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arXiv:2206.06610 · Physical review. B./Physical review. B

Nonlinear optical properties and Kerr nonlinearity of Rydberg excitons in Cu2O quantum wells

David Ziemkiewicz, G. Czajkowski, Karol Karpiński, Sylwia Zielińska‐Raczyńska

Published Aug 31, 2022• 3 citations
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Abstract

The quantum confinement of Rydberg excitons (REs) in quantum structures opens the way towards considering nonlinear interactions in such systems. We present a theoretical calculation of optical functions in the case of a nonlinear coupling between REs in a quantum well with an electromagnetic wave. Using the real density matrix approach (RDMA), the analytical expressions for a linear and nonlinear absorption are derived and numerical calculations for ${\mathrm{Cu}}_{2}\mathrm{O}$ quantum wells are performed. The results indicate the conditions in which quantum well confinement states can be observed in linear and nonlinear optical spectra. The Kerr nonlinearity and self-phase modulation in such a system are studied. The effect of the Rydberg blockade and the associated optical bleaching are also discussed and confronted with available experimental data.

DOI
10.1103/physrevb.106.085431

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The quantum confinement of Rydberg excitons (REs) in quantum structures opens the way towards considering nonlinear interactions in such systems. We present a theoretical calculation of optical functions in the case of a nonlinear coupling between REs in a quantum well with an electromagnetic wave. Using the real density matrix approach (RDMA), the analytical expressions for a linear and nonlinear absorption are derived and numerical calculations for Cu2O{\mathrm{Cu}}_{2}\mathrm{O}Cu2​O quantum wells are performed. The results indicate the conditions in which quantum well confinement states can be observed in linear and nonlinear optical spectra. The Kerr nonlinearity and self-phase modulation in such a system are studied. The effect of the Rydberg blockade and the associated optical bleaching are also discussed and confronted with available experimental data.
We show how to compute the nonlinear optical functions (absorption, reflection, and transmission) for a medium with Rydberg excitons (REs), including the effect of the coherence between the electron-hole pair and the electromagnetic field. Using the real density-matrix approach the analytical expressions for nonlinear optical functions are obtained and numerical calculations are performed for Cu2O{\mathrm{Cu}}_{2}\mathrm{O}Cu2​O crystal where REs have been observed. We report a good agreement with recently published experimental data. Propagation of the electromagnetic waves in Rydberg exciton media with nonlinear effect is also discussed and the possibility of obtaining self-phase modulation due to Kerr nonlinearity is investigated.
Abstract Rydberg excitons (analogues of Rydberg atoms in condensed matter systems) are highly excited bound electron-hole states with large Bohr radii. The interaction between them as well as exciton coupling to light may lead to strong optical nonlinearity, with applications in sensing and quantum information processing. Here, we achieve strong effective photon–photon interactions (Kerr-like optical nonlinearity) via the Rydberg blockade phenomenon and the hybridisation of excitons and photons forming polaritons in a Cu2O-filled microresonator. Under pulsed resonant excitation polariton resonance frequencies are renormalised due to the reduction of the photon-exciton coupling with increasing exciton density. Theoretical analysis shows that the Rydberg blockade plays a major role in the experimentally observed scaling of the polariton nonlinearity coefficient as ∝ n4.4±1.8 for principal quantum numbers up to n = 7. Such high principal quantum numbers studied in a polariton system for the first time are essential for realisation of high Rydberg optical nonlinearities, which paves the way towards quantum optical applications and fundamental studies of strongly correlated photonic (polaritonic) states in a solid state system.
We show how to compute the optical functions when Rydberg Excitons appear, including the effect of the coherence between the electron-hole pair and the electromagnetic field. We use the Real Density Matrix Approach (RDMA), which, combined with Green's function method, enables one to derive analytical expressions for the optical functions. Choosing the susceptibility, we performed numerical calculations appropriate to a Cu$_2$0 crystal. The effect of the coherence is displayed in the line shape. We also examine in details and explain the dependence of the oscillator strength and the resonance placement on the state number. We report a good agreement with recently published experimental data.
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.
Self-Kerr effect across the yellow Cu2O\mathrm{Cu_2O}Cu2​O Rydberg series
We investigate the nonlinear refraction induced by Rydberg excitons in Cu2O\mathrm{Cu_2O}Cu2​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 n2\mathrm{n_2}n2​, present the n2\mathrm{n_2}n2​ spectrum for n≥5n\geq 5n≥5 and report large n2\mathrm{n_2}n2​ values of order 10−310^{-3}10−3 mm2^22/mW. Moreover, we observe a rapid saturation of the Kerr nonlinearity and find that the saturation intensity Isat\mathrm{I_{sat}}Isat​ decreases as n−7n^{-7}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.