Analyzing the positivity preservation of numerical methods for the Liouville-von Neumann equation

The theory of open quantum system is one of the most important tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) Master Equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and simple as possible to be useful to readers with no previous knowledge of this field.

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“…An analysis of different methods and their advantages and disadvantages can be found at Ref. [32]. Box 6.…”

Section: Properties Of the Lindblad Master Equationmentioning

confidence: 99%

A short introduction to the Lindblad master equation

2020

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“…In order to have a CPTP splitting in Eq. (A3) and thus obtain numerical solutions compatible with physics, we must assure that all the discrete operators (namely,Ĝ t rec ,Ĝ t imp ,Ĝ t coh , andL t ) are CPTP [65]. In case of two VB levels, the recombination operator introduces recombination process from each CB level to each VB level (k = 0, k = −1) with the same characteristic time scale τ rec similarly to the case of one VB [Eq.…”

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confidence: 99%

“…which preserves all the properties of density matrixρ. There are several possibilities of obtaining a CPTP discrete Liouville-von Neumann operator, such as the Crank-Nicolson approach [66,67], Runge-Kutta methods [68][69][70], and the matrix exponential approach [26,65]. The latter technique is chosen in this paper because it adapts well to the alternatively updating nature of FDTD codes.…”

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confidence: 99%

Optical Bloch modeling of femtosecond-laser-induced electron dynamics in dielectrics

et al. 2020

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A model based on optical Bloch equations is developed to describe the interaction of femtosecond laser pulses with dielectric solids, accounting for optical-cycle-resolved electron dynamics. It includes the main physical processes at play: photoionization, impact ionization, direct and collisional laser heating, and recombination. By using an electron band structure, this approach also accounts for material optical properties as nonlinear polarization response. Various studies are performed, shedding light on the contribution of various processes to the full electron dynamics depending on laser intensity and wavelength. In particular, the standard influence of the impact ionization process is retrieved.

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“…[256] Furthermore, it was found that both the predictor-corrector and Runge-Kutta method yield negative populations in certain cases (e.g., long simulation end time combined with unfortunate choices for the time step size), while the matrix exponential method preserves the properties of the density matrix independent of the simulation settings. [273]…”

Section: Comparison Of Numerical Methods For the Bloch Equationsmentioning

confidence: 99%

Optoelectronic Device Simulations Based on Macroscopic Maxwell–Bloch Equations

Jirauschek

Riesch

Tzenov

2019

Advcd Theory and Sims

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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. of a lower bandgap material than the adjacent layers, which restricts the free electron motion in that layer to the in-plane directions and gives rise to quantized energy states in growth direction. As a consequence of the further restriction of the energy spectrum and the even stronger carrier localization, additional improvement can be expected from 2D or 3D confinement, resulting in quantum wire/dash and quantum dot (QD) structures, respectively. Indeed, QD [1][2][3] and quantum dash [4] lasers and laser amplifiers have been shown to exhibit excellent characteristics. In Figure 1, the formation of quantized states in quantum wells, wires, and dots is schematically illustrated. The term quantum dash refers to an elongated nanostructure, that is, some kind of short quantum wire. By contrast, the term nanowire does not necessarily indicate strong quantum confinement. For example, in nanowire lasers, the nanowire geometry typically serves as a single-mode optical waveguide resonator, while the active region is based on a heterostructure or quantum well, as in a conventional laser diode. [5,6] Semiconductor optoelectronic devices usually rely on electron-hole recombination, that is, optical transitions between conduction and valence band states. The associated resonance wavelength is largely determined by the semiconductor bandgap, which establishes a lower bound on the transition energy. Thus, the coverage of a certain spectral region depends on the existence of suitable semiconductor materials, which for example restricts the availability of practical optoelectronic sources and detectors in the mid-infrared and terahertz regions. An alternative concept is based on intersubband devices, which employ so-called i...

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Analyzing the positivity preservation of numerical methods for the Liouville-von Neumann equation

Cited by 16 publications

References 36 publications

A short introduction to the Lindblad master equation

A short introduction to the Lindblad master equation

Optical Bloch modeling of femtosecond-laser-induced electron dynamics in dielectrics

Optoelectronic Device Simulations Based on Macroscopic Maxwell–Bloch Equations

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