versions [arXiv:2106.15242v1] and [arXiv:2106.15242v2]. This comment claimed that a term in the model presented in the PRL had been incorrectly omitted and that, hence, the laser threshold predicted by the model in the PRL is unattainable. We show that the term in question was correctly neglected because it represents collective effects that are not observable in the devices modelled in the PRL. Moreover, even if this term were to be included, the laser threshold would still be present, contrary to what was claimed in the comment. We conclude that the model presented in PRL is correct and that its results are innovative and of wide application in laser physics and quantum optics.

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Semiconductor Lasers and Laser Dynamics•May 23, 2018•0 citations
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.
Starting from a fully quantized Hamiltonian for an ensemble of identical emitters coupled to the modes of an optical cavity, we determine analytically regimes of thermal, collective anti-bunching and laser emission that depend explicitly on the number of emitters. The lasing regime is reached for a number of emitters above a critical number-which depends on the light-matter coupling, detuning, and the dissipation rates-via a universal transition from thermal emission to collective anti-bunching to lasing as the pump increases. Cases where the second order intensity correlation fails to predict laser action are also presented.
We deconstruct and address a comment to Carroll et al. [Phys Rev Lett 126, 063902 (2021)] (PRL) that has been posted on arXiv appearing as two versions [arXiv:2106.15242v1] and [arXiv:2106.15242v2]. This comment claimed that a term in the model presented in the PRL had been incorrectly omitted and that, hence, the laser threshold predicted by the model in the PRL is unattainable. We show that the term in question was correctly neglected because it represents collective effects that are not observable in the devices modelled in the PRL. Moreover, even if this term were to be included, the laser threshold would still be present, contrary to what was claimed in the comment. We conclude that the model presented in PRL is correct and that its results are innovative and of wide application in laser physics and quantum optics.
We detail the derivation of nanolaser models that include coherent and incoherent variables and predict the existence of a laser threshold, irrespective of cavity size and emitter number, for both single- and multielectron systems. The growth in photon number in the lasing mode is driven by an increase in correlation between absorption and emission processes, leading to the onset of self-sustained stimulated emission (laser threshold), followed in turn by a correlation decrease and ending with the dominance of coherent emission. The first-order coherence g(1){g}^{(1)}g(1) steadily increases, as the pump grows towards the laser threshold value, and reaches unity at or beyond threshold. The transition toward coherent emission becomes increasingly sharp as the numbers of emitters and coupled electromagnetic cavity modes increase, continuously connecting, in the thermodynamic limit, the physics of nano- and macroscopic lasers at threshold. Our predictions are in remarkable agreement with experiments whose first-order coherence measurements have so far been explained only phenomenologically. A consistent evaluation of different threshold indicators provides a tool for a correct interpretation of experimental measurements at the onset of laser action.
We derive a model for a single-mode laser that includes all two-particle quantum correlations between photons and electrons. In contrast to the predictions of semiclassical models, we find that lasing takes place in the presence of quantum bistability between a nonlasing and a nonclassical coherent state. The coherent state is characterized by a central frequency and a finite linewidth and emerges with finite amplitude from a saddle-node bifurcation together with an unstable coherent state. Hence coherent emission in nanolasers originates through a mixing of lasing and nonlasing states. In the limit of a macrolaser with a large number of emitters and nonresonant modes, the laser threshold approaches the prediction of the semiclassical theory, but with the important difference that lasing can be achieved only in the presence of finite-size perturbations.
Traditionally, the active material in a laser is modelled as independent emitters, but in recent years it has become increasingly clear that radiative coupling between emitters can significantly change the characteristics of small lasers. Collective effects in free space such as superradiance have been studied extensively [1,2], but the effects of inter-emitter correlation in micro- and nano-cavities need further examination to be put on firm theoretical ground. Several studies of collective effects in nano-cavities have been made [3-6], but the theoretical models employed are intricate, and numerical methods are needed both to generate the dynamic equations and to solve them. We propose a model where the complexity is strongly reduced, allowing analytical solutions [7]. We consider a collection of identical two-level emitters interacting with a single cavity mode. We start from Maxwell-Bloch equations, but instead of making the typical adiabatic elimination of the polarization, we allow the polarization decay rate to be of the same magnitude or smaller than other decay rates. Hence, the traditional laser rate equations for the photon number and the population inversion must be supplemented by equations for the emitter-field correlation and the emitter-emitter correlation. This gives us four generalized laser rate equations, which we solve analytically in steady state. Comparing with the steady state results obtained from the traditional laser rate equations we see that inclusion of collective effects leads to a reduction of the photon number for small pump rates, similarly to what is found in [4]. From the generalized laser rate equations, we derive a measure of the strength of collective effects in terms of laser parameters: This describes the difference between results with and without inter-emitter correlations, and it goes smoothly to zero as we approach parameter values where the traditional laser rate equations become valid. To gain insight into the photon statistics of the laser, we construct dynamic equations for higher order correlations of operators. We derive an analytical expression for the zero-delay photon auto-correlation function, and for low pump rates we find that the interaction of emitters results in super-thermal values of the auto-correlation. This feature is observed in experiments and numerical models [4-5], and with our analytical expressions, we are able to pinpoint the parameter combinations for which the collective effects have the largest impact. Considering the same model in terms of the Fourier components of the operators, we find results for the photon number that agree well with the previous approach, while allowing computation of the linewidth. Thus, we can examine how emitter-emitter correlation affects the line broadening of the laser.