Modeling Quantum Noise in Nanolasers using Markov Chains

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.

💡 Research Summary

This paper addresses the fundamental problem of quantum noise in lasers, especially in the regime of nanolasers where the discrete nature of photons and excited carriers cannot be ignored. Traditional approaches incorporate random Langevin forces into continuous laser rate equations, which work well for macroscopic lasers with large photon numbers but break down for nanolasers. Below threshold, stochastic fluctuations in the Langevin formulation can drive populations to unphysical negative values; imposing artificial population bounds then distorts the noise statistics, leading to inaccurate predictions of linewidth and intensity fluctuations.

The authors start from a full open‑system master equation (ME) describing N identical two‑level emitters coupled to a single cavity mode via the Jaynes‑Cummings Hamiltonian, including cavity loss, background decay, pumping, and pure dephasing through Lindblad terms. By applying the standard cumulant expansion and adiabatic elimination of the off‑diagonal polarization, they recover the familiar semiclassical laser rate equations for the average photon number (n_p) and excited‑emitter number (n_e). However, these approximations discard quantum correlations and assume continuous populations, which is precisely the limitation they aim to overcome.

The core contribution is a rigorous derivation of a stochastic Laser Markov Chain (LMC) directly from the master equation. By projecting the ME onto the diagonal elements of the density matrix—i.e., the joint probability (P(n_p,n_e)) of having (n_p) photons and (n_e) excited emitters—the authors obtain a master‑type equation of the form

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