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IEEE Antennas and Propagation Magazine

Numerical Simulations of Laser Pulse Propagation in Quantum Active Media: Using a semiclassical model

Elliot Lu, Connor Glosser, Carlo Piermarocchi, B. Shanker

Published May 12, 2021• 1 citation
Publisher

Abstract

Active media are materials that consist of quantum systems, such as atoms, impurities, or quantum dots, and are characterized by strong resonant absorption and re-emission of radiation. In this article, we present a general framework for the numerical simulation and analysis of time-dependent radiation-induced phenomena in active media. The formulation used is based on the solution of semiclassical Maxwell–Bloch equations describing the evolution of each quantum element under the effect of external and re-emitted radiation. Within these Maxwell–Bloch equations, the coupling of quantum systems to classical Maxwellian fields poses computational challenges due to the strong nonlinearities involved. In contrast to traditional mesh-based solvers, we adopt an electric field integral-operator approach that can reliably account for near-field effects—including self-radiation—and is scalable to large systems. We then focus on media based on large numbers of quantum dots and demonstrate various physical effects arising from the near-field coupling, including polarization modulations and superradiance.

DOI
10.1109/map.2021.3073147

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Active media are materials that consist of quantum systems, such as atoms, impurities, or quantum dots, and are characterized by strong resonant absorption and re-emission of radiation. In this article, we present a general framework for the numerical simulation and analysis of time-dependent radiation-induced phenomena in active media. The formulation used is based on the solution of semiclassical Maxwell–Bloch equations describing the evolution of each quantum element under the effect of external and re-emitted radiation. Within these Maxwell–Bloch equations, the coupling of quantum systems to classical Maxwellian fields poses computational challenges due to the strong nonlinearities involved. In contrast to traditional mesh-based solvers, we adopt an electric field integral-operator approach that can reliably account for near-field effects—including self-radiation—and is scalable to large systems. We then focus on media based on large numbers of quantum dots and demonstrate various physical effects arising from the near-field coupling, including polarization modulations and superradiance.
We introduce a computational Maxwell-Bloch framework for investigating out-of-equilibrium optical emitters in open systems. To do so, we compute the pulse-induced dynamics of each emitter from fundamental light-matter interactions and self-consistently calculate their radiative coupling, including phase inhomogeneity from propagation effects. This semiclassical framework is applied to open quantum dots systems with different densities and dipolar coupling. We observe signatures of superradiant behavior, such as directionality and faster decay, as well as subradiant emission. We compare and discuss the computed light emission obtained with our method and a master equation approach. Our framework enables quantitative investigations of large optical ensembles in the time domain and could be used to design new systems with enhanced superradiant and subradiant properties.
We study the general problem of matter-laser interaction within the framework of nonrelativistic quantum electrodynamics, with particular emphasis on strong laser field effects. Consequently, we formulate a well-defined approximation leading in a straightforward manner toward the conventional semiclassical mode of description. Namely, we arrive naturally to two coupled equations of motion: (i) the Schr\"odinger equation which governs the quantum dynamics of an atomic system driven by classically described radiation field (composed of an incoming laser pulse plus radiation emitted from the atom), and (ii) the classical Maxwell wave equation which describes the emission of radiation from the mentioned atomic source. Employing the formalism of adiabatic Floquet theory, we derive a simple criterion of validity of the just described semiclassical approach. It shows that the semiclassical treatment is justified in most situations. On the other hand, it turns out that the semiclassical approximation breaks down completely in certain special but realistic cases, regardless of the fact that the incoming laser pulse contains a huge number of photons. Under such special circumstances, we anticipate new effects arising due to the quantized nature of the radiation field, to be observable, for example, in harmonic generation spectra. Our considerations are illustrated more explicitly using a simple model of a two-level atom strongly driven by a laser. The quantum dynamics of this model problem is resolved within the framework of quantum electrodynamics while adopting well-defined and physically justifiable approximations. As an outcome, analytic formulas are found serving as a quantitative criterion of (non)applicability of the semiclassical approach and demonstrating the breakdown of semiclassical theory under well-defined conditions. An illustrative numerical calculation is provided.
The dynamics of ultrashort pulses propagating in a quantum dot amplifier is determined by a complex nonlinear coupling and dynamic interplay of light fields and carriers in the spatially inhomogeneous quantum dot ensemble. Computational modeling shows that in spite of the large complexity the strong localization of the carrier inversion and the low amplitude phase coupling may allow the amplification and transmission of ultrahort light pulses with minimum deterioration of the pulse properties (e.g. pulse shape, duration). The theoretical description is based on spatially resolved Quantum Dot Maxwell-Bloch equations that describe the spatio-temporal light field and inter-/intra-level carrier dynamics in each quantum dot of a typical quantum dot ensemble. In particular, this includes spontaneous luminescence, counterpropagation of amplified spontaneous emission and induced recombination as well as carrier diffusion in the wetting layer of the laser. Intradot scattering via emission and absorption of phonons, as well as the scattering with the carriers and phonons of the surrounding wetting layer are dynamically included on a mesoscopic level. Spatial fluctuations in size and energy levels of the quantum dots and irregularities in the spatial distribution of the quantum dots in the active layer are simulated via statistical methods. Simulation results of the nonlinear pulse propagation in quantum dot optical amplifiers allow visualization and interpretation of fundamental nonlinear processes such as selective depletion and re-filling of quantum dot energy levels leading to a complex gain and index dynamics that affect the amplitude and phase of a propagating light pulse. Computational modelling thus may lay the foundation for an optimization and tayloring of pulse properties.
We study the Maxwell–Bloch model, which describes the propagation of a laser through a material and the associated interaction between laser and matter (polarization of the atoms through light propagation, photon emission and absorption, etc.). The laser field is described through Maxwell's equations, a classical equation, while matter is represented at a quantum level and satisfies a quantum Liouville equation known as the Bloch model. Coupling between laser and matter is described through a quadratic source term in both equations. The model also takes into account partial relaxation effects, namely the trend of matter to return to its natural thermodynamic equilibrium. The whole system involves 6 + N(N + 1)/2 unknowns, the six-dimensional electromagnetic field plus the N(N + 1)/2 unknowns describing the state of matter, where N is the number of atomic energy levels of the considered material. We consider at once a high frequency and weak coupling situation, in the general case of anisotropic electromagnetic fields that are subject to diffraction. Degenerate energy levels are allowed. The whole system is stiff and involves strong nonlinearities. We show the convergence to a nonstiff, nonlinear, coupled Schrödinger-rate model, involving 3 + N unknowns. The electromagnetic field is eventually described through its envelope, one unknown vector in ℂ3. It satisfies a Schrödinger equation that takes into account propagation and diffraction of light inside the material. Matter on the other hand is described through a N-dimensional vector describing the occupation numbers of each atomic level. It satisfies Einstein's rate equation that describes the jumps of the electrons between the various atomic energy levels, as induced by the interaction with light. The rate of exchange between the atomic levels is proportional to the intensity of the laser field. The whole system is the physically natural nonlinear model. In order to provide an important and explicit example, we completely analyze the specific (two-dimensional) Transverse Magnetic case, for which formulae turn out to be simpler. Technically speaking, our analysis does not enter the usual mathematical framework of geometric optics: it is more singular, and requires an ad hoc Ansatz.
The quantum technologies series of articles started with the basics of quantum electromagnetics and then turned to deal with various promising applications. These range from the computation of the Casimir force to the analysis of interactions between electromagnetic waves and large molecules behaving as nanoantennas. Another article outlined the advantages of more futuristic higher level applications, such as quantum radars and lidars. Recent articles in the series covered device-level performance, including the quantum electrodynamics of plasmonic waveguides and construction of transmon qubits. Similarly, the present contribution, “Numerical Simulations of Laser Pulse Propagation in Quantum Active Media,” deals with the semiclassical analysis of field interaction with a collection of quantum dots, posing a heavy computational challenge. One typically sidesteps this by deriving a homogeneous behavior. But this leaves out interesting design opportunities. This article takes a different approach to the self-consistent analysis of a large ensemble of quantum dots. It combines analysis methods that are well known in the electromagnetics community together with predictor–corrector methods to solve for the evolution of the polarization of quantum dots. Interestingly, the approach correctly predicts experimentally observed behavior.
A new semiclassical model of the active medium of a gas laser and a description of its interaction with a classical electromagnetic field are reported. The model is based upon an exact microscopic formulation of the density of the active medium in a suitably defined semiclassical state space. Field-medium interaction mechanisms are studied by coupling the equation describing the dynamics of this density with Maxwell's field equations and the Schr\"odinger wave equation. Coupled stochastic nonlinear equations are formulated and quasilinear techniques are employed to effect their solutions. Wave-dipole correlations, the dominant nonlinearity, is shown to effect stability, produce the phenomena of hole burning, mode coupling, and frequency conversion, and to provide the mechanism whereby nonresonant pump field energy is converted into resonant laser oscillations.