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arXiv:2608.08705

Interaction between Rydberg Excitons in Cuprous Oxide Revealed through Resonant Second Harmonic Generation

Andreas Farenbruch, Henje Stolz, Peter Grünwald, Dirk Semkat, Nikita Siverin, Dmitri R. Yakovlev, Dietmar Fröhlich, Manfred Bayer

cond-mat.mes-hallquant-phSubmitted Aug 9, 2026• Announced Aug 11, 2026• 0 citations
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Abstract

We report experimental and theoretical investigations of interacting excitons of the yellow series in cuprous oxide (Cu$_2$O) with principal quantum numbers up to by means of second harmonic generation (SHG). Using picosecond pulsed laser excitation up to 10 GW/cm$^2$ peak intensity we observe a pronounced change of the spectra with increasing pump laser intensity: an energetic shift to lower absolute energies and a spectral broadening. The absolute intensities of the spectral lines scale for low powers with the square of the pump power, but saturates at higher powers. At still higher powers the SHG intensity is actually reduced. To explain these results quantitively, we developed a semi-classical theory of resonant SHG where the process of SHG is fully coherent. The excitons are assumed to be bosons interacting by a distance dependent potential giving rise to both the changes in spectral line shape and the saturation by a $\it{Rydberg}$ blockade. The concomitant measurement of two-photon absorption allows to derive quantitative values for the exciton-exciton interaction. While the results agree in order of magnitude with those calculated by state-of-the art atomic-like van der Waals interaction theory, the scaling with principle quantum number is quite different. As a possible screening by an electron-hole plasma created by three-photon absorption into blue and violet band states could be ruled out, our results point toward fundamental differences between excitons and atoms.

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arXiv:2104.07344•The European Physical Journal Special Topics•Apr 19, 2021•7 citations
We report experimental and theoretical investigations of interacting excitons of the yellow series in cuprous oxide (Cu2_22​O) with principal quantum numbers up to by means of second harmonic generation (SHG). Using picosecond pulsed laser excitation up to 10 GW/cm2^22 peak intensity we observe a pronounced change of the spectra with increasing pump laser intensity: an energetic shift to lower absolute energies and a spectral broadening. The absolute intensities of the spectral lines scale for low powers with the square of the pump power, but saturates at higher powers. At still higher powers the SHG intensity is actually reduced. To explain these results quantitively, we developed a semi-classical theory of resonant SHG where the process of SHG is fully coherent. The excitons are assumed to be bosons interacting by a distance dependent potential giving rise to both the changes in spectral line shape and the saturation by a Rydberg\it{Rydberg}Rydberg blockade. The concomitant measurement of two-photon absorption allows to derive quantitative values for the exciton-exciton interaction. While the results agree in order of magnitude with those calculated by state-of-the art atomic-like van der Waals interaction theory, the scaling with principle quantum number is quite different. As a possible screening by an electron-hole plasma created by three-photon absorption into blue and violet band states could be ruled out, our results point toward fundamental differences between excitons and atoms.
We theoretically investigate the nonlinear optical transmission through a cuprous oxide crystal for wavelengths that cover the series of highly excited excitons, observed in recent experiments. Since such Rydberg excitons have strong van der Waals interactions, they can dynamically break the conditions for resonant exciton creation and dramatically modify the refractive index of the material in a nonlinear manner. We explore this mechanism theoretically and determine its effects on the optical properties of a semiconductor for the case of degenerate pair-state asymptotes of Rydberg excitons in Cu_{2}O. Upon analyzing the additional effects of a dilute residual electron-hole plasma, we find quantitative agreement with previous transmission measurements, which provides strong indications for the enhancement of Rydberg-induced nonlinearities by surrounding free charges.
Emission Dynamics of Rydberg Excitons in Cu2O\mathbf{\mathrm{Cu_2O}}Cu2​O: Distinguishing Second Harmonic Generation from Secondary Emission
Rydberg excitons in Cu2O\mathrm{Cu_2O}Cu2​O simultaneously give rise to two very different optical responses under resonant two-photon excitation: a coherent second-harmonic signal mediated by the excitonic second order susceptibility tensor χ(2)χ^{(2)}χ(2), and a secondary emission originating from the radiative decay of real exciton populations. Distinguishing these two channels is essential for interpreting nonlinear and quantum-optical experiments based on high-nnn states, yet their temporal, spectral, and power-dependent signatures often overlap. Here we use time-resolved resonant two-photon excitation to cleanly separate SHG and SE and to map how each depends on nnn, temperature, excitation power, and crystal quality. This approach reveals the markedly different sensitivities of the two processes to phonons, defects, and many-body effects, and establishes practical criteria for identifying SE and SHG in a wide range of experimental conditions. Our results provide a unified framework for interpreting emission from Rydberg excitons and offer guidelines for future studies aiming to exploit their nonlinear response and long-range interactions.
We investigate the nonlinear refraction induced by Rydberg excitons in Cu_{2}O. Using a high-precision interferometry imaging technique that spatially resolves the nonlinear phase shift, we observe significant shifts at extremely low laser intensity near each exciton resonance. From this, we derive the nonlinear index n_{2}, present the n_{2} spectrum for principal quantum numbers n≥5, and report large n_{2} values of order 10^{-3} mm^{2}/mW. Moreover, we observe a rapid saturation of the Kerr nonlinearity and find that the saturation intensity I_{sat} decreases as n^{-7}. We explain this with the Rydberg blockade mechanism, whereby giant Rydberg interactions limit the exciton density, resulting in a maximum phase shift of 0.5 rad in our setup.
We simulate the effects of a neutral electron--hole plasma on Rydberg excitons in cuprous oxide (Cu2_22​O), focusing on the validity of Debye screening and the role of plasma-induced thermalization. Unlike atomic Rydberg states, excitons in Cu2_22​O consist of quasiparticles with comparable effective masses whose orbital frequencies can exceed the plasma frequency, invalidating the assumption of a stationary screened charge. Using two complementary approaches, a classical orbit model and a harmonic-oscillator representation evolved via the truncated Wigner approximation, we study exciton lifetimes and interaction screening under realistic plasma conditions. We find numerically that plasma-induced scattering induces finite exciton lifetimes with specific scaling relations with plasma density, principal quantum number nnn and temperature, possibly providing an explanation for experimentally observed deviations from the n3n^3n3 scaling at high principal quantum numbers. By explicitly computing time-averaged electric fields, we show that Debye screening overestimates the screening of the exciton's internal field, especially for high angular momentum states. Furthermore, we demonstrate that exciton-exciton interactions are not Debye screened at separations comparable to the Debye length for Rydberg excitons that are well resolvable in absorption measurements.
The ability to generate and control strong long-range interactions via highly excited electronic states has been the foundation for recent breakthroughs in a host of areas, from atomic and molecular physics to quantum optics and technology. Rydberg excitons provide a promising solid-state realization of such highly excited states, for which record-breaking orbital sizes of up to a micrometer have indeed been observed in cuprous oxide semiconductors. Here, we demonstrate the generation and control of strong exciton interactions in this material by optically producing two distinct quantum states of Rydberg excitons. This is made possible by two-color pump-probe experiments that allow for a detailed probing of the interactions. Our experiments reveal the emergence of strong spatial correlations and an inter-state Rydberg blockade that extends over remarkably large distances of several micrometers. The generated many-body states of semiconductor excitons exhibit universal properties that only depend on the shape of the interaction potential and yield clear evidence for its vastly extended-range and power-law character.