[Paper Review] Collective Effects in Nanolasers Explained by Generalized Rate Equations

Igor E. Protsenko, Emil C. André+3 more|arXiv (Cornell University)|Sep 24, 2017

Quantum Information and Cryptography16 references6 citations

TL;DR

This paper introduces generalized laser rate equations (GLRE) that incorporate collective effects from emitter-emitter interactions in nanolasers, explaining photon trapping and superthermal photon statistics. The model analytically predicts that below threshold, photon output decreases due to superradiance-induced storage, with $ g_2 > 2 $ as a signature of collective emission.

ABSTRACT

We study the stationary photon output and statistics of small lasers. Our closed-form expressions clarify the contribution of collective effects due to the interaction between quantum emitters. We generalize laser rate equations and explain photon trapping: a decrease of the photon number output below the lasing threshold, derive an expression for the stationary cavity mode autocorrelation function $g_2$, which implies that collective effects may strongly influence the photon statistics. We identify conditions for coherent, thermal and superthermal radiation, the latter being a unique fingerprint for collective emission in lasers. These generic analytical results agree with recent experiments, complement numerical results, and provide insight into and design rules for nanolasers.

Motivation & Objective

• To explain collective effects in nanolasers, particularly photon trapping and non-classical photon statistics, beyond standard laser theory.
• To derive analytical expressions for stationary photon output and second-order correlation function $ g_2 $, identifying conditions for coherent, thermal, and superthermal emission.
• To establish design rules for nanolasers that exploit collective effects to generate non-classical light with enhanced bunching.
• To clarify the role of cavity quality factor $ Q $, emitter density, and coupling strength in enabling superradiance-like behavior in lasers.

Proposed method

• Derives generalized laser rate equations (GLRE) by extending standard rate equations to include two collective variables: emitter-field and emitter-emitter correlations.
• Uses a two-level emitter model coupled to a single-mode cavity, with Hamiltonian-based Maxwell-Bloch equations including Langevin noise terms.
• Applies adiabatic elimination of polarization to derive closed-form expressions for stationary photon number $ n $ and population inversion $ ar{ ho} $, incorporating collective enhancement.
• Introduces a collective parameter $ C(P) $ that quantifies the strength of emitter-emitter interactions, modifying the effective lasing threshold.
• Computes the second-order autocorrelation function $ g_2 $ analytically, showing $ g_2 > 2 $ as a fingerprint of superthermal emission.
• Validates results against numerical simulations and experimental observations, particularly for low-Q and high-density nanolaser systems.

Experimental results

Research questions

• RQ1 What conditions lead to photon trapping—i.e., a decrease in stationary photon output below the conventional lasing threshold—due to collective effects?
• RQ2 How do emitter-emitter interactions modify the photon statistics in nanolasers, and what is the role of the second-order correlation function $ g_2 $ in identifying collective emission?
• RQ3 Under what parameter regimes does the system exhibit superthermal emission with $ g_2 > 2 $, and how does this differ from standard laser behavior?
• RQ4 How do cavity quality factor $ Q $, emitter density, and coupling strength influence the onset of superradiance-like behavior in lasers?
• RQ5 Can a simple analytical model capture complex collective effects such as photon bunching and delayed emission without including higher-order correlations?

Key findings

• Photon trapping occurs when $ 2 au_{ ext{rad}} o au_{ ext{rad}} $, i.e., when the radiative decay time is comparable to the inverse of the collective emission rate, leading to reduced output below threshold.
• The second-order correlation function $ g_2 > 2 $ is predicted for $ 2 au_{ ext{rad}} o au_{ ext{rad}} $, indicating superthermal photon bunching, a unique signature of collective emission.
• The condition $ 2 au_{ ext{rad}} o au_{ ext{rad}} $ is equivalent to $ 2 au_{ ext{rad}} o au_{ ext{rad}} $, which requires $ 2 au_{ ext{rad}} o au_{ ext{rad}} $, or $ 2 au_{ ext{rad}} o au_{ ext{rad}} $, with $ au_{ ext{rad}} $ being the radiative decay time.
• The threshold population inversion $ ar{ ho}_{ ext{th}} $ scales as $ ar{ ho}_{ ext{th}} o ar{ ho}_{ ext{th}} $, and the condition $ ar{ ho}_{ ext{th}} o 0 $ is necessary for strong collective effects.
• The model predicts that superthermal emission occurs only below threshold, with maximum bunching when $ 2 au_{ ext{rad}} o au_{ ext{rad}} $ and $ ar{ ho}_{ ext{th}} o 0 $, consistent with experimental observations.
• The GLRE model successfully captures photon trapping and superthermal statistics without including correlations beyond fourth order, indicating that low-order approximations suffice for collective effects in nanolasers.