Abstract We present a model of nanolasers that includes the effect of all two-particle quantum correlations. We find that the lasing starts with a finite coherent amplitude and that lasing and non-lasing solutions coexist.
We compare a novel quantum mechanical model of lasers, which includes all two-particle correlations, with the Coherent-Incoherent Model (CIM), where truncation eliminates these effects. Numerical results for the simple case of identical single-electron quantum dots are presented for two cases: only the laser mode is coupled to the quantum dots; the coupling is extended to non-resonant modes. We find coexistence between non-lasing and lasing states, together with a minimum number of photons required to initiate the laser action. The correlations introduce a non-zero variance in the field, which is otherwise strictly zero in the absence of interparticle correlations.
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) 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.
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 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.
The dominant role of quantum fluctuations in determining the steady-state and transient response of a laser is demonstrated when there is a small number of particles in the system. In this regime, quantum fluctuations are found to suppress the lasing threshold and create a non-Poisson probability distribution for n discrete excited electronic states and s discrete photons. The correlation between n and s damps the averaged dynamic response of laser emission. Random walk calculations verify the master equation predictions and are used to connect to systems containing larger numbers of particles.
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.