**Project Heads**

*Uwe Bandelow*

**Project Members**

Alexander Pimenov (WIAS), Markus Kanter (WIAS, core support)

**Project Duration**

01.01.2019 – 31.12.2020

**Located at**

WIAS

The second quantum revolution aims to bring quantum communication and information processing to real world applications, and semiconductor quantum dots (QDs) were identified as ideal optically active elements for the emerging quantum technologies as they can be directly integrated into semiconductor-based photonic resonators. However, simple electrically driven quantum light emitting diodes have severe limitations as they do not allow for resonant excitation and intrinsically suffer from charge noise. A promising solution to this problem are nanolasers (see Fig. 1a), which employ only a few QDs as gain medium, embedded in a high quality optical micro-resonator with spatial dimensions down to the diffraction limit. This tremendous size reduction takes the concept of lasing to a completely new regime, where the light-matter interaction is dominated by cavity quantum electrodynamic (cQED) effects. In this respect, nanolasers operate on the exciting crossroad between quantum light emitters and conventional diode lasers. The exploitation of cQED effects facilitates thresholdless lasing (see Fig. 1b), increased modulation speed and significantly reduced energy consumption (“green photonics”). Beyond their usage in nanophotonic circuits, nanolasers are promising candidates for optical interconnects in data centers and supercomputers.

The **mathematical modeling** of nanolasers is a complex multi-physics problem, which requires a fully quantum mechanical description of the light-matter interaction in the optically active region. We employ a comprehensive hybrid quantum-classical modeling approach, which includes aspects of semi-classical transport theory in semiconductors, electromagnetic field theory and cQED in second quantization for the coupled QD-photon system. Mathematically, this involves systems of nonlinear partial differential equations (electronic transport, heat transport), complex non-Hermitian eigenvalue problems (optical fields in microcavities, Schrödinger–Poisson problem for QD electronics) and operator evolution equations for the quantum statistical operator describing the open quantum system (Lindblad master equation, Jaynes–Cummings model). Our comprehensive modeling framework allows for investigation of the device characteristics under stationary, pulsed and noisy (stochastic) injection conditions.

For the **numerical simulation** of the nanolasers we employ tools such as state-of-the-art solvers for nonlinear transport problems (finite volumes Scharfetter–Gummel method) and coupled cluster expansion techniques for the solution of the quantum optical problem. Other possibilities to truncate the huge dimensionality of the underlying Hilbert space include tensor network methods. Finally, we perform further simplifications within our framework, and study the dynamics of the reduced systems of differential equations to gain qualitative insights on the novel regimes of operation arising due to the extreme size of these lasers.

**Project Webpages**

**Selected Publications
**

- M. Kantner: “Generalized Scharfetter–Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient”,
*Journal of Computational Physics***402**, 109091 (2020). - A. V. Kovalev, P. S. Dmitriev, A. G. Vladimirov, A. Pimenov, G. Huyet and E. A. Viktorov: “Bifurcation structure of a swept-source laser”,
*Physical Review E***101**(1), p.012212 (2020). - M. Kantner, Th. Koprucki: “Non-isothermal Scharfetter-Gummel scheme for electro-thermal transport simulation in degenerate semiconductors”,
*In Finite Volumes for Complex Applications IX, Eds.: R. Klöfkorn R., E. Keilegavlen, F. Radu and J. Fuhrmann. Springer Proceedings in Mathematics and Statistics*, Vol. 323, pp. 173–182, Springer, Cham (2020). DOI: 10.1007/978-3-030-43651-3_14 - U. Gowda, A. Roche, A. Pimenov, A. G. Vladimirov, S. Slepneva, E. A. Viktorov, G. Huyet: “Turbulent coherent structures in a long cavity semiconductor laser near the lasing threshold”,
*Preprint no. 2724*, WIAS, Berlin (2020), DOI 10.20347/WIAS.PREPRINT.2724 - M. Heida, M. Kantner, A. Stephan: “Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator”,
*Preprint no. 2684*, WIAS, Berlin (2020), DOI 10.20347/WIAS.PREPRINT.2684 - M. Kantner, M. Mittnenzweig, A. Mielke and N. Rotundo: “Mathematical modeling of semiconductors: From quantum mechanics to devices.”
*In Topics in Applied Analysis and Optimisation: Partial Differential Equations, Stochastic and Numerical Analysis, Eds.: M. Hintermüller and J. Rodrigues*, pp. 269-293, Springer, Cham (2019). - M. Kantner: “Hybrid modeling of quantum light emitting diodes: Self-consistent coupling of drift-diffusion, Schrödinger–Poisson and quantum master equations,”
*Proc. SPIE Vol. 10912, Physics and Simulation of Optoelectronic Devices XXVII*, 109120U (2019).

**Selected Pictures
**

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