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arXiv:1709.08200

Collective Effects in Nanolasers Explained by Generalized Rate Equations

Игорь Евгеньевич Проценко, Emil C. André, Alexander V. Uskov, Jesper Mørk, Martijn Wubs

Published Sep 24, 2017• 7 citations
arXivHTMLPDF

Abstract

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 $g_2$, 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.

DOI
10.48550/arxiv.1709.08200

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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 g2g_2g2​, 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.
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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.
The random nature of spontaneous emission leads to unavoidable fluctuations in a laser's output. This is often included through random Langevin forces in laser rate equations, but this approach falls short for nanolasers. In this paper, we show that the laser quantum noise can be quantitatively computed for a very broad class of lasers by starting from simple and intuitive rate equations and merely assuming that the number of photons and excited electrons only takes discrete values. While the approach has seen previous success, we here derive it rigorously from an open quantum system master equation, whereas it was previously introduced only on phenomenological grounds. We further show that in the many-photon limit, the model simplifies to Langevin equations. We perform an extensive comparison of different approaches for computing quantum noise in lasers, identifying the best approach for different system sizes, ranging from nanolasers to macroscopic lasers, and different levels of excitation, i.e., cavity photon number. In particular, we show that below the laser threshold, stochastic fluctuations in the numerical solution to the Langevin equations can drive populations to unphysical negative values, requiring the introduction of population bounds, which in turn skew the noise statistics, leading to inaccuracies. The Laser Markov Chain model, on the other hand, is accurate for all pump values and laser sizes when collective emitter effects are excluded.
Using two equivalent methods, the Heisenberg-Langevin and the master equation, we investigate the properties of a single-emitter nanolaser: an incoherently pumped single two-level system interacting with a single cavity mode of finite finesse. We are interested in a stationary generation regime when the mean number of photons inside the laser cavity is much larger than unity. In this case we linearize the Heisenberg-Langevin equations around the stationary classical solution and obtain analytical results for the linewidth, the quadrature fluctuation spectra, and the intracavity Mandel QQQ parameter. As the second method we use the master equation for the density operator of the laser field. Using the coherent-state representation of the density matrix we obtain the stationary solution for the Glauber PPP function of the laser mode and investigate the quantum statistics of the laser field and its threshold behavior. We compare the results of both methods with numerical simulations and find very good agreement. In the end we use our theoretical results for analyzing experiment with single-ion nanolasers.
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.