Workshop on Mathematical Models for Quantum and Semiclassical Dynamics

While crystalline silicon dominates today’s solar market, metal halide perovskites are promising due to their high efficiency and low manufacturing costs. However, key questions remain to realize their full potential, particularly regarding empty sites in the crystalline semiconductor lattice, so-called vacancies, which contribute to the charge transport. This talk explores the impact of vacancy dynamics on device performance through a three-stage theoretical framework: modeling, (numerical) analysis, and simulation of drift-diffusion charge transport equations. Key contributions include a physically accurate model for limiting vacancy accumulation, a rigorous existence proof for weak and discrete solutions of an implicit-in-time two-point-flux finite volume scheme, and the investigation of experimentally relevant phenomena using a self-developed open-source simulation tool.

The interplay between nonlinearity and point interactions in the framework of the Schrödinger equation has been an active field of research in the last twenty years. The occurrence of several unexpected phenomena has been pointed out. Here we summarize some results in that direction, and highlight the counterintuitive creation of ground states by a repulsive point interaction. This is a joint project with F. Boni, R. Carlone, M. Gallone, and L. Tentarelli.

We prove an existence result for the Van Roosbroeck-Helmholtz model for a semiconductor laser diode. The model consists of the 2D Van Roosbroeck equations for semiconductors on the transversal section of a laser diode with an additional generation term depending on the optical electromagnetic field, and an Helmholtz eigenvalue problem for the optical field with an optical permettivity depending on the semiconductor carrier densities. To our knowledge, this is the first rigorous result in the literature dealing with this kind of coupling.

Semiconductor lasers are pivotal components in modern technologies, spanning medical procedures, manufacturing, and autonomous systems like LiDARs. Understanding their operation and developing simulation tools are paramount for advancing such technologies.

In this talk, we present a mathematical PDE model for an edge-emitting laser, combining charge transport and light propagation. We will begin by presenting the complete mathematical model where charge transport is described by a drift-diffusion system and light propagation by the Helmholtz equation. To make the problem more tractable, we simplify the coupled system using a perturbation approach.

We discretize the drift-diffusion model using the finite volume method and solve it with a Newton solver. Moreover, we discretize the Helmholtz model using the finite element method and solve the resulting matrix eigenvalue problem with ARPACK. We discuss a coupling strategy for both models and showcase numerical simulations for a benchmark example. Our implementation is done in Julia, a language that combines the ease of use of e.g. MATLAB with the performance and efficiency of C++.

We will provide a brief insight into the analytical framework underlying our approach, including existence and uniqueness considerations for the coupled system. For a more detailed mathematical analysis of the coupled PDE system, we refer to Giuseppe Alì's talk.

We  show that some quantum mechanical systems are naturally defined in spaces which are different from L^2(R). This opens the way to many mathematical questions. Among these, the possibility to extend the standard scalar product. In this talk we show that these extensions do exist in compatible spaces, and for suitable sets of tempered distributions. Some concrete applications are proposed.

Nanowires are rising much interest as a novel nanoelectronic platform for light harvesting, controlled single-photon emitters, and for the control of topological states. Core-shell nanowires, adding one or more radial layers of a different material to a central nanowire core, substantially expand the possibility to engineer the bandstructure. In case the radial hetherojunction presents an inverted-gap alignment, electron and hole states coexist at the same energy, and the hybridization of these states takes place.

We present our Python numerical library "nwkpy" (freely available online) implementing through the Finite Element Method a self-consistent 8-band k.p approach which takes into account the symmetry and crystallographic directions of the nano-crystal, its realistic hexagonal section, arbitrary doping and, possibly, external electric fields. 

We will illustrate the fundamental role of self-consistent effects and how the charge transfer at the interface can be tailored. Finally, we predict the formation of topological end states in broken-gap samples.

We study the optimization of neutral atom transport and transfer between optical tweezers, both critical steps in the implementation of quantum computers and simulators. We analyze and optimized using d-CRAB, four experimentally relevant pulse shapes (piece-wise linear, piece-wise quadratic, minimum jerk, and a combination of linear and minimum jerk) and a protocol developed using Shortcuts-to-Adiabaticity (STA) methods to incorporate the time-dependent effects of static traps. By computing a measure of the final transport error, we show that our proposed STA protocol comprehensively outperforms all the experimentally inspired pulses. Our STA results prove that modulation in the depth of the moving tweezer designed to time-dependently counteract the effect of the static traps is key to reduce errors and pulse duration. To motivate the implementation of our STA pulses in future experiments, we provide a simple analytical approximation for the moving tweezer position and depth controls.

Entropy and free energy are central concepts in both statistical physics and information theory, with quantum and classical facets. In mathematics these concepts appear quite often in different contexts (dynamical systems, probability theory, von Neumann alge bras, etc.). In this work, we study the von Neumann and Wehrl entropies from the point of view of semiclassical analysis. We first prove the semiclassical convergence of the von Neumann to the Wehrl entropy for quantum Gibbs states (thermal equilibrium), after a suitable renormalization has been taken intoaccount. Then, we show that, in the same limit, the free energy functional defined with the Wehrl entropy $ \Gamma-$converges to its classical counterpart, so implying convergence of the minima and the associated minimizers.

The Poisson-Boltzmann equation (PBE) is a critical tool for model- ing electrostatic interactions in biophysics, colloidal science, and materials science. It describes the mean-field electrostatic potential created by fixed charges in a dielectric medium with mobile ions, playing a vital role in applications such as protein interactions, enzyme activity, and the behav- ior of colloidal particles. The PBE is nonlinear and challenging to solve, even in its linearized form (LPBE), which is a reasonable approximation for systems with low surface potentials. Advances in computational meth- ods have improved the accuracy and efficiency of solving the PBE, but challenges persist, including computational cost, scalability to large sys- tems, and agreement with experimental data. Here, we analyze some ad- vancements in solving the LPBE, improving accuracy without increasing computational costs. By leveraging analytical techniques and optimizing computational approaches, this enhanced method has been validated through comparison with analytical solutions and applied to biological systems such as proteins and nucleic acids. These developments are implemented in the recently released LPBE solver NextGenPB [1]

[1] V. Di Florio, P. Ansalone, S. V. Siryk, S. Decherchi, C. de Falco, and W. Roc- chia. Nextgenpb: An analytically-enabled super resolution tool for solving the poisson-boltzmann equation featuring local (de)refinement. Computer Physics Communications, 317:109816, 2025.

We discuss stationary scattering theory for a model with zero range interactions and a confined particle.

We study a one-dimensional free quantum particle with a mass profile that contains jump discontinuities. The system's Hamiltonian is defined as a self-adjoint extension of the kinetic energy operator in divergence form, with the boundary conditions at the discontinuity points specifying the extension. Focusing on a class of scale-invariant boundary conditions, we investigate the resulting spectral problem. Our analysis reveals that the eigenfunctions depend on energy in a highly sensitive and irregular manner, giving rise to complex spectral features. Remarkably, the system admits infinitely many distinct semiclassical limits, each associated with a point on a spectral curve embedded in a two-torus. These findings highlight the intricate connections between discontinuous media, boundary conditions, and spectral asymptotics.

In this talk, I will present recent results on the well-posedness and quantitative long-term behavior of solutions to the Vlasov–Fokker–Planck equation in the presence of an external confinement potential and general self-consistent interactions.

I will describe the physical motivations stemming from plasma physics and particle accelerator physics. In the latter context, I will explain how non-symmetric interactions arise from the modeling of relativistic electron beams.

As a corollary, we extend existing results on the Vlasov–Poisson–Fokker–Planck equation in three dimensions by establishing exponential convergence to equilibrium for low-regularity initial data and arbitrarily small Debye length.

This is joint work with Pierre Gervais (Université de Toulouse).

The transfer of angular momentum from helicity-controlled laser fields to non-magnetic electronic systems can induce magnetisation via the inverse Faraday effect. This phenomenon has been investigated in metallic nanoparticles using a semi-classical model (QHD) and a full quantum approach within a many-body framework. Real-time time dependent density functional theory (TDDFT) was employed, with the nanoparticles modelled using either a spherical jellium approximation on a real-space grid or a realistic atomic structure. Circularly polarised light induces orbital magnetism, peaking at surface plasmon resonance, which confirms its plasmonic origin. The dominant magnetic contribution arises from surface currents; however, quantum-induced bulk effects, such as Friedel oscillations, also persist.

A graphene field-effect transistor (GFET) is simulated by direct numerical integration of the Boltzmann transport equation coupled with the Poisson equation for the electrostatic potential. The full electron-phonon collision operator has been considered. A discontinuous Galerkin (DG) method has been adopted for the Boltzmann equation, while a finite differences scheme has been used for the Poisson equation. The numerical results have been compared with those obtained by the drift-diffusion model. The findings are useful for the validation of the drift-diffusion and hydrodynamic models for charge transport in semiconductor devices where the active area is made of graphene. As a further extension, the same approach can be used for graphene nanoribbons, bilayer, and multi-layer graphene.

Engineering a quantum system that evolves into a target state has relevance in quantum information science. In particular, the realization of a controllable quantum system is an interesting topic in modern physical science. In recent years a new promising platform emerges in the area of quantum simulation and quantum computing based on the so-called optical tweezers, highly focused laser beams able to trap elementary quantum information carriers. In this context, we aim to formulate an optimal control problem on the transport of atoms in optical tweezers.  We derive an optimal control procedure aimed at steering the atom from an initial to a target position under the guidance of a time-dependent optical tweezers field. The main parameters are designed in order to achieve some optimality conditions by maximizing the success rate of the protocol and at the same time minimizing the energetic cost of the control. We describe the optimal control of the trajectories of flying atoms driven by the optical tweezers field at various degrees of precision.

Understanding the behaviour of classical and quantum systems when coupled to their environment is of fundamental importance at both theoretical and applied levels. In this talk I will review the so-called Caldeira-Leggett model, whose reduced dynamics account for quantum relaxation and decoherence. Further to analysing the Markovian limit, I will consider the more general non-Markovian case. I will introduce non-Markovianizable systems in terms of the relaxational properties of the memory kernel, and propose extensions of the Caldeira and Leggett model capable of capturing these dynamics.

It is presented a fourth order Schrödinger equation (SE) for the description of  ballistic charge  transport in semiconductors with the inclusion of non parabolic effects in the dispersion relation in order to go beyond the simple effective mass approximation. Similarly to the standard (second order) SE, the problem is reduced to a finite spatial domain with appropriate transparent boundary conditions to simulate charge transport in a quantum coupler  where an active region representing an electron device is coupled to leads which take the role of reservoirs. Some analytical properties are investigated and a generalized formula for the current is obtained. Numerical results show the main features of the solutions of the new model.  In particular, an effect of interference appears due to a richer wave structure than that arising for the second order SE n the effective mass approximation.

Perfectly entangling gates (PE) are crucial for various applications in quantum information. One method to realize these gates is with the help of an external control field, whose concrete shape is found using optimal control theory. Instead of optimizing the shape that realizes a specific gate, the optimization target can be extended to the full set of PE. This increases the flexibility of optimization and allows to find the best PE from the set of all PE. For unitary dynamics, the PE optimization functional can readily be evaluated. In contrast, for non-unitary dynamics, one has to approximate the unitary part of the dynamics first. We employ this technique to superconducting qubits, where we apply a cross-resonant drive to two coupled fixed-frequency transmons to generate entangled states.

We propose a theoretical model for a quantum sensor that can determine in a very simple way whether the intensity of an electric eld has an assigned value or not. It is based on the analysis of the resonances crossing in a double-well quantum system subject to an external DC electric field.

It is well known that the paper by Einstein, Podolski and Rosen (EPR) in 1935 has been of fundamental importance both in the debate on the foundations of Quantum Mechanics and in the concrete developments of quantum theory and its applications. In the seminar we briefly recall the content of the paper and then we introduce and discuss a simple mathematical model where the EPR argument can be illustrated in a quantitative and rigorous way. This is a joint work with  R. Adami and L. Barletti.