Temperature-Dependent Evolution of Coherence, Entropy, and Photon Statistics in Photoluminescence

TL;DR

This work develops a temperature-aware photoluminescence framework that treats PL with a generalized Planck law incorporating a chemical potential, enabling a Planck-like description across pumping regimes. It derives an explicit spectral rate $R_{PL}(\nu,T,T_p)$ combining thermal and pump-driven contributions and provides a closed-form for the chemical potential $\mu(T)$ in terms of $QE$, bandgap, and pump: $ \mu(T) \approx kT \ln \left[1 + QE\left(\frac{\int R_{pump}(\nu,T_p)d\nu}{\int R_{BB}(\nu,T)d\nu}-1\right)\right]$. The analysis reveals a quasi-conserved emission rate at low $T$, a universal point at $T=T_p$ where $R_{PL}$ matches the pump, and a rapid shift to thermal behavior with $\mu$ and entropy changing accordingly, while coherence time and photon statistics vary smoothly. The framework enables temperature-tunable light sources with controllable coherence and photon statistics and discusses equilibrium with non-thermal pumping.

Abstract

Photoluminescence (PL) is a fundamental light-matter interaction in which absorbed photons are re-emitted, playing a key role in science and engineering. It is commonly modeled by introducing a non-zero chemical potential into Planck's law to capture its deviation from thermal emission. In this work, we establish, for the first time to our knowledge, a fundamental relationship that expresses the chemical potential as a function of temperature, material properties, and excitation conditions, enabling a treatment of PL analogous to Planck's law with thermal radiation. This formulation allows for the analysis of temperature-dependent PL properties, including spectral emission, entropy, temporal coherence, and photon statistics, capturing the transition from narrowband pump-induced to broadband thermal emission. Notably, we identify a temperature range where the emission rate is quasi-conserved, associated with the previously reported blueshift. This is followed by a rapid transition to thermal behavior, reflected in both the chemical potential and entropy. Conversely, the coherence time and photon statistics evolve smoothly across the entire temperature range. Alongside its scientific contribution, this framework provides a foundation for designing temperature-tunable light sources, enabling control over coherence length and photon statistics.

Figures (4)

• Figure 1: Illustration of the photon spectra and total photon rate as a function of temperature. (a) Evolution of the spectral photon rate with temperature of a PL material pumped by a blackbody at $T_p$. At $T \ll T_p$, the emission is pump-dominated and spectrally narrow. As $T$ increases, the rate at higher frequencies increases at the expense of the lower (blueshift), maintaining a quasi-conserved rate. Towards the universal temperature $(T=T_p)$, the emission becomes thermally dominated, increasing across all frequencies. (b) Total photon rate versus temperature (blue). At the universal temperature, the total photon rate equals the total absorbed pump rate (green).
• Figure 2: Chemical potential and entropy vs temperature, where $T_p=750 [K]$ and $v_{bg} = 2 \times 10^{14}$ [Hz]. (a) Chemical potential of PL emission (blue) and corresponding emission by a similar unpumped material, emitting thermal radiation (dashed). $\mu >0$ signals electronics excitation above thermal, and is reduced with temperature, as zero signals the point of equilibrium (universal point). For $T>T_p$, negative $\mu$ approaches the thermal emission line at high temperatures. (b) Entropy of PL emission (blue) and unpumped PL material thermal emission (dashed). The entropy increases as thermalization takes place, eventually reaching the universal temperature where the entropy is maximal. Beyond this point, the entropy is reduced, behaving as a thermal emitter.
• Figure 3: Results for a PL body at different temperatures, where $T_p = 1000$ [k], $\nu_{bg} = 10^{14}$ [Hz], corresponding with 1.5 microns. (a) Normalized temporal coherence function of PL material for different relative temperatures, where $T/T_p=1$ is the universal temperature and the coherence time equals the full-width-half-maximum. (b) Coherence time vs. relative temperature. Inversely proportional to the temperature. At low temperatures, PL emits with high intensity and narrow bandwidth, therefore having a longer coherence time and length.
• Figure 4: Photon statistics comparison between PL body at different temperatures and a constant thermal pump. BE statistics is demonstrated at four different regimes: $T \ll Tp$, $T < T_p$, $T = T_p$ (universal temperature), and $T > Tp$. (c) is the equilibrium point since the material and pump share an identical distribution.