Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2006.05072 · Advanced Theory and Simulations

Optoelectronic Device Simulations Based on Macroscopic Maxwell–Bloch Equations

Christian Jirauschek, Michael Riesch, Petar Tzenov

Published Jun 7, 2019• 62 citations
arXivHTMLPDF

Abstract

Abstract Due to their intuitiveness, flexibility, and relative numerical efficiency, the macroscopic Maxwell–Bloch (MB) equations are a widely used semiclassical and semi‐phenomenological model to describe optical propagation and coherent light–matter interaction in media consisting of discrete‐level quantum systems. This review focuses on the application of this model to advanced optoelectronic devices, such as quantum cascade and quantum dot lasers. The Bloch equations are here treated as a density matrix model for driven quantum systems with two or multiple discrete energy levels, where dissipation is included by Lindblad terms. Furthermore, the 1D MB equations for semiconductor waveguide structures and optical fibers are rigorously derived. Special analytical solutions and suitable numerical methods are presented. Due to the importance of the MB equations in computational electrodynamics, an emphasis is placed on the comparison of different numerical schemes, both with and without the rotating wave approximation. The implementation of additional effects which can become relevant in semiconductor structures, such as spatial hole burning, inhomogeneous broadening, and local‐field corrections, is discussed. Finally, links to microscopic models and suitable extensions of the Lindblad formalism are briefly addressed.

DOI
10.1002/adts.201900018

Related papers

Modeling of fluctuations in dynamical optoelectronic device simulations within a Maxwell-density matrix Langevin approach

Johannes Popp, Johannes Stowasser, Michael A. Schreiber +5 more

We present a full-wave Maxwell-density matrix simulation tool including c-number stochastic noise terms for the modeling of the spatiotemporal dynamics in active photonic devices, such as quantum cascade lasers and quantum dot structures. The coherent light–matter interaction in such devices plays an important role in the generation of frequency combs and other nonlinear and non-classical optical phenomena. Since the emergence of nonlinear and non-classical features is directly linked to the noise properties, detailed simulations of the noise characteristics are required for the development of low-noise quantum optoelectronic sources. Our semiclassical simulation framework is based on the Lindblad equation for the electron dynamics, coupled with Maxwell’s equations for optical propagation in the laser waveguide. Fluctuations arising from the interactions of the optical field and quantum system with their reservoirs are treated within the quantum Langevin theory. Here, the fluctuations are included by adding stochastic c-number terms to the Maxwell-density matrix equations. The implementation in the mbsolve dynamic simulation framework is publicly available.

arXiv:2310.16039•APL Quantum•Mar 1, 2024•6 citations

Completely Positive Trace Preserving Numerical Methods for Long-Term Generalized Maxwell-Bloch Simulations

Michael Riesch, Christian Jirauschek

The Maxwell-Bloch equations describe the interaction of light (treated classically with Maxwell's equations) and matter, where the latter is modeled as an ensemble of quantum mechanical systems with two energy levels. The dynamical behavior of the systems is described by the optical Bloch equations. This concept can be generalized to an arbitrary number of energy levels with the help of the density matrix ∧ρ and the Lindblad master equation ∂t ∧ρ = -i ħ-1[∧H, ∧ρ] + D(∧ρ), where ∧H is the Hamiltonian of the system and D accounts for dissipation.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•Jun 1, 2019•1 citation

Self-consistent Maxwell-Bloch theory of quantum-dot-population switching in photonic crystals

Hiroyuki Takeda, Sajeev John

We theoretically demonstrate the population switching of quantum dots (QD's), modeled as two-level atoms in idealized one-dimensional (1D) and two-dimensional (2D) photonic crystals (PC's) by self-consistent solution of the Maxwell-Bloch equations. In our semiclassical theory, energy states of the electron are quantized, and electron dynamics is described by the atomic Bloch equation, while electromagnetic waves satisfy the classical Maxwell equations. Near a waveguide cutoff in a photonic band gap, the local electromagnetic density of states (LDOS) and spontaneous emission rates exhibit abrupt changes with frequency, enabling large QD population inversion driven by both continuous and pulsed optical fields. We recapture and generalize this ultrafast population switching using the Maxwell-Bloch equations. Radiative emission from the QD is obtained directly from the surrounding PC geometry using finite-difference time-domain simulation of the electromagnetic field. The atomic Bloch equations provide a source term for the electromagnetic field. The total electromagnetic field, consisting of the external input and radiated field, drives the polarization components of the atomic Bloch vector. We also include a microscopic model for phonon dephasing of the atomic polarization and nonradiative decay caused by damped phonons. Our self-consistent theory captures stimulated emission and coherent feedback effects of the atomic Mollow sidebands, neglected in earlier treatments. This leads to remarkable high-contrast QD-population switching with relatively modest (factor of 10) jump discontinuities in the electromagnetic LDOS. Switching is demonstrated in three separate models of QD's placed (i) in the vicinity of a band edge of a 1D PC, (ii) near a cutoff frequency in a bimodal waveguide channel of a 2D PC, and (iii) in the vicinity of a localized defect mode side coupled to a single-mode waveguide channel in a 2D PC.

Physical Review A•May 9, 2011•19 citations

Numerical Methods and Application for Optical Maxwell-Bloch Equations

Hua Zhang

An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations,which describe the optical pulse propagation and interaction with homogeneously broadened two-level medium.The convergence and stability of this numerical method are also proved by comparing with the analytic solutions derived under special condition,and the method maintains its erroneous exponents and is applicable.Simulations of arbitrary conditions are discussed by employing this new method,and the evolutions of pulse and Bloch vectors for different detunings,relaxation times and initial input pulse power are analyzed.The established numerical method can be used to solve the Maxwell-Bloch equations and their corrections.

ACTA PHOTONICA SINICA•Jan 1, 2009•0 citations

Numerical Method and Application for Optical Maxwell-Bloch-Equations in Inhomogeneously Broadened Medium

Cheng Li, Huarong Zhang, Xiangyang Yu

An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations that are used to describe the optical pulse propagation and interaction with inhomogeneously broadened two-level medium.The convergence and stability of this numerical method are proved by comparing with the analytic solutions and area theorem derived under special condition.The simulations of arbitrary conditions is discussed by employing this method and analyze the pulse evolution for different detunings,relaxation times and inhomogeneously broadened linewidthes.The established numerical method provides a good way to have further study about the coherent interaction of short optical pulses with resonant mediums,and can be used to solve the corrections of Maxwell-Bloch equations.

Acta Scientiarum Naturalium Universitatis Sunyatseni•Jul 25, 2009•0 citations

Mathematical Models of Dissipative Systems in Quantum Engineering

Andreea Rodica Sterian, Paul E. Sterian

The paper shows the results of theoretical research concerning the modeling and characterization of the dissipative structures generally, the dissipation being an essential property of the system with self‐organization which include the laser‐type systems also. The most important results presented are new formulae which relate the coupling parametersainfrom Lindblad equation with environment operators Γi; microscopic quantitative expressions for the dissipative coefficients of the master equations; explicit expressions which describe the changes of the environment density operator during the system evolution for fermion systems coupled with free electromagnetic field; the generalized Bloch‐Feynman equations forN‐level systems with microscopic coefficients in agreement with generally accepted physical interpretations. Based on Maxwell‐Bloch equations with consideration of the interactions between nearing atomic dipoles, for the dense optical media we have shown that in the presence of the short optical pulses, the population inversion oscillates between two extreme values, depending on the strength of the interaction and the optical pulse energy.

Mathematical Problems in Engineering•Jan 1, 2012•11 citations
Abstract Due to their intuitiveness, flexibility, and relative numerical efficiency, the macroscopic Maxwell–Bloch (MB) equations are a widely used semiclassical and semi‐phenomenological model to describe optical propagation and coherent light–matter interaction in media consisting of discrete‐level quantum systems. This review focuses on the application of this model to advanced optoelectronic devices, such as quantum cascade and quantum dot lasers. The Bloch equations are here treated as a density matrix model for driven quantum systems with two or multiple discrete energy levels, where dissipation is included by Lindblad terms. Furthermore, the 1D MB equations for semiconductor waveguide structures and optical fibers are rigorously derived. Special analytical solutions and suitable numerical methods are presented. Due to the importance of the MB equations in computational electrodynamics, an emphasis is placed on the comparison of different numerical schemes, both with and without the rotating wave approximation. The implementation of additional effects which can become relevant in semiconductor structures, such as spatial hole burning, inhomogeneous broadening, and local‐field corrections, is discussed. Finally, links to microscopic models and suitable extensions of the Lindblad formalism are briefly addressed.
We present a full-wave Maxwell-density matrix simulation tool including c-number stochastic noise terms for the modeling of the spatiotemporal dynamics in active photonic devices, such as quantum cascade lasers and quantum dot structures. The coherent light–matter interaction in such devices plays an important role in the generation of frequency combs and other nonlinear and non-classical optical phenomena. Since the emergence of nonlinear and non-classical features is directly linked to the noise properties, detailed simulations of the noise characteristics are required for the development of low-noise quantum optoelectronic sources. Our semiclassical simulation framework is based on the Lindblad equation for the electron dynamics, coupled with Maxwell’s equations for optical propagation in the laser waveguide. Fluctuations arising from the interactions of the optical field and quantum system with their reservoirs are treated within the quantum Langevin theory. Here, the fluctuations are included by adding stochastic c-number terms to the Maxwell-density matrix equations. The implementation in the mbsolve dynamic simulation framework is publicly available.
The Maxwell-Bloch equations describe the interaction of light (treated classically with Maxwell's equations) and matter, where the latter is modeled as an ensemble of quantum mechanical systems with two energy levels. The dynamical behavior of the systems is described by the optical Bloch equations. This concept can be generalized to an arbitrary number of energy levels with the help of the density matrix ∧ρ and the Lindblad master equation ∂t ∧ρ = -i ħ-1[∧H, ∧ρ] + D(∧ρ), where ∧H is the Hamiltonian of the system and D accounts for dissipation.
We theoretically demonstrate the population switching of quantum dots (QD's), modeled as two-level atoms in idealized one-dimensional (1D) and two-dimensional (2D) photonic crystals (PC's) by self-consistent solution of the Maxwell-Bloch equations. In our semiclassical theory, energy states of the electron are quantized, and electron dynamics is described by the atomic Bloch equation, while electromagnetic waves satisfy the classical Maxwell equations. Near a waveguide cutoff in a photonic band gap, the local electromagnetic density of states (LDOS) and spontaneous emission rates exhibit abrupt changes with frequency, enabling large QD population inversion driven by both continuous and pulsed optical fields. We recapture and generalize this ultrafast population switching using the Maxwell-Bloch equations. Radiative emission from the QD is obtained directly from the surrounding PC geometry using finite-difference time-domain simulation of the electromagnetic field. The atomic Bloch equations provide a source term for the electromagnetic field. The total electromagnetic field, consisting of the external input and radiated field, drives the polarization components of the atomic Bloch vector. We also include a microscopic model for phonon dephasing of the atomic polarization and nonradiative decay caused by damped phonons. Our self-consistent theory captures stimulated emission and coherent feedback effects of the atomic Mollow sidebands, neglected in earlier treatments. This leads to remarkable high-contrast QD-population switching with relatively modest (factor of 10) jump discontinuities in the electromagnetic LDOS. Switching is demonstrated in three separate models of QD's placed (i) in the vicinity of a band edge of a 1D PC, (ii) near a cutoff frequency in a bimodal waveguide channel of a 2D PC, and (iii) in the vicinity of a localized defect mode side coupled to a single-mode waveguide channel in a 2D PC.
An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations,which describe the optical pulse propagation and interaction with homogeneously broadened two-level medium.The convergence and stability of this numerical method are also proved by comparing with the analytic solutions derived under special condition,and the method maintains its erroneous exponents and is applicable.Simulations of arbitrary conditions are discussed by employing this new method,and the evolutions of pulse and Bloch vectors for different detunings,relaxation times and initial input pulse power are analyzed.The established numerical method can be used to solve the Maxwell-Bloch equations and their corrections.
An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations that are used to describe the optical pulse propagation and interaction with inhomogeneously broadened two-level medium.The convergence and stability of this numerical method are proved by comparing with the analytic solutions and area theorem derived under special condition.The simulations of arbitrary conditions is discussed by employing this method and analyze the pulse evolution for different detunings,relaxation times and inhomogeneously broadened linewidthes.The established numerical method provides a good way to have further study about the coherent interaction of short optical pulses with resonant mediums,and can be used to solve the corrections of Maxwell-Bloch equations.
The paper shows the results of theoretical research concerning the modeling and characterization of the dissipative structures generally, the dissipation being an essential property of the system with self‐organization which include the laser‐type systems also. The most important results presented are new formulae which relate the coupling parametersainfrom Lindblad equation with environment operators Γi; microscopic quantitative expressions for the dissipative coefficients of the master equations; explicit expressions which describe the changes of the environment density operator during the system evolution for fermion systems coupled with free electromagnetic field; the generalized Bloch‐Feynman equations forN‐level systems with microscopic coefficients in agreement with generally accepted physical interpretations. Based on Maxwell‐Bloch equations with consideration of the interactions between nearing atomic dipoles, for the dense optical media we have shown that in the presence of the short optical pulses, the population inversion oscillates between two extreme values, depending on the strength of the interaction and the optical pulse energy.