This is a native 2D counterpart to the 3D single-electron phonon-bottleneck simulator: instead of following one carrier down the ladder, it runs a real ensemble of 240 independent electrons through the same continuous-time Markov process and lets you watch the population statistics — the thing an experimentalist actually measures (an ensemble-averaged photoluminescence decay), not a single trajectory.
Per-carrier hazard: R_ph(ΔE) = R0 · 1 / [1 + ((ΔE − ℏω_LO)/Γ)²] (Lorentzian resonance)
R_Auger = R0,A · η (η = Auger coupling)
R = R_ph + R_Auger, transition prob. per Δt: p = 1 − e^(−R·Δt)
Since every rung has the same ΔE, R is the same at every level, so the number of
completed transitions by time t is a Poisson process: K(t) ~ Poisson(R·t), and
level(t) = max(8 − K(t), 0). This gives an exact analytic prediction for the
ensemble-mean level, ⟨n(t)⟩ = Σ_{k=0}^{7} (8−k)·P(K(t)=k), plotted live (dashed)
against the actual simulated ensemble mean (solid) — if the Monte Carlo swarm and
the closed-form Poisson curve track each other, the simulation is behaving correctly.
- Waterfall (left) — every dot is one independently-simulated electron; column height = current level. Watch the whole swarm cascade and pile up at the band edge, or stall mid-ladder when off resonance.
- Resonance curve (top right) — the real Lorentzian R_ph(ΔE) plotted against level spacing, with a marker at the current ΔE. This is the same curve every carrier's hazard rate is sampled from — the reason the swarm speeds up or stalls.
- ⟨n(t)⟩ verification (bottom right) — live ensemble mean (cyan) vs. the exact Poisson-process prediction (white dashed) computed from the rate captured at the last re-excite. Close agreement is the numerical proof that the stochastic simulation correctly reproduces the analytic bottleneck theory.
- Auger coupling — models carrier density; turning it up adds a phonon-independent hazard so the swarm keeps draining to the band edge even when ΔE and ℏω_LO are badly detuned.
Real dots (Bockelmann & Bastard 1990; Benisty et al. 1991) show exactly this population-level bottleneck in time-resolved photoluminescence, and the Auger-assisted escape route (Klimov et al., Guyot-Sionnest et al.) is what keeps real quantum-dot ensembles cooling on picosecond timescales despite it.