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For low-density plasmas, the classical limit described by the Debye-H\"uckel theory is still considered as an appropriate description even though a clear experimental proof of this paradigm is lacking due to the problems in determining the plasma-induced shift of single-particle energies in atomic systems. We show that Rydberg excitons in states with a high principal quantum number are highly sensitive probes for their surrounding making it possible to unravel accurately the basic properties of low-density non-degenerate electron-hole plasmas. To this end, we accurately measure the parameters of Rydberg excitons such as energies and linewidths in absorption spectra of bulk cuprous oxide crystals in which a tailored electron-hole plasma has been generated optically. Since from the absorption spectra exciton energies, as well as the shift of the single-particle energies given by the band edge, can be directly derived, the measurements allow us to determine the plasma density and temperature independently, which has been a notoriously hard problem in semiconductor physics. Our analysis shows unambiguously that the impact of the plasma cannot be described by the classical Debye model, but requires a quantum many-body theory, not only for the semiconductor plasma investigated here, but in general. Furthermore, it reveals a new exciton scattering mechanism with coupled plasmon-phonon modes becoming important even at very low plasma densities.

Analytical results for the dielectric function in RPA are derived for three-, two-, and one-dimensional semiconductors in the weakly-degenerate limit. Based on this limit, quantum corrections are derived. Further attention is devoted to systems with linear carrier dispersion and the resulting Dirac-cone physics., Comment: 26 pages, 9 figures, accepted for publication in J. Phys.: Condens. Matter

We investigate quantum many-body effects on Rydberg excitons in cuprous oxide induced by the surrounding electron-hole plasma. Line shifts and widths are calculated by full diagonalisation of the plasma Hamiltonian and compared to results in first order perturbation theory, and the oscillator strength of the exciton lines is analysed., Comment: 6 pages, 1 figure

In this study we analyze the transmission spectrum of a thin plate of Cuprous Oxide in the range of the absorption of the yellow exciton states with the coherent transfer matrix method. We demonstrate that, in contrast to the usual analysis using Beer's law, which turns out to be a rather good approximation only in the spectral region of high principal quantum numbers, a consistent quantitative description over the whole spectral range under consideration is possible. This leads to new and more accurate parameters not only for the Rydberg exciton states, but also for the strengths of indirect transitions. Furthermore, the results have consequences on the determination of the density of electron-hole pairs after optical excitation.

The ongoing COVID-19 pandemic is challenging every part of society. From a scientific point of view the first major task is to predict the dynamics of the pandemic, allowing governments to allocate proper resources and measures to fight it, as well as gauging the success of these measures by comparison with the predictions in hindsight. The vast majority of pandemic models are based on extensive models with large numbers of fit parameters, leading to individual descriptions for every hot spot on the world. This makes predictions and comparisons cumbersome, if not impossible. We here propose a different approach, by moving away from a description over time, and instead choosing the total number of infected people in an enclosed area as the independent variable. Analyzing a few hot spots data, we derive an empirical formula for the dynamics, dependent only on three variables. The final number of infections is strictly connected to one fit parameter we call mitigation factor, which in turn is mostly dependent only on the enclosed population. Despite its simpleness, this description applies to every of the around 50 countries we have analyzed, allows to separate different waves of the pandemic, provides a figure of merit for the overall usefulness of government measures, and shows when a pandemic is ending. Our model is robust against undetected cases, and allows all nations, in particular those with fewer resources, to reasonably predict the outcome of the pandemic in their country.

We investigate the influence of a static, uncorrelated distribution of charged impurities on the spectrum of bound excitons in the copper oxide Cu$_{\text{2}}$O. We show that the statistical distribution of Stark shifts and ionisation rates leads to the vanishing of Rydberg resonances into an apparent continuum. The appearance of additional absorption lines due to the broken rotational symmetry, together with spatially inhomogeneous Stark shifts, leads to a modification of the observed line shapes that agree qualitatively with the changes observed in the experiment., Comment: 6 pages, 4 figures

We develop a many-body approach to the behavior of exciton bound states and the conduction electron band edge in a surrounding electron-hole plasma with a focus on the absorption spectrum of Rydberg excitons in cuprous oxide. The interplay of band edge and exciton levels is analyzed numerically, whereby the self-consistent solution is compared to the semiclassical Debye approximation. Our results provide criteria which allow to verify or rule out the different band edge models against future experimental data., Comment: 7 pages, 9 figures

Using the theory of imaging with partially coherent light, we derive general expressions for different kinds of interferometric setups like double slit, shift and mirror interference. We show that in all cases the interference patterns depend not only on the point spread function of the imaging setup but also strongly on the spatial emission pattern of the sample. Taking typical experimentally observed spatial emission patterns into account, we can reproduce at least qualitatively all the observed interference structures, which have been interpreted as signatures for spontaneous long range coherence of excitons, already for incoherent emitters. This requires a critical reexamination of the previous work.

We demonstrate that the optical linewidth of Rydberg excitons in Cu2O can be completely explained by scattering with acoustical and optical phonons, whereby the dominant contributions stems from the non-polar optical modes. The consequences for the observation of polariton effects are discussed. We find that an anti-crossing of photon and exciton dispersions exists only for states with main quantum numbers n>28, so polariton effects do not play any role in the experiments reported up to now.

Photon-number measurements are a fundamental technique for the discrimination and characterization of quantum states of light. Beyond the abilities of state-of-the-art devices, we present measurements with an array of 100 avalanche photodiodes exposed to photon-numbers ranging from well below to significantly above one photon per diode. Despite each single diode only discriminating between zero and non-zero photon-numbers we are able to extract characteristic information about the quantum state. We demonstrate a vast enhancement of the applicable intensity range by two orders of magnitude relative to the standard application of such devices. It turns out that the probabilistic mapping of arbitrary photon-numbers on a finite number of registered clicks is not per se a disadvantage compared with true photon counters. Such detector arrays can bridge the gap between single-photon and linear detection, by directly using the recorded data, without the need of elaborate data reconstruction methods., Comment: 4 figures

The basic theoretical foundation for the modelling of phonon assisted absorption spectra in direct bandgap semiconductors, introduced by Elliott 60 years ago using second order perturbation theory, results in a square root shaped dependency close to the absorption edge. A careful analysis of the experiments reveals that for the yellow S excitons in Cu$_2$O the lineshape does not follow that square root dependence. The reexamination of the theory shows that the basic assumptions of constant matrix elements and constant energy denominators is invalid for semiconductors with dominant exciton effects like Cu$_2$O, where the phonon assisted absorption proceeds via intermediate exciton states. The overlap between these and the final exciton states strongly determines the dependence of the absorption on the photon energy. To describe the experimental observed line shape of the indirect absorption of the yellow S exciton states we find it necessary to assume a momentum dependent deformation potential for the optical phonons.

In this paper we present a hydrodynamic model to describe the dynamics of para- and orthoexcitons in cuprous oxide at ultralow temperatures inside a stress induced potential trap. We take into account the finite lifetime of the excitons, the excitation process and exciton-phonon as well as exciton-exciton interaction. Furthermore, we model the two-body loss mechanism assuming an Auger-like effect and compare it to an alternative explanation which relies on the formation of biexcitons. We discuss in detail the influence on the numerical results and compare the predictions to experimental data., Comment: 10 pages, 8 figures, submitted to J. Phys. B

Recent high-resolution absorption spectroscopy on excited excitons in cuprous oxide [Nature (London) 514, 343 (2014)] has revealed significant deviations of their spectrum from that of the ideal hydrogen-like series. Here we show that the complex band dispersion of the crystal, which determines the kinetic energy of electrons and holes, strongly affects the exciton binding energy. Specifically, we show that the nonparabolicity of the band dispersion is the main cause of the deviation from the hydrogen series. Experimental data collected from high-resolution absorption spectroscopy in electric fields validate the assignment of the deviation to the nonparabolicity of the band dispersion.

Highly excited atoms with an electron moved into a level with large principal quantum number are fascinating hydrogen-like objects. The giant extension of these Rydberg atoms leads to huge interaction effects. Monitoring these interactions has provided novel insights into molecular and condensed matter physics problems on a single quantum level. Excitons, the fundamental optical excitations in semiconductors consisting of a negatively charged electron and a positively charged hole, are the condensed matter analogues of hydrogen. Highly excited excitons with extensions similar to Rydberg atoms are attractive because they may be placed and moved in a crystal with high precision using microscopic potential landscapes. Their interaction may allow formation of ordered exciton phases or sensing of elementary excitations in the surrounding, also on a quantum level. Here we demonstrate the existence of Rydberg excitons in cuprous oxide, Cu2O, with principal quantum numbers as large as n=25 . These states have giant wave function extensions of more than 2 micrometers, compared to about a nanometer for the ground state. The strong dipole-dipole interaction is evidenced by a blockade effect, where the presence of an exciton prevents excitation of a further exciton in its vicinity.

We have studied theoretically the relaxation behaviour of excitons in cuprous oxide (Cu2O) at ultra low temperatures when excitons are confined within a potential trap by solving numerically the Boltzmann equation. As relaxation processes, we have included in this paper deformation potential phonon scattering, radiative and non-radiative decay and Auger decay. The relaxation kinetics has been analysed for temperatures in the range between 0.3K and 5K. Under the action of deformation potential phonon scattering only, we find for temperatures above 0.5K that the excitons reach local equilibrium with the lattice i.e. that the effective local temperature is coming down to bath temperature, while below 0.5K a non-thermal energy distribution remains. Interestingly, for all temperatures the global spatial distribution of excitons does not reach the equilibrium distribution, but stays at a much higher effective temperature. If we include further a finite lifetime of the excitons and the two-particle Auger decay, we find that both the local and the global effective temperature are not coming down to bath temperature. In the first case we find a Bose-Einstein condensation (BEC) to occur for all temperatures in the investigated range. Comparing our results with the thermal equilibrium case, we find that BEC occurs for a significantly higher number of excitons in the trap. This effect could be related to the higher global temperature, which requires an increased number of excitons within the trap to observe the BEC. In case of Auger decay, we do not find at any temperature a BEC due to the heating of the exciton gas.

We present experiments on the luminescence of excitons confined in a potential trap at milli-Kelvin bath temperatures under cw-excitation. They reveal several distinct features like a kink in the dependence of the total integrated luminescence intensity on excitation laser power and a bimodal distribution of the spatially resolved luminescence. Furthermore, we discuss the present state of the theoretical description of Bose-Einstein condensation of excitons with respect to signatures of a condensate in the luminescence. The comparison of the experimental data with theoretical results with respect to the spatially resolved as well as the integrated luminescence intensity shows the necessity of taking into account a Bose-Einstein condensed excitonic phase in order to understand the behaviour of the trapped excitons., Comment: 41 pages, 23 figures

For the observation of Bose-Einstein condensation, excitons in cuprous oxide are regarded as promising candidates due to their large binding energy and long lifetime. High particle densities may be achieved by entrapment in a stress induced potential. We consider a multi-component gas of interacting para- and orthoexcitons in cuprous oxide confined in a three-dimensional potential trap. Based on the Hartree-Fock-Bogoliubov theory, we calculate density profiles as well as decay luminescence spectra which exhibit signatures of the separation of the Bose-condensed phases., Comment: 4 pages, 1 figure, presented at NOEKS 10, Paderborn, August 2010

We calculate the spatially resolved optical emission spectrum of a weakly interacting Bose gas of excitons confined in a three dimensional potential trap due to interband transitions involving weak direct and phonon mediated exciton-photon interactions. Applying the local density approximation, we show that for a non-condensed system the spatio-spectral lineshape of the direct process reflects directly the shape of the potential. The existence of a Bose-Einstein condensate changes the spectrum in a characteristic way so that it directly reflects the constant chemical potential of the excitons and the renormalization of the quasiparticle excitation spectrum. Typical examples are given for parameters of the lowest yellow excitons in cuprous oxide., Comment: 5 pages, 2 figures