**General information**

**Project code:** № Ф-2021-440

**Project name:**

Quantum transport and quasi-particle dynamics in low-dimensional branched structures: Modeling and design of transparent quantum networks

**Period:** 01.10.2021 – 30.09.2026 years.

**Customer:** Innovative Development Agency under the Ministry of Higher Education, Science and Innovation of the Republic of Uzbekistan

**Contractor:** Kimyo International University in Tashkent

**Contract:** №ФЗ-20200928103

**Project information**

**Introduction**

Networks and network-like structures, in which the transfer of charge, energy, signal and information occur in the quantum regime, appear in many branches of modern technology. Moreover, the coverage of various devices and technologies with such systems has been growing rapidly over the past few years due to the functionalization of quantum processes in nano-electronics, nano-mechanics, optoelectronics and quantum technologies. Therefore, understanding the fundamental processes occurring in such systems, managing their development scenario are of great importance for their functionalization and optimization of technologies with their participation. The solution of such a problem is impossible without developing realistic and physically acceptable models of quantum transport processes in network-like systems and their computer visualization. This problem is considered as one of the most actual in today's world science, and the direction itself is one of the most advanced. Therefore, such research contributes to a large extent to the penetration of advanced trends in domestic science, its integration into world science, as well as the growth of its competitiveness.

**The research goal**

The main goal of this project is to study the transport of waves and quasiparticles in network-like structures that arise in quantum optics and optoelectronics, and apply the results of the study to the modeling and design of transparent and non-Hermitian (PT-symmetric) quantum networks, which allow signal transmission to be performed without loss or with minimal losses. The possibility of using such networks in quantum communications will also be explored. The project will consider quantum networks described by the Schrödinger equation on metric graphs. The transparency of graph nodes will be ensured using the so-called transparent boundary conditions. Boundary conditions at the nodes of the quantum graph, which ensure the PT-symmetry of the graph will be derived. The quantum dynamics of quasiparticles in the presence of absorbing boundary conditions at the site and their possible equivalence to PT-symmetric boundary conditions will be studied. Models of transparent networks for application in optoelectronics, quantum optics and quantum communications will be the final result of the project.

**Project objectives **

The main objectives of this project are the development of models that make it possible to study the tunable dynamics of quasiparticles in branched quantum structures that arise in optics, optoelectronics and nanophysics. Within the framework of the research, two types of quantum structures will be considered: Hermitian quantum structures and PT-symmetric quantum systems. The following specific tasks will be solved within the framework of the project:

- Modeling and design of transparent quantum graphs for nonrelativistic quasiparticles. Study of the dynamics of nonrelativistic quasiparticles in transparent branched quantum structures (as well as in discrete branched quantum structures). Derivation of transparent boundary conditions for the Schrödinger equation on graphs of arbitrary topologies. Development of an algorithm for discretization of transparent boundary conditions at a node for graphs of arbitrary topologies. Numerical implementation of transparent boundary conditions. Obtaining physically acceptable boundary conditions at the node, that are equivalent to transparent boundary conditions.
- Modeling and design of PT-symmetric quantum graphs. Studying the dynamics of nonrelativistic quasiparticles in PT-symmetric branched quantum structures. Solution of the Schrödinger equation on PT-symmetric quantum graphs. Derivation of various types of boundary conditions on graph nodes that ensure the PT-symmetry of a quantum graph. Calculation of the energy spectrum of the structure.
- Investigation of the dynamics of nonrelativistic quasiparticles in branched quantum structures interacting with PT-symmetric external electromagnetic fields. Solution of the Schrödinger equation on quantum graphs in the presence of PT-symmetric potentials. Calculation of the energy spectrum of the structure.
- Simulation of quantum transport in branched quantum structures interacting with PT-symmetric external electromagnetic fields. Derivation of transparent boundary conditions for quantum graphs in the presence of PT-symmetric potentials. Development of algorithms for discretization of transparent graphs and their numerical implementation. Computer visualization for the process of quantum transport of quasiparticles in quantum graphs interacting with PT-symmetric external electromagnetic fields.
- Modeling of quantum transport in branched quantum structures with absorbing nodes. Derivation of absorbing boundary conditions at nodes for the Schrödinger equation on quantum graphs. Finding PT-symmetric conditions at nodes that are equivalent to transparent boundary conditions. Numerical solution of the Schrödinger equation on quantum graphs with absorbing boundary conditions.
- Studying entangled quantum states arising on PT-symmetric quantum graphs. Modeling a quantum processor on PT-symmetric quantum graphs.

In all the above tasks, efficient algorithms for the numerical solution of the Schrödinger equation on quantum graphs of various topologies will be developed, and the corresponding software will be created.

**Previous projects on this topic:**

http://www-amna.math.uni-wuppertal.de/~ehrhardt/Projects/AQUAGRAPH.html