Project OPUS is supported by National Science Center, Poland grant no 25/B/ST3/00817, 20182020
The Rydberg excitons are unique structures, a bridge joining macroscopic and quantum world. Let’s start from the very bottom of the scale’s size, from one of the smallest fundamental particles – the electron. In semiconductors, most of the electrons are highly confined and strongly binding to their atoms. The valence electrons, populating the outer atomic orbitals, can be freed by providing them with the sufficient amount of the energy in the form of a light quanta – a photon. Even then, the newly deteched electron is not totally free. The atom it has left now has a single, missing negative elementary charge. This empty space is called a hole, and it can be described as a sort of positively charged particle. The opposite charges are attracting each other; the electron establishes a stable orbit around the positive center. As a matter of fact, we get something resembling the hydrogen atom, but much bigger. The exciton is created. The orbits are quantized with the principal number n, e. g. they have discrete numbers n = 1,2,3… and corresponding radius proportional to n^{2}. For very high n >> 1, we obtain the socalled Rydberg excitons. Electrons on those high energy levels are weakly connected to the atoms and react strongly to the external electric and magnetic fields. Moreover, these orbits are gigantic – on the order of micrometres, many times larger than the wavelength of light that creates them. A single exciton spans hundreds of atoms across – we are talking about a quantum structure that is comparable to the diameter of spider silk strand! This is the key property making the Rydberg excitons so interesting – they give us an unprecedented opportunity to observe quantum mechanical phenomena on macroscopic scales, bringing together two completely different worlds.
Fig.1 Simplified model of RE 
In our project we intend to study theoretically various aspects of lightmatter interaction in media containing Rydberg excitons. We will take advantage of their unique properties – sensitivity to external fields, long range interactions between excitons, closely spaced energy levels and their long lifetimes. One of the interesting phenomena that can occur in a medium containing Rydberg Excitons is Franz – Keldysh effect: the external electric field modifies the absorption coefficient of the semiconductor. We are going to study this effect in Cu_{2}O crystal. Strong, longrange interactions between excitons lead to the nonlinear effects: the propagation of electromagnetic wave in the medium becomes dependent on its amplitude and multiple external factors such as temperature. In our project we will concentrate on the optical properties stemming from intra and interband transitions.
The exceptionally large size of the exciton means that it is „aware” of its surroundings. In particular, if we try to constrain its mobility, for example by confining it inside a nanowire, then the properties of exciton will change considerably. As opposed to typical quantum nanostructures (dots, wells, wires) which, as the name suggests, have dimensions on the order of nanometres, we can operate on a much bigger scale. This makes them more convenient to study and easier to manufacture.
The electromagnetically induced transparency (EIT) is a spectacular quantum effect that allows one to slow down the light, or even to stop completlly and store it in the form of medium excitations. Light storage is a key functionality of quantum memories – a foundation of a full quantum computer. However, there is one significant obstacle – up till now the traditional EIT media are mostly gaseous, not very suitable for miniaturized components. The long „ladder” of energy levels available with Rydberg excitons allows one to choose convenient states for realization of EIT in crystal, for example in Cu_{2}O. We will investigate the influence of nonlinear effects on this process in order to find the optimal conditions for storing the light, taking advantage of long lifetimes of Rydberg excitons.
Among many available energy levels, there are highly excited ones, which are located closely together. The transition between them is accompanied by emission of microwave photon. By choosing appropriate transitions, we will investigate the possibility of constructing a maser, e. g. a microwave laser. The high sensitivity of Rydberg excitons to the external field is highly desirable here – even a small number of them is sufficient to initiate stimulated emission, creating high intensity, coherent beam of light.
Research Project Objectives
The continuous feedback between the progress in the scientific research and the technological advancement forms the basis for the civilizational breakthroughs over the last century. On the one side the progress in the experimental techniques gives the possibility to discover new physical phenomena, on the other side the development of theoretical tools to describe it. The Rydberg atoms and Rydberg excitons (RE) are examples of such feedback. New materials and new experimental techniques paved the way for discovery of Rydberg excitons in 2014. This discovery was a challenge for theoretical physics to described this phenomenon. In the years 20142017 a number of properties, first of all, the optical properties, has been studied and the main effort was focussed on the linear optical properties of a medium with RE, including the excitonic resonance positions, line shapes, selection rules etc., in the discrete part of the spectrum. The main aim of this project is to investigate both quasistationary and dynamical optical properties of semiconductor media in which Rydberg excitons exist and the unexplored yet properties of RE. Namely, we want to clarify the role of the effect of confinement of RE in nanowires as well as to give deeper insight into how nonlinear interactions between RE modify the signal propagation under EIT condition and explore the possibility of manufacture the maser working on RE. This includes:
 the investigation of linear and nonlinear optical properties of RE, both for discrete spectrum and the continuum, also taking into account the intraband processes;
 the dynamical aspects of RE interacting with light. Here we will investigate evolution of RE medium excitations’ and possibility of their modifications associated with dynamical changes of Electromagnetically Induced Transparency (EIT) conditions.
Task 1. FranzKeldysh effect in a medium with Rydberg Excitons.
 The main effort of the investigation of the optical properties of RE was focussed on the excitation energies below the fundamental gap, where the multiplicity of resonances occurs. We intend to extend the research for the energetic region above the gap, where the dissociation of the electronhole pair occurs. When applying an external electric field, some oscillations in the spectra appear, known as the FranzKeldysh effect. For RE in Cu_{2}O, one can expect differences regarding electroabsorption periodicity and amplitudes, mostly due to the different type of symmetry. It will be also interesting to see how multiplicity of exciton resonances below the gap influences the behaviour of FranzKeldysh oscillations.
Task 2. Nonlinear effects in a medium with Rydberg Excitons, related to inter and intraband electronic excitations.
 The pertinent theory describing the dynamics of the semiconductor electrons under the influence of the driving laser field has the structure of an infinite open hierarchy of equations of motion for the npoint density matrices. It has been shown that a systematic truncation of the hierarchy of density matrices can be achieved using a classification according to powers in the driving field. The lowest level of this hierarchy consists of the twopoint density matrices representing transitions between valence and conduction bands, and occupation densities of these bands. In our project we use the equations of motion for these quantities (the constitutive equations), solve it , and derive expressions for the nonlinear susceptibility χ(3), which is responsible for dependence of nonlinear optical functions on intensity and on temperature in a system with RE
Task 3. The influence of the confinement on Rydberg Excitons in nanowires.
 The excitons in lowdimensional structures (e.g., Quantum Wells, Quantum Wires, Quantum Dots) have a larger binding energy due to the confinement effects. Controlling on demand the composition and confinement shapes one can manipulate the position of energetic levels. Here we aim to analyse the impact of the confinement on the optical properties of a confined system with RE, taking as example a Cu_{2}O nanowire.
Task 4. Dynamical processes in RE media connected with electromagnetically induced transparency.
 The goal of this task is to analyse the dynamical aspects and properties of Rydberg excitons in Cu2O and show, that the Rydberg excitonic states can be used to perform the modified electromagnetically induced transparency (EIT). The detail analysis of relations between laser fields intensity, RE density and their influence on transparency conditions will be applied to study the medium dynamics in the conditions of population inversion. The aim is to explore the possibility of creating a maser based on Cu2O, taking advantage of unique properties of Rydberg excitons.
Each of the above tasks will combine a theoretical and numerical analysis. The theoretical part of tasks 13 will involve calculations based on the socalled Real Density Matrix Approach, while in task 4 we will apply the MaxwellBloch equations for RE medium+light density matrix modified by inclusion of strong nonlinear interactions between RE.
Team
Project coordinator:
prof. UTP dr hab. Sylwia RaczyńskaZielińska
Sylwia.ZielinskaRaczynska@utp.edu.pl
tel. 52 340 86 88
dr inż. David Ziemkiewicz
David.Ziemkiewicz@utp.edu.pl
tel. 52 340 86 20
prof. dr hab. Gerard Czajkowski
Gerard.Czajkowski@utp.edu.pl
mgr inż. Karol Karpiński
Karol.Karpinski@utp.edu.pl
Publications

Electrooptical properties of Cu_{2}O for P excitons in the regime of FranzKeldysh oscillations,
Sylwia ZielińskaRaczyńska, David Ziemkiewicz, Gerard Czajkowski,
Physical Review B, Volume 97, Issue 16, p.165205165216, (2018)
DOI:10.1103/PhysRevB.97.165205  Proposal of tunable Rydberg exciton maser,
Sylwia ZielińskaRaczyńska, David Ziemkiewicz,
Opt. Lett. 43, 37423745 (2018)
DOI:10.1103/PhysRevB.97.165205 
Towards HighlyPrecise Tunable ElectroModulator Based on Franz–Keldysh Effect,
Sylwia ZielińskaRaczyńska, David Ziemkiewicz, Gerard Czajkowski, Karol Karpiński,
Phys. Status Solidi B2019, Volume 256, Issue 6, 1800502, (2019)
DOI:10.1002/pssb.201800502 
Dynamically Steered Maser Action of Rydberg Excitons in Cu_{2}O,
D. Ziemkiewicz and S. ZielińskaRaczyńska,
Phys. Status Solidi B 2019, Volume 256, Issue 6, 1800503, (2019)
DOI:10.1002/pssb.201800503 
Solidstate pulsed microwave emitter based on Rydberg excitons,
D. Ziemkiewicz and S. ZielińskaRaczyńska,
Optics Express 27(12):16983, (2019)
DOI:10.1364/OE.27.016983 
Nonlinear optical properties and selfKerr effect of Rydberg excitons,
Sylwia ZielińskaRaczyńska, David Ziemkiewicz, Gerard Czajkowski, Karol Karpiński,
Phys. Rev. B 99, 245206, (2019)
DOI:10.1103/PhysRevB.99.245206 
Magnetoexcitons in Cu_{2}O: theoretical model from weak to high magnetic fields,
Sylwia ZielińskaRaczyńska, Dmitry A Fishman, Clément Faugeras, Marek M P Potemski, Paul H M van Loosdrecht, Karol Karpiński, Gerard Czajkowski, David Ziemkiewicz,
New J. Phys. 21 103012, (2019)
DOI:10.1088/13672630/ab4633 
Electromagnetically Induced Transparency in Media with Rydberg Excitons 1: Slow Light,
David Ziemkiewicz,
Entropy, 22(2), 177 (2020)
DOI:10.3390/e22020177 
Electromagnetically Induced Transparency in Media with Rydberg Excitons 2: CrossKerr Modulation,
Sylwia ZielińskaRaczyńska, David Ziemkiewicz,
Entropy, 22(2), 160 (2020)
DOI:10.3390/e22020160 
Excitons in Cu_{2}O: From quantum dots to bulk crystals and additional boundary conditions for Rydberg excitonpolaritons,
David Ziemkiewicz, Karol Karpiński, Gerard Czajkowski, and Sylwia ZielińskaRaczyńska,
Phys. Rev. B 101, 205202 (2020)
DOI:10.1103/PhysRevB.101.205202