Quantum Langevin equations for semiconductor light-emitting devices and the photon statistics at a low-injection level

From the microscopic quantum Langevin equations (QLEs) we derive the effective semiconductor QLEs and the associated noise correlations which are valid at a low-injection level and in real devices. Applying the semiconductor QLEs to semiconductor light-emitting devices (LEDs), we obtain a formula for the Fano factor of photons that gives the photon-number statistics as a function of the pump statistics and several parameters of LEDs. Key ingredients are nonradiative processes, carrier-number dependence of the radiative and nonradiative lifetimes, and multimodeness of LEDs. The formula is applicable to the actual cases where the quantum efficiency η differs from the differential quantum efficiency η_d, whereas previous theories implicitly assumed η=η_d. It is also applicable to the cases where photons in each mode of the cavity are emitted and/or detected inhomogeneously. When η_d<η at a running point, in particular, our formula predicts that even a Poissonian pump can produce sub-Poissonian light. This mechanism for generation of sub-Poissonian light is completely different from those of previous theories, which assumed sub-Poissonian statistics for the current injected into the active layers of LEDs. Our results agree with recent experiments. We also discuss frequency dependence of the photon statistics.