Bilateral German-Uzbek Project

AQUAGRAPH: Absorbing boundary conditions for quantum graphs

Programme Re-invitation Programme for Former Scholarship Holders, 2018 financed by DAAD

(09/2018)

Summary

This project aims at the study of wave dynamics in quantum networks with transparent branching points. The latter means that wave/particle transmission through the network branching points (vertices) occurs without reflection. This will be done by solving linear and nonlinear Schrödinger equations on metric graphs for which so-called absorbing (transparent) vertex boundary conditions at the graph vertices are imposed in order to provide the transparency of the network. Applications of the results for modeling and design different optical fiber networks and branched nanoscale systems will be discussed concisely.

Introduction

Quantum graphs have been shown to serve as accurate models for the study of quantum transport and spectral statistics in nanosized systems. Recently quantum graphs are studied experimentally by using microwave networks consisting of coaxial waveguides. Zero point energy in quantum graphs may play important role in various systems (e.g., polymers, molecular networks, microwave networks and other supramolecular structures), whose dynamics can be modelled by quantum graphs, as well as in nanomechanics.

The problem of absorbing (transparent) boundary conditions (ABCs) for wave equations has attracted much attention in different practically important contexts. Such boundary conditions describe absorption of particles and waves in their transmission from one domain to another one. From the mathematical point of view, absorbing boundary conditions are the same as those describing reflectionless transmission of particles (waves) through the boundary of a given domain. Therefore, one uses similar terminology for the boundary conditions of both types of processes, calling them absorbing or transparent boundary conditions. For both processes, the boundary conditions can be derived by factorization of the differential operator, corresponding to a wave equation, which in general lead to a complicated equations for the boundary conditions. Explicit form of such boundary conditions are much complicated than those of Dirichlet, Neumann and Robin conditions.

For absorbing boundary conditions the wave equation cannot be solved analytically and always requires using numerical methods. Depending on the type of the wave equation, they require different discretization schemes, which vary also for the types of the process. Here we will apply absorbing boundary conditions to quantum graphs considering the problem of reflectionless transmission of particle through the graph branching point (vertex).

In this project we will extend the classical derivation of ABCs for Schrödinger equations to the case of quantum graphs. First we will consider the simplest, star-shaped graph.

Scientific Objectives

Within this research project we address two problems: the first problem can be called "linear wave dynamics in quantum graphs with transparent vertices", while the second one is called "soliton dynamics in transparent networks".
The specific objectives of the project includes the following:

Particle dynamics in transparent quantum networks. Linear Schrödinger equation on metric graphs will be solved by imposing transparent vertex boundary conditions.
Soliton transport in networks with transparent networks. Nonlinear Schrödinger equation will be solved by imposing transparent vertex boundary conditions. Soliton dynamics in such system will be modeled. Constraints providing reflectionless transport of solitons in networks will be derived.
Computation of conductance and optical conductivity of transparent networks interacting with external time-periodic field.

Uzbek Institution:

State Testing Center, Uzbekistan, 100084, Tashkent, Uzbekistan.
Laboratory for Advanced Studies, Turin Polytechnic University in Tashkent, 17 Niyazov Str. 100095 Tashkent, Uzbekistan.

Publications related to the Project

J.R. Yusupov, M. Ehrhardt, Kh.Sh. Matyokubov and D.U. Matrasulov, Driven transparent quantum graphs, Preprint 23/19, November 2023.
J.R. Yusupov, Kh.Sh. Matyokubov, M. Ehrhardt and D.U. Matrasulov, Manakov system on metric graphs: Modeling the reflectionless propagation of vector solitons in networks, Physics Letters A, Volume 479 (2023), 128928. DOI: 10.1016/j.physleta.2023.128928.
M.E. Akramov, J.R. Yusupov, M. Ehrhardt, H. Susanto and D.U. Matrasulov, Transparent boundary conditions for the nonlocal nonlinear Schrödinger equation: A model for reflectionless propagation of PT-symmetric solitons, Physics Letters A, Volume 459, January 2023, 128611. DOI: 10.1016/j.physleta.2022.128611.
K.K. Sabirov, J.R. Yusupov, M. Ehrhardt and D.U. Matrasulov, Transparent boundary conditions for the sine-Gordon equation: Modeling the reflectionless propagation of kink solitons on a line, Physics Letters A, Vol. 423 (2022), 127822. DOI: 10.1016/j.physleta.2021.127822.
K.K. Sabirov, J.R. Yusupov, M.M. Aripov, M. Ehrhardt and D.U. Matrasulov, Reflectionless propagation of Manakov solitons on a line: A model based on the concept of transparent boundary conditions, Physical Review E, Vol. 103, No. 4 (2021), 043305. DOI: 10.1103/PhysRevE.103.043305
J.R. Yusupov, K.K. Sabirov, Q.U. Asadov, M. Ehrhardt and D.U. Matrasulov, Dirac Particles in Transparent Quantum Graphs: Tunable transport of relativistic quasiparticles in branched structures, Physical Review E, Volume 101, Number 6 (June 2020), 062208. DOI: 10.1103/PhysRevE.101.062208
J.R. Yusupov, K.K. Sabirov, M. Ehrhardt and D.U. Matrasulov, Transparent nonlinear networks, Physical Review E, Vol. 100, No. 3 (September 2019), 032204. DOI: 10.1103/PhysRevE.100.032204
J.R. Yusupov, K.K. Sabirov, M. Ehrhardt and D.U. Matrasulov, Transparent Quantum Graphs, Physics Letters A, Volume 383, Issue 20 (July 2019), 2382-2388. DOI:10.1016/j.physleta.2019.04.059