Using a fully stochastic numerical scheme, we theoretically investigate the behavior of a nanolaser in the lasing transition region under the influence of intensity feedback. Studying the input-output curves as well as the second-order correlations for different feedback fractions, we obtain an insight into the role played by the fraction of photons reinjected into the cavity. Our results reveal that the transition shrinks and moves to lower pump values with the feedback strength, and irregular pulses can be generated when the feedback is sufficiently large. The interpretation of the observation is strengthened through the comparison with the temporal traces of the emitted photons and with the radio-frequency power spectra. These results provide insight into the physics of nanolasers as well as validating the use of the second-order autocorrelation as a sufficient tool for interpretation of the dynamics. This confirmation offers a solid basis for the reliance on autocorrelations in experiments studying the effects of feedback in nanodevices.
Using a fully stochastic numerical scheme, we theoretically investigate the behavior of a nanolaser in the lasing transition region under the influence of intensity feedback. Studying the input-output curves as well as the second-order correlations for different feedback fractions, we obtain an insight into the role played by the fraction of photons reinjected into the cavity. Our results reveal that the transition shrinks and moves to lower pump values with the feedback strength, and irregular pulses can be generated when the feedback is sufficiently large. The interpretation of the observation is strengthened through the comparison with the temporal traces of the emitted photons and with the radio-frequency power spectra. These results provide insight into the physics of nanolasers as well as validating the use of the second-order autocorrelation as a sufficient tool for interpretation of the dynamics. This confirmation offers a solid basis for the reliance on autocorrelations in experiments studying the effects of feedback in nanodevices.
It is shown that the quantum intensity noise of a single-emitter nanolaser can be accurately computed by adopting a stochastic interpretation of the standard rate equation model. The only assumption made is that the emitter excitation and photon number are stochastic variables with integer values. This extends the validity of rate equations beyond the mean-field limit and avoids using the standard Langevin approach, which is shown to fail for few emitters. The model is validated by comparison to full quantum simulations of the relative intensity noise and second-order intensity correlation function, g^{(2)}(0). Surprisingly, even when the full quantum model displays vacuum Rabi oscillations, which are not accounted for by rate equations, the intensity quantum noise is correctly predicted by the stochastic approach. Adopting a simple discretization of the emitter and photon populations, thus, goes a long way in describing quantum noise in lasers. Besides providing a versatile and easy-to-use tool for modeling emerging nanolasers, these results provide insight into the fundamental nature of quantum noise in lasers.
We study the stationary photon output and statistics of small lasers. Our closed-form expressions clarify the contribution of collective effects due to the interaction between quantum emitters. We generalize laser rate equations and explain photon trapping: a decrease of the photon number output below the lasing threshold, derive an expression for the stationary cavity mode autocorrelation function g2, which implies that collective effects may strongly influence the photon statistics. We identify conditions for coherent, thermal and superthermal radiation, the latter being a unique fingerprint for collective emission in lasers. These generic analytical results agree with recent experiments, complement numerical results, and provide insight into and design rules for nanolasers.
Laser amplification kicks off with just a few photons in the cavity, a regime where standard additive Langevin noise cannot capture the real fluctuations. To understand how this amplification begins, we map out a single-mode photon-carrier model that links macroscopic lasers directly into nanolasers. Stationary rate equations reveal that stimulated emission takes over across a tight pump window. In nanolasers, however, spontaneous processes coexist across a much broader range. Because steady-state averages miss the individual events that lead to the macroscopic photon number, we frame the stochastic dynamics around the actual discrete transitions, deriving a state-dependent diffusion approximation. The destruction of an excited carrier concurrent with the generation of a photon creates a negative photon-carrier cross-correlation. This unavoidably leads to a correlated, multiplicative Gaussian-noise framework. We track the resulting discrete dynamics with an established, event-based Laser Markov Chain (LMC) which stochastically samples pump, relaxation, emission, and cavity-loss events keeping all physical quantities as non-negative integers. Comparing its predictions to those of our multiplicative model proves that the latter reproduces excellently well the statistical behavior of photons and carrier, as long as a continuous diffusion description holds. Once the emission drops into the few-photon regime, however, the discrete model becomes indispensable. On the other hand, at large photon counts relative fluctuations shrink to a level where the state-dependence of the diffusion matrix can be ignored, to leading order, thus returning to the classic Langevin description of the system's stochastic dynamics.
The laser has evolved from table size apparatuses to truly nano sized devices, in much the same way that computer chips have been continuously minimized. The few-emitter nanolaser represents an extreme in terms of size. The emitters can be either atoms or quantum dots, and they are coupled to a high-Q optical cavity. This system has previously been studied for the one and two emitters case, see e.g. [1, 2]. The emitters and cavity are modelled as two-level systems and a harmonic oscillator, respectively. The coupled system is modelled using the Jaynes-Cummings Hamiltonian and the two-level systems are pumped incoherently by a rate P. Solutions are found using the corresponding master equation. However, with cavity populations exceeding 100 and several emitters, the dimension of the Hilbert space of the system becomes too large to handle efficiently on a conventional computer. E.g. for four emitters and 100 photon states the density matrix has more than 2.5 × 106 elements. We have been able to simplify the problem significantly by adiabatically eliminating the photon-assisted polarizations and the correlations between emitters and cavity [3]. This results in a set of rate equations for the population of the cavity, na, and the occupation of the emitters. Fig. 1a) shows na for up to 4 emitters coupled to a cavity, as a function of the pumping rate P. The figure also shows results from a full master equation solution, and the correspondence is very good for large values of P. In Fig. 1b) the second order correlation functions g(2)(0) are shown, as obtained using the full model. The second order correlation functions become 1 for sufficiently large values of P signifying the onset of lasing. However, for larger values of P the laser quenches and the emission becomes chaotic (g(2)(0) = 2). This quenching effect is well-known in two-level models [3], but might not be realistic for semiconductor quantum dots. A proposed application for these nanolasers is in on-chip optical interconnects. In this application the modulation bandwidth is important. We have calculated the modulation bandwidth using a small-signal analysis of the rate equations, which has the advantage compared the full model of yielding a semianalytical result. The results are shown in Fig. 1c). The nanolaser has a high bandwidth, which is mostly limited by the cavity lifetime (/ 1/), but shows a prominent drop during the lasing regime. The modulation bandwidth again becomes limited by as the output becomes chaotic.
Do Nanolasers Benefit From a High β-Factor?
The realization of high-β 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-Q 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 factors in terms of power reduction, as a significant amount of stimulated emission is required to quieten the spontaneous emission noise.