Modeling Quantum Noise in Nanolasers using Markov Chains

Abstract

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. The success of the model is explained by showing that it constitutes a Markov chain, which can be derived from the full master equations. We 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 find that the numerical solution to the Langevin equations is inaccurate below the laser threshold, while the laser Markov chain model, on the other hand, is accurate for all pump values and laser sizes when collective emitter effects are excluded.

Figures (7)

• Figure 1: Sketch of the model considered: A cavity containing $n_0$ emitters, which experience various decay and decoherence processes, labeled with $\gamma_i$ (see Table \ref{['tab:variables']} for the definitions). We show the system in various regimes together with the appropriate model for that regime. MEs capture quantum coherence, but their feasibility for numerical simulation is limited to microscopic cavities. The LMC describes quantized populations well, while LRE are more efficient for large systems where the laser noise is Gaussian (white noise). In the figure $\expval{\sigma_i a^\dagger}$ denotes the average emitter-photon polarization (see Eq. \ref{['eq:polarization']}), $\epsilon$ is the fractional population change due to a stochastic event (Eq. \ref{['eq:tau_leaping_step']}), and $a_j(\mathbf{x})$ represent the rates of the stochastic events (Eq. \ref{['eq:cme']}).
• Figure 2: The photon number $n_p$, second-order correlation function $g^{(2)}(0)$, and RIN as function of pump rate for different number of emitters $n_0$ as computed by the analytical small-signal solution to the LRE, the numerical solution to the LRE, the trajectories of the LMC computed using Gillespie First Reaction Method or tau-leaping, and finally for $n_0=1$ and $n_0=10$, the solution obtained using the ME. The parameter values used are light-matter coupling $g=0.1\ \mathrm{ps}^{-1}$, cavity decay rate $\gamma_c = 0.04 \ \mathrm{ps}^{-1}$, pure dephasing rate of the emitters $\gamma_D=1 \ \mathrm{ps}^{-1}$, and a varying non-radiative decay rate for the different emitter numbers $n_0$: $\gamma_A = [0.0, 0.263941, 1.51458, 19.4566] \ \mathrm{ps}^{-1}$. The choice of parameters ensures $\beta = 1/n_0$ as appropriate for investigating the changes from microscopic to macroscopic lasers.
• Figure 3: Overview of the validity of methods to compute laser quantum noise as a function of the number of photons and emitters in the laser. The boxes of different colors indicate the areas in which the different methods are valid or computationally feasible. The crossed-out area marks a combination of photon and emitter numbers, which with the parameters used in this paper, were not obtainable.
• Figure 4: The relative error $\delta$ (largest relative deviation of the three quantities $\expval{n_p}$, $g^{(2)}(0)$, and $\mathrm{RIN}$ over 5 runs) as a function of simulation wall-clock time for different emitter numbers $n_0$ and pump rates $\gamma_P$. The vertical dashed lines indicate the simulation time used for the ME calculations (only for $n_0=1$ and $n_0=10$). For $n_0=1$ and $n_0=10$, the ME results are used as ground truth, while for $n_0=100$ and $n_0=10000$, the analytical Langevin solution is used. Parameters are the same as in Fig. \ref{['fig:comparisons']}.
• Figure 5: The imaginary part of the off-diagonal element $\rho_{n_p+1,g;n_p,e}$ as calculated using the ME (solid lines), with superimposed dots corresponding to the approximation $\rho_{n_p+1,g;n_p,e} \approx \left( (n_p-1) / \sqrt{n_p^2+n_p} \right ) \rho_{n_p,g;n_p-1,e}$. For the approximation, the previous matrix element $\rho_{n_p,g;n_p-1,e}$ is obtained from the ME. Parameters are the same as in Fig. \ref{['fig:comparisons']} with $n_0 = 1$.