The semiconductor luminescence equations (SLEs) describe luminescence of semiconductors resulting from spontaneous recombination of electronic excitations, producing a flux of spontaneously emitted light. This description established the first step toward semiconductor quantum optics because the SLEs simultaneously includes the quantized light–matter interaction and the Coulomb-interaction coupling among electronic excitations within a semiconductor. The SLEs are one of the most accurate methods to describe light emission in semiconductors and they are suited for a systematic modeling of semiconductor emission ranging from excitonic luminescence to lasers. Due to randomness of the vacuum-field fluctuations, semiconductor luminescence is incoherent whereas the extensions of the SLEs include the possibility to study resonance fluorescence resulting from optical pumping with coherent laser light. At this level, one is often interested to control and access higher-order photon-correlation effects, distinct many-body states, as well as light–semiconductor entanglement. Such investigations are the basis of realizing and developing the field of quantum-optical spectroscopy which is a branch of quantum optics.
Starting point The derivation of the SLEs starts from a system Hamiltonian that fully includes many-body interactions, quantized light field, and quantized light–matter interaction. Like almost always in many-body physics, it is most convenient to apply the second-quantization formalism. For example, a light field corresponding to frequency ω {\displaystyle \omega } is then described through Boson creation and annihilation operators B ^ ω † {\displaystyle {\hat {B}}_{\omega }^{\dagger }} and B ^ ω {\displaystyle {\hat {B}}_{\omega }} , respectively, where the "hat" over B {\displaystyle B} signifies the operator nature of the quantity. The operator-combination B ^ ω † B ^ ω {\displaystyle {\hat {B}}_{\omega }^{\dagger }\,{\hat {B}}_{\omega }} determines the photon-number operator. When the photon coherences, here the expectation value ⟨ B ^ ω ⟩ {\displaystyle \langle {\hat {B}}_{\omega }\rangle } , vanish and the system becomes quasistationary, semiconductors emit incoherent light spontaneously, commonly referred to as luminescence (L). (This is the underlying principle behind light-emitting diodes.) The corresponding luminescence flux is proportional to the temporal change in photon number,
L ( ω ) = ∂ ∂ t ⟨ B ^ ω † B ^ ω ⟩ = 2 R e [ ∑ k F ω ⋆ Π k , ω ] . {\displaystyle \mathrm {L} (\omega )={\frac {\partial }{\partial t}}\langle {\hat {B}}_{\omega }^{\dagger }{\hat {B}}_{\omega }\rangle =2\,\mathrm {Re} \left[\sum _{\mathbf {k} }{\mathcal {F}}_{\omega }^{\star }\,\Pi _{\mathbf {k} ,\omega }\right]\,.}
As a result, the luminescence becomes directly generated by a photon-assisted electron–hole recombination,
Π k , ω ≡ Δ ⟨ B ^ ω † P ^ k ⟩ {\displaystyle \Pi _{\mathbf {k} ,\omega }\equiv \Delta \langle {\hat {B}}_{\omega }^{\dagger }{\hat {P}}_{\mathbf {k} }\rangle }
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